Wavelet Transform in Image Regions Classification

Transcription

1 Wavelet Transform in Image Regions Classification By Aleš Procházka, Eva Hošťálková, Andrea Gavlasová Institute of Chemical Technology in Prague epartment of Computing Control Engineering Technická Street 5, 66 8 Prague 6, Czech Republic Phone: + 98 * Fax: A.Prochazka@ieee.org * Web: Abstract Texture segmentation classification form very important topics of the interdisciplinary area of digital processing with many applications in different areas including analysis of microscopic s, biomedical processing or detection of satellite components. The paper presents selected mathematical tools for pattern recognition applied in crystalography to detect individual objects, to analyze their properties to find the percentage of desired individuals. Proposed methods have been verified for simulated structures then used for analysis of microscopic s of crystals of different shapes sizes.. Introduction A basic problem encountered in digital processing is in de-noising [], restoration of their missing or corrupted components [] their enhancement [], feature extraction [] classification [5]. The paper is devoted to segmentation to the application of discrete wavelet transform for segments feature extraction using their boundary s [6], [7]. Image components detection [8] feature extraction can then be followed by classification using self-organizing neural networks [9]. The paper presents their use in this case estimating class boundaries as well. An example of the application of proposed methods in crystallography is presented in Fig. (a) showing the microscopic of crystals of different shapes, textures sizes. Similar methods can be also used in other ares including biomedical imaging, environmental processing or analysis of satellite observations. (a) GIVEN IMAGE (b) COMPLEMENTARY BINARY IMAGE (c) RIGE LINES Fig.. Crystallographic processing presenting (a) microscopic, (b) its binary representation, (c) ridge lines detection resulting from its watershed transform

2 7 (a) IMAGE CONTOR (b) BONARY SIGNAL (a) GIVEN SIGNAL (b) SIGNAL SCALOGRAM Index (c) SIGNAL SPECTRM Frequency Level WAVELET: db (c) SIGNAL SCALING AN WAVELET COEFFICIENTS: LEVELS Fig.. Selected segment analysis showing (a) its boundary in three dimensions, (b) the boundary in two dimensions, (c) its discrete Fourier transform Fig.. Selected segment analysis presenting (a) two dimensional segment boundary, (b) scalogram presenting coefficients of its decomposition into three levels, (c) wavelet transform coefficients organized in a row vector. Image Segmentation Segmentation represents an important initial step of processing. Figs (b),(c) present an example of a such a process for components of different shapes textures their segmentation using the watershed method [5]. The proposed algorithm consists of these steps thresholding to convert it to the black white form distance watershed transform use to find ridge lines presented in Fig. (b) ridge lines detection presented in Fig. (c) The process of classification assumes definition of a pattern matrix containing features of separate segments. Many possibilities of their extraction [] include analysis of statistical properties of the boundary (see Fig. (a)) or segment structure in the space domain transform of boundary or segment structure allowing feature specification in the transform domain using discrete Fourier transform or discrete wavelet transform among others The boundary of a segment in Figs (a),(b) in three or two dimensions its discrete Fourier transform are shown in Fig. (c). Image features used for classification can be based upon the mean value the variance of discrete Fourier transform coefficients. Wavelet transform discussed further can be even more efficient allowing multiresolution or analysis.. Image Wavelet ecomposition Reconstruction Signal wavelet decomposition using iscrete Wavelet Transform (WT) provides an alternative to the iscrete Fourier Transform (FT) for analysis resulting in decomposition into two-dimensional functions of time scale. The main benefit of WT over FT is in its multi-resolution time-scale analysis ability according to Fig.5. The principle of wavelet decomposition reconstruction [], [] is presented in Fig. for an matrix [g(n, m)] N,M or one-dimensional x[n] N, =[g(n, )] N, considered as a special case of an having one column only.

3 Shannon Wavelet Function Spectrum Estimation ECOMPOSITION STAGE RECONSTRCTION STAGE Original or [g(n,m)] Final Final or or or [z(n,m)] Column Row Row Column downsampling downsampling upsampling upsampling.. R. R. Scale Time Scale Frequency.5 Fig.. Wavelet transform use in decomposition reconstruction Fig. 5. Shannon wavelet the effect of its dilation to spectrum compression The high-pass filter the complementary low-pass filter is applied to columns then to its rows followed by downsampling after each processing stage. Resulting coefficients can be used for or analysis or for its reconstruction again. Fig. presents wavelet decomposition of a selected segment boundary. Studying the first decomposition step let us denote a column of the matrix [g(n, m)] N,M as {x(n)} N n= =[x(),x(),..., x(n )]. This can be analyzed by half-b low-pass high-pass filters with their impulse responses {s(n)} n= L =[s(),s(),,s(l )], {w(n)}l n= =[w(),w(),,w(l )] () The first stage (see Fig. ) assumes the of a given the appropriate filter for decomposition of all columns at first at first by relations L L xl(n) = s(k)x(n k), xh(n) = w(k)x(n k) () k= for all values of n followed by subsampling by factor c=. In the following decomposition stage., the same process is applied to rows followed by row downsampling to form four separate s. The low-low sub can be further decomposed results used for reconstruction again. In the case of one-dimensional processing, steps. R. are omitted. The whole process can be used for or ) decomposition perfect reconstruction allowing compression de-noising ) resolution enhancement interpolation ) feature extraction using the variance of summed squared coefficients In all these cases the multiresolution properties of wavelet transform are used.. Image Regions Classification Classification of Q segments using R features organized in pattern matrix P R,Q can be realized by application of self-organizing neural networks. The number S of output layer elements is equal to classes must be either defined in advance or it can be automatically increased to create new classes. uring the learning process network weights forming matrix W S,R are changed to minimize distances between each input vector corresponding weights of a winning neuron characterized by its coefficients closest to the current pattern. In case that the learning process is successfully completed network weights belonging to separate output elements represent typical class individuals. k=

4 TEXTRE (a) CLASSIFICATION (b) CLASSIFICATION (a) (b) B. 7 (c) (d) W(:,), P(,:) W(:,), P(,:) C C 9 A A B6 5 Fig. 6. Results of simulated texture segmentation W(:,), P(,:) W(:,), P(,:) Fig. 7. Results of (a) the discrete Fourier transform, (b) the discrete wavelet transform for the simulated classification into three classes Class boundaries presented in Fig. 7 are defined by values of matrix W S,R for R =have been evaluated by a special algorithm dividing the plane into the number of regions equal to classes. Results of classification of 9 segments given in Fig. 6(a) into three classes are presented in Fig. 7 for two selected features obtained (i) as the mean variance of discrete Fourier transform (FT) coefficients (ii) by the variance of discrete wavelet transform (WT) coefficients at the first the second decomposition level. Location of features is given by position of segment numbers. Results obtained by both methods are the same but the variance of features evaluated by the WT is smaller comparing to that obtained by the FT. 5. Results The method presented has been applied for several kinds if real s. Results of processing of the real biomedical MR are presented in Fig. 8. Image segmentation using watershed transform is able to detect most of segments even though the problem of fault class boundaries can arise in some cases. The results of classification of Q segments with feature matrix P R,Q = [p, p,, p Q ] for the selection of different sets of R =features C classes have (a) MR IMAGE (b) RIGE LINES (c) SEGMENT Fig. 8. An example of MR segmentation presenting (a) original MR area, (b) ridge lines resulting from its watershed segmentation, (c) a selected segment sting for a vein

5 been numerically compared by the proposed criteria function. Each class i =,,, C can be characterized by the mean distance of column feature vectors p jk belonging to class segments j k for k =,,,N i from the class centre in the i th row of matrix W C,R =[w, w,, w C ] by relation Classist(i) = N i dist(p jk, w i ) () N i k= where N i represents the number of segments belonging to class i function dist is used for evaluation of the Euclidean distance between two vectors. Results of classification can be numerically characterized by the mean value of average class distances related to the mean value of class centers distances obtained after the learning process according to relation crit = mean(classist)/mean(dist(w, W )) () This proposed Cluster Segmentation Criterion (CSC) provides low values for compact well separated clusters while close clusters with extensive dispersion of cluster vectors provide high values of this criterion. 6. Conclusion The paper presents some aspects of components classification using wavelet transform which provides many possibilities of detection of segment features owing to its multiresolution properties. Segments boundary s were used for classification even though there is possible to use two dimensional wavelet transform for detection of patterns of individual segments texture, too. Methods discussed in the paper have been applied to shape analysis of microscopic s of crystals. Similar methods can be used in other applications in a wide range of interdisciplinary problems of texture analysis including biomedical imaging, processing of satellite s, communications remote earth observations. Acknowledgments The work has been supported by the research grant of the Faculty of Chemical Engineering of the Institute of Chemical Technology, Prague No. MSM References [] Saeed V. Vaseghi, Advanced igital Signal Processing Noise Reduction, Wiley & Teubner, West Sussex,.K., second edition,. [] Onur G. Guleryuz, Iterated enoising for Image Recovery, in ata Compression Conference (CC ), Snao Bird, tah., IEEE. [] J. Ptáček, I. Šindelářová, A. Procházka, J. Smith, Wavelet Transforms In Signal And Image Resolution Enhancement, in International Conference Algoritmy, Podbanske., ST. [] M. Nixon A. Aguado, Feature Extraction & Image Processing, NewNes Elsevier,. [5] R. C. Gonzales, R. E. Woods, S. L. Eddins, igital Image Processing sing MATLAB, Prentice Hall,. [6] S. Arivazhagan L. Ganesan, Texture Classification sing Wavelet Transform, Pattern Recogn. Lett., vol., no. 9-, pp. 5 5,. [7] S. Li J. Shawe-Taylor, Comparison fusion of multiresolution features for texture classification, Pattern Recogn. Lett., vol. 5,. [8] T. Ren J. H. Husoy, Filtering for Texture Classification: A Comparative Study, IEEE Trans. on PAMI, vol., no., pp. 9,. [9] C. M. Bishop, Neural Networks for Pattern Recognition, Oxford niversity Press, 995. [] N. G. Kingsbury, Complex Wavelets for Shift Invariant Analysis Filtering of Signals, Journal of Applied Computational Harmonic Analysis, vol., no., pp. 5, May.