Three-dimensional co-occurrence matrices & Gabor filters:Current progress

Gray-level co-occurrence matricesCarl Philips

Gabor filtersDaniel Li

Supervisor: Jacob D. Furst, Ph.D.

Goals

Comparison of co-occurrence matrices and Gabor filters

2D GLCM vs. 3D GLCM 2D Gabor vs. 3D Gabor Linear Discriminate Analysis vs.

Decision Tree

3D image

Co-occurrence matrices

1 6 3 4 2

5 6 5 1 5

2 4 4 3 5

4 3 6 2 2

1 3 2 1 1

Original Image

o Co-occurrence Matrix•Distance =2•Angle = 0° 1 2 3 4 5 6

1 0 1 1 0 0 0

2 1 0 0 1 0 0

3 1 2 0 0 0 0

4 0 0 1 0 0 1

5 1 1 0 1 0 0

6 1 1 0 1 0 0

Energy

Entropy

Correlation

Contrast

Inverse Difference Moment

Variance

Sum Mean

Co-occurrence matrices

Output: 13 Haralick texture descriptors

Inertia

Cluster Shade

Cluster Prominence

Max Probability

Inverse Variance

Mode Probability

Co-occurrence matrices

Global Features extracted are for the

entire cube 13 Directions

Four original 2D directions Nine new 3D directions

4 Distances 1, 2, 4, and 8 pixels

13 features extracted per distance per direction

13*4*13=676 features per cube

Principle Component Analysis

676 is far to many features Computers unable to perform LDA

Retain 1.0000% variability with 5 component Main variable within each component is

Cluster Tendency

Linear Discriminate Analysis

Decision Tree

Linear Discriminate Analysis results

Classification of cubes using 2D Co-Occurrence Matrices and Linear Discriminate Analysis

Predicted GroupLiver TRUE FALSE Total

Training Count TRUE 2145 325 2470Set FALSE 1769 703 2472

% TRUE 86.8 13.2 100FALSE 71.6 28.4 100

Testing Count TRUE 1095 143 1238Set FALSE 890 346 1236

% TRUE 88.4 11.6 100FALSE 72 28 100

57.6% of the training set correctly classified58.2 % of the testing set correctly classified

Linear Discriminate Analysis results

Classification of cubes using 3D Co-Occurrence Matrices and Linear Discriminate Analysis

Predicted GroupLiver TRUE FALSE Total

Training Count TRUE 2162 306 2468Set FALSE 1771 700 2471

% TRUE 87.6 12.4 100FALSE 71.7 28.3 100

Testing Count TRUE 896 339 1238Set FALSE 809 309 1236

% TRUE 72.6 27.4 100FALSE 72.4 27.6 100

57.9% of the training set correctly classified51.2 % of the testing set correctly classified

Decision Tree results

Classification of cubes using 2D Co-Occurrence Matrices and Decision Tree

Predicted GroupLiver TRUE FALSE Total

Training Count TRUE 2307 137 2444Set FALSE 188 2260 2448

% TRUE 94.4 5.6 100FALSE 7.7 92.3 100

Testing Count TRUE 1143 123 1266Set FALSE 160 1100 1260

% TRUE 90.3 9.7 100FALSE 12.7 87.3 100

93.4% of the training set correctly classified88.8 % of the testing set correctly classified

Decision Tree results

Classification of cubes using 3D Co-Occurrence Matrices and Decision Tree

Predicted GroupLiver TRUE FALSE Total

Training Count TRUE 2298 146 2444Set FALSE 260 2187 2447

% TRUE 94 6 100FALSE 10.6 89.4 100

Testing Count TRUE 1141 125 1266Set FALSE 149 1110 1259

% TRUE 90.1 9.9 100FALSE 11.8 88.2 100

91.7% of the training set correctly classified89.1 % of the testing set correctly classified

Gabor filters, introduction

×

Sinusoid Gaussian

Gabor

Gabor filters, a 2D example

Gabor filters, a 2D example

Gabor filters, a 2D example

Gabor filter: Construction

Gaussian2

Gaussian3

Sinusoid1In2D

Sinusoid2In2D

Sinusoid1In3D

Sinusoid2In3D

Sinusoid3In3D

Gabor1In2D

Gabor2In2D

Gabor1In3D

Gabor2In3D

Gabor3In3D

Gabor filters: The five filters

2D: 1-dir and 2-dir

3D: 1-dir, 2-dir, 3-dir

Gabor filters: Our tests

x 3710

Liver cube

x 3710

Non-liver cube

Results for Gabor filters

0102030405060708090

100

1in2D(1)

1In2D(3)

2in2D(1)

2in2D(3)

1in3D 2In3D

LDA

DT

Results for Gabor filters

76.9

98

76.2

92.1

74.8

98.3

77

92.9

78.6

89

81

87

70

75

80

85

90

95

100

1in2D(1)

1In2D(3)

2in2D(1)

2in2D(3)

1in3D 2In3D

LDA

DT

Hypotheses for results

2D data sets were 20x / 60x larger than 3D data sets

Scans were not isotropic, and distance in Z-direction not uniform across patients

Future work

Reduce cases of 2D to be same as 3D and compare results

Complete 3in3D testing Use isotropic data if possible

Questions

Any questions?

Sum Mean

Entropy

Correlation

Variance

Co-occurrence matrices

Output: 13 Haralick texture descriptors

2

,

{ ( , )}i j

P i j

,

( , )*log( ( , ))i j

P i j P i j

,

( )( ) ( , )i j

i j i j

i j P i j

2

,

( , )i j

i j P i j

2 2

,

( ) ( , ) ( ) ( , )i ji j

i P i j j P i j

,

1( ( , ) ( , ))

2 i j

iP i j jP i j Inertia

Contrast

Energy

2

,

( ) * ( , )i j

i j P i j

Co-occurrence matrices

Output: 13 Haralick texture descriptors

Homogeneity

Cluster Shade

Cluster Prominence

3

,

( ) * ( , )x yi j

i j P i j

4,

* ( , )x yi j

i j P i j

max( )P

2,

( , )

( )i j

P i j

i j

mode( )P

,

( , )

1i j

P i j

i j

Max Probability

Inverse Variance

Mode Probability