Hindawi Publishing CorporationAdvances in Fuzzy SystemsVolume 2013 Article ID 760704 10 pageshttpdxdoiorg1011552013760704
Research ArticleImage Matching by Using Fuzzy Transforms
Ferdinando Di Martino and Salvatore Sessa
Dipartimento di Architettura Universita degli Studi di Napoli Federico II Via Monteoliveto 3 80134 Napoli Italy
Correspondence should be addressed to Salvatore Sessa sessauninait
Received 3 March 2013 Accepted 9 May 2013
Academic Editor Irina Perfilieva
Copyright copy 2013 F Di Martino and S Sessa This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
We apply the concept of Fuzzy Transform (for short F-transform) for improving the results of the image matching based on theGreatest Eigen Fuzzy Set (for short GEFS) with respect tomax-min composition and the Smallest Eigen Fuzzy Set (for short SEFS)with respect to min-max composition already studied in the literature The direct F-transform of an image can be compared withthe direct F-transform of a sample image to be matched and we use suitable indexes to measure the grade of similarity between thetwo images We make our experiments on the image dataset extracted from the well-known Prima Project View Sphere Databasecomparing the results obtained with this method with that one based on the GEFS and SEFS Other experiments are performed onframes of videos extracted from the Ohio State University dataset
1 Introduction
Solution methods of fuzzy relational equations have beenwell studied in the literature (cf eg [1ndash15]) and applied toimage processing problems like image compression [16ndash19]and image reconstruction [7 8 20ndash22] In particular EigenFuzzy Sets [23ndash25] have been applied to image processingand medical diagnosis [2 6 7 16] If an image 119868 of sizes119873 times 119873 (pixels) is interpreted as a fuzzy relation 119877 on theset 1 2 119873 times 1 2 119873 rarr [0 1] the concepts of theGreatest Eigen Fuzzy Set (for short GEFS) of 119868 with respectto the max-min composition and of the Smallest Eigen FuzzySet (for short SEFS) of 119877 with respect to the min-maxdecomposition [2 24 25] were studied and used in [26 27]for an image matching process defined over square imagesThe GEFS and SEFS of the original image are compared withthe GEFS and SEFS of the image to be matched by using asimilarity measure based on the RootMean Square Error (forshort RMSE) The advantage of using GEFS and SEFS is interms of memory storage is that we can indeed compress animage dataset (in which each image has sizes 119873 times 119873) in adataset in which each image is stored by means of its GEFSand SEFS which have total dimension 2119873
The main disadvantage of using GEFS and SEFS is thatwe cannot compare images in which the number of rows isdifferent from the number of columns Our aim is to show
that we can use an F-transform for imagematching problemsreducing an image dataset of sizes 119873 times 119872 (in general 119872is not necessarily equal to 119873) into a dataset of dimensionscomparable with that one obtained by using GEFS and SEFSif119872 = 119873 so having convenience in terms ofmemory storage
The F-transform based method [28ndash30] is used in theliterature for image and video compression [29 31ndash33] imagesegmentation [20] and data analysis [22 34] indeed in[31 32] the quality of the decoded images obtained by usingthe F-transform compression method is shown to be betterthan that one obtained with the fuzzy relation equations andfully comparable with the JPEG technique
The main characteristic of the F-transform method is tomaintain an acceptable quality in the reconstructed imageeven under strong compression rates indeed in [20] theauthors show that the segmentation process can be applieddirectly over the compressed images Here we use the directF-transform in image matching analysis with the aim ofreducing the memory used to store the image dataset Infact we compress a monochromatic image (or a band of amultiband image) 119868 of sizes119873times119872 via the direct F-transformto a matrix 119865 of sizes 119899 times 119898 using a compression rate 120588 =(119899 times 119898)(119873 times119872)
By using a distance we compare the F-transform of eachimage with the F-transform of the sample image We also
2 Advances in Fuzzy Systems
Normalization
Extract
Store
Pixel value Normalized value
Image dataset
Compressed image dataset
Direct F-transform
F-transformcomponent
R
GB
RR
GG
BB
Figure 1 The preprocessing phase
Normalization
Similarity
YesNo
Extract
Normalized valueF-transformcomponent
Direct F-transform
Retrieved imagePixel value
Compressed image dataset F-transform component
= max similarity
Calculate similarity
gt max similarity
R
GB
R
GB
RR
G G
B B
Figure 2 F-transform image matching process
adopt a preprocessing phase for compressing each imagewithseveral compression rates In Figure 1 we show the prepro-cessing phase on a dataset of color images We compresseach color image in the three monochromatic componentscorresponding to the three bands 119877 119866 and 119861
At the end of the preprocessing phase we can usethe compressed image dataset for image matching analysisSupposing that the original image dataset was composed bys color images of sizes 119873 times 119872 using a compression rate120588 = (119899 times 119898)(119873 times 119872) we obtain that the dimension of thecompressed image dataset is constituted totally of 3119904(119899 times 119898)pixels
In Figure 2 we schematize the image matching processThe sample image is compressed by the F-transformmethodthen we compare the three compressed bands of each imageobtained via F-transform with those ones deduced for thesample image by using the Peak Signal to Noise Ratio (forshort PSNR) At the end of this process we determine the
image in the dataset with the greatest overall PSNR withrespect to the sample image
Here a monochromatic image or a band of a color image119868 of sizes 119873 times 119872 is interpreted as a fuzzy relation 119877 whoseentries 119877(119909 119910) are obtained by normalizing the intensity119868(119909 119910) of each pixel with respect to the length 119871 of the scalethat is 119877(119909 119910) = 119868(119909 119910)119871 We show that our F-transformapproach can be also applied in image matching processes toimages of sizes119872times119873 (eventually119872 =119873) giving analogousresults with respect to that one obtained with GEFS andSEFS based method The comparison tests are made on the256 times 256 color image dataset extracted from View SphereDatabase an image dataset consisting in a set of images ofobjects in which an object is photographed from variousdirections by using a camera placed on a semisphere whosecenter is the same considered object We also use the OhioState University color video datasets sample for our testsEach video is composed by frames consisting of color images
Advances in Fuzzy Systems 3
R2
y
x
R1
z
Φ2
12057921205791
Φ1
Figure 3 The angles for two directions 1198771and 119877
2(the object is in
the origin)
we show the results for the Mom-Daughter and sflowgmotions In Section 2we recall the concepts of F-transform intwo variables In Section 3we recall theGEFS and SEFS basedmethod in Section 4wepropose our imagematchingmethodbased on the F-transforms Our experiments are illustrated inSection 5 and Section 6 is conclusive
2 F-Transforms in Two Variables
Following [29] and limiting ourselves to the discrete case let119899 ge 2 and 119909
1 1199092 119909
119899be a increasing sequence of points
(nodes) of [119886 119887] 1199091= 119886 lt 119909
119894(119909119894+ 119909) for every 119909 isin [0 ℎ] and 119894 =
2 119899 minus 1
(8) 119860119894+1(119909) = 119860
119894(119909 minus ℎ) for every 119909 isin [119909
119894 119909119894+1] and 119894 =
1 2 119899 minus 1
Let 119898 ge 2 1199101 1199102 119910
119898isin [119888 119889] be 119898 nodes such that
1199101= 119888 lt sdot sdot sdot lt 119910
119898= 119889 Furthermore let119861
1 119861
119898 [119888 119889] rarr
[0 1] be a fuzzy partition of [119888 119889] and let 119891 119875 times 119876 rarr
reals be an assigned function 119875 times 119876 sube [119886 119887] times [119888 119889] with119875 = 119901
1 119901
119873 and 119876 = 119902
1 119902
119872 being ldquosufficiently
denserdquo sets with respect to the chosen partitions that is foreach 119894 = 1 119873 (resp 119895 = 1 119872) there exists an index119896 isin 1 119899 (resp 119897 isin 1 119898) such that 119860
119896(119901119894) gt 0
0 20 40 60 8014
16
18
20
22
24
26
PSNR120588(R1 R2)
PSN
R 120588(R
1R
2)
D (R1 R2)
Figure 4 Trend of PSNR with respect to 119863 index for the eraserobtained from the comparison with the sample image at 120579 = 11
∘
and 120601 = 36∘
(resp 119861119897(119902119895) gt 0) The matrix [119865
119896119897] is said to be the direct F-
transform of 119891 with respect to 1198601 119860
119899 and 119861
1 119861
119898
if we have for each 119896 = 1 119899 and 119897 = 1 119898
Theorem 1 Let 119891 119875 times 119876 rarr reals be a given function119875 times 119876 sube [119886 119887] times [119888 119889] with 119875 = 119901
1 119901
119873 and 119876 =
1199021 119902
119872 Then for every 120576 gt 0 there exist two integers
119899(120576) 119898(120576) and related fuzzy partitions 1198601 1198602 119860
119899(120576) of
[119886 119887] and 1198611 1198612 119861
119898(120576) of [119888 119889] such that the sets 119875119876 are
sufficiently dense with respect to such partitions and |119891(119901119894 119902119895)minus
119891119865
119899(120576)119898(120576)(119901119894 119902119895)| lt 120576 is satisfied for every 119894 isin 1 119873 and
119895 isin 1 119872
Let 119877 be a gray image of sizes 119873 times 119872 seen as 119877
(119894 119895) isin 1 119873 times 1 119872 rarr [0 1] with 119877(119894 119895) beingthe normalized value of the pixel 119875(119894 119895) given by 119877(119894 119895) =119875(119894 119895)119871119905 if 119871119905 is the length of the gray scale In [27] 119877 is
4 Advances in Fuzzy Systems
Figure 5 Eraser at 120579 = 11∘ and 120601 = 36∘
Figure 6 Eraser at 120579 = 10∘ and 120601 = 54∘
compressed via the F-transform defined for each 119896 = 1 119899and 119897 = 1 119898 as
which approximates 119877119861up to an arbitrary quantity 120576 in the
sense of Theorem 1 which unfortunately does not give amethod for finding two integers 119899(119861) and 119898(119861) such that|119877119861(119901119894 119902119895) minus 119877
119865
119899(120576)119898(120576)(119901119894 119902119895)| lt 120576 Then we prove several
values of 119899(119861) and 119898(119861) For every compression rate 120588 weevaluate the quality of the reconstructed image via the PSNRdefined as
(PSNR)120588= 20 log
10
119871
(RMSE)120588
(9)
where (RMSE)120588is
(RMSE)120588=radicsum119873
119894=1sum119872
119895=1(119877 (119894 119895) minus 119877
119865
119873119872(119894 119895))
2
119873 times119872
(10)
Here 119877119865119873119872
is the reconstructed image obtained by recompos-ing the blocks 119877119865
119899(119861)119898(119861)
1015840
3 Max-Min and Min-Max Eigen Fuzzy Sets
Let 119883 be a nonempty finite set 119877 119883 times 119883 rarr [0 1] and119860 119883 rarr [0 1] such that
119877 ∘ 119860 = 119860 (11)
where ldquo∘rdquo is the max-min composition In terms of member-ship functions we have that
119860 (119910) = max119909isin119883
min (119860 (119909) 119877 (119909 119910) (12)
for all 119909 119910 isin 119883 and 119860 is defined as an Eigen Fuzzy Set of 119877Let 119860
119894 119883 rarr [0 1] 119894 = 1 2 be defined iteratively by
1198601(119911) = max
119909isin119883
119877 (119909 119911)
1198602= 119877 ∘ 119860
1 119860
119899+1= 119877 ∘ 119860
119899 119911 isin 119883
(13)
6 Advances in Fuzzy Systems
It is known [2 24 25] that there exists an integer 119901 isin
1 card119883 such that 119860119901is the GEFS of 119877 with respect to
the max-min composition We also consider the following
119877◻119860 = 119860 (14)
where ldquo◻rdquo denotes themin-max composition that is in termsof membership functions
119860 (119910) = min119909isin119883
max (119860 (119909) 119877 (119909 119910) (15)
for all 119909 119910 isin 119883 and 119860 is also defined to be an Eigen FuzzySet of 119877 with respect to the min-max composition It is easilyseen that (14) is equivalent to the following
119877◻119860 = 119860 (16)
where 119877 and119860 are pointwise defined as 119877(119909 119910) = 1 minus119877(119909 119910)and 119860(119909) = 1 minus 119860(119909) for all 119909 119910 isin 119883 Since 119860
119901for some 119901 isin
1 card119883 is the GEFS of 119877 with respect to the max-mincomposition it is immediately proved that the fuzzy set 119861 119883 rarr [0 1] defined as 119861(119909) = 1 minus 119860
119901(119909) for every 119909 isin [0 1]
is the SEFS of 119877 with respect to the min-max compositionIn [27] a distance based on GEFS and SEFS for image
matching is used over images of sizes 119873 times 119873 Indeedconsidering two single-band images of sizes 119873 times 119873 say 119877
1
and 1198772 such distance is given by
119889 (1198771 1198772) = sum
119909isin119883
((1198601(119909) minus 119860
2(119909))2
+ (1198611(119909) minus 119861
2(119909))2
)
(17)
where 119883 = 1 2 119873 119860119894 119861119894are the GEFS and SEFS of
the fuzzy relation 119877119894 respectively obtained by normalizing
in [0 1] the pixels of the image 119868119894 119894 = 1 2
In [26 27] experiments are presented over color imagesof sizes 256 times 256 concerning two objects (an eraser anda pen) extracted from View Sphere Database Each objectis put in the center of a semisphere on which a camera isplaced in 91 different directions The camera establishes animage (photography) of the object for each direction whichcan be identified from two angles 120579 (0∘ lt 120579 lt 90
∘) and
Φ (minus180∘lt Φ lt 180
∘) as illustrated in Figure 3
A sample image 1198771(with given 120579 = 11
∘ Φ = 36∘
for the eraser and 120579 = 10∘ Φ = 54
∘ for the pen) isto be compared with another image 119877
2chosen among the
remaining 90 directions GEFS and SEFS are calculated in thethree components of each image in the RGB space for whichit is natural to assume the following extension of (17)
where 119889119877(1198771 1198772) 119889119866(1198771 1198772) 119889119861(1198771 1198772) are the measures
(17) calculated in each band 119877 119866 119861 For image matchingthe GEFS and SEFS components in each band are extractedfrom each image thus forming a dataset with reduced storagememoryAn image is comparedwith the images in the datasetusing (18) If the dataset contains 119904 color images of sizes119873times119873
Table 1 Best distances fromGEFS and SEFS basedmethod with 120588 =0007813 for the eraser image dataset obtained from the comparisonwith the sample image at 120579 = 11∘ and 120601 = 36∘
and the dimension of the original dataset is 31199041198732 then thedimension of the GEFS and SEFS dataset is 6119904119873 so we havea compression rate given by
120588 =6119904119873
31199041198732=2
119873 (19)
So we obtain a compression rate 120588 = 0007813 if119873 = 256
4 The Image Matching Process viaF-Transforms
We consider an image dataset formed by color images of sizes119873 times119872 In the preprocessing phase we compress each imageof the dataset using the direct F-transform Each image isdivided in blocks of sizes 119873(119861) times 119872(119861) and each block iscompressed in a block of sizes 119899(119861) times 119898(119861) Thus the imagesare coded with a compression rate 120588 = (119899(119861)times119898(119861))(119873(119861)times119872(119861)) In our experiments we set the sizes of the originaland compressed blocks so that 120588 is comparable with (18) Forexample for 119873 = 119872 = 256 we use 119873(119861) = 119872(119861) = 24 and119899(119861) = 119898(119861) = 2 so 120588 = 0006944
In the reduced dataset we store the F-transform compo-nents of each image We use the PSNR between a sampleimage 119877
1and an image 119877
2defined for every compression rate
120588 (cf (9)) as
PSNR120588(1198771 1198772) = 20 log
10
119871
RMSE120588(1198771 1198772) (20)
Advances in Fuzzy Systems 7
Table 2 Best PSNR from the F-transform based method with 119871 = 255 and 120588 = 0006944 for the eraser image dataset obtained from thecomparison with the sample image at 120579 = 11∘ and 120601 = 36∘
Figure 12 Trend of PSNR (120588 = 0006944) with respect to framenumber for the video Mom-Daughter
Figure 13 Frame 1 in sflowg
where RMSE (Root Mean Square Error) is given by (cf (10))
RMSE120588(1198771 1198772) =
radicsum119873
119894=1sum119872
119895=1(1198771(119894 119895) minus 119877
2(119894 119895))2
119873 times119872
(21)
If we have color images we define an overall PSNR as
PSNR120588(1198771 1198772)
=1
3[PSNR
120588119877(1198771 1198772) + PSNR
120588119866(1198771 1198772)
+PSNR120588119861(1198771 1198772)]
(22)
Figure 14 Frame 2 in sflowg
0 10 20 30 40 50
PSNR
Frame n∘
20
25
30
35
40
45
PSN
R
Figure 15 Trend of PSNR (120588 = 0006944) with respect to framenumber for the video sflowg
where PSNR120588119877(1198771 1198772) PSNR
120588119866(1198771 1198772) PSNR
120588119861(1198771 1198772) are
the similarity measures (20) calculated in each band 119877 119866119861 compression rate 120588 In our experiments we compare theresults obtained by using the F-transforms (resp GEFS andSEFS) based method with the PSNR (20) (resp (18)) Weuse the color image datasets of 256 gray levels and of sizes256 times 256 pixels available in the View Sphere Database foreach object considered the best image 119877
2of the object itself
maximizes the PSNR (22) In other experiments we use ourF-transform method over color video datasets in which eachframe is formed by images of 256 gray levels and of sizes
8 Advances in Fuzzy Systems
20 30 40 50 60
PSNR0
PSN
R di
ffere
nce
PSNR difference
0
02
04
06
08
1
12
14
16
18
2
Figure 16 Trend of PSNR difference with respect to PSNR0(120588 =
0006944)
Table 3 Best distances from GEFS and SEFS based method with 120588= 0007813 for the pen image dataset obtained from the comparisonwith the sample image at 120579 = 10∘ and 120601 = 54∘
360times24 available in the Ohio University sample digital colorvideo database A color video is schematically formed by asequence of frames If we consider a frame in a video as thesample image we prove that the image with greatest PSNRwith respect to the sample image is an image with framenumber close to the frame number of the sample image
5 Results of Tests
We compare results obtained by using the GEFS and SEFSand F-transform based methods for image matching onall the image datasets each of sizes 256 times 256 extractedfrom the View Sphere Database In the first image datasetconcerning an eraser we consider as sample image 119877
1 the
image obtained from the camera in the direction with angles120579 = 11
∘ and 120601 = 36∘ For brevity we consider a dataset of 40test images andwe compare119877
1with the images considered in
the remaining 40 other directions In Table 1 (resp Table 2)we report the distances (17) and (18) (resp PSNR (20) and
(22) with 119871 = 255) obtained using the GEFS and SEFS (respF-transform) based method
In Figure 4we show the trend of the index PSNRobtainedby the F-Transform method with respect to the distance (18)obtained using the GEFS and SEFS method
As we can see from Tables 1 and 2 both methods give thesame reply the better image similar with the image eraser inthe direction 120579 = 11∘ and120601 = 36∘ (Figure 5) is given from thatone at 120579 = 10∘ and 120601 = 54∘ (Figure 6) The trend in Figure 4shows that the value of the distance (18) increases as the PSNRdecreases
In order to have a further confirmation of our approachwe have considered a second object a pen contained in theView Sphere Database whose sample image 119877
1is obtained
from the camera in the direction with angles 120579 = 10∘ and
120601 = 54∘ We also limit the problem to a dataset of 40 test
images whose best distances (17) and (18) (resp (20) and (22)with 119871 = 255) under the SEFS and GEFS (resp F-transform)based method are reported in Table 3 (resp Table 4)
In Figure 7we show the trend of the index PSNRobtainedby the F-transform method with respect to the distance 119863obtained by using the GEFS and SEFS method As we cansee from Tables 3 and 4 in both methods the best imagesimilar to the original image in the directions 120579 = 10∘ and120601 = 54
∘ (Figure 8) is given from that one at 120579 = 10∘ and
120601 = 18∘ (Figure 9) Also in this example the trend in Figure 7
shows that the value of the distance (18) increases as the PSNRdecreases
Now we present the results over a sequence of framesof a video Mom-Daughter available in the Ohio Universitysample digital color video database Each frame is a colorimage of sizes 360 times 240 with 256 gray levels for each bandWe use our method with a compression rate 120588 = 0006944that is in each band every frame is decomposed in 150 blocksand each block has sizes 24 times 24 compressed to a block ofsizes 2 times 2 Since119872 =119873 the GEFS and SEFS based methodis not applicable We set the sample image as the imagecorresponding to the first frame of the video We expectthat the frame number of the image with higher PSNR withrespect to the sample image is the image with frame numberclose to the frame number of sample image In Table 5 wereport the best results obtained using the F-transform basedmethod in terms of the (20) and (22) with 119871 = 255 Asexpected albeit with slight variations all the PSNRs diminishby increasing of the frame number and the second frame(Figure 11) is the frame with the greatest PSNR w r t the firstframe (Figure 10) containing the sample image
In Figure 12 we show the trend of the PSNR (22) with theframe numberThis trend is obtained for all the sample videoframes in the video dataset For reasons of brevity now wereport only the results obtained for another test performedon the sequence of frames of another video in the Ohiosample digital video database the video sflowg The PSNR inFigure 15 diminishes by increasing the frame number and thesecond frame (Figure 14) is the frame with the greatest PSNRw r t the first frame (Figure 13) containing the sample image
For supporting the validity of the F-transform methodfor all the sample frames we measure for the frame withthe greatest PSNR wrt the sample frame the correspondent
Advances in Fuzzy Systems 9
Table 4 Best PSNR from the F-transform based method with 119871 = 255 and 120588 = 0006944 for the pen image dataset obtained from thecomparison with the sample image at 120579 = 10∘ and 120601 = 54∘
value PSNR0obtained by using the original frame instead
of the correspondent compressed frame decoded via theinverse F-Transform In Figure 16 we show the trend of thedifference PSNR
0minus PSNR with respect to PSNR
0 The trend
indicates that this difference is always less than 2 This resultshows that if we compress the images in the dataset with rate120588 = 0006944 by using the F-transform method we can usethe compressed image dataset for image matching processescomparing the decompressed image with respect to a sampleimage despite the loss of information due to the compression
6 Conclusions
The results on the images of sizes 119873 times 119872 (119872 = 119873 = 256)
of the View Sphere Image Database show that using our F-transform based method we obtain the same results in termsof image matching and in terms of reduced memory storagereached also via the GEFS and SEFS based method which isapplicable only over images with 119873 = 119872 while our methodconcerns images of any sizes
Moreover our tests executed on color video frames of sizes119873times119872 (119872 = 360119873 = 240 pixels with 256 gray levels) of theOhio University color videos dataset show that by choosingthe first frame as the sample image we obtain as image withthe highest PSNR that one corresponding to the successiveframe as expected although a loss of information on thedecoded images because of the compression process
References
[1] I Bartolini P Ciaccia and M Patella ldquoQuery processing issuesin region-based image databasesrdquo Knowledge and InformationSystems vol 25 no 2 pp 389ndash420 2010
[2] B De Baets ldquoAnalytical solution methods for fuzzy relationalequationsrdquo in Fundamentals of Fuzzy Sets D Dubois and HPrade Eds vol 1 ofTheHandbooks of Fuzzy Sets Series pp 291ndash340 Kluwer Academic Publishers DordrechtTheNetherlands2000
[3] A Di NolaW Pedrycz S Sessa and E Sanchez Fuzzy RelationEquations and Their Application to Knowledge EngineeringKluwer Academic Publishers Dordrecht The Netherlands1989
[4] M Higashi and G J Klir ldquoResolution of finite fuzzy relationequationsrdquo Fuzzy Sets and Systems vol 13 no 1 pp 65ndash82 1984
[5] P Li and S C Fang ldquoA survey on fuzzy relational equationspart I classification and solvabilityrdquo Fuzzy Optimization andDecision Making vol 8 no 2 pp 179ndash229 2009
[6] V Loia W Pedrycz and S Sessa ldquoFuzzy relation calculusin the compression and decompression of fuzzy relationsrdquoInternational Journal of Image and Graphics vol 2 no 4 pp617ndash631 2002
[7] A VMarkovskii ldquoOn the relation between equationswithmax-product composition and the covering problemrdquo Fuzzy Sets andSystems vol 153 no 2 pp 261ndash273 2005
[8] M Miyakoshi and M Shimbo ldquoLower solutions of systems offuzzy equationsrdquo Fuzzy Sets and Systems vol 19 no 1 pp 37ndash461986
[9] K Peeva ldquoResolution of min-max fuzzy relational equationsrdquoin Fuzzy Partial Differential Equations and Relational EquationsM Nikravesh L A Zadeh and V Korotkikh Eds pp 153ndash166Springer New York NY USA 2004
[10] K Peeva ldquoUniversal algorithm for solving fuzzy relationalequationsrdquo Italian Jour-Nal of Pure and Applied Mathematicsvol 19 pp 9ndash20 2006
[11] K Peeva and Y Kyosev ldquoAlgorithm for solving max-productfuzzy relational equationsrdquo Soft Computing vol 11 no 7 pp593ndash605 2007
[12] K Peeva andY Kyosev Fuzzy Relational Calculus-Theory Appli-cations and Software vol 22 of Advances in Fuzzy SystemsmdashApplications and Theory World Scientific Publishing RiverEdge NJ USA 2004
[13] E Sanchez ldquoResolution of composite fuzzy relation equationsrdquoInformation and Control vol 30 no 1 pp 38ndash48 1976
[14] B S Shieh ldquoNew resolution of finite fuzzy relation equationswith max-min compositionrdquo International Journal of Uncer-tainty Fuzziness and Knowlege-Based Systems vol 16 no 1 pp19ndash33 2008
10 Advances in Fuzzy Systems
[15] Y K Wu and S M Guu ldquoAn efficient procedure for solvinga fuzzy relational equation with max-Archimedean t-normcompositionrdquo IEEE Transactions on Fuzzy Systems vol 16 no1 pp 73ndash84 2008
[16] F Di Martino V Loia and S Sessa ldquoA method in the com-pressiondecompression of images using fuzzy equations andfuzzy similaritiesrdquo in Proceedings of IFSA Conference pp 524ndash527 2003
[17] F Di Martino V Loia and S Sessa ldquoA method for cod-ingdecoding images by using fuzzy relation equationsrdquo in10th International Fuzzy Systems Association World Congress TBilgic B De Baets and O Kaynak Eds vol 2715 of LectureNotes in Artificial Intelligence pp 436ndash441 Springer BerlinGermany July 2003
[18] V Loia and S Sessa ldquoFuzzy relation equations for cod-ingdecoding processes of images and videosrdquo InformationSciences vol 171 no 1-3 pp 145ndash172 2005
[19] H Nobuhara W Pedrycz and K Hirota ldquoFast solving methodof fuzzy relational equation and its application to lossy imagecompressionreconstructionrdquo IEEE Transactions on Fuzzy Sys-tems vol 8 no 3 pp 325ndash334 2000
[20] F Di Martino V Loia and S Sessa ldquoA segmentation methodfor images compressed by fuzzy transformsrdquo Fuzzy Sets andSystems vol 161 no 1 pp 56ndash74 2010 Special section NewTrends on Pattern Recognition with Fuzzy Models
[21] J Lu R Li Y Zhang T Zhao and Z Lu ldquoImage annotationtechniques based on feature selection for class-pairsrdquo Knowl-edge and Information Systems vol 24 no 2 pp 325ndash337 2010
[22] I Perfilieva V Novak and A Dvorak ldquoFuzzy transform inthe analysis of datardquo International Journal of ApproximateReasoning vol 48 no 1 pp 36ndash46 2008
[23] L Chen andPWang ldquoFuzzy relational equations (I) the generaland specialized solving algorithmsrdquo Soft Computing vol 6 pp428ndash435 2002
[24] E Sanchez ldquoResolution of Eigen fuzzy sets equationsrdquo FuzzySets and Systems vol 1 no 1 pp 69ndash74 1978
[25] E Sanchez ldquoEigen fuzzy sets and fuzzy relationsrdquo Journal ofMathematical Analysis and Applications vol 81 no 2 pp 399ndash421 1981
[26] F Di Martino S Sessa and H Nobuhara ldquoEigen fuzzy sets andimage information retrievalrdquo in Proceedings of IEEE Interna-tional Conference on Fuzzy Systems vol 3 pp 1385ndash1390 IEEEPress July 2004
[27] F Di Martino H Nobuhara and S Sessa ldquoEigen fuzzy setsand image information retrievalrdquo in Handbook of GranularComputIng W Pedrycz A Skowron and V Kreinovich Edspp 863ndash872 John Wiley amp Sons Chichester UK 2008
[28] I Perfilieva ldquoFuzzy transforms application to reef growthproblemrdquo in Fuzzy Logic in Geology R B Demicco and GJ Klir Eds pp 275ndash300 Academic Press Amsterdam TheNetherlands 2003
[29] I Perfilieva ldquoFuzzy transforms theory and applicationsrdquo FuzzySets and Systems vol 157 no 8 pp 993ndash1023 2006
[30] I Perfilieva and E Chaldeeva ldquoFuzzy transformationrdquo inProceedings of 9th IFSA World Congress and 20th NAFIPSInternational Conference pp 1946ndash1948 2001
[31] F Di Martino V Loia I Perfilieva and S Sessa ldquoAn imagecodingdecoding method based on direct and inverse fuzzytransformsrdquo International Journal of Approximate Reasoningvol 48 no 1 pp 110ndash131 2008
[32] F Di Martino and S Sessa ldquoCompression and decompressionof images with discrete fuzzy transformsrdquo Information Sciencesvol 177 no 11 pp 2349ndash2362 2007
[33] I Perfilieva and B De Baets ldquoFuzzy transforms of monotonefunctions with application to image compressionrdquo InformationSciences vol 180 no 17 pp 3304ndash3315 2010
[34] F Di Martino V Loia and S Sessa ldquoFuzzy transforms methodand attribute dependency in data analysisrdquo Information Sci-ences vol 180 no 4 pp 493ndash505 2010
adopt a preprocessing phase for compressing each imagewithseveral compression rates In Figure 1 we show the prepro-cessing phase on a dataset of color images We compresseach color image in the three monochromatic componentscorresponding to the three bands 119877 119866 and 119861
At the end of the preprocessing phase we can usethe compressed image dataset for image matching analysisSupposing that the original image dataset was composed bys color images of sizes 119873 times 119872 using a compression rate120588 = (119899 times 119898)(119873 times 119872) we obtain that the dimension of thecompressed image dataset is constituted totally of 3119904(119899 times 119898)pixels
In Figure 2 we schematize the image matching processThe sample image is compressed by the F-transformmethodthen we compare the three compressed bands of each imageobtained via F-transform with those ones deduced for thesample image by using the Peak Signal to Noise Ratio (forshort PSNR) At the end of this process we determine the
image in the dataset with the greatest overall PSNR withrespect to the sample image
Here a monochromatic image or a band of a color image119868 of sizes 119873 times 119872 is interpreted as a fuzzy relation 119877 whoseentries 119877(119909 119910) are obtained by normalizing the intensity119868(119909 119910) of each pixel with respect to the length 119871 of the scalethat is 119877(119909 119910) = 119868(119909 119910)119871 We show that our F-transformapproach can be also applied in image matching processes toimages of sizes119872times119873 (eventually119872 =119873) giving analogousresults with respect to that one obtained with GEFS andSEFS based method The comparison tests are made on the256 times 256 color image dataset extracted from View SphereDatabase an image dataset consisting in a set of images ofobjects in which an object is photographed from variousdirections by using a camera placed on a semisphere whosecenter is the same considered object We also use the OhioState University color video datasets sample for our testsEach video is composed by frames consisting of color images
Advances in Fuzzy Systems 3
R2
y
x
R1
z
Φ2
12057921205791
Φ1
Figure 3 The angles for two directions 1198771and 119877
2(the object is in
the origin)
we show the results for the Mom-Daughter and sflowgmotions In Section 2we recall the concepts of F-transform intwo variables In Section 3we recall theGEFS and SEFS basedmethod in Section 4wepropose our imagematchingmethodbased on the F-transforms Our experiments are illustrated inSection 5 and Section 6 is conclusive
2 F-Transforms in Two Variables
Following [29] and limiting ourselves to the discrete case let119899 ge 2 and 119909
1 1199092 119909
119899be a increasing sequence of points
(nodes) of [119886 119887] 1199091= 119886 lt 119909
119894(119909119894+ 119909) for every 119909 isin [0 ℎ] and 119894 =
2 119899 minus 1
(8) 119860119894+1(119909) = 119860
119894(119909 minus ℎ) for every 119909 isin [119909
119894 119909119894+1] and 119894 =
1 2 119899 minus 1
Let 119898 ge 2 1199101 1199102 119910
119898isin [119888 119889] be 119898 nodes such that
1199101= 119888 lt sdot sdot sdot lt 119910
119898= 119889 Furthermore let119861
1 119861
119898 [119888 119889] rarr
[0 1] be a fuzzy partition of [119888 119889] and let 119891 119875 times 119876 rarr
reals be an assigned function 119875 times 119876 sube [119886 119887] times [119888 119889] with119875 = 119901
1 119901
119873 and 119876 = 119902
1 119902
119872 being ldquosufficiently
denserdquo sets with respect to the chosen partitions that is foreach 119894 = 1 119873 (resp 119895 = 1 119872) there exists an index119896 isin 1 119899 (resp 119897 isin 1 119898) such that 119860
119896(119901119894) gt 0
0 20 40 60 8014
16
18
20
22
24
26
PSNR120588(R1 R2)
PSN
R 120588(R
1R
2)
D (R1 R2)
Figure 4 Trend of PSNR with respect to 119863 index for the eraserobtained from the comparison with the sample image at 120579 = 11
∘
and 120601 = 36∘
(resp 119861119897(119902119895) gt 0) The matrix [119865
119896119897] is said to be the direct F-
transform of 119891 with respect to 1198601 119860
119899 and 119861
1 119861
119898
if we have for each 119896 = 1 119899 and 119897 = 1 119898
Theorem 1 Let 119891 119875 times 119876 rarr reals be a given function119875 times 119876 sube [119886 119887] times [119888 119889] with 119875 = 119901
1 119901
119873 and 119876 =
1199021 119902
119872 Then for every 120576 gt 0 there exist two integers
119899(120576) 119898(120576) and related fuzzy partitions 1198601 1198602 119860
119899(120576) of
[119886 119887] and 1198611 1198612 119861
119898(120576) of [119888 119889] such that the sets 119875119876 are
sufficiently dense with respect to such partitions and |119891(119901119894 119902119895)minus
119891119865
119899(120576)119898(120576)(119901119894 119902119895)| lt 120576 is satisfied for every 119894 isin 1 119873 and
119895 isin 1 119872
Let 119877 be a gray image of sizes 119873 times 119872 seen as 119877
(119894 119895) isin 1 119873 times 1 119872 rarr [0 1] with 119877(119894 119895) beingthe normalized value of the pixel 119875(119894 119895) given by 119877(119894 119895) =119875(119894 119895)119871119905 if 119871119905 is the length of the gray scale In [27] 119877 is
4 Advances in Fuzzy Systems
Figure 5 Eraser at 120579 = 11∘ and 120601 = 36∘
Figure 6 Eraser at 120579 = 10∘ and 120601 = 54∘
compressed via the F-transform defined for each 119896 = 1 119899and 119897 = 1 119898 as
which approximates 119877119861up to an arbitrary quantity 120576 in the
sense of Theorem 1 which unfortunately does not give amethod for finding two integers 119899(119861) and 119898(119861) such that|119877119861(119901119894 119902119895) minus 119877
119865
119899(120576)119898(120576)(119901119894 119902119895)| lt 120576 Then we prove several
values of 119899(119861) and 119898(119861) For every compression rate 120588 weevaluate the quality of the reconstructed image via the PSNRdefined as
(PSNR)120588= 20 log
10
119871
(RMSE)120588
(9)
where (RMSE)120588is
(RMSE)120588=radicsum119873
119894=1sum119872
119895=1(119877 (119894 119895) minus 119877
119865
119873119872(119894 119895))
2
119873 times119872
(10)
Here 119877119865119873119872
is the reconstructed image obtained by recompos-ing the blocks 119877119865
119899(119861)119898(119861)
1015840
3 Max-Min and Min-Max Eigen Fuzzy Sets
Let 119883 be a nonempty finite set 119877 119883 times 119883 rarr [0 1] and119860 119883 rarr [0 1] such that
119877 ∘ 119860 = 119860 (11)
where ldquo∘rdquo is the max-min composition In terms of member-ship functions we have that
119860 (119910) = max119909isin119883
min (119860 (119909) 119877 (119909 119910) (12)
for all 119909 119910 isin 119883 and 119860 is defined as an Eigen Fuzzy Set of 119877Let 119860
119894 119883 rarr [0 1] 119894 = 1 2 be defined iteratively by
1198601(119911) = max
119909isin119883
119877 (119909 119911)
1198602= 119877 ∘ 119860
1 119860
119899+1= 119877 ∘ 119860
119899 119911 isin 119883
(13)
6 Advances in Fuzzy Systems
It is known [2 24 25] that there exists an integer 119901 isin
1 card119883 such that 119860119901is the GEFS of 119877 with respect to
the max-min composition We also consider the following
119877◻119860 = 119860 (14)
where ldquo◻rdquo denotes themin-max composition that is in termsof membership functions
119860 (119910) = min119909isin119883
max (119860 (119909) 119877 (119909 119910) (15)
for all 119909 119910 isin 119883 and 119860 is also defined to be an Eigen FuzzySet of 119877 with respect to the min-max composition It is easilyseen that (14) is equivalent to the following
119877◻119860 = 119860 (16)
where 119877 and119860 are pointwise defined as 119877(119909 119910) = 1 minus119877(119909 119910)and 119860(119909) = 1 minus 119860(119909) for all 119909 119910 isin 119883 Since 119860
119901for some 119901 isin
1 card119883 is the GEFS of 119877 with respect to the max-mincomposition it is immediately proved that the fuzzy set 119861 119883 rarr [0 1] defined as 119861(119909) = 1 minus 119860
119901(119909) for every 119909 isin [0 1]
is the SEFS of 119877 with respect to the min-max compositionIn [27] a distance based on GEFS and SEFS for image
matching is used over images of sizes 119873 times 119873 Indeedconsidering two single-band images of sizes 119873 times 119873 say 119877
1
and 1198772 such distance is given by
119889 (1198771 1198772) = sum
119909isin119883
((1198601(119909) minus 119860
2(119909))2
+ (1198611(119909) minus 119861
2(119909))2
)
(17)
where 119883 = 1 2 119873 119860119894 119861119894are the GEFS and SEFS of
the fuzzy relation 119877119894 respectively obtained by normalizing
in [0 1] the pixels of the image 119868119894 119894 = 1 2
In [26 27] experiments are presented over color imagesof sizes 256 times 256 concerning two objects (an eraser anda pen) extracted from View Sphere Database Each objectis put in the center of a semisphere on which a camera isplaced in 91 different directions The camera establishes animage (photography) of the object for each direction whichcan be identified from two angles 120579 (0∘ lt 120579 lt 90
∘) and
Φ (minus180∘lt Φ lt 180
∘) as illustrated in Figure 3
A sample image 1198771(with given 120579 = 11
∘ Φ = 36∘
for the eraser and 120579 = 10∘ Φ = 54
∘ for the pen) isto be compared with another image 119877
2chosen among the
remaining 90 directions GEFS and SEFS are calculated in thethree components of each image in the RGB space for whichit is natural to assume the following extension of (17)
where 119889119877(1198771 1198772) 119889119866(1198771 1198772) 119889119861(1198771 1198772) are the measures
(17) calculated in each band 119877 119866 119861 For image matchingthe GEFS and SEFS components in each band are extractedfrom each image thus forming a dataset with reduced storagememoryAn image is comparedwith the images in the datasetusing (18) If the dataset contains 119904 color images of sizes119873times119873
Table 1 Best distances fromGEFS and SEFS basedmethod with 120588 =0007813 for the eraser image dataset obtained from the comparisonwith the sample image at 120579 = 11∘ and 120601 = 36∘
and the dimension of the original dataset is 31199041198732 then thedimension of the GEFS and SEFS dataset is 6119904119873 so we havea compression rate given by
120588 =6119904119873
31199041198732=2
119873 (19)
So we obtain a compression rate 120588 = 0007813 if119873 = 256
4 The Image Matching Process viaF-Transforms
We consider an image dataset formed by color images of sizes119873 times119872 In the preprocessing phase we compress each imageof the dataset using the direct F-transform Each image isdivided in blocks of sizes 119873(119861) times 119872(119861) and each block iscompressed in a block of sizes 119899(119861) times 119898(119861) Thus the imagesare coded with a compression rate 120588 = (119899(119861)times119898(119861))(119873(119861)times119872(119861)) In our experiments we set the sizes of the originaland compressed blocks so that 120588 is comparable with (18) Forexample for 119873 = 119872 = 256 we use 119873(119861) = 119872(119861) = 24 and119899(119861) = 119898(119861) = 2 so 120588 = 0006944
In the reduced dataset we store the F-transform compo-nents of each image We use the PSNR between a sampleimage 119877
1and an image 119877
2defined for every compression rate
120588 (cf (9)) as
PSNR120588(1198771 1198772) = 20 log
10
119871
RMSE120588(1198771 1198772) (20)
Advances in Fuzzy Systems 7
Table 2 Best PSNR from the F-transform based method with 119871 = 255 and 120588 = 0006944 for the eraser image dataset obtained from thecomparison with the sample image at 120579 = 11∘ and 120601 = 36∘
Figure 12 Trend of PSNR (120588 = 0006944) with respect to framenumber for the video Mom-Daughter
Figure 13 Frame 1 in sflowg
where RMSE (Root Mean Square Error) is given by (cf (10))
RMSE120588(1198771 1198772) =
radicsum119873
119894=1sum119872
119895=1(1198771(119894 119895) minus 119877
2(119894 119895))2
119873 times119872
(21)
If we have color images we define an overall PSNR as
PSNR120588(1198771 1198772)
=1
3[PSNR
120588119877(1198771 1198772) + PSNR
120588119866(1198771 1198772)
+PSNR120588119861(1198771 1198772)]
(22)
Figure 14 Frame 2 in sflowg
0 10 20 30 40 50
PSNR
Frame n∘
20
25
30
35
40
45
PSN
R
Figure 15 Trend of PSNR (120588 = 0006944) with respect to framenumber for the video sflowg
where PSNR120588119877(1198771 1198772) PSNR
120588119866(1198771 1198772) PSNR
120588119861(1198771 1198772) are
the similarity measures (20) calculated in each band 119877 119866119861 compression rate 120588 In our experiments we compare theresults obtained by using the F-transforms (resp GEFS andSEFS) based method with the PSNR (20) (resp (18)) Weuse the color image datasets of 256 gray levels and of sizes256 times 256 pixels available in the View Sphere Database foreach object considered the best image 119877
2of the object itself
maximizes the PSNR (22) In other experiments we use ourF-transform method over color video datasets in which eachframe is formed by images of 256 gray levels and of sizes
8 Advances in Fuzzy Systems
20 30 40 50 60
PSNR0
PSN
R di
ffere
nce
PSNR difference
0
02
04
06
08
1
12
14
16
18
2
Figure 16 Trend of PSNR difference with respect to PSNR0(120588 =
0006944)
Table 3 Best distances from GEFS and SEFS based method with 120588= 0007813 for the pen image dataset obtained from the comparisonwith the sample image at 120579 = 10∘ and 120601 = 54∘
360times24 available in the Ohio University sample digital colorvideo database A color video is schematically formed by asequence of frames If we consider a frame in a video as thesample image we prove that the image with greatest PSNRwith respect to the sample image is an image with framenumber close to the frame number of the sample image
5 Results of Tests
We compare results obtained by using the GEFS and SEFSand F-transform based methods for image matching onall the image datasets each of sizes 256 times 256 extractedfrom the View Sphere Database In the first image datasetconcerning an eraser we consider as sample image 119877
1 the
image obtained from the camera in the direction with angles120579 = 11
∘ and 120601 = 36∘ For brevity we consider a dataset of 40test images andwe compare119877
1with the images considered in
the remaining 40 other directions In Table 1 (resp Table 2)we report the distances (17) and (18) (resp PSNR (20) and
(22) with 119871 = 255) obtained using the GEFS and SEFS (respF-transform) based method
In Figure 4we show the trend of the index PSNRobtainedby the F-Transform method with respect to the distance (18)obtained using the GEFS and SEFS method
As we can see from Tables 1 and 2 both methods give thesame reply the better image similar with the image eraser inthe direction 120579 = 11∘ and120601 = 36∘ (Figure 5) is given from thatone at 120579 = 10∘ and 120601 = 54∘ (Figure 6) The trend in Figure 4shows that the value of the distance (18) increases as the PSNRdecreases
In order to have a further confirmation of our approachwe have considered a second object a pen contained in theView Sphere Database whose sample image 119877
1is obtained
from the camera in the direction with angles 120579 = 10∘ and
120601 = 54∘ We also limit the problem to a dataset of 40 test
images whose best distances (17) and (18) (resp (20) and (22)with 119871 = 255) under the SEFS and GEFS (resp F-transform)based method are reported in Table 3 (resp Table 4)
In Figure 7we show the trend of the index PSNRobtainedby the F-transform method with respect to the distance 119863obtained by using the GEFS and SEFS method As we cansee from Tables 3 and 4 in both methods the best imagesimilar to the original image in the directions 120579 = 10∘ and120601 = 54
∘ (Figure 8) is given from that one at 120579 = 10∘ and
120601 = 18∘ (Figure 9) Also in this example the trend in Figure 7
shows that the value of the distance (18) increases as the PSNRdecreases
Now we present the results over a sequence of framesof a video Mom-Daughter available in the Ohio Universitysample digital color video database Each frame is a colorimage of sizes 360 times 240 with 256 gray levels for each bandWe use our method with a compression rate 120588 = 0006944that is in each band every frame is decomposed in 150 blocksand each block has sizes 24 times 24 compressed to a block ofsizes 2 times 2 Since119872 =119873 the GEFS and SEFS based methodis not applicable We set the sample image as the imagecorresponding to the first frame of the video We expectthat the frame number of the image with higher PSNR withrespect to the sample image is the image with frame numberclose to the frame number of sample image In Table 5 wereport the best results obtained using the F-transform basedmethod in terms of the (20) and (22) with 119871 = 255 Asexpected albeit with slight variations all the PSNRs diminishby increasing of the frame number and the second frame(Figure 11) is the frame with the greatest PSNR w r t the firstframe (Figure 10) containing the sample image
In Figure 12 we show the trend of the PSNR (22) with theframe numberThis trend is obtained for all the sample videoframes in the video dataset For reasons of brevity now wereport only the results obtained for another test performedon the sequence of frames of another video in the Ohiosample digital video database the video sflowg The PSNR inFigure 15 diminishes by increasing the frame number and thesecond frame (Figure 14) is the frame with the greatest PSNRw r t the first frame (Figure 13) containing the sample image
For supporting the validity of the F-transform methodfor all the sample frames we measure for the frame withthe greatest PSNR wrt the sample frame the correspondent
Advances in Fuzzy Systems 9
Table 4 Best PSNR from the F-transform based method with 119871 = 255 and 120588 = 0006944 for the pen image dataset obtained from thecomparison with the sample image at 120579 = 10∘ and 120601 = 54∘
value PSNR0obtained by using the original frame instead
of the correspondent compressed frame decoded via theinverse F-Transform In Figure 16 we show the trend of thedifference PSNR
0minus PSNR with respect to PSNR
0 The trend
indicates that this difference is always less than 2 This resultshows that if we compress the images in the dataset with rate120588 = 0006944 by using the F-transform method we can usethe compressed image dataset for image matching processescomparing the decompressed image with respect to a sampleimage despite the loss of information due to the compression
6 Conclusions
The results on the images of sizes 119873 times 119872 (119872 = 119873 = 256)
of the View Sphere Image Database show that using our F-transform based method we obtain the same results in termsof image matching and in terms of reduced memory storagereached also via the GEFS and SEFS based method which isapplicable only over images with 119873 = 119872 while our methodconcerns images of any sizes
Moreover our tests executed on color video frames of sizes119873times119872 (119872 = 360119873 = 240 pixels with 256 gray levels) of theOhio University color videos dataset show that by choosingthe first frame as the sample image we obtain as image withthe highest PSNR that one corresponding to the successiveframe as expected although a loss of information on thedecoded images because of the compression process
References
[1] I Bartolini P Ciaccia and M Patella ldquoQuery processing issuesin region-based image databasesrdquo Knowledge and InformationSystems vol 25 no 2 pp 389ndash420 2010
[2] B De Baets ldquoAnalytical solution methods for fuzzy relationalequationsrdquo in Fundamentals of Fuzzy Sets D Dubois and HPrade Eds vol 1 ofTheHandbooks of Fuzzy Sets Series pp 291ndash340 Kluwer Academic Publishers DordrechtTheNetherlands2000
[3] A Di NolaW Pedrycz S Sessa and E Sanchez Fuzzy RelationEquations and Their Application to Knowledge EngineeringKluwer Academic Publishers Dordrecht The Netherlands1989
[4] M Higashi and G J Klir ldquoResolution of finite fuzzy relationequationsrdquo Fuzzy Sets and Systems vol 13 no 1 pp 65ndash82 1984
[5] P Li and S C Fang ldquoA survey on fuzzy relational equationspart I classification and solvabilityrdquo Fuzzy Optimization andDecision Making vol 8 no 2 pp 179ndash229 2009
[6] V Loia W Pedrycz and S Sessa ldquoFuzzy relation calculusin the compression and decompression of fuzzy relationsrdquoInternational Journal of Image and Graphics vol 2 no 4 pp617ndash631 2002
[7] A VMarkovskii ldquoOn the relation between equationswithmax-product composition and the covering problemrdquo Fuzzy Sets andSystems vol 153 no 2 pp 261ndash273 2005
[8] M Miyakoshi and M Shimbo ldquoLower solutions of systems offuzzy equationsrdquo Fuzzy Sets and Systems vol 19 no 1 pp 37ndash461986
[9] K Peeva ldquoResolution of min-max fuzzy relational equationsrdquoin Fuzzy Partial Differential Equations and Relational EquationsM Nikravesh L A Zadeh and V Korotkikh Eds pp 153ndash166Springer New York NY USA 2004
[10] K Peeva ldquoUniversal algorithm for solving fuzzy relationalequationsrdquo Italian Jour-Nal of Pure and Applied Mathematicsvol 19 pp 9ndash20 2006
[11] K Peeva and Y Kyosev ldquoAlgorithm for solving max-productfuzzy relational equationsrdquo Soft Computing vol 11 no 7 pp593ndash605 2007
[12] K Peeva andY Kyosev Fuzzy Relational Calculus-Theory Appli-cations and Software vol 22 of Advances in Fuzzy SystemsmdashApplications and Theory World Scientific Publishing RiverEdge NJ USA 2004
[13] E Sanchez ldquoResolution of composite fuzzy relation equationsrdquoInformation and Control vol 30 no 1 pp 38ndash48 1976
[14] B S Shieh ldquoNew resolution of finite fuzzy relation equationswith max-min compositionrdquo International Journal of Uncer-tainty Fuzziness and Knowlege-Based Systems vol 16 no 1 pp19ndash33 2008
10 Advances in Fuzzy Systems
[15] Y K Wu and S M Guu ldquoAn efficient procedure for solvinga fuzzy relational equation with max-Archimedean t-normcompositionrdquo IEEE Transactions on Fuzzy Systems vol 16 no1 pp 73ndash84 2008
[16] F Di Martino V Loia and S Sessa ldquoA method in the com-pressiondecompression of images using fuzzy equations andfuzzy similaritiesrdquo in Proceedings of IFSA Conference pp 524ndash527 2003
[17] F Di Martino V Loia and S Sessa ldquoA method for cod-ingdecoding images by using fuzzy relation equationsrdquo in10th International Fuzzy Systems Association World Congress TBilgic B De Baets and O Kaynak Eds vol 2715 of LectureNotes in Artificial Intelligence pp 436ndash441 Springer BerlinGermany July 2003
[18] V Loia and S Sessa ldquoFuzzy relation equations for cod-ingdecoding processes of images and videosrdquo InformationSciences vol 171 no 1-3 pp 145ndash172 2005
[19] H Nobuhara W Pedrycz and K Hirota ldquoFast solving methodof fuzzy relational equation and its application to lossy imagecompressionreconstructionrdquo IEEE Transactions on Fuzzy Sys-tems vol 8 no 3 pp 325ndash334 2000
[20] F Di Martino V Loia and S Sessa ldquoA segmentation methodfor images compressed by fuzzy transformsrdquo Fuzzy Sets andSystems vol 161 no 1 pp 56ndash74 2010 Special section NewTrends on Pattern Recognition with Fuzzy Models
[21] J Lu R Li Y Zhang T Zhao and Z Lu ldquoImage annotationtechniques based on feature selection for class-pairsrdquo Knowl-edge and Information Systems vol 24 no 2 pp 325ndash337 2010
[22] I Perfilieva V Novak and A Dvorak ldquoFuzzy transform inthe analysis of datardquo International Journal of ApproximateReasoning vol 48 no 1 pp 36ndash46 2008
[23] L Chen andPWang ldquoFuzzy relational equations (I) the generaland specialized solving algorithmsrdquo Soft Computing vol 6 pp428ndash435 2002
[24] E Sanchez ldquoResolution of Eigen fuzzy sets equationsrdquo FuzzySets and Systems vol 1 no 1 pp 69ndash74 1978
[25] E Sanchez ldquoEigen fuzzy sets and fuzzy relationsrdquo Journal ofMathematical Analysis and Applications vol 81 no 2 pp 399ndash421 1981
[26] F Di Martino S Sessa and H Nobuhara ldquoEigen fuzzy sets andimage information retrievalrdquo in Proceedings of IEEE Interna-tional Conference on Fuzzy Systems vol 3 pp 1385ndash1390 IEEEPress July 2004
[27] F Di Martino H Nobuhara and S Sessa ldquoEigen fuzzy setsand image information retrievalrdquo in Handbook of GranularComputIng W Pedrycz A Skowron and V Kreinovich Edspp 863ndash872 John Wiley amp Sons Chichester UK 2008
[28] I Perfilieva ldquoFuzzy transforms application to reef growthproblemrdquo in Fuzzy Logic in Geology R B Demicco and GJ Klir Eds pp 275ndash300 Academic Press Amsterdam TheNetherlands 2003
[29] I Perfilieva ldquoFuzzy transforms theory and applicationsrdquo FuzzySets and Systems vol 157 no 8 pp 993ndash1023 2006
[30] I Perfilieva and E Chaldeeva ldquoFuzzy transformationrdquo inProceedings of 9th IFSA World Congress and 20th NAFIPSInternational Conference pp 1946ndash1948 2001
[31] F Di Martino V Loia I Perfilieva and S Sessa ldquoAn imagecodingdecoding method based on direct and inverse fuzzytransformsrdquo International Journal of Approximate Reasoningvol 48 no 1 pp 110ndash131 2008
[32] F Di Martino and S Sessa ldquoCompression and decompressionof images with discrete fuzzy transformsrdquo Information Sciencesvol 177 no 11 pp 2349ndash2362 2007
[33] I Perfilieva and B De Baets ldquoFuzzy transforms of monotonefunctions with application to image compressionrdquo InformationSciences vol 180 no 17 pp 3304ndash3315 2010
[34] F Di Martino V Loia and S Sessa ldquoFuzzy transforms methodand attribute dependency in data analysisrdquo Information Sci-ences vol 180 no 4 pp 493ndash505 2010
Figure 3 The angles for two directions 1198771and 119877
2(the object is in
the origin)
we show the results for the Mom-Daughter and sflowgmotions In Section 2we recall the concepts of F-transform intwo variables In Section 3we recall theGEFS and SEFS basedmethod in Section 4wepropose our imagematchingmethodbased on the F-transforms Our experiments are illustrated inSection 5 and Section 6 is conclusive
2 F-Transforms in Two Variables
Following [29] and limiting ourselves to the discrete case let119899 ge 2 and 119909
1 1199092 119909
119899be a increasing sequence of points
(nodes) of [119886 119887] 1199091= 119886 lt 119909
119894(119909119894+ 119909) for every 119909 isin [0 ℎ] and 119894 =
2 119899 minus 1
(8) 119860119894+1(119909) = 119860
119894(119909 minus ℎ) for every 119909 isin [119909
119894 119909119894+1] and 119894 =
1 2 119899 minus 1
Let 119898 ge 2 1199101 1199102 119910
119898isin [119888 119889] be 119898 nodes such that
1199101= 119888 lt sdot sdot sdot lt 119910
119898= 119889 Furthermore let119861
1 119861
119898 [119888 119889] rarr
[0 1] be a fuzzy partition of [119888 119889] and let 119891 119875 times 119876 rarr
reals be an assigned function 119875 times 119876 sube [119886 119887] times [119888 119889] with119875 = 119901
1 119901
119873 and 119876 = 119902
1 119902
119872 being ldquosufficiently
denserdquo sets with respect to the chosen partitions that is foreach 119894 = 1 119873 (resp 119895 = 1 119872) there exists an index119896 isin 1 119899 (resp 119897 isin 1 119898) such that 119860
119896(119901119894) gt 0
0 20 40 60 8014
16
18
20
22
24
26
PSNR120588(R1 R2)
PSN
R 120588(R
1R
2)
D (R1 R2)
Figure 4 Trend of PSNR with respect to 119863 index for the eraserobtained from the comparison with the sample image at 120579 = 11
∘
and 120601 = 36∘
(resp 119861119897(119902119895) gt 0) The matrix [119865
119896119897] is said to be the direct F-
transform of 119891 with respect to 1198601 119860
119899 and 119861
1 119861
119898
if we have for each 119896 = 1 119899 and 119897 = 1 119898
Theorem 1 Let 119891 119875 times 119876 rarr reals be a given function119875 times 119876 sube [119886 119887] times [119888 119889] with 119875 = 119901
1 119901
119873 and 119876 =
1199021 119902
119872 Then for every 120576 gt 0 there exist two integers
119899(120576) 119898(120576) and related fuzzy partitions 1198601 1198602 119860
119899(120576) of
[119886 119887] and 1198611 1198612 119861
119898(120576) of [119888 119889] such that the sets 119875119876 are
sufficiently dense with respect to such partitions and |119891(119901119894 119902119895)minus
119891119865
119899(120576)119898(120576)(119901119894 119902119895)| lt 120576 is satisfied for every 119894 isin 1 119873 and
119895 isin 1 119872
Let 119877 be a gray image of sizes 119873 times 119872 seen as 119877
(119894 119895) isin 1 119873 times 1 119872 rarr [0 1] with 119877(119894 119895) beingthe normalized value of the pixel 119875(119894 119895) given by 119877(119894 119895) =119875(119894 119895)119871119905 if 119871119905 is the length of the gray scale In [27] 119877 is
4 Advances in Fuzzy Systems
Figure 5 Eraser at 120579 = 11∘ and 120601 = 36∘
Figure 6 Eraser at 120579 = 10∘ and 120601 = 54∘
compressed via the F-transform defined for each 119896 = 1 119899and 119897 = 1 119898 as
which approximates 119877119861up to an arbitrary quantity 120576 in the
sense of Theorem 1 which unfortunately does not give amethod for finding two integers 119899(119861) and 119898(119861) such that|119877119861(119901119894 119902119895) minus 119877
119865
119899(120576)119898(120576)(119901119894 119902119895)| lt 120576 Then we prove several
values of 119899(119861) and 119898(119861) For every compression rate 120588 weevaluate the quality of the reconstructed image via the PSNRdefined as
(PSNR)120588= 20 log
10
119871
(RMSE)120588
(9)
where (RMSE)120588is
(RMSE)120588=radicsum119873
119894=1sum119872
119895=1(119877 (119894 119895) minus 119877
119865
119873119872(119894 119895))
2
119873 times119872
(10)
Here 119877119865119873119872
is the reconstructed image obtained by recompos-ing the blocks 119877119865
119899(119861)119898(119861)
1015840
3 Max-Min and Min-Max Eigen Fuzzy Sets
Let 119883 be a nonempty finite set 119877 119883 times 119883 rarr [0 1] and119860 119883 rarr [0 1] such that
119877 ∘ 119860 = 119860 (11)
where ldquo∘rdquo is the max-min composition In terms of member-ship functions we have that
119860 (119910) = max119909isin119883
min (119860 (119909) 119877 (119909 119910) (12)
for all 119909 119910 isin 119883 and 119860 is defined as an Eigen Fuzzy Set of 119877Let 119860
119894 119883 rarr [0 1] 119894 = 1 2 be defined iteratively by
1198601(119911) = max
119909isin119883
119877 (119909 119911)
1198602= 119877 ∘ 119860
1 119860
119899+1= 119877 ∘ 119860
119899 119911 isin 119883
(13)
6 Advances in Fuzzy Systems
It is known [2 24 25] that there exists an integer 119901 isin
1 card119883 such that 119860119901is the GEFS of 119877 with respect to
the max-min composition We also consider the following
119877◻119860 = 119860 (14)
where ldquo◻rdquo denotes themin-max composition that is in termsof membership functions
119860 (119910) = min119909isin119883
max (119860 (119909) 119877 (119909 119910) (15)
for all 119909 119910 isin 119883 and 119860 is also defined to be an Eigen FuzzySet of 119877 with respect to the min-max composition It is easilyseen that (14) is equivalent to the following
119877◻119860 = 119860 (16)
where 119877 and119860 are pointwise defined as 119877(119909 119910) = 1 minus119877(119909 119910)and 119860(119909) = 1 minus 119860(119909) for all 119909 119910 isin 119883 Since 119860
119901for some 119901 isin
1 card119883 is the GEFS of 119877 with respect to the max-mincomposition it is immediately proved that the fuzzy set 119861 119883 rarr [0 1] defined as 119861(119909) = 1 minus 119860
119901(119909) for every 119909 isin [0 1]
is the SEFS of 119877 with respect to the min-max compositionIn [27] a distance based on GEFS and SEFS for image
matching is used over images of sizes 119873 times 119873 Indeedconsidering two single-band images of sizes 119873 times 119873 say 119877
1
and 1198772 such distance is given by
119889 (1198771 1198772) = sum
119909isin119883
((1198601(119909) minus 119860
2(119909))2
+ (1198611(119909) minus 119861
2(119909))2
)
(17)
where 119883 = 1 2 119873 119860119894 119861119894are the GEFS and SEFS of
the fuzzy relation 119877119894 respectively obtained by normalizing
in [0 1] the pixels of the image 119868119894 119894 = 1 2
In [26 27] experiments are presented over color imagesof sizes 256 times 256 concerning two objects (an eraser anda pen) extracted from View Sphere Database Each objectis put in the center of a semisphere on which a camera isplaced in 91 different directions The camera establishes animage (photography) of the object for each direction whichcan be identified from two angles 120579 (0∘ lt 120579 lt 90
∘) and
Φ (minus180∘lt Φ lt 180
∘) as illustrated in Figure 3
A sample image 1198771(with given 120579 = 11
∘ Φ = 36∘
for the eraser and 120579 = 10∘ Φ = 54
∘ for the pen) isto be compared with another image 119877
2chosen among the
remaining 90 directions GEFS and SEFS are calculated in thethree components of each image in the RGB space for whichit is natural to assume the following extension of (17)
where 119889119877(1198771 1198772) 119889119866(1198771 1198772) 119889119861(1198771 1198772) are the measures
(17) calculated in each band 119877 119866 119861 For image matchingthe GEFS and SEFS components in each band are extractedfrom each image thus forming a dataset with reduced storagememoryAn image is comparedwith the images in the datasetusing (18) If the dataset contains 119904 color images of sizes119873times119873
Table 1 Best distances fromGEFS and SEFS basedmethod with 120588 =0007813 for the eraser image dataset obtained from the comparisonwith the sample image at 120579 = 11∘ and 120601 = 36∘
and the dimension of the original dataset is 31199041198732 then thedimension of the GEFS and SEFS dataset is 6119904119873 so we havea compression rate given by
120588 =6119904119873
31199041198732=2
119873 (19)
So we obtain a compression rate 120588 = 0007813 if119873 = 256
4 The Image Matching Process viaF-Transforms
We consider an image dataset formed by color images of sizes119873 times119872 In the preprocessing phase we compress each imageof the dataset using the direct F-transform Each image isdivided in blocks of sizes 119873(119861) times 119872(119861) and each block iscompressed in a block of sizes 119899(119861) times 119898(119861) Thus the imagesare coded with a compression rate 120588 = (119899(119861)times119898(119861))(119873(119861)times119872(119861)) In our experiments we set the sizes of the originaland compressed blocks so that 120588 is comparable with (18) Forexample for 119873 = 119872 = 256 we use 119873(119861) = 119872(119861) = 24 and119899(119861) = 119898(119861) = 2 so 120588 = 0006944
In the reduced dataset we store the F-transform compo-nents of each image We use the PSNR between a sampleimage 119877
1and an image 119877
2defined for every compression rate
120588 (cf (9)) as
PSNR120588(1198771 1198772) = 20 log
10
119871
RMSE120588(1198771 1198772) (20)
Advances in Fuzzy Systems 7
Table 2 Best PSNR from the F-transform based method with 119871 = 255 and 120588 = 0006944 for the eraser image dataset obtained from thecomparison with the sample image at 120579 = 11∘ and 120601 = 36∘
Figure 12 Trend of PSNR (120588 = 0006944) with respect to framenumber for the video Mom-Daughter
Figure 13 Frame 1 in sflowg
where RMSE (Root Mean Square Error) is given by (cf (10))
RMSE120588(1198771 1198772) =
radicsum119873
119894=1sum119872
119895=1(1198771(119894 119895) minus 119877
2(119894 119895))2
119873 times119872
(21)
If we have color images we define an overall PSNR as
PSNR120588(1198771 1198772)
=1
3[PSNR
120588119877(1198771 1198772) + PSNR
120588119866(1198771 1198772)
+PSNR120588119861(1198771 1198772)]
(22)
Figure 14 Frame 2 in sflowg
0 10 20 30 40 50
PSNR
Frame n∘
20
25
30
35
40
45
PSN
R
Figure 15 Trend of PSNR (120588 = 0006944) with respect to framenumber for the video sflowg
where PSNR120588119877(1198771 1198772) PSNR
120588119866(1198771 1198772) PSNR
120588119861(1198771 1198772) are
the similarity measures (20) calculated in each band 119877 119866119861 compression rate 120588 In our experiments we compare theresults obtained by using the F-transforms (resp GEFS andSEFS) based method with the PSNR (20) (resp (18)) Weuse the color image datasets of 256 gray levels and of sizes256 times 256 pixels available in the View Sphere Database foreach object considered the best image 119877
2of the object itself
maximizes the PSNR (22) In other experiments we use ourF-transform method over color video datasets in which eachframe is formed by images of 256 gray levels and of sizes
8 Advances in Fuzzy Systems
20 30 40 50 60
PSNR0
PSN
R di
ffere
nce
PSNR difference
0
02
04
06
08
1
12
14
16
18
2
Figure 16 Trend of PSNR difference with respect to PSNR0(120588 =
0006944)
Table 3 Best distances from GEFS and SEFS based method with 120588= 0007813 for the pen image dataset obtained from the comparisonwith the sample image at 120579 = 10∘ and 120601 = 54∘
360times24 available in the Ohio University sample digital colorvideo database A color video is schematically formed by asequence of frames If we consider a frame in a video as thesample image we prove that the image with greatest PSNRwith respect to the sample image is an image with framenumber close to the frame number of the sample image
5 Results of Tests
We compare results obtained by using the GEFS and SEFSand F-transform based methods for image matching onall the image datasets each of sizes 256 times 256 extractedfrom the View Sphere Database In the first image datasetconcerning an eraser we consider as sample image 119877
1 the
image obtained from the camera in the direction with angles120579 = 11
∘ and 120601 = 36∘ For brevity we consider a dataset of 40test images andwe compare119877
1with the images considered in
the remaining 40 other directions In Table 1 (resp Table 2)we report the distances (17) and (18) (resp PSNR (20) and
(22) with 119871 = 255) obtained using the GEFS and SEFS (respF-transform) based method
In Figure 4we show the trend of the index PSNRobtainedby the F-Transform method with respect to the distance (18)obtained using the GEFS and SEFS method
As we can see from Tables 1 and 2 both methods give thesame reply the better image similar with the image eraser inthe direction 120579 = 11∘ and120601 = 36∘ (Figure 5) is given from thatone at 120579 = 10∘ and 120601 = 54∘ (Figure 6) The trend in Figure 4shows that the value of the distance (18) increases as the PSNRdecreases
In order to have a further confirmation of our approachwe have considered a second object a pen contained in theView Sphere Database whose sample image 119877
1is obtained
from the camera in the direction with angles 120579 = 10∘ and
120601 = 54∘ We also limit the problem to a dataset of 40 test
images whose best distances (17) and (18) (resp (20) and (22)with 119871 = 255) under the SEFS and GEFS (resp F-transform)based method are reported in Table 3 (resp Table 4)
In Figure 7we show the trend of the index PSNRobtainedby the F-transform method with respect to the distance 119863obtained by using the GEFS and SEFS method As we cansee from Tables 3 and 4 in both methods the best imagesimilar to the original image in the directions 120579 = 10∘ and120601 = 54
∘ (Figure 8) is given from that one at 120579 = 10∘ and
120601 = 18∘ (Figure 9) Also in this example the trend in Figure 7
shows that the value of the distance (18) increases as the PSNRdecreases
Now we present the results over a sequence of framesof a video Mom-Daughter available in the Ohio Universitysample digital color video database Each frame is a colorimage of sizes 360 times 240 with 256 gray levels for each bandWe use our method with a compression rate 120588 = 0006944that is in each band every frame is decomposed in 150 blocksand each block has sizes 24 times 24 compressed to a block ofsizes 2 times 2 Since119872 =119873 the GEFS and SEFS based methodis not applicable We set the sample image as the imagecorresponding to the first frame of the video We expectthat the frame number of the image with higher PSNR withrespect to the sample image is the image with frame numberclose to the frame number of sample image In Table 5 wereport the best results obtained using the F-transform basedmethod in terms of the (20) and (22) with 119871 = 255 Asexpected albeit with slight variations all the PSNRs diminishby increasing of the frame number and the second frame(Figure 11) is the frame with the greatest PSNR w r t the firstframe (Figure 10) containing the sample image
In Figure 12 we show the trend of the PSNR (22) with theframe numberThis trend is obtained for all the sample videoframes in the video dataset For reasons of brevity now wereport only the results obtained for another test performedon the sequence of frames of another video in the Ohiosample digital video database the video sflowg The PSNR inFigure 15 diminishes by increasing the frame number and thesecond frame (Figure 14) is the frame with the greatest PSNRw r t the first frame (Figure 13) containing the sample image
For supporting the validity of the F-transform methodfor all the sample frames we measure for the frame withthe greatest PSNR wrt the sample frame the correspondent
Advances in Fuzzy Systems 9
Table 4 Best PSNR from the F-transform based method with 119871 = 255 and 120588 = 0006944 for the pen image dataset obtained from thecomparison with the sample image at 120579 = 10∘ and 120601 = 54∘
value PSNR0obtained by using the original frame instead
of the correspondent compressed frame decoded via theinverse F-Transform In Figure 16 we show the trend of thedifference PSNR
0minus PSNR with respect to PSNR
0 The trend
indicates that this difference is always less than 2 This resultshows that if we compress the images in the dataset with rate120588 = 0006944 by using the F-transform method we can usethe compressed image dataset for image matching processescomparing the decompressed image with respect to a sampleimage despite the loss of information due to the compression
6 Conclusions
The results on the images of sizes 119873 times 119872 (119872 = 119873 = 256)
of the View Sphere Image Database show that using our F-transform based method we obtain the same results in termsof image matching and in terms of reduced memory storagereached also via the GEFS and SEFS based method which isapplicable only over images with 119873 = 119872 while our methodconcerns images of any sizes
Moreover our tests executed on color video frames of sizes119873times119872 (119872 = 360119873 = 240 pixels with 256 gray levels) of theOhio University color videos dataset show that by choosingthe first frame as the sample image we obtain as image withthe highest PSNR that one corresponding to the successiveframe as expected although a loss of information on thedecoded images because of the compression process
References
[1] I Bartolini P Ciaccia and M Patella ldquoQuery processing issuesin region-based image databasesrdquo Knowledge and InformationSystems vol 25 no 2 pp 389ndash420 2010
[2] B De Baets ldquoAnalytical solution methods for fuzzy relationalequationsrdquo in Fundamentals of Fuzzy Sets D Dubois and HPrade Eds vol 1 ofTheHandbooks of Fuzzy Sets Series pp 291ndash340 Kluwer Academic Publishers DordrechtTheNetherlands2000
[3] A Di NolaW Pedrycz S Sessa and E Sanchez Fuzzy RelationEquations and Their Application to Knowledge EngineeringKluwer Academic Publishers Dordrecht The Netherlands1989
[4] M Higashi and G J Klir ldquoResolution of finite fuzzy relationequationsrdquo Fuzzy Sets and Systems vol 13 no 1 pp 65ndash82 1984
[5] P Li and S C Fang ldquoA survey on fuzzy relational equationspart I classification and solvabilityrdquo Fuzzy Optimization andDecision Making vol 8 no 2 pp 179ndash229 2009
[6] V Loia W Pedrycz and S Sessa ldquoFuzzy relation calculusin the compression and decompression of fuzzy relationsrdquoInternational Journal of Image and Graphics vol 2 no 4 pp617ndash631 2002
[7] A VMarkovskii ldquoOn the relation between equationswithmax-product composition and the covering problemrdquo Fuzzy Sets andSystems vol 153 no 2 pp 261ndash273 2005
[8] M Miyakoshi and M Shimbo ldquoLower solutions of systems offuzzy equationsrdquo Fuzzy Sets and Systems vol 19 no 1 pp 37ndash461986
[9] K Peeva ldquoResolution of min-max fuzzy relational equationsrdquoin Fuzzy Partial Differential Equations and Relational EquationsM Nikravesh L A Zadeh and V Korotkikh Eds pp 153ndash166Springer New York NY USA 2004
[10] K Peeva ldquoUniversal algorithm for solving fuzzy relationalequationsrdquo Italian Jour-Nal of Pure and Applied Mathematicsvol 19 pp 9ndash20 2006
[11] K Peeva and Y Kyosev ldquoAlgorithm for solving max-productfuzzy relational equationsrdquo Soft Computing vol 11 no 7 pp593ndash605 2007
[12] K Peeva andY Kyosev Fuzzy Relational Calculus-Theory Appli-cations and Software vol 22 of Advances in Fuzzy SystemsmdashApplications and Theory World Scientific Publishing RiverEdge NJ USA 2004
[13] E Sanchez ldquoResolution of composite fuzzy relation equationsrdquoInformation and Control vol 30 no 1 pp 38ndash48 1976
[14] B S Shieh ldquoNew resolution of finite fuzzy relation equationswith max-min compositionrdquo International Journal of Uncer-tainty Fuzziness and Knowlege-Based Systems vol 16 no 1 pp19ndash33 2008
10 Advances in Fuzzy Systems
[15] Y K Wu and S M Guu ldquoAn efficient procedure for solvinga fuzzy relational equation with max-Archimedean t-normcompositionrdquo IEEE Transactions on Fuzzy Systems vol 16 no1 pp 73ndash84 2008
[16] F Di Martino V Loia and S Sessa ldquoA method in the com-pressiondecompression of images using fuzzy equations andfuzzy similaritiesrdquo in Proceedings of IFSA Conference pp 524ndash527 2003
[17] F Di Martino V Loia and S Sessa ldquoA method for cod-ingdecoding images by using fuzzy relation equationsrdquo in10th International Fuzzy Systems Association World Congress TBilgic B De Baets and O Kaynak Eds vol 2715 of LectureNotes in Artificial Intelligence pp 436ndash441 Springer BerlinGermany July 2003
[18] V Loia and S Sessa ldquoFuzzy relation equations for cod-ingdecoding processes of images and videosrdquo InformationSciences vol 171 no 1-3 pp 145ndash172 2005
[19] H Nobuhara W Pedrycz and K Hirota ldquoFast solving methodof fuzzy relational equation and its application to lossy imagecompressionreconstructionrdquo IEEE Transactions on Fuzzy Sys-tems vol 8 no 3 pp 325ndash334 2000
[20] F Di Martino V Loia and S Sessa ldquoA segmentation methodfor images compressed by fuzzy transformsrdquo Fuzzy Sets andSystems vol 161 no 1 pp 56ndash74 2010 Special section NewTrends on Pattern Recognition with Fuzzy Models
[21] J Lu R Li Y Zhang T Zhao and Z Lu ldquoImage annotationtechniques based on feature selection for class-pairsrdquo Knowl-edge and Information Systems vol 24 no 2 pp 325ndash337 2010
[22] I Perfilieva V Novak and A Dvorak ldquoFuzzy transform inthe analysis of datardquo International Journal of ApproximateReasoning vol 48 no 1 pp 36ndash46 2008
[23] L Chen andPWang ldquoFuzzy relational equations (I) the generaland specialized solving algorithmsrdquo Soft Computing vol 6 pp428ndash435 2002
[24] E Sanchez ldquoResolution of Eigen fuzzy sets equationsrdquo FuzzySets and Systems vol 1 no 1 pp 69ndash74 1978
[25] E Sanchez ldquoEigen fuzzy sets and fuzzy relationsrdquo Journal ofMathematical Analysis and Applications vol 81 no 2 pp 399ndash421 1981
[26] F Di Martino S Sessa and H Nobuhara ldquoEigen fuzzy sets andimage information retrievalrdquo in Proceedings of IEEE Interna-tional Conference on Fuzzy Systems vol 3 pp 1385ndash1390 IEEEPress July 2004
[27] F Di Martino H Nobuhara and S Sessa ldquoEigen fuzzy setsand image information retrievalrdquo in Handbook of GranularComputIng W Pedrycz A Skowron and V Kreinovich Edspp 863ndash872 John Wiley amp Sons Chichester UK 2008
[28] I Perfilieva ldquoFuzzy transforms application to reef growthproblemrdquo in Fuzzy Logic in Geology R B Demicco and GJ Klir Eds pp 275ndash300 Academic Press Amsterdam TheNetherlands 2003
[29] I Perfilieva ldquoFuzzy transforms theory and applicationsrdquo FuzzySets and Systems vol 157 no 8 pp 993ndash1023 2006
[30] I Perfilieva and E Chaldeeva ldquoFuzzy transformationrdquo inProceedings of 9th IFSA World Congress and 20th NAFIPSInternational Conference pp 1946ndash1948 2001
[31] F Di Martino V Loia I Perfilieva and S Sessa ldquoAn imagecodingdecoding method based on direct and inverse fuzzytransformsrdquo International Journal of Approximate Reasoningvol 48 no 1 pp 110ndash131 2008
[32] F Di Martino and S Sessa ldquoCompression and decompressionof images with discrete fuzzy transformsrdquo Information Sciencesvol 177 no 11 pp 2349ndash2362 2007
[33] I Perfilieva and B De Baets ldquoFuzzy transforms of monotonefunctions with application to image compressionrdquo InformationSciences vol 180 no 17 pp 3304ndash3315 2010
[34] F Di Martino V Loia and S Sessa ldquoFuzzy transforms methodand attribute dependency in data analysisrdquo Information Sci-ences vol 180 no 4 pp 493ndash505 2010
which approximates 119877119861up to an arbitrary quantity 120576 in the
sense of Theorem 1 which unfortunately does not give amethod for finding two integers 119899(119861) and 119898(119861) such that|119877119861(119901119894 119902119895) minus 119877
119865
119899(120576)119898(120576)(119901119894 119902119895)| lt 120576 Then we prove several
values of 119899(119861) and 119898(119861) For every compression rate 120588 weevaluate the quality of the reconstructed image via the PSNRdefined as
(PSNR)120588= 20 log
10
119871
(RMSE)120588
(9)
where (RMSE)120588is
(RMSE)120588=radicsum119873
119894=1sum119872
119895=1(119877 (119894 119895) minus 119877
119865
119873119872(119894 119895))
2
119873 times119872
(10)
Here 119877119865119873119872
is the reconstructed image obtained by recompos-ing the blocks 119877119865
119899(119861)119898(119861)
1015840
3 Max-Min and Min-Max Eigen Fuzzy Sets
Let 119883 be a nonempty finite set 119877 119883 times 119883 rarr [0 1] and119860 119883 rarr [0 1] such that
119877 ∘ 119860 = 119860 (11)
where ldquo∘rdquo is the max-min composition In terms of member-ship functions we have that
119860 (119910) = max119909isin119883
min (119860 (119909) 119877 (119909 119910) (12)
for all 119909 119910 isin 119883 and 119860 is defined as an Eigen Fuzzy Set of 119877Let 119860
119894 119883 rarr [0 1] 119894 = 1 2 be defined iteratively by
1198601(119911) = max
119909isin119883
119877 (119909 119911)
1198602= 119877 ∘ 119860
1 119860
119899+1= 119877 ∘ 119860
119899 119911 isin 119883
(13)
6 Advances in Fuzzy Systems
It is known [2 24 25] that there exists an integer 119901 isin
1 card119883 such that 119860119901is the GEFS of 119877 with respect to
the max-min composition We also consider the following
119877◻119860 = 119860 (14)
where ldquo◻rdquo denotes themin-max composition that is in termsof membership functions
119860 (119910) = min119909isin119883
max (119860 (119909) 119877 (119909 119910) (15)
for all 119909 119910 isin 119883 and 119860 is also defined to be an Eigen FuzzySet of 119877 with respect to the min-max composition It is easilyseen that (14) is equivalent to the following
119877◻119860 = 119860 (16)
where 119877 and119860 are pointwise defined as 119877(119909 119910) = 1 minus119877(119909 119910)and 119860(119909) = 1 minus 119860(119909) for all 119909 119910 isin 119883 Since 119860
119901for some 119901 isin
1 card119883 is the GEFS of 119877 with respect to the max-mincomposition it is immediately proved that the fuzzy set 119861 119883 rarr [0 1] defined as 119861(119909) = 1 minus 119860
119901(119909) for every 119909 isin [0 1]
is the SEFS of 119877 with respect to the min-max compositionIn [27] a distance based on GEFS and SEFS for image
matching is used over images of sizes 119873 times 119873 Indeedconsidering two single-band images of sizes 119873 times 119873 say 119877
1
and 1198772 such distance is given by
119889 (1198771 1198772) = sum
119909isin119883
((1198601(119909) minus 119860
2(119909))2
+ (1198611(119909) minus 119861
2(119909))2
)
(17)
where 119883 = 1 2 119873 119860119894 119861119894are the GEFS and SEFS of
the fuzzy relation 119877119894 respectively obtained by normalizing
in [0 1] the pixels of the image 119868119894 119894 = 1 2
In [26 27] experiments are presented over color imagesof sizes 256 times 256 concerning two objects (an eraser anda pen) extracted from View Sphere Database Each objectis put in the center of a semisphere on which a camera isplaced in 91 different directions The camera establishes animage (photography) of the object for each direction whichcan be identified from two angles 120579 (0∘ lt 120579 lt 90
∘) and
Φ (minus180∘lt Φ lt 180
∘) as illustrated in Figure 3
A sample image 1198771(with given 120579 = 11
∘ Φ = 36∘
for the eraser and 120579 = 10∘ Φ = 54
∘ for the pen) isto be compared with another image 119877
2chosen among the
remaining 90 directions GEFS and SEFS are calculated in thethree components of each image in the RGB space for whichit is natural to assume the following extension of (17)
where 119889119877(1198771 1198772) 119889119866(1198771 1198772) 119889119861(1198771 1198772) are the measures
(17) calculated in each band 119877 119866 119861 For image matchingthe GEFS and SEFS components in each band are extractedfrom each image thus forming a dataset with reduced storagememoryAn image is comparedwith the images in the datasetusing (18) If the dataset contains 119904 color images of sizes119873times119873
Table 1 Best distances fromGEFS and SEFS basedmethod with 120588 =0007813 for the eraser image dataset obtained from the comparisonwith the sample image at 120579 = 11∘ and 120601 = 36∘
and the dimension of the original dataset is 31199041198732 then thedimension of the GEFS and SEFS dataset is 6119904119873 so we havea compression rate given by
120588 =6119904119873
31199041198732=2
119873 (19)
So we obtain a compression rate 120588 = 0007813 if119873 = 256
4 The Image Matching Process viaF-Transforms
We consider an image dataset formed by color images of sizes119873 times119872 In the preprocessing phase we compress each imageof the dataset using the direct F-transform Each image isdivided in blocks of sizes 119873(119861) times 119872(119861) and each block iscompressed in a block of sizes 119899(119861) times 119898(119861) Thus the imagesare coded with a compression rate 120588 = (119899(119861)times119898(119861))(119873(119861)times119872(119861)) In our experiments we set the sizes of the originaland compressed blocks so that 120588 is comparable with (18) Forexample for 119873 = 119872 = 256 we use 119873(119861) = 119872(119861) = 24 and119899(119861) = 119898(119861) = 2 so 120588 = 0006944
In the reduced dataset we store the F-transform compo-nents of each image We use the PSNR between a sampleimage 119877
1and an image 119877
2defined for every compression rate
120588 (cf (9)) as
PSNR120588(1198771 1198772) = 20 log
10
119871
RMSE120588(1198771 1198772) (20)
Advances in Fuzzy Systems 7
Table 2 Best PSNR from the F-transform based method with 119871 = 255 and 120588 = 0006944 for the eraser image dataset obtained from thecomparison with the sample image at 120579 = 11∘ and 120601 = 36∘
Figure 12 Trend of PSNR (120588 = 0006944) with respect to framenumber for the video Mom-Daughter
Figure 13 Frame 1 in sflowg
where RMSE (Root Mean Square Error) is given by (cf (10))
RMSE120588(1198771 1198772) =
radicsum119873
119894=1sum119872
119895=1(1198771(119894 119895) minus 119877
2(119894 119895))2
119873 times119872
(21)
If we have color images we define an overall PSNR as
PSNR120588(1198771 1198772)
=1
3[PSNR
120588119877(1198771 1198772) + PSNR
120588119866(1198771 1198772)
+PSNR120588119861(1198771 1198772)]
(22)
Figure 14 Frame 2 in sflowg
0 10 20 30 40 50
PSNR
Frame n∘
20
25
30
35
40
45
PSN
R
Figure 15 Trend of PSNR (120588 = 0006944) with respect to framenumber for the video sflowg
where PSNR120588119877(1198771 1198772) PSNR
120588119866(1198771 1198772) PSNR
120588119861(1198771 1198772) are
the similarity measures (20) calculated in each band 119877 119866119861 compression rate 120588 In our experiments we compare theresults obtained by using the F-transforms (resp GEFS andSEFS) based method with the PSNR (20) (resp (18)) Weuse the color image datasets of 256 gray levels and of sizes256 times 256 pixels available in the View Sphere Database foreach object considered the best image 119877
2of the object itself
maximizes the PSNR (22) In other experiments we use ourF-transform method over color video datasets in which eachframe is formed by images of 256 gray levels and of sizes
8 Advances in Fuzzy Systems
20 30 40 50 60
PSNR0
PSN
R di
ffere
nce
PSNR difference
0
02
04
06
08
1
12
14
16
18
2
Figure 16 Trend of PSNR difference with respect to PSNR0(120588 =
0006944)
Table 3 Best distances from GEFS and SEFS based method with 120588= 0007813 for the pen image dataset obtained from the comparisonwith the sample image at 120579 = 10∘ and 120601 = 54∘
360times24 available in the Ohio University sample digital colorvideo database A color video is schematically formed by asequence of frames If we consider a frame in a video as thesample image we prove that the image with greatest PSNRwith respect to the sample image is an image with framenumber close to the frame number of the sample image
5 Results of Tests
We compare results obtained by using the GEFS and SEFSand F-transform based methods for image matching onall the image datasets each of sizes 256 times 256 extractedfrom the View Sphere Database In the first image datasetconcerning an eraser we consider as sample image 119877
1 the
image obtained from the camera in the direction with angles120579 = 11
∘ and 120601 = 36∘ For brevity we consider a dataset of 40test images andwe compare119877
1with the images considered in
the remaining 40 other directions In Table 1 (resp Table 2)we report the distances (17) and (18) (resp PSNR (20) and
(22) with 119871 = 255) obtained using the GEFS and SEFS (respF-transform) based method
In Figure 4we show the trend of the index PSNRobtainedby the F-Transform method with respect to the distance (18)obtained using the GEFS and SEFS method
As we can see from Tables 1 and 2 both methods give thesame reply the better image similar with the image eraser inthe direction 120579 = 11∘ and120601 = 36∘ (Figure 5) is given from thatone at 120579 = 10∘ and 120601 = 54∘ (Figure 6) The trend in Figure 4shows that the value of the distance (18) increases as the PSNRdecreases
In order to have a further confirmation of our approachwe have considered a second object a pen contained in theView Sphere Database whose sample image 119877
1is obtained
from the camera in the direction with angles 120579 = 10∘ and
120601 = 54∘ We also limit the problem to a dataset of 40 test
images whose best distances (17) and (18) (resp (20) and (22)with 119871 = 255) under the SEFS and GEFS (resp F-transform)based method are reported in Table 3 (resp Table 4)
In Figure 7we show the trend of the index PSNRobtainedby the F-transform method with respect to the distance 119863obtained by using the GEFS and SEFS method As we cansee from Tables 3 and 4 in both methods the best imagesimilar to the original image in the directions 120579 = 10∘ and120601 = 54
∘ (Figure 8) is given from that one at 120579 = 10∘ and
120601 = 18∘ (Figure 9) Also in this example the trend in Figure 7
shows that the value of the distance (18) increases as the PSNRdecreases
Now we present the results over a sequence of framesof a video Mom-Daughter available in the Ohio Universitysample digital color video database Each frame is a colorimage of sizes 360 times 240 with 256 gray levels for each bandWe use our method with a compression rate 120588 = 0006944that is in each band every frame is decomposed in 150 blocksand each block has sizes 24 times 24 compressed to a block ofsizes 2 times 2 Since119872 =119873 the GEFS and SEFS based methodis not applicable We set the sample image as the imagecorresponding to the first frame of the video We expectthat the frame number of the image with higher PSNR withrespect to the sample image is the image with frame numberclose to the frame number of sample image In Table 5 wereport the best results obtained using the F-transform basedmethod in terms of the (20) and (22) with 119871 = 255 Asexpected albeit with slight variations all the PSNRs diminishby increasing of the frame number and the second frame(Figure 11) is the frame with the greatest PSNR w r t the firstframe (Figure 10) containing the sample image
In Figure 12 we show the trend of the PSNR (22) with theframe numberThis trend is obtained for all the sample videoframes in the video dataset For reasons of brevity now wereport only the results obtained for another test performedon the sequence of frames of another video in the Ohiosample digital video database the video sflowg The PSNR inFigure 15 diminishes by increasing the frame number and thesecond frame (Figure 14) is the frame with the greatest PSNRw r t the first frame (Figure 13) containing the sample image
For supporting the validity of the F-transform methodfor all the sample frames we measure for the frame withthe greatest PSNR wrt the sample frame the correspondent
Advances in Fuzzy Systems 9
Table 4 Best PSNR from the F-transform based method with 119871 = 255 and 120588 = 0006944 for the pen image dataset obtained from thecomparison with the sample image at 120579 = 10∘ and 120601 = 54∘
value PSNR0obtained by using the original frame instead
of the correspondent compressed frame decoded via theinverse F-Transform In Figure 16 we show the trend of thedifference PSNR
0minus PSNR with respect to PSNR
0 The trend
indicates that this difference is always less than 2 This resultshows that if we compress the images in the dataset with rate120588 = 0006944 by using the F-transform method we can usethe compressed image dataset for image matching processescomparing the decompressed image with respect to a sampleimage despite the loss of information due to the compression
6 Conclusions
The results on the images of sizes 119873 times 119872 (119872 = 119873 = 256)
of the View Sphere Image Database show that using our F-transform based method we obtain the same results in termsof image matching and in terms of reduced memory storagereached also via the GEFS and SEFS based method which isapplicable only over images with 119873 = 119872 while our methodconcerns images of any sizes
Moreover our tests executed on color video frames of sizes119873times119872 (119872 = 360119873 = 240 pixels with 256 gray levels) of theOhio University color videos dataset show that by choosingthe first frame as the sample image we obtain as image withthe highest PSNR that one corresponding to the successiveframe as expected although a loss of information on thedecoded images because of the compression process
References
[1] I Bartolini P Ciaccia and M Patella ldquoQuery processing issuesin region-based image databasesrdquo Knowledge and InformationSystems vol 25 no 2 pp 389ndash420 2010
[2] B De Baets ldquoAnalytical solution methods for fuzzy relationalequationsrdquo in Fundamentals of Fuzzy Sets D Dubois and HPrade Eds vol 1 ofTheHandbooks of Fuzzy Sets Series pp 291ndash340 Kluwer Academic Publishers DordrechtTheNetherlands2000
[3] A Di NolaW Pedrycz S Sessa and E Sanchez Fuzzy RelationEquations and Their Application to Knowledge EngineeringKluwer Academic Publishers Dordrecht The Netherlands1989
[4] M Higashi and G J Klir ldquoResolution of finite fuzzy relationequationsrdquo Fuzzy Sets and Systems vol 13 no 1 pp 65ndash82 1984
[5] P Li and S C Fang ldquoA survey on fuzzy relational equationspart I classification and solvabilityrdquo Fuzzy Optimization andDecision Making vol 8 no 2 pp 179ndash229 2009
[6] V Loia W Pedrycz and S Sessa ldquoFuzzy relation calculusin the compression and decompression of fuzzy relationsrdquoInternational Journal of Image and Graphics vol 2 no 4 pp617ndash631 2002
[7] A VMarkovskii ldquoOn the relation between equationswithmax-product composition and the covering problemrdquo Fuzzy Sets andSystems vol 153 no 2 pp 261ndash273 2005
[8] M Miyakoshi and M Shimbo ldquoLower solutions of systems offuzzy equationsrdquo Fuzzy Sets and Systems vol 19 no 1 pp 37ndash461986
[9] K Peeva ldquoResolution of min-max fuzzy relational equationsrdquoin Fuzzy Partial Differential Equations and Relational EquationsM Nikravesh L A Zadeh and V Korotkikh Eds pp 153ndash166Springer New York NY USA 2004
[10] K Peeva ldquoUniversal algorithm for solving fuzzy relationalequationsrdquo Italian Jour-Nal of Pure and Applied Mathematicsvol 19 pp 9ndash20 2006
[11] K Peeva and Y Kyosev ldquoAlgorithm for solving max-productfuzzy relational equationsrdquo Soft Computing vol 11 no 7 pp593ndash605 2007
[12] K Peeva andY Kyosev Fuzzy Relational Calculus-Theory Appli-cations and Software vol 22 of Advances in Fuzzy SystemsmdashApplications and Theory World Scientific Publishing RiverEdge NJ USA 2004
[13] E Sanchez ldquoResolution of composite fuzzy relation equationsrdquoInformation and Control vol 30 no 1 pp 38ndash48 1976
[14] B S Shieh ldquoNew resolution of finite fuzzy relation equationswith max-min compositionrdquo International Journal of Uncer-tainty Fuzziness and Knowlege-Based Systems vol 16 no 1 pp19ndash33 2008
10 Advances in Fuzzy Systems
[15] Y K Wu and S M Guu ldquoAn efficient procedure for solvinga fuzzy relational equation with max-Archimedean t-normcompositionrdquo IEEE Transactions on Fuzzy Systems vol 16 no1 pp 73ndash84 2008
[16] F Di Martino V Loia and S Sessa ldquoA method in the com-pressiondecompression of images using fuzzy equations andfuzzy similaritiesrdquo in Proceedings of IFSA Conference pp 524ndash527 2003
[17] F Di Martino V Loia and S Sessa ldquoA method for cod-ingdecoding images by using fuzzy relation equationsrdquo in10th International Fuzzy Systems Association World Congress TBilgic B De Baets and O Kaynak Eds vol 2715 of LectureNotes in Artificial Intelligence pp 436ndash441 Springer BerlinGermany July 2003
[18] V Loia and S Sessa ldquoFuzzy relation equations for cod-ingdecoding processes of images and videosrdquo InformationSciences vol 171 no 1-3 pp 145ndash172 2005
[19] H Nobuhara W Pedrycz and K Hirota ldquoFast solving methodof fuzzy relational equation and its application to lossy imagecompressionreconstructionrdquo IEEE Transactions on Fuzzy Sys-tems vol 8 no 3 pp 325ndash334 2000
[20] F Di Martino V Loia and S Sessa ldquoA segmentation methodfor images compressed by fuzzy transformsrdquo Fuzzy Sets andSystems vol 161 no 1 pp 56ndash74 2010 Special section NewTrends on Pattern Recognition with Fuzzy Models
[21] J Lu R Li Y Zhang T Zhao and Z Lu ldquoImage annotationtechniques based on feature selection for class-pairsrdquo Knowl-edge and Information Systems vol 24 no 2 pp 325ndash337 2010
[22] I Perfilieva V Novak and A Dvorak ldquoFuzzy transform inthe analysis of datardquo International Journal of ApproximateReasoning vol 48 no 1 pp 36ndash46 2008
[23] L Chen andPWang ldquoFuzzy relational equations (I) the generaland specialized solving algorithmsrdquo Soft Computing vol 6 pp428ndash435 2002
[24] E Sanchez ldquoResolution of Eigen fuzzy sets equationsrdquo FuzzySets and Systems vol 1 no 1 pp 69ndash74 1978
[25] E Sanchez ldquoEigen fuzzy sets and fuzzy relationsrdquo Journal ofMathematical Analysis and Applications vol 81 no 2 pp 399ndash421 1981
[26] F Di Martino S Sessa and H Nobuhara ldquoEigen fuzzy sets andimage information retrievalrdquo in Proceedings of IEEE Interna-tional Conference on Fuzzy Systems vol 3 pp 1385ndash1390 IEEEPress July 2004
[27] F Di Martino H Nobuhara and S Sessa ldquoEigen fuzzy setsand image information retrievalrdquo in Handbook of GranularComputIng W Pedrycz A Skowron and V Kreinovich Edspp 863ndash872 John Wiley amp Sons Chichester UK 2008
[28] I Perfilieva ldquoFuzzy transforms application to reef growthproblemrdquo in Fuzzy Logic in Geology R B Demicco and GJ Klir Eds pp 275ndash300 Academic Press Amsterdam TheNetherlands 2003
[29] I Perfilieva ldquoFuzzy transforms theory and applicationsrdquo FuzzySets and Systems vol 157 no 8 pp 993ndash1023 2006
[30] I Perfilieva and E Chaldeeva ldquoFuzzy transformationrdquo inProceedings of 9th IFSA World Congress and 20th NAFIPSInternational Conference pp 1946ndash1948 2001
[31] F Di Martino V Loia I Perfilieva and S Sessa ldquoAn imagecodingdecoding method based on direct and inverse fuzzytransformsrdquo International Journal of Approximate Reasoningvol 48 no 1 pp 110ndash131 2008
[32] F Di Martino and S Sessa ldquoCompression and decompressionof images with discrete fuzzy transformsrdquo Information Sciencesvol 177 no 11 pp 2349ndash2362 2007
[33] I Perfilieva and B De Baets ldquoFuzzy transforms of monotonefunctions with application to image compressionrdquo InformationSciences vol 180 no 17 pp 3304ndash3315 2010
[34] F Di Martino V Loia and S Sessa ldquoFuzzy transforms methodand attribute dependency in data analysisrdquo Information Sci-ences vol 180 no 4 pp 493ndash505 2010
which approximates 119877119861up to an arbitrary quantity 120576 in the
sense of Theorem 1 which unfortunately does not give amethod for finding two integers 119899(119861) and 119898(119861) such that|119877119861(119901119894 119902119895) minus 119877
119865
119899(120576)119898(120576)(119901119894 119902119895)| lt 120576 Then we prove several
values of 119899(119861) and 119898(119861) For every compression rate 120588 weevaluate the quality of the reconstructed image via the PSNRdefined as
(PSNR)120588= 20 log
10
119871
(RMSE)120588
(9)
where (RMSE)120588is
(RMSE)120588=radicsum119873
119894=1sum119872
119895=1(119877 (119894 119895) minus 119877
119865
119873119872(119894 119895))
2
119873 times119872
(10)
Here 119877119865119873119872
is the reconstructed image obtained by recompos-ing the blocks 119877119865
119899(119861)119898(119861)
1015840
3 Max-Min and Min-Max Eigen Fuzzy Sets
Let 119883 be a nonempty finite set 119877 119883 times 119883 rarr [0 1] and119860 119883 rarr [0 1] such that
119877 ∘ 119860 = 119860 (11)
where ldquo∘rdquo is the max-min composition In terms of member-ship functions we have that
119860 (119910) = max119909isin119883
min (119860 (119909) 119877 (119909 119910) (12)
for all 119909 119910 isin 119883 and 119860 is defined as an Eigen Fuzzy Set of 119877Let 119860
119894 119883 rarr [0 1] 119894 = 1 2 be defined iteratively by
1198601(119911) = max
119909isin119883
119877 (119909 119911)
1198602= 119877 ∘ 119860
1 119860
119899+1= 119877 ∘ 119860
119899 119911 isin 119883
(13)
6 Advances in Fuzzy Systems
It is known [2 24 25] that there exists an integer 119901 isin
1 card119883 such that 119860119901is the GEFS of 119877 with respect to
the max-min composition We also consider the following
119877◻119860 = 119860 (14)
where ldquo◻rdquo denotes themin-max composition that is in termsof membership functions
119860 (119910) = min119909isin119883
max (119860 (119909) 119877 (119909 119910) (15)
for all 119909 119910 isin 119883 and 119860 is also defined to be an Eigen FuzzySet of 119877 with respect to the min-max composition It is easilyseen that (14) is equivalent to the following
119877◻119860 = 119860 (16)
where 119877 and119860 are pointwise defined as 119877(119909 119910) = 1 minus119877(119909 119910)and 119860(119909) = 1 minus 119860(119909) for all 119909 119910 isin 119883 Since 119860
119901for some 119901 isin
1 card119883 is the GEFS of 119877 with respect to the max-mincomposition it is immediately proved that the fuzzy set 119861 119883 rarr [0 1] defined as 119861(119909) = 1 minus 119860
119901(119909) for every 119909 isin [0 1]
is the SEFS of 119877 with respect to the min-max compositionIn [27] a distance based on GEFS and SEFS for image
matching is used over images of sizes 119873 times 119873 Indeedconsidering two single-band images of sizes 119873 times 119873 say 119877
1
and 1198772 such distance is given by
119889 (1198771 1198772) = sum
119909isin119883
((1198601(119909) minus 119860
2(119909))2
+ (1198611(119909) minus 119861
2(119909))2
)
(17)
where 119883 = 1 2 119873 119860119894 119861119894are the GEFS and SEFS of
the fuzzy relation 119877119894 respectively obtained by normalizing
in [0 1] the pixels of the image 119868119894 119894 = 1 2
In [26 27] experiments are presented over color imagesof sizes 256 times 256 concerning two objects (an eraser anda pen) extracted from View Sphere Database Each objectis put in the center of a semisphere on which a camera isplaced in 91 different directions The camera establishes animage (photography) of the object for each direction whichcan be identified from two angles 120579 (0∘ lt 120579 lt 90
∘) and
Φ (minus180∘lt Φ lt 180
∘) as illustrated in Figure 3
A sample image 1198771(with given 120579 = 11
∘ Φ = 36∘
for the eraser and 120579 = 10∘ Φ = 54
∘ for the pen) isto be compared with another image 119877
2chosen among the
remaining 90 directions GEFS and SEFS are calculated in thethree components of each image in the RGB space for whichit is natural to assume the following extension of (17)
where 119889119877(1198771 1198772) 119889119866(1198771 1198772) 119889119861(1198771 1198772) are the measures
(17) calculated in each band 119877 119866 119861 For image matchingthe GEFS and SEFS components in each band are extractedfrom each image thus forming a dataset with reduced storagememoryAn image is comparedwith the images in the datasetusing (18) If the dataset contains 119904 color images of sizes119873times119873
Table 1 Best distances fromGEFS and SEFS basedmethod with 120588 =0007813 for the eraser image dataset obtained from the comparisonwith the sample image at 120579 = 11∘ and 120601 = 36∘
and the dimension of the original dataset is 31199041198732 then thedimension of the GEFS and SEFS dataset is 6119904119873 so we havea compression rate given by
120588 =6119904119873
31199041198732=2
119873 (19)
So we obtain a compression rate 120588 = 0007813 if119873 = 256
4 The Image Matching Process viaF-Transforms
We consider an image dataset formed by color images of sizes119873 times119872 In the preprocessing phase we compress each imageof the dataset using the direct F-transform Each image isdivided in blocks of sizes 119873(119861) times 119872(119861) and each block iscompressed in a block of sizes 119899(119861) times 119898(119861) Thus the imagesare coded with a compression rate 120588 = (119899(119861)times119898(119861))(119873(119861)times119872(119861)) In our experiments we set the sizes of the originaland compressed blocks so that 120588 is comparable with (18) Forexample for 119873 = 119872 = 256 we use 119873(119861) = 119872(119861) = 24 and119899(119861) = 119898(119861) = 2 so 120588 = 0006944
In the reduced dataset we store the F-transform compo-nents of each image We use the PSNR between a sampleimage 119877
1and an image 119877
2defined for every compression rate
120588 (cf (9)) as
PSNR120588(1198771 1198772) = 20 log
10
119871
RMSE120588(1198771 1198772) (20)
Advances in Fuzzy Systems 7
Table 2 Best PSNR from the F-transform based method with 119871 = 255 and 120588 = 0006944 for the eraser image dataset obtained from thecomparison with the sample image at 120579 = 11∘ and 120601 = 36∘
Figure 12 Trend of PSNR (120588 = 0006944) with respect to framenumber for the video Mom-Daughter
Figure 13 Frame 1 in sflowg
where RMSE (Root Mean Square Error) is given by (cf (10))
RMSE120588(1198771 1198772) =
radicsum119873
119894=1sum119872
119895=1(1198771(119894 119895) minus 119877
2(119894 119895))2
119873 times119872
(21)
If we have color images we define an overall PSNR as
PSNR120588(1198771 1198772)
=1
3[PSNR
120588119877(1198771 1198772) + PSNR
120588119866(1198771 1198772)
+PSNR120588119861(1198771 1198772)]
(22)
Figure 14 Frame 2 in sflowg
0 10 20 30 40 50
PSNR
Frame n∘
20
25
30
35
40
45
PSN
R
Figure 15 Trend of PSNR (120588 = 0006944) with respect to framenumber for the video sflowg
where PSNR120588119877(1198771 1198772) PSNR
120588119866(1198771 1198772) PSNR
120588119861(1198771 1198772) are
the similarity measures (20) calculated in each band 119877 119866119861 compression rate 120588 In our experiments we compare theresults obtained by using the F-transforms (resp GEFS andSEFS) based method with the PSNR (20) (resp (18)) Weuse the color image datasets of 256 gray levels and of sizes256 times 256 pixels available in the View Sphere Database foreach object considered the best image 119877
2of the object itself
maximizes the PSNR (22) In other experiments we use ourF-transform method over color video datasets in which eachframe is formed by images of 256 gray levels and of sizes
8 Advances in Fuzzy Systems
20 30 40 50 60
PSNR0
PSN
R di
ffere
nce
PSNR difference
0
02
04
06
08
1
12
14
16
18
2
Figure 16 Trend of PSNR difference with respect to PSNR0(120588 =
0006944)
Table 3 Best distances from GEFS and SEFS based method with 120588= 0007813 for the pen image dataset obtained from the comparisonwith the sample image at 120579 = 10∘ and 120601 = 54∘
360times24 available in the Ohio University sample digital colorvideo database A color video is schematically formed by asequence of frames If we consider a frame in a video as thesample image we prove that the image with greatest PSNRwith respect to the sample image is an image with framenumber close to the frame number of the sample image
5 Results of Tests
We compare results obtained by using the GEFS and SEFSand F-transform based methods for image matching onall the image datasets each of sizes 256 times 256 extractedfrom the View Sphere Database In the first image datasetconcerning an eraser we consider as sample image 119877
1 the
image obtained from the camera in the direction with angles120579 = 11
∘ and 120601 = 36∘ For brevity we consider a dataset of 40test images andwe compare119877
1with the images considered in
the remaining 40 other directions In Table 1 (resp Table 2)we report the distances (17) and (18) (resp PSNR (20) and
(22) with 119871 = 255) obtained using the GEFS and SEFS (respF-transform) based method
In Figure 4we show the trend of the index PSNRobtainedby the F-Transform method with respect to the distance (18)obtained using the GEFS and SEFS method
As we can see from Tables 1 and 2 both methods give thesame reply the better image similar with the image eraser inthe direction 120579 = 11∘ and120601 = 36∘ (Figure 5) is given from thatone at 120579 = 10∘ and 120601 = 54∘ (Figure 6) The trend in Figure 4shows that the value of the distance (18) increases as the PSNRdecreases
In order to have a further confirmation of our approachwe have considered a second object a pen contained in theView Sphere Database whose sample image 119877
1is obtained
from the camera in the direction with angles 120579 = 10∘ and
120601 = 54∘ We also limit the problem to a dataset of 40 test
images whose best distances (17) and (18) (resp (20) and (22)with 119871 = 255) under the SEFS and GEFS (resp F-transform)based method are reported in Table 3 (resp Table 4)
In Figure 7we show the trend of the index PSNRobtainedby the F-transform method with respect to the distance 119863obtained by using the GEFS and SEFS method As we cansee from Tables 3 and 4 in both methods the best imagesimilar to the original image in the directions 120579 = 10∘ and120601 = 54
∘ (Figure 8) is given from that one at 120579 = 10∘ and
120601 = 18∘ (Figure 9) Also in this example the trend in Figure 7
shows that the value of the distance (18) increases as the PSNRdecreases
Now we present the results over a sequence of framesof a video Mom-Daughter available in the Ohio Universitysample digital color video database Each frame is a colorimage of sizes 360 times 240 with 256 gray levels for each bandWe use our method with a compression rate 120588 = 0006944that is in each band every frame is decomposed in 150 blocksand each block has sizes 24 times 24 compressed to a block ofsizes 2 times 2 Since119872 =119873 the GEFS and SEFS based methodis not applicable We set the sample image as the imagecorresponding to the first frame of the video We expectthat the frame number of the image with higher PSNR withrespect to the sample image is the image with frame numberclose to the frame number of sample image In Table 5 wereport the best results obtained using the F-transform basedmethod in terms of the (20) and (22) with 119871 = 255 Asexpected albeit with slight variations all the PSNRs diminishby increasing of the frame number and the second frame(Figure 11) is the frame with the greatest PSNR w r t the firstframe (Figure 10) containing the sample image
In Figure 12 we show the trend of the PSNR (22) with theframe numberThis trend is obtained for all the sample videoframes in the video dataset For reasons of brevity now wereport only the results obtained for another test performedon the sequence of frames of another video in the Ohiosample digital video database the video sflowg The PSNR inFigure 15 diminishes by increasing the frame number and thesecond frame (Figure 14) is the frame with the greatest PSNRw r t the first frame (Figure 13) containing the sample image
For supporting the validity of the F-transform methodfor all the sample frames we measure for the frame withthe greatest PSNR wrt the sample frame the correspondent
Advances in Fuzzy Systems 9
Table 4 Best PSNR from the F-transform based method with 119871 = 255 and 120588 = 0006944 for the pen image dataset obtained from thecomparison with the sample image at 120579 = 10∘ and 120601 = 54∘
value PSNR0obtained by using the original frame instead
of the correspondent compressed frame decoded via theinverse F-Transform In Figure 16 we show the trend of thedifference PSNR
0minus PSNR with respect to PSNR
0 The trend
indicates that this difference is always less than 2 This resultshows that if we compress the images in the dataset with rate120588 = 0006944 by using the F-transform method we can usethe compressed image dataset for image matching processescomparing the decompressed image with respect to a sampleimage despite the loss of information due to the compression
6 Conclusions
The results on the images of sizes 119873 times 119872 (119872 = 119873 = 256)
of the View Sphere Image Database show that using our F-transform based method we obtain the same results in termsof image matching and in terms of reduced memory storagereached also via the GEFS and SEFS based method which isapplicable only over images with 119873 = 119872 while our methodconcerns images of any sizes
Moreover our tests executed on color video frames of sizes119873times119872 (119872 = 360119873 = 240 pixels with 256 gray levels) of theOhio University color videos dataset show that by choosingthe first frame as the sample image we obtain as image withthe highest PSNR that one corresponding to the successiveframe as expected although a loss of information on thedecoded images because of the compression process
References
[1] I Bartolini P Ciaccia and M Patella ldquoQuery processing issuesin region-based image databasesrdquo Knowledge and InformationSystems vol 25 no 2 pp 389ndash420 2010
[2] B De Baets ldquoAnalytical solution methods for fuzzy relationalequationsrdquo in Fundamentals of Fuzzy Sets D Dubois and HPrade Eds vol 1 ofTheHandbooks of Fuzzy Sets Series pp 291ndash340 Kluwer Academic Publishers DordrechtTheNetherlands2000
[3] A Di NolaW Pedrycz S Sessa and E Sanchez Fuzzy RelationEquations and Their Application to Knowledge EngineeringKluwer Academic Publishers Dordrecht The Netherlands1989
[4] M Higashi and G J Klir ldquoResolution of finite fuzzy relationequationsrdquo Fuzzy Sets and Systems vol 13 no 1 pp 65ndash82 1984
[5] P Li and S C Fang ldquoA survey on fuzzy relational equationspart I classification and solvabilityrdquo Fuzzy Optimization andDecision Making vol 8 no 2 pp 179ndash229 2009
[6] V Loia W Pedrycz and S Sessa ldquoFuzzy relation calculusin the compression and decompression of fuzzy relationsrdquoInternational Journal of Image and Graphics vol 2 no 4 pp617ndash631 2002
[7] A VMarkovskii ldquoOn the relation between equationswithmax-product composition and the covering problemrdquo Fuzzy Sets andSystems vol 153 no 2 pp 261ndash273 2005
[8] M Miyakoshi and M Shimbo ldquoLower solutions of systems offuzzy equationsrdquo Fuzzy Sets and Systems vol 19 no 1 pp 37ndash461986
[9] K Peeva ldquoResolution of min-max fuzzy relational equationsrdquoin Fuzzy Partial Differential Equations and Relational EquationsM Nikravesh L A Zadeh and V Korotkikh Eds pp 153ndash166Springer New York NY USA 2004
[10] K Peeva ldquoUniversal algorithm for solving fuzzy relationalequationsrdquo Italian Jour-Nal of Pure and Applied Mathematicsvol 19 pp 9ndash20 2006
[11] K Peeva and Y Kyosev ldquoAlgorithm for solving max-productfuzzy relational equationsrdquo Soft Computing vol 11 no 7 pp593ndash605 2007
[12] K Peeva andY Kyosev Fuzzy Relational Calculus-Theory Appli-cations and Software vol 22 of Advances in Fuzzy SystemsmdashApplications and Theory World Scientific Publishing RiverEdge NJ USA 2004
[13] E Sanchez ldquoResolution of composite fuzzy relation equationsrdquoInformation and Control vol 30 no 1 pp 38ndash48 1976
[14] B S Shieh ldquoNew resolution of finite fuzzy relation equationswith max-min compositionrdquo International Journal of Uncer-tainty Fuzziness and Knowlege-Based Systems vol 16 no 1 pp19ndash33 2008
10 Advances in Fuzzy Systems
[15] Y K Wu and S M Guu ldquoAn efficient procedure for solvinga fuzzy relational equation with max-Archimedean t-normcompositionrdquo IEEE Transactions on Fuzzy Systems vol 16 no1 pp 73ndash84 2008
[16] F Di Martino V Loia and S Sessa ldquoA method in the com-pressiondecompression of images using fuzzy equations andfuzzy similaritiesrdquo in Proceedings of IFSA Conference pp 524ndash527 2003
[17] F Di Martino V Loia and S Sessa ldquoA method for cod-ingdecoding images by using fuzzy relation equationsrdquo in10th International Fuzzy Systems Association World Congress TBilgic B De Baets and O Kaynak Eds vol 2715 of LectureNotes in Artificial Intelligence pp 436ndash441 Springer BerlinGermany July 2003
[18] V Loia and S Sessa ldquoFuzzy relation equations for cod-ingdecoding processes of images and videosrdquo InformationSciences vol 171 no 1-3 pp 145ndash172 2005
[19] H Nobuhara W Pedrycz and K Hirota ldquoFast solving methodof fuzzy relational equation and its application to lossy imagecompressionreconstructionrdquo IEEE Transactions on Fuzzy Sys-tems vol 8 no 3 pp 325ndash334 2000
[20] F Di Martino V Loia and S Sessa ldquoA segmentation methodfor images compressed by fuzzy transformsrdquo Fuzzy Sets andSystems vol 161 no 1 pp 56ndash74 2010 Special section NewTrends on Pattern Recognition with Fuzzy Models
[21] J Lu R Li Y Zhang T Zhao and Z Lu ldquoImage annotationtechniques based on feature selection for class-pairsrdquo Knowl-edge and Information Systems vol 24 no 2 pp 325ndash337 2010
[22] I Perfilieva V Novak and A Dvorak ldquoFuzzy transform inthe analysis of datardquo International Journal of ApproximateReasoning vol 48 no 1 pp 36ndash46 2008
[23] L Chen andPWang ldquoFuzzy relational equations (I) the generaland specialized solving algorithmsrdquo Soft Computing vol 6 pp428ndash435 2002
[24] E Sanchez ldquoResolution of Eigen fuzzy sets equationsrdquo FuzzySets and Systems vol 1 no 1 pp 69ndash74 1978
[25] E Sanchez ldquoEigen fuzzy sets and fuzzy relationsrdquo Journal ofMathematical Analysis and Applications vol 81 no 2 pp 399ndash421 1981
[26] F Di Martino S Sessa and H Nobuhara ldquoEigen fuzzy sets andimage information retrievalrdquo in Proceedings of IEEE Interna-tional Conference on Fuzzy Systems vol 3 pp 1385ndash1390 IEEEPress July 2004
[27] F Di Martino H Nobuhara and S Sessa ldquoEigen fuzzy setsand image information retrievalrdquo in Handbook of GranularComputIng W Pedrycz A Skowron and V Kreinovich Edspp 863ndash872 John Wiley amp Sons Chichester UK 2008
[28] I Perfilieva ldquoFuzzy transforms application to reef growthproblemrdquo in Fuzzy Logic in Geology R B Demicco and GJ Klir Eds pp 275ndash300 Academic Press Amsterdam TheNetherlands 2003
[29] I Perfilieva ldquoFuzzy transforms theory and applicationsrdquo FuzzySets and Systems vol 157 no 8 pp 993ndash1023 2006
[30] I Perfilieva and E Chaldeeva ldquoFuzzy transformationrdquo inProceedings of 9th IFSA World Congress and 20th NAFIPSInternational Conference pp 1946ndash1948 2001
[31] F Di Martino V Loia I Perfilieva and S Sessa ldquoAn imagecodingdecoding method based on direct and inverse fuzzytransformsrdquo International Journal of Approximate Reasoningvol 48 no 1 pp 110ndash131 2008
[32] F Di Martino and S Sessa ldquoCompression and decompressionof images with discrete fuzzy transformsrdquo Information Sciencesvol 177 no 11 pp 2349ndash2362 2007
[33] I Perfilieva and B De Baets ldquoFuzzy transforms of monotonefunctions with application to image compressionrdquo InformationSciences vol 180 no 17 pp 3304ndash3315 2010
[34] F Di Martino V Loia and S Sessa ldquoFuzzy transforms methodand attribute dependency in data analysisrdquo Information Sci-ences vol 180 no 4 pp 493ndash505 2010
It is known [2 24 25] that there exists an integer 119901 isin
1 card119883 such that 119860119901is the GEFS of 119877 with respect to
the max-min composition We also consider the following
119877◻119860 = 119860 (14)
where ldquo◻rdquo denotes themin-max composition that is in termsof membership functions
119860 (119910) = min119909isin119883
max (119860 (119909) 119877 (119909 119910) (15)
for all 119909 119910 isin 119883 and 119860 is also defined to be an Eigen FuzzySet of 119877 with respect to the min-max composition It is easilyseen that (14) is equivalent to the following
119877◻119860 = 119860 (16)
where 119877 and119860 are pointwise defined as 119877(119909 119910) = 1 minus119877(119909 119910)and 119860(119909) = 1 minus 119860(119909) for all 119909 119910 isin 119883 Since 119860
119901for some 119901 isin
1 card119883 is the GEFS of 119877 with respect to the max-mincomposition it is immediately proved that the fuzzy set 119861 119883 rarr [0 1] defined as 119861(119909) = 1 minus 119860
119901(119909) for every 119909 isin [0 1]
is the SEFS of 119877 with respect to the min-max compositionIn [27] a distance based on GEFS and SEFS for image
matching is used over images of sizes 119873 times 119873 Indeedconsidering two single-band images of sizes 119873 times 119873 say 119877
1
and 1198772 such distance is given by
119889 (1198771 1198772) = sum
119909isin119883
((1198601(119909) minus 119860
2(119909))2
+ (1198611(119909) minus 119861
2(119909))2
)
(17)
where 119883 = 1 2 119873 119860119894 119861119894are the GEFS and SEFS of
the fuzzy relation 119877119894 respectively obtained by normalizing
in [0 1] the pixels of the image 119868119894 119894 = 1 2
In [26 27] experiments are presented over color imagesof sizes 256 times 256 concerning two objects (an eraser anda pen) extracted from View Sphere Database Each objectis put in the center of a semisphere on which a camera isplaced in 91 different directions The camera establishes animage (photography) of the object for each direction whichcan be identified from two angles 120579 (0∘ lt 120579 lt 90
∘) and
Φ (minus180∘lt Φ lt 180
∘) as illustrated in Figure 3
A sample image 1198771(with given 120579 = 11
∘ Φ = 36∘
for the eraser and 120579 = 10∘ Φ = 54
∘ for the pen) isto be compared with another image 119877
2chosen among the
remaining 90 directions GEFS and SEFS are calculated in thethree components of each image in the RGB space for whichit is natural to assume the following extension of (17)
where 119889119877(1198771 1198772) 119889119866(1198771 1198772) 119889119861(1198771 1198772) are the measures
(17) calculated in each band 119877 119866 119861 For image matchingthe GEFS and SEFS components in each band are extractedfrom each image thus forming a dataset with reduced storagememoryAn image is comparedwith the images in the datasetusing (18) If the dataset contains 119904 color images of sizes119873times119873
Table 1 Best distances fromGEFS and SEFS basedmethod with 120588 =0007813 for the eraser image dataset obtained from the comparisonwith the sample image at 120579 = 11∘ and 120601 = 36∘
and the dimension of the original dataset is 31199041198732 then thedimension of the GEFS and SEFS dataset is 6119904119873 so we havea compression rate given by
120588 =6119904119873
31199041198732=2
119873 (19)
So we obtain a compression rate 120588 = 0007813 if119873 = 256
4 The Image Matching Process viaF-Transforms
We consider an image dataset formed by color images of sizes119873 times119872 In the preprocessing phase we compress each imageof the dataset using the direct F-transform Each image isdivided in blocks of sizes 119873(119861) times 119872(119861) and each block iscompressed in a block of sizes 119899(119861) times 119898(119861) Thus the imagesare coded with a compression rate 120588 = (119899(119861)times119898(119861))(119873(119861)times119872(119861)) In our experiments we set the sizes of the originaland compressed blocks so that 120588 is comparable with (18) Forexample for 119873 = 119872 = 256 we use 119873(119861) = 119872(119861) = 24 and119899(119861) = 119898(119861) = 2 so 120588 = 0006944
In the reduced dataset we store the F-transform compo-nents of each image We use the PSNR between a sampleimage 119877
1and an image 119877
2defined for every compression rate
120588 (cf (9)) as
PSNR120588(1198771 1198772) = 20 log
10
119871
RMSE120588(1198771 1198772) (20)
Advances in Fuzzy Systems 7
Table 2 Best PSNR from the F-transform based method with 119871 = 255 and 120588 = 0006944 for the eraser image dataset obtained from thecomparison with the sample image at 120579 = 11∘ and 120601 = 36∘
Figure 12 Trend of PSNR (120588 = 0006944) with respect to framenumber for the video Mom-Daughter
Figure 13 Frame 1 in sflowg
where RMSE (Root Mean Square Error) is given by (cf (10))
RMSE120588(1198771 1198772) =
radicsum119873
119894=1sum119872
119895=1(1198771(119894 119895) minus 119877
2(119894 119895))2
119873 times119872
(21)
If we have color images we define an overall PSNR as
PSNR120588(1198771 1198772)
=1
3[PSNR
120588119877(1198771 1198772) + PSNR
120588119866(1198771 1198772)
+PSNR120588119861(1198771 1198772)]
(22)
Figure 14 Frame 2 in sflowg
0 10 20 30 40 50
PSNR
Frame n∘
20
25
30
35
40
45
PSN
R
Figure 15 Trend of PSNR (120588 = 0006944) with respect to framenumber for the video sflowg
where PSNR120588119877(1198771 1198772) PSNR
120588119866(1198771 1198772) PSNR
120588119861(1198771 1198772) are
the similarity measures (20) calculated in each band 119877 119866119861 compression rate 120588 In our experiments we compare theresults obtained by using the F-transforms (resp GEFS andSEFS) based method with the PSNR (20) (resp (18)) Weuse the color image datasets of 256 gray levels and of sizes256 times 256 pixels available in the View Sphere Database foreach object considered the best image 119877
2of the object itself
maximizes the PSNR (22) In other experiments we use ourF-transform method over color video datasets in which eachframe is formed by images of 256 gray levels and of sizes
8 Advances in Fuzzy Systems
20 30 40 50 60
PSNR0
PSN
R di
ffere
nce
PSNR difference
0
02
04
06
08
1
12
14
16
18
2
Figure 16 Trend of PSNR difference with respect to PSNR0(120588 =
0006944)
Table 3 Best distances from GEFS and SEFS based method with 120588= 0007813 for the pen image dataset obtained from the comparisonwith the sample image at 120579 = 10∘ and 120601 = 54∘
360times24 available in the Ohio University sample digital colorvideo database A color video is schematically formed by asequence of frames If we consider a frame in a video as thesample image we prove that the image with greatest PSNRwith respect to the sample image is an image with framenumber close to the frame number of the sample image
5 Results of Tests
We compare results obtained by using the GEFS and SEFSand F-transform based methods for image matching onall the image datasets each of sizes 256 times 256 extractedfrom the View Sphere Database In the first image datasetconcerning an eraser we consider as sample image 119877
1 the
image obtained from the camera in the direction with angles120579 = 11
∘ and 120601 = 36∘ For brevity we consider a dataset of 40test images andwe compare119877
1with the images considered in
the remaining 40 other directions In Table 1 (resp Table 2)we report the distances (17) and (18) (resp PSNR (20) and
(22) with 119871 = 255) obtained using the GEFS and SEFS (respF-transform) based method
In Figure 4we show the trend of the index PSNRobtainedby the F-Transform method with respect to the distance (18)obtained using the GEFS and SEFS method
As we can see from Tables 1 and 2 both methods give thesame reply the better image similar with the image eraser inthe direction 120579 = 11∘ and120601 = 36∘ (Figure 5) is given from thatone at 120579 = 10∘ and 120601 = 54∘ (Figure 6) The trend in Figure 4shows that the value of the distance (18) increases as the PSNRdecreases
In order to have a further confirmation of our approachwe have considered a second object a pen contained in theView Sphere Database whose sample image 119877
1is obtained
from the camera in the direction with angles 120579 = 10∘ and
120601 = 54∘ We also limit the problem to a dataset of 40 test
images whose best distances (17) and (18) (resp (20) and (22)with 119871 = 255) under the SEFS and GEFS (resp F-transform)based method are reported in Table 3 (resp Table 4)
In Figure 7we show the trend of the index PSNRobtainedby the F-transform method with respect to the distance 119863obtained by using the GEFS and SEFS method As we cansee from Tables 3 and 4 in both methods the best imagesimilar to the original image in the directions 120579 = 10∘ and120601 = 54
∘ (Figure 8) is given from that one at 120579 = 10∘ and
120601 = 18∘ (Figure 9) Also in this example the trend in Figure 7
shows that the value of the distance (18) increases as the PSNRdecreases
Now we present the results over a sequence of framesof a video Mom-Daughter available in the Ohio Universitysample digital color video database Each frame is a colorimage of sizes 360 times 240 with 256 gray levels for each bandWe use our method with a compression rate 120588 = 0006944that is in each band every frame is decomposed in 150 blocksand each block has sizes 24 times 24 compressed to a block ofsizes 2 times 2 Since119872 =119873 the GEFS and SEFS based methodis not applicable We set the sample image as the imagecorresponding to the first frame of the video We expectthat the frame number of the image with higher PSNR withrespect to the sample image is the image with frame numberclose to the frame number of sample image In Table 5 wereport the best results obtained using the F-transform basedmethod in terms of the (20) and (22) with 119871 = 255 Asexpected albeit with slight variations all the PSNRs diminishby increasing of the frame number and the second frame(Figure 11) is the frame with the greatest PSNR w r t the firstframe (Figure 10) containing the sample image
In Figure 12 we show the trend of the PSNR (22) with theframe numberThis trend is obtained for all the sample videoframes in the video dataset For reasons of brevity now wereport only the results obtained for another test performedon the sequence of frames of another video in the Ohiosample digital video database the video sflowg The PSNR inFigure 15 diminishes by increasing the frame number and thesecond frame (Figure 14) is the frame with the greatest PSNRw r t the first frame (Figure 13) containing the sample image
For supporting the validity of the F-transform methodfor all the sample frames we measure for the frame withthe greatest PSNR wrt the sample frame the correspondent
Advances in Fuzzy Systems 9
Table 4 Best PSNR from the F-transform based method with 119871 = 255 and 120588 = 0006944 for the pen image dataset obtained from thecomparison with the sample image at 120579 = 10∘ and 120601 = 54∘
value PSNR0obtained by using the original frame instead
of the correspondent compressed frame decoded via theinverse F-Transform In Figure 16 we show the trend of thedifference PSNR
0minus PSNR with respect to PSNR
0 The trend
indicates that this difference is always less than 2 This resultshows that if we compress the images in the dataset with rate120588 = 0006944 by using the F-transform method we can usethe compressed image dataset for image matching processescomparing the decompressed image with respect to a sampleimage despite the loss of information due to the compression
6 Conclusions
The results on the images of sizes 119873 times 119872 (119872 = 119873 = 256)
of the View Sphere Image Database show that using our F-transform based method we obtain the same results in termsof image matching and in terms of reduced memory storagereached also via the GEFS and SEFS based method which isapplicable only over images with 119873 = 119872 while our methodconcerns images of any sizes
Moreover our tests executed on color video frames of sizes119873times119872 (119872 = 360119873 = 240 pixels with 256 gray levels) of theOhio University color videos dataset show that by choosingthe first frame as the sample image we obtain as image withthe highest PSNR that one corresponding to the successiveframe as expected although a loss of information on thedecoded images because of the compression process
References
[1] I Bartolini P Ciaccia and M Patella ldquoQuery processing issuesin region-based image databasesrdquo Knowledge and InformationSystems vol 25 no 2 pp 389ndash420 2010
[2] B De Baets ldquoAnalytical solution methods for fuzzy relationalequationsrdquo in Fundamentals of Fuzzy Sets D Dubois and HPrade Eds vol 1 ofTheHandbooks of Fuzzy Sets Series pp 291ndash340 Kluwer Academic Publishers DordrechtTheNetherlands2000
[3] A Di NolaW Pedrycz S Sessa and E Sanchez Fuzzy RelationEquations and Their Application to Knowledge EngineeringKluwer Academic Publishers Dordrecht The Netherlands1989
[4] M Higashi and G J Klir ldquoResolution of finite fuzzy relationequationsrdquo Fuzzy Sets and Systems vol 13 no 1 pp 65ndash82 1984
[5] P Li and S C Fang ldquoA survey on fuzzy relational equationspart I classification and solvabilityrdquo Fuzzy Optimization andDecision Making vol 8 no 2 pp 179ndash229 2009
[6] V Loia W Pedrycz and S Sessa ldquoFuzzy relation calculusin the compression and decompression of fuzzy relationsrdquoInternational Journal of Image and Graphics vol 2 no 4 pp617ndash631 2002
[7] A VMarkovskii ldquoOn the relation between equationswithmax-product composition and the covering problemrdquo Fuzzy Sets andSystems vol 153 no 2 pp 261ndash273 2005
[8] M Miyakoshi and M Shimbo ldquoLower solutions of systems offuzzy equationsrdquo Fuzzy Sets and Systems vol 19 no 1 pp 37ndash461986
[9] K Peeva ldquoResolution of min-max fuzzy relational equationsrdquoin Fuzzy Partial Differential Equations and Relational EquationsM Nikravesh L A Zadeh and V Korotkikh Eds pp 153ndash166Springer New York NY USA 2004
[10] K Peeva ldquoUniversal algorithm for solving fuzzy relationalequationsrdquo Italian Jour-Nal of Pure and Applied Mathematicsvol 19 pp 9ndash20 2006
[11] K Peeva and Y Kyosev ldquoAlgorithm for solving max-productfuzzy relational equationsrdquo Soft Computing vol 11 no 7 pp593ndash605 2007
[12] K Peeva andY Kyosev Fuzzy Relational Calculus-Theory Appli-cations and Software vol 22 of Advances in Fuzzy SystemsmdashApplications and Theory World Scientific Publishing RiverEdge NJ USA 2004
[13] E Sanchez ldquoResolution of composite fuzzy relation equationsrdquoInformation and Control vol 30 no 1 pp 38ndash48 1976
[14] B S Shieh ldquoNew resolution of finite fuzzy relation equationswith max-min compositionrdquo International Journal of Uncer-tainty Fuzziness and Knowlege-Based Systems vol 16 no 1 pp19ndash33 2008
10 Advances in Fuzzy Systems
[15] Y K Wu and S M Guu ldquoAn efficient procedure for solvinga fuzzy relational equation with max-Archimedean t-normcompositionrdquo IEEE Transactions on Fuzzy Systems vol 16 no1 pp 73ndash84 2008
[16] F Di Martino V Loia and S Sessa ldquoA method in the com-pressiondecompression of images using fuzzy equations andfuzzy similaritiesrdquo in Proceedings of IFSA Conference pp 524ndash527 2003
[17] F Di Martino V Loia and S Sessa ldquoA method for cod-ingdecoding images by using fuzzy relation equationsrdquo in10th International Fuzzy Systems Association World Congress TBilgic B De Baets and O Kaynak Eds vol 2715 of LectureNotes in Artificial Intelligence pp 436ndash441 Springer BerlinGermany July 2003
[18] V Loia and S Sessa ldquoFuzzy relation equations for cod-ingdecoding processes of images and videosrdquo InformationSciences vol 171 no 1-3 pp 145ndash172 2005
[19] H Nobuhara W Pedrycz and K Hirota ldquoFast solving methodof fuzzy relational equation and its application to lossy imagecompressionreconstructionrdquo IEEE Transactions on Fuzzy Sys-tems vol 8 no 3 pp 325ndash334 2000
[20] F Di Martino V Loia and S Sessa ldquoA segmentation methodfor images compressed by fuzzy transformsrdquo Fuzzy Sets andSystems vol 161 no 1 pp 56ndash74 2010 Special section NewTrends on Pattern Recognition with Fuzzy Models
[21] J Lu R Li Y Zhang T Zhao and Z Lu ldquoImage annotationtechniques based on feature selection for class-pairsrdquo Knowl-edge and Information Systems vol 24 no 2 pp 325ndash337 2010
[22] I Perfilieva V Novak and A Dvorak ldquoFuzzy transform inthe analysis of datardquo International Journal of ApproximateReasoning vol 48 no 1 pp 36ndash46 2008
[23] L Chen andPWang ldquoFuzzy relational equations (I) the generaland specialized solving algorithmsrdquo Soft Computing vol 6 pp428ndash435 2002
[24] E Sanchez ldquoResolution of Eigen fuzzy sets equationsrdquo FuzzySets and Systems vol 1 no 1 pp 69ndash74 1978
[25] E Sanchez ldquoEigen fuzzy sets and fuzzy relationsrdquo Journal ofMathematical Analysis and Applications vol 81 no 2 pp 399ndash421 1981
[26] F Di Martino S Sessa and H Nobuhara ldquoEigen fuzzy sets andimage information retrievalrdquo in Proceedings of IEEE Interna-tional Conference on Fuzzy Systems vol 3 pp 1385ndash1390 IEEEPress July 2004
[27] F Di Martino H Nobuhara and S Sessa ldquoEigen fuzzy setsand image information retrievalrdquo in Handbook of GranularComputIng W Pedrycz A Skowron and V Kreinovich Edspp 863ndash872 John Wiley amp Sons Chichester UK 2008
[28] I Perfilieva ldquoFuzzy transforms application to reef growthproblemrdquo in Fuzzy Logic in Geology R B Demicco and GJ Klir Eds pp 275ndash300 Academic Press Amsterdam TheNetherlands 2003
[29] I Perfilieva ldquoFuzzy transforms theory and applicationsrdquo FuzzySets and Systems vol 157 no 8 pp 993ndash1023 2006
[30] I Perfilieva and E Chaldeeva ldquoFuzzy transformationrdquo inProceedings of 9th IFSA World Congress and 20th NAFIPSInternational Conference pp 1946ndash1948 2001
[31] F Di Martino V Loia I Perfilieva and S Sessa ldquoAn imagecodingdecoding method based on direct and inverse fuzzytransformsrdquo International Journal of Approximate Reasoningvol 48 no 1 pp 110ndash131 2008
[32] F Di Martino and S Sessa ldquoCompression and decompressionof images with discrete fuzzy transformsrdquo Information Sciencesvol 177 no 11 pp 2349ndash2362 2007
[33] I Perfilieva and B De Baets ldquoFuzzy transforms of monotonefunctions with application to image compressionrdquo InformationSciences vol 180 no 17 pp 3304ndash3315 2010
[34] F Di Martino V Loia and S Sessa ldquoFuzzy transforms methodand attribute dependency in data analysisrdquo Information Sci-ences vol 180 no 4 pp 493ndash505 2010
Table 2 Best PSNR from the F-transform based method with 119871 = 255 and 120588 = 0006944 for the eraser image dataset obtained from thecomparison with the sample image at 120579 = 11∘ and 120601 = 36∘
Figure 12 Trend of PSNR (120588 = 0006944) with respect to framenumber for the video Mom-Daughter
Figure 13 Frame 1 in sflowg
where RMSE (Root Mean Square Error) is given by (cf (10))
RMSE120588(1198771 1198772) =
radicsum119873
119894=1sum119872
119895=1(1198771(119894 119895) minus 119877
2(119894 119895))2
119873 times119872
(21)
If we have color images we define an overall PSNR as
PSNR120588(1198771 1198772)
=1
3[PSNR
120588119877(1198771 1198772) + PSNR
120588119866(1198771 1198772)
+PSNR120588119861(1198771 1198772)]
(22)
Figure 14 Frame 2 in sflowg
0 10 20 30 40 50
PSNR
Frame n∘
20
25
30
35
40
45
PSN
R
Figure 15 Trend of PSNR (120588 = 0006944) with respect to framenumber for the video sflowg
where PSNR120588119877(1198771 1198772) PSNR
120588119866(1198771 1198772) PSNR
120588119861(1198771 1198772) are
the similarity measures (20) calculated in each band 119877 119866119861 compression rate 120588 In our experiments we compare theresults obtained by using the F-transforms (resp GEFS andSEFS) based method with the PSNR (20) (resp (18)) Weuse the color image datasets of 256 gray levels and of sizes256 times 256 pixels available in the View Sphere Database foreach object considered the best image 119877
2of the object itself
maximizes the PSNR (22) In other experiments we use ourF-transform method over color video datasets in which eachframe is formed by images of 256 gray levels and of sizes
8 Advances in Fuzzy Systems
20 30 40 50 60
PSNR0
PSN
R di
ffere
nce
PSNR difference
0
02
04
06
08
1
12
14
16
18
2
Figure 16 Trend of PSNR difference with respect to PSNR0(120588 =
0006944)
Table 3 Best distances from GEFS and SEFS based method with 120588= 0007813 for the pen image dataset obtained from the comparisonwith the sample image at 120579 = 10∘ and 120601 = 54∘
360times24 available in the Ohio University sample digital colorvideo database A color video is schematically formed by asequence of frames If we consider a frame in a video as thesample image we prove that the image with greatest PSNRwith respect to the sample image is an image with framenumber close to the frame number of the sample image
5 Results of Tests
We compare results obtained by using the GEFS and SEFSand F-transform based methods for image matching onall the image datasets each of sizes 256 times 256 extractedfrom the View Sphere Database In the first image datasetconcerning an eraser we consider as sample image 119877
1 the
image obtained from the camera in the direction with angles120579 = 11
∘ and 120601 = 36∘ For brevity we consider a dataset of 40test images andwe compare119877
1with the images considered in
the remaining 40 other directions In Table 1 (resp Table 2)we report the distances (17) and (18) (resp PSNR (20) and
(22) with 119871 = 255) obtained using the GEFS and SEFS (respF-transform) based method
In Figure 4we show the trend of the index PSNRobtainedby the F-Transform method with respect to the distance (18)obtained using the GEFS and SEFS method
As we can see from Tables 1 and 2 both methods give thesame reply the better image similar with the image eraser inthe direction 120579 = 11∘ and120601 = 36∘ (Figure 5) is given from thatone at 120579 = 10∘ and 120601 = 54∘ (Figure 6) The trend in Figure 4shows that the value of the distance (18) increases as the PSNRdecreases
In order to have a further confirmation of our approachwe have considered a second object a pen contained in theView Sphere Database whose sample image 119877
1is obtained
from the camera in the direction with angles 120579 = 10∘ and
120601 = 54∘ We also limit the problem to a dataset of 40 test
images whose best distances (17) and (18) (resp (20) and (22)with 119871 = 255) under the SEFS and GEFS (resp F-transform)based method are reported in Table 3 (resp Table 4)
In Figure 7we show the trend of the index PSNRobtainedby the F-transform method with respect to the distance 119863obtained by using the GEFS and SEFS method As we cansee from Tables 3 and 4 in both methods the best imagesimilar to the original image in the directions 120579 = 10∘ and120601 = 54
∘ (Figure 8) is given from that one at 120579 = 10∘ and
120601 = 18∘ (Figure 9) Also in this example the trend in Figure 7
shows that the value of the distance (18) increases as the PSNRdecreases
Now we present the results over a sequence of framesof a video Mom-Daughter available in the Ohio Universitysample digital color video database Each frame is a colorimage of sizes 360 times 240 with 256 gray levels for each bandWe use our method with a compression rate 120588 = 0006944that is in each band every frame is decomposed in 150 blocksand each block has sizes 24 times 24 compressed to a block ofsizes 2 times 2 Since119872 =119873 the GEFS and SEFS based methodis not applicable We set the sample image as the imagecorresponding to the first frame of the video We expectthat the frame number of the image with higher PSNR withrespect to the sample image is the image with frame numberclose to the frame number of sample image In Table 5 wereport the best results obtained using the F-transform basedmethod in terms of the (20) and (22) with 119871 = 255 Asexpected albeit with slight variations all the PSNRs diminishby increasing of the frame number and the second frame(Figure 11) is the frame with the greatest PSNR w r t the firstframe (Figure 10) containing the sample image
In Figure 12 we show the trend of the PSNR (22) with theframe numberThis trend is obtained for all the sample videoframes in the video dataset For reasons of brevity now wereport only the results obtained for another test performedon the sequence of frames of another video in the Ohiosample digital video database the video sflowg The PSNR inFigure 15 diminishes by increasing the frame number and thesecond frame (Figure 14) is the frame with the greatest PSNRw r t the first frame (Figure 13) containing the sample image
For supporting the validity of the F-transform methodfor all the sample frames we measure for the frame withthe greatest PSNR wrt the sample frame the correspondent
Advances in Fuzzy Systems 9
Table 4 Best PSNR from the F-transform based method with 119871 = 255 and 120588 = 0006944 for the pen image dataset obtained from thecomparison with the sample image at 120579 = 10∘ and 120601 = 54∘
value PSNR0obtained by using the original frame instead
of the correspondent compressed frame decoded via theinverse F-Transform In Figure 16 we show the trend of thedifference PSNR
0minus PSNR with respect to PSNR
0 The trend
indicates that this difference is always less than 2 This resultshows that if we compress the images in the dataset with rate120588 = 0006944 by using the F-transform method we can usethe compressed image dataset for image matching processescomparing the decompressed image with respect to a sampleimage despite the loss of information due to the compression
6 Conclusions
The results on the images of sizes 119873 times 119872 (119872 = 119873 = 256)
of the View Sphere Image Database show that using our F-transform based method we obtain the same results in termsof image matching and in terms of reduced memory storagereached also via the GEFS and SEFS based method which isapplicable only over images with 119873 = 119872 while our methodconcerns images of any sizes
Moreover our tests executed on color video frames of sizes119873times119872 (119872 = 360119873 = 240 pixels with 256 gray levels) of theOhio University color videos dataset show that by choosingthe first frame as the sample image we obtain as image withthe highest PSNR that one corresponding to the successiveframe as expected although a loss of information on thedecoded images because of the compression process
References
[1] I Bartolini P Ciaccia and M Patella ldquoQuery processing issuesin region-based image databasesrdquo Knowledge and InformationSystems vol 25 no 2 pp 389ndash420 2010
[2] B De Baets ldquoAnalytical solution methods for fuzzy relationalequationsrdquo in Fundamentals of Fuzzy Sets D Dubois and HPrade Eds vol 1 ofTheHandbooks of Fuzzy Sets Series pp 291ndash340 Kluwer Academic Publishers DordrechtTheNetherlands2000
[3] A Di NolaW Pedrycz S Sessa and E Sanchez Fuzzy RelationEquations and Their Application to Knowledge EngineeringKluwer Academic Publishers Dordrecht The Netherlands1989
[4] M Higashi and G J Klir ldquoResolution of finite fuzzy relationequationsrdquo Fuzzy Sets and Systems vol 13 no 1 pp 65ndash82 1984
[5] P Li and S C Fang ldquoA survey on fuzzy relational equationspart I classification and solvabilityrdquo Fuzzy Optimization andDecision Making vol 8 no 2 pp 179ndash229 2009
[6] V Loia W Pedrycz and S Sessa ldquoFuzzy relation calculusin the compression and decompression of fuzzy relationsrdquoInternational Journal of Image and Graphics vol 2 no 4 pp617ndash631 2002
[7] A VMarkovskii ldquoOn the relation between equationswithmax-product composition and the covering problemrdquo Fuzzy Sets andSystems vol 153 no 2 pp 261ndash273 2005
[8] M Miyakoshi and M Shimbo ldquoLower solutions of systems offuzzy equationsrdquo Fuzzy Sets and Systems vol 19 no 1 pp 37ndash461986
[9] K Peeva ldquoResolution of min-max fuzzy relational equationsrdquoin Fuzzy Partial Differential Equations and Relational EquationsM Nikravesh L A Zadeh and V Korotkikh Eds pp 153ndash166Springer New York NY USA 2004
[10] K Peeva ldquoUniversal algorithm for solving fuzzy relationalequationsrdquo Italian Jour-Nal of Pure and Applied Mathematicsvol 19 pp 9ndash20 2006
[11] K Peeva and Y Kyosev ldquoAlgorithm for solving max-productfuzzy relational equationsrdquo Soft Computing vol 11 no 7 pp593ndash605 2007
[12] K Peeva andY Kyosev Fuzzy Relational Calculus-Theory Appli-cations and Software vol 22 of Advances in Fuzzy SystemsmdashApplications and Theory World Scientific Publishing RiverEdge NJ USA 2004
[13] E Sanchez ldquoResolution of composite fuzzy relation equationsrdquoInformation and Control vol 30 no 1 pp 38ndash48 1976
[14] B S Shieh ldquoNew resolution of finite fuzzy relation equationswith max-min compositionrdquo International Journal of Uncer-tainty Fuzziness and Knowlege-Based Systems vol 16 no 1 pp19ndash33 2008
10 Advances in Fuzzy Systems
[15] Y K Wu and S M Guu ldquoAn efficient procedure for solvinga fuzzy relational equation with max-Archimedean t-normcompositionrdquo IEEE Transactions on Fuzzy Systems vol 16 no1 pp 73ndash84 2008
[16] F Di Martino V Loia and S Sessa ldquoA method in the com-pressiondecompression of images using fuzzy equations andfuzzy similaritiesrdquo in Proceedings of IFSA Conference pp 524ndash527 2003
[17] F Di Martino V Loia and S Sessa ldquoA method for cod-ingdecoding images by using fuzzy relation equationsrdquo in10th International Fuzzy Systems Association World Congress TBilgic B De Baets and O Kaynak Eds vol 2715 of LectureNotes in Artificial Intelligence pp 436ndash441 Springer BerlinGermany July 2003
[18] V Loia and S Sessa ldquoFuzzy relation equations for cod-ingdecoding processes of images and videosrdquo InformationSciences vol 171 no 1-3 pp 145ndash172 2005
[19] H Nobuhara W Pedrycz and K Hirota ldquoFast solving methodof fuzzy relational equation and its application to lossy imagecompressionreconstructionrdquo IEEE Transactions on Fuzzy Sys-tems vol 8 no 3 pp 325ndash334 2000
[20] F Di Martino V Loia and S Sessa ldquoA segmentation methodfor images compressed by fuzzy transformsrdquo Fuzzy Sets andSystems vol 161 no 1 pp 56ndash74 2010 Special section NewTrends on Pattern Recognition with Fuzzy Models
[21] J Lu R Li Y Zhang T Zhao and Z Lu ldquoImage annotationtechniques based on feature selection for class-pairsrdquo Knowl-edge and Information Systems vol 24 no 2 pp 325ndash337 2010
[22] I Perfilieva V Novak and A Dvorak ldquoFuzzy transform inthe analysis of datardquo International Journal of ApproximateReasoning vol 48 no 1 pp 36ndash46 2008
[23] L Chen andPWang ldquoFuzzy relational equations (I) the generaland specialized solving algorithmsrdquo Soft Computing vol 6 pp428ndash435 2002
[24] E Sanchez ldquoResolution of Eigen fuzzy sets equationsrdquo FuzzySets and Systems vol 1 no 1 pp 69ndash74 1978
[25] E Sanchez ldquoEigen fuzzy sets and fuzzy relationsrdquo Journal ofMathematical Analysis and Applications vol 81 no 2 pp 399ndash421 1981
[26] F Di Martino S Sessa and H Nobuhara ldquoEigen fuzzy sets andimage information retrievalrdquo in Proceedings of IEEE Interna-tional Conference on Fuzzy Systems vol 3 pp 1385ndash1390 IEEEPress July 2004
[27] F Di Martino H Nobuhara and S Sessa ldquoEigen fuzzy setsand image information retrievalrdquo in Handbook of GranularComputIng W Pedrycz A Skowron and V Kreinovich Edspp 863ndash872 John Wiley amp Sons Chichester UK 2008
[28] I Perfilieva ldquoFuzzy transforms application to reef growthproblemrdquo in Fuzzy Logic in Geology R B Demicco and GJ Klir Eds pp 275ndash300 Academic Press Amsterdam TheNetherlands 2003
[29] I Perfilieva ldquoFuzzy transforms theory and applicationsrdquo FuzzySets and Systems vol 157 no 8 pp 993ndash1023 2006
[30] I Perfilieva and E Chaldeeva ldquoFuzzy transformationrdquo inProceedings of 9th IFSA World Congress and 20th NAFIPSInternational Conference pp 1946ndash1948 2001
[31] F Di Martino V Loia I Perfilieva and S Sessa ldquoAn imagecodingdecoding method based on direct and inverse fuzzytransformsrdquo International Journal of Approximate Reasoningvol 48 no 1 pp 110ndash131 2008
[32] F Di Martino and S Sessa ldquoCompression and decompressionof images with discrete fuzzy transformsrdquo Information Sciencesvol 177 no 11 pp 2349ndash2362 2007
[33] I Perfilieva and B De Baets ldquoFuzzy transforms of monotonefunctions with application to image compressionrdquo InformationSciences vol 180 no 17 pp 3304ndash3315 2010
[34] F Di Martino V Loia and S Sessa ldquoFuzzy transforms methodand attribute dependency in data analysisrdquo Information Sci-ences vol 180 no 4 pp 493ndash505 2010
Figure 16 Trend of PSNR difference with respect to PSNR0(120588 =
0006944)
Table 3 Best distances from GEFS and SEFS based method with 120588= 0007813 for the pen image dataset obtained from the comparisonwith the sample image at 120579 = 10∘ and 120601 = 54∘
360times24 available in the Ohio University sample digital colorvideo database A color video is schematically formed by asequence of frames If we consider a frame in a video as thesample image we prove that the image with greatest PSNRwith respect to the sample image is an image with framenumber close to the frame number of the sample image
5 Results of Tests
We compare results obtained by using the GEFS and SEFSand F-transform based methods for image matching onall the image datasets each of sizes 256 times 256 extractedfrom the View Sphere Database In the first image datasetconcerning an eraser we consider as sample image 119877
1 the
image obtained from the camera in the direction with angles120579 = 11
∘ and 120601 = 36∘ For brevity we consider a dataset of 40test images andwe compare119877
1with the images considered in
the remaining 40 other directions In Table 1 (resp Table 2)we report the distances (17) and (18) (resp PSNR (20) and
(22) with 119871 = 255) obtained using the GEFS and SEFS (respF-transform) based method
In Figure 4we show the trend of the index PSNRobtainedby the F-Transform method with respect to the distance (18)obtained using the GEFS and SEFS method
As we can see from Tables 1 and 2 both methods give thesame reply the better image similar with the image eraser inthe direction 120579 = 11∘ and120601 = 36∘ (Figure 5) is given from thatone at 120579 = 10∘ and 120601 = 54∘ (Figure 6) The trend in Figure 4shows that the value of the distance (18) increases as the PSNRdecreases
In order to have a further confirmation of our approachwe have considered a second object a pen contained in theView Sphere Database whose sample image 119877
1is obtained
from the camera in the direction with angles 120579 = 10∘ and
120601 = 54∘ We also limit the problem to a dataset of 40 test
images whose best distances (17) and (18) (resp (20) and (22)with 119871 = 255) under the SEFS and GEFS (resp F-transform)based method are reported in Table 3 (resp Table 4)
In Figure 7we show the trend of the index PSNRobtainedby the F-transform method with respect to the distance 119863obtained by using the GEFS and SEFS method As we cansee from Tables 3 and 4 in both methods the best imagesimilar to the original image in the directions 120579 = 10∘ and120601 = 54
∘ (Figure 8) is given from that one at 120579 = 10∘ and
120601 = 18∘ (Figure 9) Also in this example the trend in Figure 7
shows that the value of the distance (18) increases as the PSNRdecreases
Now we present the results over a sequence of framesof a video Mom-Daughter available in the Ohio Universitysample digital color video database Each frame is a colorimage of sizes 360 times 240 with 256 gray levels for each bandWe use our method with a compression rate 120588 = 0006944that is in each band every frame is decomposed in 150 blocksand each block has sizes 24 times 24 compressed to a block ofsizes 2 times 2 Since119872 =119873 the GEFS and SEFS based methodis not applicable We set the sample image as the imagecorresponding to the first frame of the video We expectthat the frame number of the image with higher PSNR withrespect to the sample image is the image with frame numberclose to the frame number of sample image In Table 5 wereport the best results obtained using the F-transform basedmethod in terms of the (20) and (22) with 119871 = 255 Asexpected albeit with slight variations all the PSNRs diminishby increasing of the frame number and the second frame(Figure 11) is the frame with the greatest PSNR w r t the firstframe (Figure 10) containing the sample image
In Figure 12 we show the trend of the PSNR (22) with theframe numberThis trend is obtained for all the sample videoframes in the video dataset For reasons of brevity now wereport only the results obtained for another test performedon the sequence of frames of another video in the Ohiosample digital video database the video sflowg The PSNR inFigure 15 diminishes by increasing the frame number and thesecond frame (Figure 14) is the frame with the greatest PSNRw r t the first frame (Figure 13) containing the sample image
For supporting the validity of the F-transform methodfor all the sample frames we measure for the frame withthe greatest PSNR wrt the sample frame the correspondent
Advances in Fuzzy Systems 9
Table 4 Best PSNR from the F-transform based method with 119871 = 255 and 120588 = 0006944 for the pen image dataset obtained from thecomparison with the sample image at 120579 = 10∘ and 120601 = 54∘
value PSNR0obtained by using the original frame instead
of the correspondent compressed frame decoded via theinverse F-Transform In Figure 16 we show the trend of thedifference PSNR
0minus PSNR with respect to PSNR
0 The trend
indicates that this difference is always less than 2 This resultshows that if we compress the images in the dataset with rate120588 = 0006944 by using the F-transform method we can usethe compressed image dataset for image matching processescomparing the decompressed image with respect to a sampleimage despite the loss of information due to the compression
6 Conclusions
The results on the images of sizes 119873 times 119872 (119872 = 119873 = 256)
of the View Sphere Image Database show that using our F-transform based method we obtain the same results in termsof image matching and in terms of reduced memory storagereached also via the GEFS and SEFS based method which isapplicable only over images with 119873 = 119872 while our methodconcerns images of any sizes
Moreover our tests executed on color video frames of sizes119873times119872 (119872 = 360119873 = 240 pixels with 256 gray levels) of theOhio University color videos dataset show that by choosingthe first frame as the sample image we obtain as image withthe highest PSNR that one corresponding to the successiveframe as expected although a loss of information on thedecoded images because of the compression process
References
[1] I Bartolini P Ciaccia and M Patella ldquoQuery processing issuesin region-based image databasesrdquo Knowledge and InformationSystems vol 25 no 2 pp 389ndash420 2010
[2] B De Baets ldquoAnalytical solution methods for fuzzy relationalequationsrdquo in Fundamentals of Fuzzy Sets D Dubois and HPrade Eds vol 1 ofTheHandbooks of Fuzzy Sets Series pp 291ndash340 Kluwer Academic Publishers DordrechtTheNetherlands2000
[3] A Di NolaW Pedrycz S Sessa and E Sanchez Fuzzy RelationEquations and Their Application to Knowledge EngineeringKluwer Academic Publishers Dordrecht The Netherlands1989
[4] M Higashi and G J Klir ldquoResolution of finite fuzzy relationequationsrdquo Fuzzy Sets and Systems vol 13 no 1 pp 65ndash82 1984
[5] P Li and S C Fang ldquoA survey on fuzzy relational equationspart I classification and solvabilityrdquo Fuzzy Optimization andDecision Making vol 8 no 2 pp 179ndash229 2009
[6] V Loia W Pedrycz and S Sessa ldquoFuzzy relation calculusin the compression and decompression of fuzzy relationsrdquoInternational Journal of Image and Graphics vol 2 no 4 pp617ndash631 2002
[7] A VMarkovskii ldquoOn the relation between equationswithmax-product composition and the covering problemrdquo Fuzzy Sets andSystems vol 153 no 2 pp 261ndash273 2005
[8] M Miyakoshi and M Shimbo ldquoLower solutions of systems offuzzy equationsrdquo Fuzzy Sets and Systems vol 19 no 1 pp 37ndash461986
[9] K Peeva ldquoResolution of min-max fuzzy relational equationsrdquoin Fuzzy Partial Differential Equations and Relational EquationsM Nikravesh L A Zadeh and V Korotkikh Eds pp 153ndash166Springer New York NY USA 2004
[10] K Peeva ldquoUniversal algorithm for solving fuzzy relationalequationsrdquo Italian Jour-Nal of Pure and Applied Mathematicsvol 19 pp 9ndash20 2006
[11] K Peeva and Y Kyosev ldquoAlgorithm for solving max-productfuzzy relational equationsrdquo Soft Computing vol 11 no 7 pp593ndash605 2007
[12] K Peeva andY Kyosev Fuzzy Relational Calculus-Theory Appli-cations and Software vol 22 of Advances in Fuzzy SystemsmdashApplications and Theory World Scientific Publishing RiverEdge NJ USA 2004
[13] E Sanchez ldquoResolution of composite fuzzy relation equationsrdquoInformation and Control vol 30 no 1 pp 38ndash48 1976
[14] B S Shieh ldquoNew resolution of finite fuzzy relation equationswith max-min compositionrdquo International Journal of Uncer-tainty Fuzziness and Knowlege-Based Systems vol 16 no 1 pp19ndash33 2008
10 Advances in Fuzzy Systems
[15] Y K Wu and S M Guu ldquoAn efficient procedure for solvinga fuzzy relational equation with max-Archimedean t-normcompositionrdquo IEEE Transactions on Fuzzy Systems vol 16 no1 pp 73ndash84 2008
[16] F Di Martino V Loia and S Sessa ldquoA method in the com-pressiondecompression of images using fuzzy equations andfuzzy similaritiesrdquo in Proceedings of IFSA Conference pp 524ndash527 2003
[17] F Di Martino V Loia and S Sessa ldquoA method for cod-ingdecoding images by using fuzzy relation equationsrdquo in10th International Fuzzy Systems Association World Congress TBilgic B De Baets and O Kaynak Eds vol 2715 of LectureNotes in Artificial Intelligence pp 436ndash441 Springer BerlinGermany July 2003
[18] V Loia and S Sessa ldquoFuzzy relation equations for cod-ingdecoding processes of images and videosrdquo InformationSciences vol 171 no 1-3 pp 145ndash172 2005
[19] H Nobuhara W Pedrycz and K Hirota ldquoFast solving methodof fuzzy relational equation and its application to lossy imagecompressionreconstructionrdquo IEEE Transactions on Fuzzy Sys-tems vol 8 no 3 pp 325ndash334 2000
[20] F Di Martino V Loia and S Sessa ldquoA segmentation methodfor images compressed by fuzzy transformsrdquo Fuzzy Sets andSystems vol 161 no 1 pp 56ndash74 2010 Special section NewTrends on Pattern Recognition with Fuzzy Models
[21] J Lu R Li Y Zhang T Zhao and Z Lu ldquoImage annotationtechniques based on feature selection for class-pairsrdquo Knowl-edge and Information Systems vol 24 no 2 pp 325ndash337 2010
[22] I Perfilieva V Novak and A Dvorak ldquoFuzzy transform inthe analysis of datardquo International Journal of ApproximateReasoning vol 48 no 1 pp 36ndash46 2008
[23] L Chen andPWang ldquoFuzzy relational equations (I) the generaland specialized solving algorithmsrdquo Soft Computing vol 6 pp428ndash435 2002
[24] E Sanchez ldquoResolution of Eigen fuzzy sets equationsrdquo FuzzySets and Systems vol 1 no 1 pp 69ndash74 1978
[25] E Sanchez ldquoEigen fuzzy sets and fuzzy relationsrdquo Journal ofMathematical Analysis and Applications vol 81 no 2 pp 399ndash421 1981
[26] F Di Martino S Sessa and H Nobuhara ldquoEigen fuzzy sets andimage information retrievalrdquo in Proceedings of IEEE Interna-tional Conference on Fuzzy Systems vol 3 pp 1385ndash1390 IEEEPress July 2004
[27] F Di Martino H Nobuhara and S Sessa ldquoEigen fuzzy setsand image information retrievalrdquo in Handbook of GranularComputIng W Pedrycz A Skowron and V Kreinovich Edspp 863ndash872 John Wiley amp Sons Chichester UK 2008
[28] I Perfilieva ldquoFuzzy transforms application to reef growthproblemrdquo in Fuzzy Logic in Geology R B Demicco and GJ Klir Eds pp 275ndash300 Academic Press Amsterdam TheNetherlands 2003
[29] I Perfilieva ldquoFuzzy transforms theory and applicationsrdquo FuzzySets and Systems vol 157 no 8 pp 993ndash1023 2006
[30] I Perfilieva and E Chaldeeva ldquoFuzzy transformationrdquo inProceedings of 9th IFSA World Congress and 20th NAFIPSInternational Conference pp 1946ndash1948 2001
[31] F Di Martino V Loia I Perfilieva and S Sessa ldquoAn imagecodingdecoding method based on direct and inverse fuzzytransformsrdquo International Journal of Approximate Reasoningvol 48 no 1 pp 110ndash131 2008
[32] F Di Martino and S Sessa ldquoCompression and decompressionof images with discrete fuzzy transformsrdquo Information Sciencesvol 177 no 11 pp 2349ndash2362 2007
[33] I Perfilieva and B De Baets ldquoFuzzy transforms of monotonefunctions with application to image compressionrdquo InformationSciences vol 180 no 17 pp 3304ndash3315 2010
[34] F Di Martino V Loia and S Sessa ldquoFuzzy transforms methodand attribute dependency in data analysisrdquo Information Sci-ences vol 180 no 4 pp 493ndash505 2010
Table 4 Best PSNR from the F-transform based method with 119871 = 255 and 120588 = 0006944 for the pen image dataset obtained from thecomparison with the sample image at 120579 = 10∘ and 120601 = 54∘
value PSNR0obtained by using the original frame instead
of the correspondent compressed frame decoded via theinverse F-Transform In Figure 16 we show the trend of thedifference PSNR
0minus PSNR with respect to PSNR
0 The trend
indicates that this difference is always less than 2 This resultshows that if we compress the images in the dataset with rate120588 = 0006944 by using the F-transform method we can usethe compressed image dataset for image matching processescomparing the decompressed image with respect to a sampleimage despite the loss of information due to the compression
6 Conclusions
The results on the images of sizes 119873 times 119872 (119872 = 119873 = 256)
of the View Sphere Image Database show that using our F-transform based method we obtain the same results in termsof image matching and in terms of reduced memory storagereached also via the GEFS and SEFS based method which isapplicable only over images with 119873 = 119872 while our methodconcerns images of any sizes
Moreover our tests executed on color video frames of sizes119873times119872 (119872 = 360119873 = 240 pixels with 256 gray levels) of theOhio University color videos dataset show that by choosingthe first frame as the sample image we obtain as image withthe highest PSNR that one corresponding to the successiveframe as expected although a loss of information on thedecoded images because of the compression process
References
[1] I Bartolini P Ciaccia and M Patella ldquoQuery processing issuesin region-based image databasesrdquo Knowledge and InformationSystems vol 25 no 2 pp 389ndash420 2010
[2] B De Baets ldquoAnalytical solution methods for fuzzy relationalequationsrdquo in Fundamentals of Fuzzy Sets D Dubois and HPrade Eds vol 1 ofTheHandbooks of Fuzzy Sets Series pp 291ndash340 Kluwer Academic Publishers DordrechtTheNetherlands2000
[3] A Di NolaW Pedrycz S Sessa and E Sanchez Fuzzy RelationEquations and Their Application to Knowledge EngineeringKluwer Academic Publishers Dordrecht The Netherlands1989
[4] M Higashi and G J Klir ldquoResolution of finite fuzzy relationequationsrdquo Fuzzy Sets and Systems vol 13 no 1 pp 65ndash82 1984
[5] P Li and S C Fang ldquoA survey on fuzzy relational equationspart I classification and solvabilityrdquo Fuzzy Optimization andDecision Making vol 8 no 2 pp 179ndash229 2009
[6] V Loia W Pedrycz and S Sessa ldquoFuzzy relation calculusin the compression and decompression of fuzzy relationsrdquoInternational Journal of Image and Graphics vol 2 no 4 pp617ndash631 2002
[7] A VMarkovskii ldquoOn the relation between equationswithmax-product composition and the covering problemrdquo Fuzzy Sets andSystems vol 153 no 2 pp 261ndash273 2005
[8] M Miyakoshi and M Shimbo ldquoLower solutions of systems offuzzy equationsrdquo Fuzzy Sets and Systems vol 19 no 1 pp 37ndash461986
[9] K Peeva ldquoResolution of min-max fuzzy relational equationsrdquoin Fuzzy Partial Differential Equations and Relational EquationsM Nikravesh L A Zadeh and V Korotkikh Eds pp 153ndash166Springer New York NY USA 2004
[10] K Peeva ldquoUniversal algorithm for solving fuzzy relationalequationsrdquo Italian Jour-Nal of Pure and Applied Mathematicsvol 19 pp 9ndash20 2006
[11] K Peeva and Y Kyosev ldquoAlgorithm for solving max-productfuzzy relational equationsrdquo Soft Computing vol 11 no 7 pp593ndash605 2007
[12] K Peeva andY Kyosev Fuzzy Relational Calculus-Theory Appli-cations and Software vol 22 of Advances in Fuzzy SystemsmdashApplications and Theory World Scientific Publishing RiverEdge NJ USA 2004
[13] E Sanchez ldquoResolution of composite fuzzy relation equationsrdquoInformation and Control vol 30 no 1 pp 38ndash48 1976
[14] B S Shieh ldquoNew resolution of finite fuzzy relation equationswith max-min compositionrdquo International Journal of Uncer-tainty Fuzziness and Knowlege-Based Systems vol 16 no 1 pp19ndash33 2008
10 Advances in Fuzzy Systems
[15] Y K Wu and S M Guu ldquoAn efficient procedure for solvinga fuzzy relational equation with max-Archimedean t-normcompositionrdquo IEEE Transactions on Fuzzy Systems vol 16 no1 pp 73ndash84 2008
[16] F Di Martino V Loia and S Sessa ldquoA method in the com-pressiondecompression of images using fuzzy equations andfuzzy similaritiesrdquo in Proceedings of IFSA Conference pp 524ndash527 2003
[17] F Di Martino V Loia and S Sessa ldquoA method for cod-ingdecoding images by using fuzzy relation equationsrdquo in10th International Fuzzy Systems Association World Congress TBilgic B De Baets and O Kaynak Eds vol 2715 of LectureNotes in Artificial Intelligence pp 436ndash441 Springer BerlinGermany July 2003
[18] V Loia and S Sessa ldquoFuzzy relation equations for cod-ingdecoding processes of images and videosrdquo InformationSciences vol 171 no 1-3 pp 145ndash172 2005
[19] H Nobuhara W Pedrycz and K Hirota ldquoFast solving methodof fuzzy relational equation and its application to lossy imagecompressionreconstructionrdquo IEEE Transactions on Fuzzy Sys-tems vol 8 no 3 pp 325ndash334 2000
[20] F Di Martino V Loia and S Sessa ldquoA segmentation methodfor images compressed by fuzzy transformsrdquo Fuzzy Sets andSystems vol 161 no 1 pp 56ndash74 2010 Special section NewTrends on Pattern Recognition with Fuzzy Models
[21] J Lu R Li Y Zhang T Zhao and Z Lu ldquoImage annotationtechniques based on feature selection for class-pairsrdquo Knowl-edge and Information Systems vol 24 no 2 pp 325ndash337 2010
[22] I Perfilieva V Novak and A Dvorak ldquoFuzzy transform inthe analysis of datardquo International Journal of ApproximateReasoning vol 48 no 1 pp 36ndash46 2008
[23] L Chen andPWang ldquoFuzzy relational equations (I) the generaland specialized solving algorithmsrdquo Soft Computing vol 6 pp428ndash435 2002
[24] E Sanchez ldquoResolution of Eigen fuzzy sets equationsrdquo FuzzySets and Systems vol 1 no 1 pp 69ndash74 1978
[25] E Sanchez ldquoEigen fuzzy sets and fuzzy relationsrdquo Journal ofMathematical Analysis and Applications vol 81 no 2 pp 399ndash421 1981
[26] F Di Martino S Sessa and H Nobuhara ldquoEigen fuzzy sets andimage information retrievalrdquo in Proceedings of IEEE Interna-tional Conference on Fuzzy Systems vol 3 pp 1385ndash1390 IEEEPress July 2004
[27] F Di Martino H Nobuhara and S Sessa ldquoEigen fuzzy setsand image information retrievalrdquo in Handbook of GranularComputIng W Pedrycz A Skowron and V Kreinovich Edspp 863ndash872 John Wiley amp Sons Chichester UK 2008
[28] I Perfilieva ldquoFuzzy transforms application to reef growthproblemrdquo in Fuzzy Logic in Geology R B Demicco and GJ Klir Eds pp 275ndash300 Academic Press Amsterdam TheNetherlands 2003
[29] I Perfilieva ldquoFuzzy transforms theory and applicationsrdquo FuzzySets and Systems vol 157 no 8 pp 993ndash1023 2006
[30] I Perfilieva and E Chaldeeva ldquoFuzzy transformationrdquo inProceedings of 9th IFSA World Congress and 20th NAFIPSInternational Conference pp 1946ndash1948 2001
[31] F Di Martino V Loia I Perfilieva and S Sessa ldquoAn imagecodingdecoding method based on direct and inverse fuzzytransformsrdquo International Journal of Approximate Reasoningvol 48 no 1 pp 110ndash131 2008
[32] F Di Martino and S Sessa ldquoCompression and decompressionof images with discrete fuzzy transformsrdquo Information Sciencesvol 177 no 11 pp 2349ndash2362 2007
[33] I Perfilieva and B De Baets ldquoFuzzy transforms of monotonefunctions with application to image compressionrdquo InformationSciences vol 180 no 17 pp 3304ndash3315 2010
[34] F Di Martino V Loia and S Sessa ldquoFuzzy transforms methodand attribute dependency in data analysisrdquo Information Sci-ences vol 180 no 4 pp 493ndash505 2010
[15] Y K Wu and S M Guu ldquoAn efficient procedure for solvinga fuzzy relational equation with max-Archimedean t-normcompositionrdquo IEEE Transactions on Fuzzy Systems vol 16 no1 pp 73ndash84 2008
[16] F Di Martino V Loia and S Sessa ldquoA method in the com-pressiondecompression of images using fuzzy equations andfuzzy similaritiesrdquo in Proceedings of IFSA Conference pp 524ndash527 2003
[17] F Di Martino V Loia and S Sessa ldquoA method for cod-ingdecoding images by using fuzzy relation equationsrdquo in10th International Fuzzy Systems Association World Congress TBilgic B De Baets and O Kaynak Eds vol 2715 of LectureNotes in Artificial Intelligence pp 436ndash441 Springer BerlinGermany July 2003
[18] V Loia and S Sessa ldquoFuzzy relation equations for cod-ingdecoding processes of images and videosrdquo InformationSciences vol 171 no 1-3 pp 145ndash172 2005
[19] H Nobuhara W Pedrycz and K Hirota ldquoFast solving methodof fuzzy relational equation and its application to lossy imagecompressionreconstructionrdquo IEEE Transactions on Fuzzy Sys-tems vol 8 no 3 pp 325ndash334 2000
[20] F Di Martino V Loia and S Sessa ldquoA segmentation methodfor images compressed by fuzzy transformsrdquo Fuzzy Sets andSystems vol 161 no 1 pp 56ndash74 2010 Special section NewTrends on Pattern Recognition with Fuzzy Models
[21] J Lu R Li Y Zhang T Zhao and Z Lu ldquoImage annotationtechniques based on feature selection for class-pairsrdquo Knowl-edge and Information Systems vol 24 no 2 pp 325ndash337 2010
[22] I Perfilieva V Novak and A Dvorak ldquoFuzzy transform inthe analysis of datardquo International Journal of ApproximateReasoning vol 48 no 1 pp 36ndash46 2008
[23] L Chen andPWang ldquoFuzzy relational equations (I) the generaland specialized solving algorithmsrdquo Soft Computing vol 6 pp428ndash435 2002
[24] E Sanchez ldquoResolution of Eigen fuzzy sets equationsrdquo FuzzySets and Systems vol 1 no 1 pp 69ndash74 1978
[25] E Sanchez ldquoEigen fuzzy sets and fuzzy relationsrdquo Journal ofMathematical Analysis and Applications vol 81 no 2 pp 399ndash421 1981
[26] F Di Martino S Sessa and H Nobuhara ldquoEigen fuzzy sets andimage information retrievalrdquo in Proceedings of IEEE Interna-tional Conference on Fuzzy Systems vol 3 pp 1385ndash1390 IEEEPress July 2004
[27] F Di Martino H Nobuhara and S Sessa ldquoEigen fuzzy setsand image information retrievalrdquo in Handbook of GranularComputIng W Pedrycz A Skowron and V Kreinovich Edspp 863ndash872 John Wiley amp Sons Chichester UK 2008
[28] I Perfilieva ldquoFuzzy transforms application to reef growthproblemrdquo in Fuzzy Logic in Geology R B Demicco and GJ Klir Eds pp 275ndash300 Academic Press Amsterdam TheNetherlands 2003
[29] I Perfilieva ldquoFuzzy transforms theory and applicationsrdquo FuzzySets and Systems vol 157 no 8 pp 993ndash1023 2006
[30] I Perfilieva and E Chaldeeva ldquoFuzzy transformationrdquo inProceedings of 9th IFSA World Congress and 20th NAFIPSInternational Conference pp 1946ndash1948 2001
[31] F Di Martino V Loia I Perfilieva and S Sessa ldquoAn imagecodingdecoding method based on direct and inverse fuzzytransformsrdquo International Journal of Approximate Reasoningvol 48 no 1 pp 110ndash131 2008
[32] F Di Martino and S Sessa ldquoCompression and decompressionof images with discrete fuzzy transformsrdquo Information Sciencesvol 177 no 11 pp 2349ndash2362 2007
[33] I Perfilieva and B De Baets ldquoFuzzy transforms of monotonefunctions with application to image compressionrdquo InformationSciences vol 180 no 17 pp 3304ndash3315 2010
[34] F Di Martino V Loia and S Sessa ldquoFuzzy transforms methodand attribute dependency in data analysisrdquo Information Sci-ences vol 180 no 4 pp 493ndash505 2010
