IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 41, NO. 8, AUGUST 2003 1765
Contribution of the Fractal Dimension to MultiscaleAdaptive Filtering of SAR Imagery
Mickaël Germain, Goze B. Bénié, Jean-Marc Boucher, Member, IEEE, Samuel Foucher, Associate Member, IEEE,Ko Fung, and Kalifa Goïta, Member, IEEE
Abstract—Radar images can show great variability from pixel topixel, which is an obstacle to effective processing. This variability,due to speckle created by the radar wave coherence, necessitatesthe use of more adapted filters. Previous studies have shown thatmultiresolution wavelet analysis yields better results but producesartefacts due to multiscale decomposition. This paper proposes amethod that reduces these effects by introducing the fractal di-mension. The resultant filter combines wavelet decomposition andvariance change model based on the level of variance estimated bystudying the fractal dimension of the image.
RADAR IMAGING systems produce complementaryinformations to optical sensors in various fields like
geology, forestry, hydrology, and agriculture [2]. They can beused at any time and through any athmospheric conditionswithout altering image quality. But the major problem of theseactive systems is the image perturbation due to a particularnoise called speckle. The speckle results from the randominterference of electromagnetic waves scattered by reflectorsinside a resolution cell. Remote sensing specialists try todevelop efficient methods to reduce the speckle effects inradar images. Multiresolution wavelet-based methods yieldinteresting results but produce artefacts sometimes, due tomultiscale decomposition. This study proposes a method thatreduces these effects by introducing the fractal dimension.
Many algorithms have been developed in attempting to re-duce speckle effect. Most of them have been based on a prelim-inary segmentation of the image into homogeneous or hetero-geneous areas according to different criteria. In the case of themaximuma posteriori(MAP) filter, surface reflectivity is esti-mated using a MAP criterion. In the case of homogeneous sur-faces, reflectivity can be evaluated with a simple local average.
Manuscript received April 11, 2002; revised January 22, 2003. This work wassupported by the Canada Centre for Remote Sensing and Natural Science andEngineering Research Council of Canada.
M. Germain, G. B. Bénié, S. Foucher, and K. Goïta are with the Estritel,Centre d’Applications et de Recherches en Télédétection (CARTEL), Uni-versité de Sherbrooke, Sherbrooke, QC, J1K 2R1 Canada (e-mail: [email protected]).
J.-M. Boucher is with the Ecole Nationale Supérieure des Télécommunica-tions de Bretagne, 29238 Bretagne, France (e-mail: [email protected]).
K. Fung is with the Centre Canadien de Télédétection, Ressources Naturelles,Ottawa, ON, K1A 0Y7 Canada (e-mail: [email protected]).
Digital Object Identifier 10.1109/TGRS.2003.811695
However, for heterogeneous surfaces, the pixel value must beestimated from a local statistical study.
The appearance of wavelets and time-frequency imageanalysis developed by Mallat [12] improved the frequencyand spatial analysis of imagery. In this approach, the detailscorresponding to high frequencies are represented as reducedspatial resolution wavelets, whereas larger variations are pro-jected onto spatially dilated version of basis wavelets. Mallat’smethod, however, based on a pyramidal representation of theimage, does not preserve the translation invariance that ensuresthe independence of the wavelet coefficients on the image. Thisaccounts for the interest in other wavelet decomposition algo-rithms such as the “algorithme à trous” [1] or stationary wavelettransform. This algorithm provides redundant information ofuse in image analysis.
Wavelet decomposition algorithms have been proposed forreducing speckle effect [5], [6]. The one proposed by Foucheret al. [5], based on an extension of the Gamma MAP filter ap-plied on wavelet decomposition image, takes into account radarreflectivity and speckle effect, thereby yielding good-quality re-sults. However, the reconstruction process gives some artefactsdue to a biased estimation of statistical parameters for somelevels of decomposition and for a direction. One solution tosolve this problem is to use a separable wavelet decompositionsuch as “Quincux” [17] or others [9], [18]. But some tests withthis sort of wavelet show that they are not well suited for imageanalysis.
This paper proposes an enhancement of Gamma MAP fil-tering with multiscale analysis, which takes into account thefractal components of the image. The role of this new input is toprovide finer analysis of the textural features of the image and tomodel multiscale decomposition in order to eliminate artefacts.
Following a summary of the traditional hypotheses aboutmultiresolution analysis of radar images (Section II) andGamma MAP filtering with multiscale analysis (Section III),we will introduce the fractal dimension of an image (Sec-tion IV), and finally we will show how to integrate this newparameter into the processing procedure in order to eliminateartefacts resulting from wavelet decomposition (Section V).
II. M ULTIRESOLUTION ANALYSIS
A. Principle
In this paper, we use to represent the Hilbert space offunctions of summable squares with the scalar product
1766 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 41, NO. 8, AUGUST 2003
Multiresolution analysis to levels of the signal ofsummable squares is the projection ofon a base of theform [3], [4]. The basic functions
are yielded by translationsand dilations of the function , called the scale function,verifying . The set engenders asubspace of written as . The projection of onto
yields an approximation ofon the 2 scale.
Similarly, the basic functionsare obtained by dilations and translations of the function ,called the mother wavelet, verifying . The set
engenders a subspace of written as . Theprojection of on yields the wavelet coefficients
containing the details aboutthat makeit possible to move from an approximation on the scale 2to afiner approximation on the scale 2 . Consequently, the sub-spaces and are complementary in
(1)
From a -level multiresolution analysis, we can obtain a de-composition of such that
(2)
All functions of can be derived in the following way:
(3)
The dual functions and must be defined in order to obtaina perfect reconstruction.
B. Filter Bank
The link between the filter bank and the wavelet originates inthe dilation equations that make it possible to change scales [3]
(4)
(5)
with and .The normalization of the scale function implies ,
whereas implies . The multiresolutionanalysis of a signal can, therefore, be carried out through a filterbank of a lowpass analysis filter {} and a highpass analysisfilter { }
(6)
(7)
Consequently, the successive approximations ofincreasingscales result from successive lowpass filtering (each filtering isfollowed by decimation). The wavelet coefficients at a scale 2are obtained by the highpass filtering of an approximation ofat the scale 2 , followed by decimation.
The signal reconstruction is directly derived from relation (1),which is equivalent to
(8)
where the coefficients { } and { } define the synthesis filters.
C. Stationary Wavelet Transform
The preceding algorithms do not maintain invariance undertranslation, which means that translation of the original signaldoes not necessarily involve translation of the correspondingwavelet coefficients. This property is essential for image pro-cessing. If this were not the case, the wavelet coefficients de-rived by a discontinuity in the image could disappear artificially.This nonstationary detection is a direct consequence of the sub-sampling after each instance of filtering. In order to maintain in-variance under translation, certain authors have introduced theconcept of the stationary wavelet transform, also called the “al-gorithme à trous” [15]. As a result, the subsampling operationis dropped, but the filters are dilated by inserting 2 zerosbetween the coefficients of the lowpass and highpass filters foreach level in order to reduce the bandwidth by a factor of twofrom one level to the next
ifelse
(9)
ifelse.
(10)
D. Stationary Multiresolution Image Analysis
Mallat [13] was the first to apply multiresolution analysis toan image. The bank of one-dimensional filters in stationary mul-tiresolution analysis can be extended to two dimensions. Thelines and columns of an image are filtered separately. The fil-tering equations for changing from levelto level are asfollows [ designates pixel position]:
(11)
(12)
(13)
(14)
The approximation of the original image at the reso-lution 2 , provides the low-frequency contents in the [0,2 ]band. Image details are contained in three high-frequency im-ages , , and , corresponding to the details ofhorizontal, vertical, and diagonal orientation, respectively. The
GERMAIN et al.: CONTRIBUTION OF FRACTAL DIMENSION TO MULTISCALE ADAPTIVE FILTERING OF SAR IMAGERY 1767
wavelet coefficients for level provide the information for theimage in the [ 2 2 ] band. Images at the levelmain-tain an identical size because the subsampling operation aftereach filtering was suppressed.
III. M ULTISCALE MAP FILTERING OF SYNTHETIC
APETURERADAR IMAGERY
This section briefly describes the theory underlying MAPmultiscale filtering developed by Foucheret al. [5]. Since thesquare root of the power of an intensity image gives the ampli-tude, the result for an amplitude image can easily be deduced bysimply changing variables.
A. Multiplicative Model
Let the random process of observed intensity and surface re-flectivity be represented by and , respectively.
Generally, when the surface reflectivity is known, the proba-bility of the observed intensity for an looks intensity image isof the form
(15)
Normally, however, the speckle random process is normal-ized, which yields a random process meanhaving the form
(16)
This normalization leads to a multiplicative model of the form
(17)
This model, valid for homogeneous low-texture surfaces, en-tails the following relations established between the standarddeviations of the surface reflectivity, speckle, and intensity:
(18)
(19)
where , ,The local estimate of makes it possible to distinguish ho-
mogeneous regions ( ) where from heteroge-neous regions where .
B. Influence of Speckle on Wavelet Coefficients
Relation (19) can be extended to the wavelet coefficients [5]in order to segment high-frequency images
(20)
with , ( ),
and corresponding to the second-order cumulant of levelof the high frequencies. .
Therefore, for a homogeneous area ( ) , we have
(21)
with .In order to detect heterogeneous areas and, therefore, preserve
significant pixels (contours, targets, and so on), we have chosento threshold the image at a value of [10],[16]. Beyond this value, the region is considered to be highlyheterogeneous. A similar threshold can be defined for
(22)
C. Filter Implementation
and are estimated in a neighborhood ofaround each pixel in the high-frequency and original images.The value of measures the degree of localhomogeneity, thereby indicating the estimator to be applied.
• If : The neighborhood ofaround each pixel is textured, and the normalized stan-
dard deviation of can be estimated with
(23)
Therefore, we have the point estimated by theFoucheret al.’s algorithm [5]
• If : The neighborhood ofaround each pixel is highly textured and may, therefore,contain a target or steep contour. The point is thereforemaintained
(24)
• If : The neighborhood of is consid-ered homogeneous; therefore
(25)
The resulting filtered image is obtained by applying thereconstruction algorithm based on the filtered high-frequencyimages.
IV. STUDY OF THE FRACTAL DIMENSION OF AN IMAGE
The purpose of this section is to study and analyze the fractaldimension of an image and define the mathematical conceptsinvolved. We will also present calculations for the dimensionof processes based on a preliminary orthogonal waveletsdecomposed image.
A. Processes
processes are generally defined as having a spectralpower density over many frequency decadesof the form
(26)
where is a parameter in the range . Despite this fre-quency limitation, there is an enormous amount of phenomena
1768 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 41, NO. 8, AUGUST 2003
Fig. 1. Flowchart of MAP multiscale filtering applied to SAR images.
for which the recorded data justify a spectrum having theform of (26) on all accessible frequencies [8], [11], [14]. Even ifthis density is not integrable (and, therefore, does not constitutea valid spectral power density within the theory of stationaryrandom processes), it corresponds to a significant range of nat-ural and man-made phenomena that have a spectral behavior ofthe type over many decades of frequency. To illustrate, thetexture variation of a natural terrain verifies (26) [19].
In the case of , the variance becomes infinitein the neighborhood of the spectral origin. In such cases, theprocess is nonstationary. For , the signal is said tobe stationary. For example, stationary white noise has avaluethat tends toward zero.
As defined in Mandelbrot’s work, the fractal dimension isexpressed as
(27)
This value quantitatively describes the roughness of the ob-jects in an image. It is, therefore, an important tool in classifyingfractal objects and analyzing textures.
But the main feature of processes is invariance with scalechange. Wornell and Oppenheim [19] demonstrate that the sto-chastic processes are statistically invariant with a scale changeof factor two. If is a stochastic process with the parameters
and , then 2 is statistically similar to if, where is the similarity parameter.
B. Estimation of the Parameter
The wavelet variance must be calculated in order to estimatethe parameter
(28)
Therefore, the exponentcan be deduced by linear regressionof the logarithm of as a function of the scale.
Let be the coefficients of a wavelet such that. Let be the realization of a process
whose variance, for the unit scale only, is sought. Consider, which is the output filter for each
wavelet applied to signal . Therefore, the variance of waveletcoefficients at the unit scale is of the form [7]
(29)
This holds true provided that .Let be the output wavelet filter at
the scale . Then the wavelet variance is of the form
(30)
C. Simulation for Estimating Fractal Dimension
In order to verify the preceding theory, we will simulate dif-ferent random processes for whichis known. Indeed, whitenoise has a constant spectral density (therefore, ), whereasBrownian motion and random walks have spectral densities of
.In order to estimate, the process is decomposed by a wavelet
transform, and the variance is estimated for each high-frequencyimage. Once the images of the variances for each level have beenobtained, linear regression of the images makes it possible toestimate the coefficient, as shown in Fig. 1.
For two-dimensional random processes, we simulated fourBrownian motions that were statistically identical but had dif-ferent fractal dimensions (Fig. 2). The estimation followed bysegmentation yielded the image in Fig. 3. The fractal dimen-sions measured are of the same order of magnitude as those inthe four Brownian motions. It is, therefore, possible to carry outsegmentation by estimating the fractal dimension.
GERMAIN et al.: CONTRIBUTION OF FRACTAL DIMENSION TO MULTISCALE ADAPTIVE FILTERING OF SAR IMAGERY 1769
Fig. 2. Image comprised of four different Brownian motions.
Fig. 3. Segmentation of the fractal dimension.
Fig. 4. Image comprised of four different Brodatz textures.
To highlight the effectiveness of segmentation, we alsocreated an image composed of four different Brodatz textures(Fig. 4). These textures are generally used to show the qualitiesof segmentation and correspond to the textures most oftenobserved in nature. The results depicted in Fig. 5 show thequality of the segmentation.
Fig. 5. Segmentation of the fractal dimension.
Fig. 6. Estimating the coefficient for calculating the fractal dimension.
V. APPLICATION OFFRACTAL DIMENSION IN IMAGE FILTERING
This section discusses the filtering of SAR images, takinginto account the fractal dimension as defined above. Since thisdimension characterizes texture and object detection, it wouldappear interesting to take it into account when analyzing andfiltering images by modeling statistical parameters.
A. Multiscale Filtering and Fractal Dimension
The filtering algorithm developed by Foucheret al. [5] es-timates the variance and mean in a neighborhood of
around each pixel for both the high-frequency imageand original image. The value of , there-fore, measures the degree of local homogeneity and determineswhich estimator to apply.
The purpose of the filtering is to replace the variancecalculated for each level with the variance estimated by cal-culating the fractal dimension as shown in Fig. 6. To illustrate,in level 2, the variance is replaced by the variance estimated bythe fractal dimension estimation. The estimated variance for alevel takes into account the variance of the other levels, whichmakes it possible to eliminate false-alarm phenomena at a givenlevel. Indeed, calculating the fractal dimension makes it pos-sible to model the variance characteristic at a specific point inthe image, as we mentioned in Section IV. By taking into ac-count the levels of decomposition carried out by the wavelet,this modeling makes it possible to characterize variance changeand, thereby, control parasitic phenomena in an image level.
1770 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 41, NO. 8, AUGUST 2003
Fig. 7. Radar image of farmland in Altona (Manitoba).
Fig. 8. (a) MAP filtering without fractal dimension. (b) MAP filtering withfractal dimension. (c) SEM segmentation after multiscale filtering. (d) SEMsegmentation after multiscale filtering with fractal dimension.
VI. FILTERING APPLICATIONS
We filtered the fractal dimension of many synthetic aper-ture radar (SAR) images and have compared them to outputfrom conventional MAP filtering. The results confirmed thepreceding theory about the disappearance of false alarms inthe image. However, having a relatively coarse analysis of thefractal dimension on the image generates areas in which MAPfiltering is inactive. These areas are generally located aroundheterogeneous regions.
The image in Fig. 7, comprised mainly of homogeneous re-gions delimited by clear contours, highlights the disappearanceof false alarms. Indeed, the image filtered without taking intoaccount the estimation of the fractal dimension brings out falsealarms in the homogeneous areas [Fig. 8(a)], whereas the imagefiltered with the fractal dimension eliminates these parasitic el-ements [Fig. 8(b)]. On the other hand, estimating the fractaldimension generates areas around the heterogeneous regionsthat are difficult to filter, as are evident around the heteroge-neous area corresponding to the road located in the middle ofthe image. This phenomenon can be especially troublesome inurban scenes. The SEM segmentation gives better results intaking fractal dimension into account [Fig. 8(c) and (d)].
Fig. 9. Original image. St. Barthelemy with RADARSAT 1.
Fig. 10. Multiscale MAP filtering.
Conversely to Fig. 8, Fig. 9 shows a urban region and, there-fore,containssignificantheterogeneousareas.The imagefilteredwithout using the estimate of the fractal dimension reveals falsealarms in the homogeneous areas and tries to preserve the hetero-geneousareascorrespondingtothetown.Whenthefractaldimen-sion is taken into account, the filtering eliminates the false alarmsin the homogeneous areas, while leaving the heterogeneous areascorresponding to the city relatively unfiltered. Indeed, the detailscorresponding to town streets are difficult to identify by calcu-lating the fractal dimension. This phenomenon brings out con-tours that lack clarity, which means that they are less readilyevident through the ambient noise (Figs. 9–11).
GERMAIN et al.: CONTRIBUTION OF FRACTAL DIMENSION TO MULTISCALE ADAPTIVE FILTERING OF SAR IMAGERY 1771
Fig. 11. Multiscale MAP filtering with fractal dimension.
In this section, we have demonstrated that estimating thefractal dimension can be used to link variances to each levelof image decomposition, thereby resolving the phenomenon offalse alarms at a particular level. The main drawback with thismethod, however, is the low precision of the fractal dimensionin heterogeneous areas and around contours, which reducesquality in modeling the change of variance in these areas.
VII. CONCLUSION
The initial purpose of this paper consisted in finding solutionsfor improving the filtering of remote sensing images throughwavelet processing. The images provided by the multiscaleMAP filtering developed by Foucheret al. yields good resultsbut creates decomposition-related artefacts due to a statisticalestimation errors.
The solution used was to model the change of the variance ondecomposition levels in order to process false alarms detectedon a given level. In actuality, implementing this is the same thingas calculating the fractal dimension of the image. The resultsshow the disappearance of false alarms in homogeneous areas,contrasting with less defective filtering around heterogeneousareas.
We were able to significantly improve the processing of SARimages with respect to initial processing. Nevertheless, theproper validation of these new filtering techniques, in particularthe use of the fractal dimension, requires in-depth investigation.
ACKNOWLEDGMENT
The authors would like to thank M. Garriss for his linguisticcontribution.
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Mickaël Germain was born in Bressuire, France, in 1974. He received thetelecommunication engineering degree from the Ecole Nationale Supérieuredes Télécommunications de Bretagne, Bretagne, France, and the M.S. degreein image processing from the University of Rennes, Rennes, France, in 1998.He is currently pursuing the Ph.D degree at the University of Sherbrooke, Sher-brooke, QC, Canada.
His research interests include multispectral image fusion and segmentation.
Goze Bertin Béniéwas born in Daloa, Côte d’Ivoire. He received the B.A.S.degree in surveying and the M.S. and the Ph.D. degrees in photogrammetry andremote sensing from Universite Laval, Sainte-Foy, QC, Canada, in 1977 and1987, respectively.
He was a Postdoctoral Fellow at the Canada Centre for Remote Sensing,Digim, Inc., Lavalin, Montreal, QC, and at Intera Information Technologies,Inc., Calgary, AB, Canada, from 1987 to 1990. In 1990, he joined the De-partment of Geography and Remote Sensing and the Centre d’Applications etde Recherches en Télédétection (CARTEL), Université de Sherbrooke, Sher-brooke, QC, Canada, as an Assistant Professor. He was the Head of CARTELfrom 1995 to 2000. He is currently the Head of the Department of Geographyand Remote Sensing, and Full Professor in image processing and geomatics atthe Université de Sherbrooke. His research interests include image filtering, seg-mentation, and classification methodology and spatial modeling in GIS.
1772 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 41, NO. 8, AUGUST 2003
Jean-Marc Boucher (M’84) was born in 1952. He received the engineeringdegree in telecommunications from the Ecole Nationale Supérieure desTélécommunications, Paris, France, in 1975, and the Habilitationà Diriger desRecherches degree from the University of Rennes 1, Rennes, France, in 1995.
He is currently a Professor with the Department of Signal and Communi-cations, Ecole Nationale Supérieure des Télécommunications de Bretagne,Bretagne, France, where he is also Education Deputy Director. His currentresearch interests include estimation theory, Markov models and Gibbs fields,blind deconvolution, wavelets and multiscale image analysis with applicationsto radar and sonar image filtering and classification, multisensor seismic signaldeconvolution, electrocardiographic signal processing, and speech codeing. Hehas published 100 technical articles in these areas in international journals andconferences.
Samuel Foucher(S’00–A’01) was born in Nantes, France, in 1969. He re-ceived the B.S. degree in physics from the University of Nantes, Nantes, France,in 1989, the telecommunication engineering degree from the Ecole NationaleSupérieure des Télécommunications de Bretagne, Bretagne, France, the M.S.degree in image processing from the University of Rennes, Rennes, France, in1996, and the Ph.D degree in radar filtering and segmentation from the Univer-sity of Sherbrooke, Sherbrooke, QC, Canada, in 2001.
He is currently with with the Estritel, Centre d’Applications et de Recherchesen Télédétection (CARTEL), Université de Sherbrooke. His research interestsinclude image processing.
Ko Fung received the B.S., M.S., and Ph.D. degrees in electrical engineeringand computer science.
He joined the Canada Centre for Remote Sensing, Natural Resources Canada,Ottawa, ON, in 1977. He has participated in the design and implementation ofthe first Digital Image Correction System (DICS) in Canada. Then he movedon to research and development activities related to methodology and integra-tion of geospatial data with remotely sensed imagery. In the early 1990s, he wasHead of the Knowledge Based Methods and Systems Section. His team wasinvolved in the implementation of the System of Hierarchical Experts for Re-sources Inventorying (SHERI), and the other entitled Photo Interpretation KeyedExpert System (PIKES). In 1996, he was the Manager of the Local Environ-mental Application Program (LEAP) and was responsible for the applicationof remotely sensed techniques for local environmental issues. Since 2001, hehas been involved in the development of information systems for disaster andemergency responses. His interests are in information extraction, integration,and visualization.
Kalifa Goïta (M’99) received the engineering degree in surveying engineeringfrom the École Nationale d’Ingénieurs, Bamako, Mali, in 1987, and the M.S. andPh.D. degrees in remote sensing from the Université de Sherbrooke, Sherbrooke,QC, Canada, in 1991 and 1995, respectively.
He was a Postdoctoral Fellow with the Climate Research Branch of Environ-ment Canada, Toronto, ON, from 1995 to 1997. From 1997 to 2002, he waswith the Faculté de Foresterie, Université de Moncton, Moncton, NB, Canada,as Professor of remote sensing and GIS. Since June 2002, he has been a Pro-fessor of geomatics with the Université de Sherbrooke and a Researcher with theCentre d’Applications et de Recherche en Télédétection (CARTEL), Universitéde Sherbrooke. His research interests include passive microwave remote sensingof snow, remote sensing of ecological indicators for forestry applications, andthe development of dynamic geographic information systems for environmentalmonitoring.
Dr. Goïta is a Member of the Association Québécoise de Télédétection.
