Texture ClassificationUsing Wavelets

Lindsay Semler

Research Goal:

Organ/Tissue segmentation in

CT images

Wavelet CoefficientsClassification rules for tissue/organs in

CT images

Decision Trees Texture Descriptors

To investigate the use of the Haar wavelet in the texture classification of human organs in CT scans.

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

Organs Backbone Heart Liver Kidney Spleen

Segmented 140 50 56 55 39

Cropped 665 103 122 183 55

Sample Decision Tree

Step 2: Texture Analysis and ClassificationTexture DescriptorsMean, 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)

filename distance resolution HStandard DeviationHMean HEntropy HEnergy HContrast HHomogeneity HSumMean HVariance HMaximum ProbabilityHID Moment HCluster Tendency2_hea_33_1_crop.dcm [0,1] 1 5.55E+09 2352 -3939.132 93164 2326 632.73333 20241 4.06E+11 222 430.48886 1.62E+122_hea_33_1_crop.dcm [1,0] 1 5.55E+09 2352 -3829.523 80134 3032 587.32738 20252 4.06E+11 185 422.81874 1.62E+122_hea_33_1_crop.dcm [1,1] 1 5.55E+09 2352 -3675.531 74631 3854 558.12738 19577 3.68E+11 175 405.50709 1.47E+122_hea_33_1_crop.dcm [1,-1] 1 5.55E+09 2352 -3724.958 78683 2490 580.74802 19583 3.68E+11 187 418.54187 1.47E+122_hea_33_1_crop.dcm [0,1] 2 59001100 500 -611.9883 5258 1114 149.5214 4915 5.75E+09 45 101.61562 2.3E+102_hea_33_1_crop.dcm [1,0] 2 59001100 500 -608.0232 4874 1148 141.10162 4915 5.75E+09 39 105.45074 2.3E+102_hea_33_1_crop.dcm [1,1] 2 59001100 500 -551.4408 4075 1108 128.87107 4591 4.7E+09 31 105.95544 1.88E+102_hea_33_1_crop.dcm [1,-1] 2 59001100 500 -570.0992 4577 663 140.54405 4594.5 4.71E+09 40 101.47735 1.88E+102_hea_33_1_crop.dcm [0,1] 3 44324 32 -74.1895 412 462 30.61389 1157 72311314 18 16.73761 2.89E+082_hea_33_1_crop.dcm [1,0] 3 44324 32 -62.46166 308 435 33.00278 1164.5 73251825 15 14.39755 2.93E+082_hea_33_1_crop.dcm [1,1] 3 44324 32 -53.79896 241 691 25.95684 1011.5 48108399 13 12.36299 1.92E+082_hea_33_1_crop.dcm [1,-1] 3 44324 32 -60.13391 317 237 29.58333 1008.5 47823378 16 11.73806 1.91E+08filename distance resolution DStandard DeviationDMean DEntropy DEnergy DContrast DHomogeneity DSumMean DVariance DMaximum ProbabilityDID Moment DCluster Tendency2_hea_33_1_crop.dcm [0,1] 1 9.79E+09 3181 -5545.14 408826 2368 754.01591 21404 4.54E+11 606 75.55591 1.81E+122_hea_33_1_crop.dcm [1,0] 1 9.79E+09 3181 -5565.141 425474 2135 771.65833 21388.5 4.53E+11 625 70.67911 1.81E+122_hea_33_1_crop.dcm [1,1] 1 9.79E+09 3181 -5383.14 392081 1973 732.65758 20706.5 4.11E+11 593 75.68637 1.64E+122_hea_33_1_crop.dcm [1,-1] 1 9.79E+09 3181 -5376.553 390867 2239 728.725 20704.5 4.11E+11 592 76.42687 1.64E+122_hea_33_1_crop.dcm [0,1] 2 97253317 671 -645.019 5290 1990 120.82778 5493 7.18E+09 45 49.3589 2.87E+102_hea_33_1_crop.dcm [1,0] 2 97253317 671 -644.396 5282 1981 124.87266 5501.5 7.2E+09 45 41.78323 2.88E+102_hea_33_1_crop.dcm [1,1] 2 97253317 671 -603.3629 4653 1577 110.67103 5140.5 5.89E+09 38 41.84095 2.36E+102_hea_33_1_crop.dcm [1,-1] 2 97253317 671 -591.8402 4481 1915 111.02222 5143.5 5.9E+09 37 40.43049 2.36E+102_hea_33_1_crop.dcm [0,1] 3 8819 -21 -50.59539 176 1310 25.17345 1330 95552825 6 13.74144 3.82E+082_hea_33_1_crop.dcm [1,0] 3 8819 -21 -50.00292 172 1456 22.81095 1346 97865818 6 14.05578 3.91E+082_hea_33_1_crop.dcm [1,1] 3 8819 -21 -39.43581 163 1006 24.23307 1163 63598952 9 13.0534 2.54E+082_hea_33_1_crop.dcm [1,-1] 3 8819 -21 -41.04768 137 1377 18.73698 1162.5 63544272 5 11.38172 2.54E+08filename distance resolution VStandard DeviationVMean VEntropy VEnergy VContrast VHomogeneity VSumMean VVariance VMaximum ProbabilityVID Moment VCluster Tendency2_hea_33_1_crop.dcm [0,1] 1 3.27E+09 1805 -4588.165 211898 2008 718.66591 19694 3.84E+11 413 356.98088 1.54E+122_hea_33_1_crop.dcm [1,0] 1 3.27E+09 1805 -4694.086 224924 1038 753.93333 19724 3.85E+11 427 374.25528 1.54E+122_hea_33_1_crop.dcm [1,1] 1 3.27E+09 1805 -4416.922 196121 2408 692.48285 19048 3.48E+11 396 344.65273 1.39E+122_hea_33_1_crop.dcm [1,-1] 1 3.27E+09 1805 -4482.175 208795 1355 717.47024 19053.5 3.48E+11 413 338.5868 1.39E+122_hea_33_1_crop.dcm [0,1] 2 30416101 359 -773.428 13754 814 171.29042 4770 5.42E+09 112 64.10451 2.17E+102_hea_33_1_crop.dcm [1,0] 2 30416101 359 -788.9697 14762 501 179.63377 4781.5 5.44E+09 116 71.73137 2.18E+102_hea_33_1_crop.dcm [1,1] 2 30416101 359 -725.5789 11221 741 153.68373 4453.5 4.42E+09 98 75.53501 1.77E+102_hea_33_1_crop.dcm [1,-1] 2 30416101 359 -726.3755 12743 565 163.41944 4457.5 4.43E+09 108 62.93576 1.77E+102_hea_33_1_crop.dcm [0,1] 3 -298 -4 -104.0122 644 465 39.35519 1123.5 68184421 23 12.20703 2.73E+082_hea_33_1_crop.dcm [1,0] 3 -298 -4 -106.414 614 279 37.30202 1126.5 68549049 21 20.40757 2.74E+082_hea_33_1_crop.dcm [1,1] 3 -298 -4 -89.48374 471 390 32.63501 976 44790709 18 17.44141 1.79E+082_hea_33_1_crop.dcm [1,-1] 3 -298 -4 -85.8321 449 247 32.72424 973.5 44561470 18 15.24444 1.78E+08

Wavelet coefficientsHaar

Other Possible Descriptors:

* Run-Length Statistics * Spectral Measures * Fractal Dimension * Statistical Moments * Co-occurrence

Matrices

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.

Haar Wavelet

8 4

9 7 3 5

6

1 -1

2 1 -1

4

2

1

Original image

Wavelet coefficients 6 2 1 -1

Resolution

Averages Details

Calculate one resolution of wavelet coefficients horizontally

Calculate one resolution of wavelet coefficients vertically

A D

A D

A

D

A

D

AA AD

DA DD

Haar Wavelet

AD

DA DD

AA AD

DA DDRepeat process on averages (AA) until desired resolution level is reached

Haar Wavelet

Averages Horizontal Activity

Vertical Activity

Diagonal Activity

Texture DescriptorsCalculate 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

Texture Descriptors

MeanStandard Deviation

Histogram

Energy, Entropy, ContrastHomogeneity, Summean, Variance

Maximum ProbailityID Moment, Cluster Tendency

Co-occurence (4 directions)

Horizontal

MeanStandard Deviation

Histogram

Energy, Entropy, ContrastHomogeneity, Summean, Variance

Maximum ProbailityID Moment, Cluster Tendency

Co-occurence (4 directions)

Diagonal

MeanStandard Deviation

Histogram

Energy, Entropy, ContrastHomogeneity, Summean, Variance

Maximum ProbailityID Moment, Cluster Tendency

Co-occurence (4 directions)

Vertical

Wavelet coefficientsResoution Level

Total Descriptors Per Resolution Level: 114

3 Levels of Resolution: 342

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)

HistogramCo-occurrence

(Directions averaged)

Standard DeviationMean

Energy, Entropy, Contrast, Homegeneity, Summean, Variance, Max. Probability, ID

Moment, Cluster Tendency

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: 1Depth: 10

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

Misclassification Matrix (Haar 33)

Measure Definition

Sensitivity True Positives / Total Positives

Specificity True Negatives / Total Negatives

Precision True Positives / (True Positives + False Positives)

Accuracy (True Positives + True Negatives) / Total Samples

Actual Category

Backbone Heart Liver Kidney Spleen TotalPredicted Backbone 182 6 1 6 0 195Category Heart 3 18 4 0 0 25

Liver 0 3 30 1 7 41Kidney 10 4 0 49 1 64Spleen 0 0 4 0 8 12Total 195 31 39 56 16 337

Results (Haar)

Organ Sensitivity Specificity Precision Accuracy

Backbone 93.3333 90.8451 93.3333 92.2849Heart 58.0645 97.7124 72 94.0653Liver 76.9231 96.3087 73.1707 94.0653Kidney 87.5 94.6619 76.5625 93.4718Spleen 50 98.7539 66.6667 96.4392

Depth: 10

Parent Node: 20

Child Node: 4

Actual Category

Backbone Heart Liver Kidney Spleen TotalPredicted Backbone 182 6 1 6 0 195Category Heart 3 18 4 0 0 25

Liver 0 3 30 1 7 41Kidney 10 4 0 49 1 64Spleen 0 0 4 0 8 12Total 195 31 39 56 16 337

ReferencesMallat, 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 JournalMulcahy, 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 (2nd 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.

Texture DescriptorsFeature Definition Interpretation

EntropyEntropy = - P [i, j]× log P [i, j] i j

Measures the randomness of a gray-level distributionThe Entropy is expected high if the gray levels are distributed randomly through

out the image

EnergyEnergy = P² [i, j] i j

Measures the number of repeated pairsThe Energy is expected high if the occurrence of repeated pixel pairs are high.

ContrastContrast = (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.

HomogeneityHomogeneity = (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 imageThe SumMean is expected large if the sum of the gray levels of the image is high

VarianceVariance = (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 ProbabilityMax P [i, j] i,j

Provides the pixel pair that is most predominant in the imageThe 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 homogeneityThe IDM is expected high if the gray levels of the pixel pairs are similar

Cluster TendencyCluster 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)