1. Texture Classification Using Wavelets Lindsay Semler

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

3. Step 1: Segmentation and Cropping Segmented Heart Slice segmentation Active Contour Mapping (Snakes) – a boundary based segmentation algorithm Data: 340 Dicom Images The image must be cropped, since wavelets are extremely sensitive to areas of high contrast (background) cropping OrgansBackboneHeartLiverKidneySpleen Segmented14050565539 Cropped66510312218355

4. Sample Decision Tree Step 2: Texture Analysis and Classification 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) Wavelet coefficients Haar Other Possible Descriptors: * Run-Length Statistics * Spectral Measures * Fractal Dimension * Statistical Moments * Co-occurrence Matrices

5. Wavelets A mathematical function that can decompose a signal or an image with a series of averaging and differencing calculations. Wavelets calculate average intensity properties as well as several detailed contrast levels distributed throughout the image. 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.

6. Haar Wavelet 84 9735 6 1 21 421421 Original image Wavelet coefficients 621 Resolution AveragesDetails

7. Calculate one resolution of wavelet coefficients horizontally Calculate one resolution of wavelet coefficients vertically AD AD A D A D AAAD DADD Haar Wavelet AD DADD AAAD DADD Repeat process on averages (AA) until desired resolution level is reached

8. Haar Wavelet AveragesHorizontal Activity Vertical Activity Diagonal Activity

9. Texture Descriptors Calculate one resolution level of wavelet coefficients Results: horizontal, vertical, and diagonal details Calculate mean and standard deviation of each histogram Repeat the process for each resolution level Calculate histograms for each wavelet detail 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

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

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

12. Decision Trees: (Classification and Regression Tree) 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 Parent Node: 20 Child Node: 1 Depth: 10

13. Decision Trees: Haar Wavelets 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 Training Data Testing Data Sample Decision Tree

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

15. Results (Haar) OrganSensitivitySpecificityPrecisionAccuracy Backbone 93.333390.845193.333392.2849 Heart 58.064597.71247294.0653 Liver 76.923196.308773.170794.0653 Kidney 87.594.661976.562593.4718 Spleen 5098.753966.666796.4392 Depth: 10 Parent Node: 20 Child Node: 4 Actual Category BackboneHeartLiverKidneySpleenTotal Predicted Backbone 182 6 1 6 0 195 CategoryHeart 3 18 4 0 0 25 Liver 0 3 30 1 7 41 Kidney 10 4 0 49 1 64 Spleen 0 0 4 0 8 12 Total 195 31 39 56 16 337

16. 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. DeRose, 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. DeRose, 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 7th 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.

17. Texture Descriptors FeatureDefinitionInterpretation Entropy 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 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 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 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 SumMean SumMean = (1/2)[   iP [i, j] +   jP [i, j] ] i j i j Provides the mean of the gray levels in the image The SumMean is expected large if the sum of the gray levels of the image is high Variance Variance = (1/2)[   (i-µ)²P [i, j] +   (j-µ)² P [i, j] 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. InversDifference 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 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)