Biomedical Image Enhancement, Segmentation andClassification Using Wavelet Transform

A. PROCHAZKA, J. MARES, M. YADOLLAHI, and O. VYSATAInstitute of Chemical Technology, Department of Computing and Control Engineering

Technicka 5, 166 28 Prague 6, CZECH REPUBLICA.Prochazka@ieee.org, {Jan.Mares, Mahammadreza1.Yadollahi, Oldrich.Vysata}@vscht.cz

Abstract: Mathematical methods used for analysis of biomedical data include many topics of interdisciplinarygeneral area of digital signal and image processing and they cover algorithmic tools for image enhancement, imagecomponents detection and their segmentation using feature vectors estimated either in the space and frequencydomains by selected statistical methods and functional transforms. The paper is devoted to specific topics ofbiomedical image processing based upon the time-scale image decomposition using the set of dilated and translatedwavelet functions. Topics covered include (i) the use of wavelet transform for modification of image resolution,(ii) wavelet coefficients thresholding used for image de-nosing, (iii) evaluation of image components features fortheir classification into the given number of classes using neural networks. Methods proposed are applied forbiomedical images to allow another view to their analysis and to contribute to early diagnostics of serious diseases.

Key–Words: Biomedical data processing, wavelet transform, resolution enhancement, contour detection, textureanalysis, data de-noising, image components analysis, classification, cluster analysis, neural networks

1 IntroductionA common problem of biomedical multi-dimensionalsignal processing is in image enhancement, its seg-mentation and analysis of its components closely con-nected with object analysis and early diagnostics ofserious diseases. These problems are often relatedalso to the selection of appropriate image resolutionenabling (i) detection of object details with the speci-fied precision and (ii) compression of information forefficient data processing and their transmission overcommunication links. Signal resolution choice and

(a) GIVEN MRI IMAGE (b) SELECTED SUB−IMAGE

(c) DE−NOISED SUB−IMAGE (d) ENHANCED SUB−IMAGE

Figure 1: Selected steps of the brain MRI enhance-ment presenting (a) observed image, (b) its sub-region, (c) de-noised sub-image, and (d) its enhance-ment using gradient

the appropriate transform selection is therefore one ofbasic tasks of signal processing.

Mathematical topics related to biomedical dataanalysis [7] include general problems of functionaltransforms in image processing [15, 4] and especiallyproblems of signal de-noising [20], restoration of cor-rupted regions [9], multi-resolution analysis [17], de-tection of image objects and evaluation of associatedpattern matrix for their classification. Figs 1 and2 present application of these methods in neurologyand analysis of biomedical images using gradient en-hancement and wavelet transform studied before [16].

The following paper presents in its main part theuse of wavelet transform for image de-noising, itsenhancement and analysis to evaluate image compo-nents features using their contour signals. The qualityof data clusters is then analysed by the proposed cri-terion with neural networks [5] applied for cluster el-ements classification using different methods for fea-ture vectors selection.

2 Principle of Image DecompositionFunctional transforms represent basic mathematicaltools for multi-dimensional signal processing as theyenable signal representation in different spaces [1,2, 3]. These transforms include traditional DiscreteFourier Transform (DFT) and various further pro-cessing tools including Discrete Wavelet Transform

(a) SELECTED MRI SUB−IMAGE (b) SUB−IMAGE ENHANCEMENT

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Figure 2: A selected part of the MRI resonance im-age of the brain presenting (a) chosen sub-image withveins, (b) sub-image resolution enhancement usingthe wavelet transform, and (c), (d) 3D visualization

(DWT) or the multiscale Dual Tree Complex WaveletTransform (DTCWT) [19] allowing to analyze signalson different scales with a selected resolution.

The set of wavelet functions can be the same bothfor signals and images using as a template the motherfunction h(t) with the possibility of its modificationby parameters a and b in the form

hm,k(t)=1√a

h (1a

(t−b)) (1)

In case of often used selection of dilation a=2m andtranslation b = k 2m the set of these functions is de-fined by relation

D

D

D

D

D

D

U

U

U

U

U

U

(a) IMAGE DECOMPOSITION AND RECONTRUCTION

Columnconvolution

Rowconvolution

Rowconvolution

Columnconvolution

anddownsampling

anddownsampling

andupsampling

andupsampling

D.1 D.2 R.1 R.2

Originalsignal

orimage

[g(n,m)]

Finalsignal

orimage

[z(n,m)]

D

D

D

D

D

D

U

U

U

U

U

U

(a) IMAGE DECOMPOSITION AND RECONSTRUCTION

Columnconvolution

Rowconvolution

Rowconvolution

Columnconvolution

anddownsampling

anddownsampling

andupsampling

andupsampling

D.1 D.2 R.1 R.2

Originalsignal

orimage

[g(n,m)]

Finalsignal

orimage

[z(n,m)]

D

D

D

D

D

D

U

U

U

U

U

U

(a) IMAGE DECOMPOSITION AND RECONSTRUCTION

Columnprocessing

Rowprocessing

Rowprocessing

Columnprocessing

D.1 D.2 R.1 R.2

Inputimage

[g(n,m)]

Outputsignal

[z(n,m)]

(b) DILATED SHANNON WAVELETS

12

34

TimeScale

(c) WAVELET SPECTRA

0

0.5

12

34

FrequencyScale

Figure 3: Image analysis by the DWT presenting(a) the decomposition tree with image processedby columns and rows, (b) the set of dilated Shan-non wavelet functions, and (c) associated compressedspectra corresponding to dilated wavelet functions

hm,k(t)=1√2m

h (2−mt−k) (2)

forming the set of functions which for the different di-lation parameters enable both the local and the globalsignal view allowing to analyse either global signalproperties or its details.

The similar approach can be applied for a multi-dimensional signal with its values stored in the multi-dimensional matrix. Using this approach it is possibleto describe the one-dimensional signal as its specialcase with its values in one column of such a matrixonly. Fig. 3 presents the decomposition tree for animage matrix

[g(n, m)]N,M = [s1, s2, · · · , sM ] (3)

formed by column vectors (signals) {sk(n)}N−1n=0 =

[sk(0), sk(1), ..., sk(N − 1)]T and k = 1, 2, · · · , M .The image decomposition assumes the convolu-

tion of image matrix with wavelet and scaling func-tions by columns at first and downsampling by valueD=2 in the first stage. Decomposition functions arerepresented by the half-band low-pass scaling func-tion

{l(k)}L−1k=0 = [l(0), l(1), · · · , l(L − 1)] (4)

and complementary high-pass wavelet function

{h(k)}L−1k=0 = [h(0), h(1), · · · , h(L − 1)] (5)

used for convolution in the form

sl(n)=L−1∑

k=0

l(k) s(n−k) sh(n)=L−1∑

k=0

h(k) s(n−k)

for all values of n. The following subsampling byvalue D=2 imply that the number of resulting coeffi-cients is equal to its original value. The next decom-position stage is applied to rows with row downsam-pling. Resulting multi-dimensional signal is formedby four images for all combinations of the low-passand high-pass processing stages of the initial imageaccording to Fig. 4(b) presenting decomposition intothe second level.

Reconstruction processing blocks (Fig. 3) includethe row upsampling by value U =2 followed by rowconvolution, summation of results and column upsam-pling by value U =2 followed by column convolutionand summation.

The complete DWT can be applied for processingof multi-dimensional signals in the following steps

1. decomposition of an image to allow image anal-ysis and possible compression [13]

2. increase of image resolution [16] for the down-sampling coefficient D=1 and upsampling byU=2 with results presented in Fig. 2(b)

(a) GIVEN IMAGE (b) IMAGE DECOMPOSITION

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(d) SCALING AND WAVELET COEFFICIENTS WITH THRESHOLD VALUES

WA

VE

LE

T:

db

2

2,0 2,1 2,2 2,3 1,1 1,2 1,3

(c) RESULTING IMAGE

Figure 4: Principle of wavelet denoising presenting(a) simulated noisy image, (b) its wavelet decom-position into the second level by the DB2 functions,(c) resulting image obtained from (d) DWT coeffi-cients after their local thresholding different in eachdecomposition level

3. image de-noising using image decompositionand the following modification of image coeffi-cients for given threshold limits [13] and sampleresults presented in Fig. 4

4. interpolation of corrupted image regions [3]

5. extraction of subimage bodies using statisticalproperties of wavelet coefficients for selected de-composition functions and a selected level [15]

Fig. 3(a) presents the decomposition and reconstruc-tion tree for image decomposition into the first levelallowing to decompose the low-low subimage again.The decomposition function can be chosen accordingto the given application. Figs 3(b) and 3(c) show theset of Shannon wavelet functions in the time and fre-quency domains presenting changes of their resolu-tion.

(a) SIM IMAGE (b) RIDGE LINES (c) SEGMENT

Figure 5: Image segmentation presenting (a) an imagewith simulated structures, (b) their watershed segmen-tation, and (c) selected image segment area

3 Image SegmentationImage segmentation is a very common problem inmany biomedical applications allowing detection ofimage or volume components, evaluation of theirproperties and the time evolution study. Fig. 5(a)presents an example of the simulated image composedof regions with different shapes and textures.

Image segmentation [8] can be based upon thewatershed transform and the proposed method forclassification of image segments consists of the fol-lowing steps

• thresholding of image pixels to transform the im-age to the black and white form

• application of the distance and watershed trans-forms and evaluation of ridge lines (Fig. 5(b))

• detection of boundary values of individual seg-ments and their textures (Fig. 5(c)) to define thepattern matrix for their classification

Further possibilities include the application of the re-gion growing method for image components segmen-tation. In all cases it is necessary to solve problems of(i) oversegmentation [8, 21], (ii) overlapping objects,and (iii) their separation.

Fig. 6 presents selected results of segmentationof real orthodontic objects with false components ob-tained after the initial segmentation removed in thenext step completing edges in the overlapping region.

4 Feature Matrix EstimationFig. 7 presents the principle of image segmentationto evaluate image components properties and to de-fine feature vectors for each image segment. Patternmatrix formed by feature column vectors is then usedfor image segments classification. Different ways ofimage subregions feature extraction [15] include pos-sibilities to analyze contour signals properties used inthis study. The proposed method includes

• detection of image boundary and its inner texturefor each image segment

(a) ORIGINAL IMAGE (b) SEGMENTED IMAGE (c) FINAL SEGMENTATION

Figure 6: Segmentation of the orthodontic object withoverlapping presenting (a) original image, (b) its ini-tial segmentation, and (c) its final segmentation withadded line for object separation

• analysis of the edge signal for each image seg-ment and evaluation of its statistical properties

• functional transform of the edge signal or seg-ment texture using translation independent meth-ods to detect signal features

Both discrete wavelet transform and discrete Fouriertransform have been used for signal features detection.Coefficients obtained after selected functional trans-forms have been used to define feature vectors includ-ing the distribution of energy and statistical proper-ties of coefficients in the transform domain. Discretewavelet transform proved in general its flexibility al-lowing to use different wavelet functions and selecteddecomposition levels.

The DFT and the DWT have been applied both tothe boundary values of the selected object and its in-ner area pixels as presented in Fig. 7(c) and 7(d) forthe simulated structure. Fig. 8 presents the applicationof the discrete wavelet transform to the edge analysisof each image segment selected in Fig. 5(c) allowingstatistical analysis of resulting coefficients and featurematrix selection. Further problems include rotation-invariant texture analysis [12], the space objects de-tection and their volumetric features estimation usefulin many biomedical applications.

5 Feature Vectors ClassificationEach region after image decomposition into Q seg-ments can be described by R features formingcolumns of the pattern matrix PR,Q. Each col-umn feature vector represents coordinates in the R-dimensional space with possible clustering into Sgroups. The proposed set of algorithms for classifi-cation of feature vectors is based on artificial neural

TEXTURE

(a) (b)

(c) (d)

Figure 7: Principle of image segment analysis pre-senting (a) texture of the subimage object, (b) its dis-tance transform, (c) edge values of the subimage ob-ject, and (d) its inner area pixels

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−1

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1

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(a) EDGE SIGNAL

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(c) EDGE SIGNAL WAVELET COEFFICIENTS FOR DECOMPOSITION INTO LEVEL 3

WA

VE

LET

: db2

0 3 2 1

Leve

l

(b) SCALOGRAM OF THE EDGE SIGNAL

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3

2

1−2

−1

0

1

Figure 8: Discrete wavelet transform analysis appliedto a selected image segment presenting (a) the edgesignal, (b) the scalogram of wavelet coefficients eval-uated after the signal decomposition into the thirdlevel and the DB2 wavelet function, and (c) discretewavelet transform decomposition coefficients

networks [10, 18, 14] modified for the automatic se-lection of number of classes using the self-organizedstructure [6]. The learning process includes the op-timization of neural network coefficients minimizingdistances of the winning neuron weights and corre-sponding column feature vector. Final values of out-put neurons coefficients point to typical class ele-ments.

Classification of Q image components with fea-ture matrix PR,Q formed by feature column vectors[p1,p2, · · · ,pQ] has been analyzed for sets with dif-ferent number of features R and visualized for R = 2features and S classes. To compare results for dif-ferent features a specific criterion has been designed.The mean distance of each column vector pjk

of aclass segment jk from the i−th class centre in thei−th row of matrix WS,R = [w1,w2, · · · ,wS ]′ canbe evaluated by relation

MD(i) =1Ni

Ni∑

k=1

dist(pjk,wi) (6)

for i = 1, 2, · · · , S and k = 1, 2, · · · , Ni where thevalue Ni stands for the number of segments of class iand function dist evaluates the Euclidean distance be-tween given vectors. The proposed Cluster Segmen-tation Criterion (CSC) defined by relation

CSC=mean(MD)/mean(dist(W,W′)) (7)

relates the mean value of average class distances tothe mean value of class centers distances at the endof the learning process and it results in low values forcompact clusters.

Table 1: MEAN CLASS DISTANCES OF IMAGE COM-PONENTS CLASSIFIED INTO 3 CLASSES USING THE

DFT AND DWT DB2 FEATURE VECTORS EVALU-ATED FROM IMAGE EDGE SIGNALS

FeatureClass Distances / Typical Element

Class A Class B Class C

(i) DFT 0.003 / 2 0.006 / 4 0.012 / 9

(ii) DWT 0.002 / 3 0.001 / 6 0.007 / 8

6 ResultsFig. 9 presents processing of feature vectors associ-ated with the simulated image presented in Fig. 5(a)and obtained in the different way using neural net-works for their classification. The proposed algo-rithm is able to visualize projections in case of morethen two features. Fig. 9 presents the distribution oftwo features and their classification into three classes.Weight coefficients of the output neurons stand fortypical class features at the end of the learning pro-cedure and they represent gravity centers of individ-ual classes. The set algorithms is able to visualizeclass boundaries as well and to choose typical classelements close to class centers.

Results of classification presented in Fig. 9 arecompared in Table 1 summarizing the mean class dis-tances evaluated for individual classes. Features ofimage edge signals have been estimated (i) by the DFTas means and variances of its coefficients in selectedfrequency regions and (ii) variances of the DWT co-efficients using Daubechies wavelet functions of thesecond order and decomposition into the first and thesecond level. Table 1 presents indices of typical imageobject elements with the lowest distance of its featurevector elements from the neuron weights detecting thesame class as well.

0.5 0.55 0.6 0.65 0.7

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9

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(a) DFT CLASSIFICATION

W(:,2), P(2,:)

W(:

,1),

P(1

,:)

0.065 0.07 0.075 0.08 0.085 0.09 0.095 0.1 0.105

0.08

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234

56

7

89

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(b) DWT CLASSIFICATION

W(:,2), P(2,:)

W(:

,1),

P(1

,:)

Figure 9: Distribution of feature values evaluatedfrom edge signals of simulated image components us-ing (a) the DFT and (b) the DWT, classification offeature vectors into 3 classes and visualization of typ-ical features and boundaries of individual classes

The proposed algorithm of object segmentation,feature vectors definition and their classification hasbeen used for analysis of selected biomedical images.Fig. 10(a) presents an example of the selected brainimage after its de-noising to reduce problems of over-segmentation and after its resolution enhancement.The whole algorithm includes (a) image preprocess-ing, (b) detection of subimage edges, and (c) extrac-tion of the vein in the image. The next steps include(d) the detection of edge signal, (e) its 2D visualiza-tion, and (f) its analysis either by the DFT or DWT tofind its feature vector used for their classification.

Both watershed and region growing methods canbe used to detect image segments in selected appli-cations. In case of real signal processing it is neces-sary to choose appropriate preprocessing methods toreduce problems of oversegmentation and detection ofoverlapping objects. The appropriate segmentation al-lows the following efficient feature extraction.

7 ConclusionThe paper is devoted to specific topics and prob-lems of biomedical image segmentation and classi-fication using wavelet transforms. A special atten-tion is paid to image enhancement and denoising byimage wavelet decomposition and reconstruction fol-lowed by segmentation of image components and theuse of functional transforms to estimate image featurevectors. The final part of the work is devoted to clas-sification of image objects using the pattern matrix.The proposed criterion has been used to compare re-sults of clustering by the FFT and DWT for feature

(a) MR SUBIMAGE (b) EDGE LINES (c) SELECTED OBJECT

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(e) EDGE IMAGE SIGNAL

Index

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(f) EDGE SIGNAL SPECTRUM

Frequency

Figure 10: Analysis of a selected part of the brain MRimage presenting (a) given MR subimage, (b) edgesof individual image structures, (c) an image objectstanding for a vein, (d) the edge signal of the vein inthe 3D space, (e) the edge signal, and (f) the DFT ofthe edge signal used for estimation of object features

vectors definition with better results obtained for thewavelet transform owing to its flexibility and possibil-ity to choose different wavelet functions.

Proposed algorithms of image components seg-mentation and classification form a library allowingto use artificial neural networks for (i) classificationof feature vectors, (ii) suggestion of the number ofclasses with visualization of class boundaries, and(iii) the choice of typical image segments. Methodspresented have been used for analysis of shapes ofbiomedical images with application in neurology anddetection of specific objects in the brain.

The future work will be devoted to furtherbiomedical applications including research related toorthodontic images [21, 11] and lung granuloma studywith emphasis to segmentation of overlapping ob-jects. A specific research will be devoted to theirthree-dimensional modelling, visualization and studyto contribute to problems of early diagnostics of seri-ous diseases.

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