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Texture Classification Using Wavelets Lindsay Semler

Research Goal: To investigate the use of the Haar wavelet in the texture classification of human organs in CT scans. Organ/Tissue segmentation in CT images Classification rules for tissue/organs in CT images Decision Trees Wavelet Coefficients Texture Descriptors

Step 1: Segmentation and Cropping cropping segmentation Data: 340 Dicom Images Segmented Heart Slice The image must be cropped, since wavelets are extremely sensitive to areas of high contrast (background) Active Contour Mapping (Snakes) – a boundary based segmentation algorithm Organs Backbone Heart Liver Kidney Spleen Segmented 140 50 56 55 39 Cropped 665 103 122 183 55

Step 2: Texture Analysis and Classification Wavelet coefficients Haar Texture Descriptors Mean, Standard Deviation, Energy, Entropy, Contrast, Homogeniety, Variance, Maximum Probability, Inverse Difference Moment, Cluster Tendency, and Summean Classification The process of identifying a region as part of a class based on its texture properties. (Decision Trees) Other Possible Descriptors: * Run-Length Statistics * Spectral Measures * Fractal Dimension * Statistical Moments * Co-occurrence Matrices Sample Decision Tree

Wavelets n A mathematical function that can decompose a signal or an image with a series of averaging and differencing calculations. n Wavelets calculate average intensity properties as well as several detailed contrast levels distributed throughout the image. n They are sensitive to the spatial distribution of grey level pixels, but are also able to differentiate and preserve details at various scales or resolutions.

Haar Wavelet Resolution Original image Wavelet coefficients 7 3 5 4 8 4 1 -1 2 6 Averages 9 2 1 -1 6 2 1 -1 Details 1

Haar Wavelet n n Calculate one resolution of wavelet coefficients horizontally Calculate one resolution of wavelet coefficients vertically A D AD DA DD DA AD DD Repeat process on averages (AA) until desired resolution level is reached A D AA AD DA DD

Haar Wavelet Averages Horizontal Activity Vertical Activity Diagonal Activity

Texture Descriptors Calculate one resolution level of wavelet coefficients Results: horizontal, vertical, and diagonal details Calculate histograms for each wavelet detail Calculate mean and standard deviation of each histogram Calculate four co-occurrence matrices for each wavelet detail based on the four directions: 0, 45, 90, 135 Calculate Energy, Entropy, Contrast, Homogeneity, Summean, Variance, Maximum Probability, Inverse Difference Moment, Cluster Tendency for each co-occurrence matrix Repeat the process for each resolution level

Texture Descriptors Total Descriptors Per Resolution Level: 114 3 Levels of Resolution: 342

Feature Reduction Average over co-occurrence directions: 99 Descriptors total 342 Descriptors Total Average over wavelet details: 33 Descriptors Total Wavelet coefficients per resolution level (Details averaged) Histogram Co-occurrence (Directions averaged) Standard Deviation Mean Energy, Entropy, Contrast, Homegeneity, Summean, Variance, Max. Probability, ID Moment, Cluster Tendency

Decision Trees: (Classification and Regression Tree) n n A decision tree predicts the class of an object from values of predictor variables Depth: the depth of the decision tree Parent Nodes: the # of possible roots per node Child Nodes: the number of possible stems per root node Depth: 10 Parent Node: 20 Child Node: 1

Decision Trees: Haar Wavelets Sample Decision Tree Training Data Testing Data Resulting Tree Total nodes: 49 Total levels: 10 Total terminal nodes: 25 Optimal Parameters Depth of Decision Tree: 10 Parent Node: 20 Child Node: 4

Misclassification Matrix (Haar 33) Actual Category Predicted Category Backbone Heart Liver Kidney Spleen Total Backbone 182 3 0 10 0 195 Heart 6 18 3 4 0 31 Liver 1 4 30 0 4 39 Kidney 6 0 1 49 0 56 Spleen 0 0 7 1 8 16 Total 195 25 41 64 12 337

Results (Haar) Child Node: 4 Sensitivity Specificity Precision Accuracy Backbone 93. 3333 90. 8451 93. 3333 92. 2849 Heart 58. 0645 97. 7124 72 94. 0653 Liver 76. 9231 96. 3087 73. 1707 94. 0653 87. 5 94. 6619 76. 5625 93. 4718 Spleen Parent Node: 20 Organ Kidney Depth: 10 50 98. 7539 66. 6667 96. 4392 Actual Category Predicted Category Backbone Heart Liver Kidney Spleen Total Backbone 182 3 0 10 0 195 Heart 6 18 3 4 0 31 Liver 1 4 30 0 4 39 Kidney 6 0 1 49 0 56 Spleen 0 0 7 1 8 16 Total 195 25 41 64 12 337

References Mallat, Stephane G. . A Theory for Multiresolution Signal Decomposition. IEEE Transactions on Pattern Analysis and Machine Intelligence, VOL 11. NO. 7. July 1989. Mulcahy, Colm. Image Compression using the Haar Wavelet Transform. Spleman Science and Math Journal Mulcahy, Colm. Plotting and Scheming with Wavelets. Mathematics Magazine 69, 5, (1996), 323 -343. Stollnitz, Eric J. , Tony D. De. Rose, David H. Salesin. Wavelets for Computer Graphics: A Primer 1. IEEE Computer Graphics and Applications, 15(4): 75 -85, July 1995. Stollnitz, Eric J. , Tony D. De. Rose, David H. Salesin. Wavelets for Computer Graphics: A Primer 2. IEEE Computer Graphics and Applications, 15(4): 75 -85, July 1995. Tomita, Fumiaki, and Saburo Tsuji. Computer Analysis of Visual Textures. Kluwer Academic Publishers: Norwell, Massachusetts, 1990. Tuceryan, Mihran and Anil K. Jain. Texture Analysis. The Handbook of Pattern Recognition and Computer Vision (2 nd Edition). World Scientific Publishing Co, 1998. Van de Wouwer, G. , P. Scheunders, and D. Van Dyck. Statistical Texture Characterization from Discrete Wavelet Representations. University of Antwerp: Antwerpen, Belgium. Weeks, Arthur R. Jr. , Fundamentals of Electronic Image Processing. The Society for Optical Engineering: Bellingham, Washington, 1996. D. Xu, J. Lee, D. S. Raicu, J. D. Furst, D. Channin. "Texture Classification of Normal Tissues in Computed Tomography", The 2005 Annual Meeting of the Society for Computer Applications in Radiology, Orlando, Florida, June 2 -5, 2005. A. Kurani, D. H. Xu, J. D. Furst, & D. S. Raicu, "Co-occurrence matrices for volumetric data", The 7 th IASTED International Conference on Computer Graphics and Imaging - CGIM 2004, Kauai, Hawaii, USA, in August 16 -18, 2004 Walker, James S. A Primer on Wavelets and their Scientific Applications. CRC Press LLC: Boca Raton, Florida, 1999. Gonzalez, Rafael C. , and Richard E. Woods. Digital Image Processing. Pearson Education: Singapore, 2003.

Texture Descriptors Feature Definition Interpretation Entropy = - P [i, j]× log P [i, j] i j Measures the randomness of a gray-level distribution The Entropy is expected high if the gray levels are distributed randomly through out the image Energy = P² [i, j] i j Measures the number of repeated pairs The Energy is expected high if the occurrence of repeated pixel pairs are high. Contrast = (i -j)²P [i, j] i j Measures the local contrast of an image (how different the gray-level values in the pixel pair are) The Contrast is expect low if the gray levels of each pixel pair are similar. Homogeneity = (P [i, j] / (1 + |i – j| )) i j Measures the local homogeneity of a pixel pair (how similar the gray-level values in the pixel pair are) The Homogeneity is expect large if the gray levels of each pixel pair are similar Sum. Mean = (1/2)[ i. P [i, j] + j. P [i, j] ] i j Provides the mean of the gray levels in the image The Sum. Mean is expected large if the sum of the gray levels of the image is high Variance = (1/2)[ (i-µ)²P [i, j] + (j-µ)² P [i, j] i j Variance tells us how spread out the distribution of gray-levels is The Variance is expect large if the gray levels of the image are spread out greatly. Maximum Probability Max P [i, j] i, j Provides the pixel pair that is most predominant in the image The MP is expected high if the occurrence of the most predominant pixel pair is high. Invers. Difference Moment Inverse Difference Moment = (P [i, j] ) l i, j |i-j|k i≠j Provides the smoothness of the image, just like homogeneity The IDM is expected high if the gray levels of the pixel pairs are similar Cluster Tendency = (i + j - 2µ ) k P [i, j] i, j Measures the grouping of pixels that have similar gray-level values (an image of a black and white cow would result in a higher value for cluster tendency)