FractalDimensionforClassifying3DBrainMRIUsingImprovedTriangle

Box‐CountingMethod

Yothin KaewaramsriSyukron Abu Ishaq AlfaroziKuntpong WoraratpanyaYoshimitsu Kuroki

Faculty of Information Technology, King Mongkut’s Institute of Technology Ladkrabang

(KMITL), Bangkok, Thailand

Outline

▪ Introduction

▪ Background

▪ Proposed method

▪ Experimental results

▪ Conclusion

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Outline

▪ Introduction

▪ Background

▪ Proposed method

▪ Experimental results

▪ Conclusion

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Alzheimer's disease (AD)

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Alzheimer's disease (AD) is the most common causes of dementia in the elder people.The increasing population of elderly society affects to the increasing number of AD patients.

Brain Magnetic Resonance Imaging (MRI)

sagittal coronal axial

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Fractal dimension (FD)

0 321

FD ≈ [1-2]

FD ≈ [2-3]

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Outline

▪ Introduction

▪ Background▪ Box-counting

▪ Proposed method

▪ Experimental results

▪ Conclusion

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Box-Counting (BC)

▪ In 1980, D. A. Russell et al proposed the FD estimation method for binary images as so-called the box-counting (BC)

FD ≈ log (N(s))log (1/s)

Number of box count

Box size

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s = 512 N(s) = 1 s = 256 N(s) = 4

s = 128 N(s) = 6 s = 64 N(s) = 18

Box-Counting (BC)

1. Partition image into square grid

2. Count a number of boxes on each grid

3. Sum the total boxes on each size

4. Take logarithm to the data

5. Plot graph to estimate FD

6. FD ≈Slope of regression line

Image size 512x512

size 2 4 8 16 32 64 128 256 512

N(size) 687 345 166 76 38 18 6 4 19

FD ≈ 1.262

Box-Counting (BC)

FD ≈ log (N(s))log (1/s)

size 2 4 8 16 32 64 128 256 512

N(size) 687 345 166 76 38 18 6 4 110

1. Partition image into square grid

2. Count a number of boxes on each grid

3. Sum the total boxes on each size

4. Take logarithm to the data

5. Plot graph to estimate FD

6. FD ≈Slope of regression line

Differential Box-Counting (DBC)

BC

DBC

nd is the number of box-count in each grid

bmax and bmin the order of boxes that contain the maximum and minimum intensities

nd u,v = bmax−bmin+1

Maximum intensity

Minimum intensity

Sarkar et al proposed DBC to estimate FD in gray level images.

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Improved Differential Box-Counting (IDBC)

BC

DBC IDBC

J. Li. et al proposed IDBC to reduce the undercounting problem occurred in the DBC method.

This method modifies the conventional DBC method in three parts as follows.

1. Box-height selection

2. Box-number calculation

3. Image partition

h′=h

1 2aσ

s

s

h

n′d = ceill kh′

1

;l k;l k

nd u,v = bmax bmin+1

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Improved Triangle Box-Counting (ITBC)Y. Kaewaramsri et al proposed an ITBC to reduce the effect of over-counting problem.

This method provides the two triangle box patterns

pattern1 - right diagonals

pattern2 - left diagonals

to increase the precision of box-counts.

BC

DBC IDBC ITDBC

Pattern 1Right diagonal

Pattern 2left diagonal

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Outline

▪ Introduction

▪ Background

▪ Proposed method▪ FD estimation in 3D images

▪ Experimental results

▪ Conclusion

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FD Estimation in 3D Images

BC

DBC IDBC ITBC

• Zhang, Luduan, Jing Z. Liu, David Dean, Vinod Sahgal, and Guang H. Yue. “A Three-Dimensional Fractal Analysis Method for Quantifying White Matter Structure in Human Brain.” Journal of Neuroscience Methods 150, no. 2 (January 30, 2006): 242–53.

• Ruiz de Miras, J., J. Navas, P. Villoslada, and F. J. Esteban. “UJA-3DFD: A Program to Compute the 3D Fractal Dimension from MRI Data.” Computer Methods and Programs in Biomedicine 104, no. 3 (December 2011): 452–60.

BC for 3D image

3D-to-slice image

DBC IDBC ITBC3D 3D 3D

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Equivalent property of FD

Definition:

“Let α be a square image and let β be summation of subimages. The fractal dimension of β is equivalent to that of α if the image sizes β (β1+β2+β3+…+βn) and α are identical.”

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Proof of Definition of equivalent property Brodatz

DatasetOriginal Image Size, α

(512×512)β Image

(4×256×256)d08 2.5470 2.5459d11 2.7270 2.7256d23 2.5980 2.5960d38 2.7182 2.7174d55 2.7221 2.7198d56 2.6136 2.6116d62 2.5467 2.5444d69 2.6409 2.6382d71 2.6478 2.6463d89 2.5491 2.5507d90 2.5147 2.5126d91 2.3759 2.3753d93 2.7539 2.7520d98 2.5318 2.5306d99 2.5171 2.5161d100 2.7497 2.7489

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statistically significant correlation at the 99% (2-tailed).

FD Estimation in 3D Images

FD estimation in 3D-MRI have two main steps

1. 3D-to-slice image transformation

2. FD estimation

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FD Estimation in 3D Images

1. 3D-to-slice image transformation

Isag, i = I x,y,zx i;for1 i Mx,

y = 1,My ,z = 1,Mz

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3D-to-slice image transformation

FD vector 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5

FD Estimation in 3D Images

1. 3D-to-slice image transformation

2. FD estimationFD′ = FDsag,q FDcor,q FDaxi,q

FDsag, q log Nssag,q d

log 1/d

Nssag, q= ∑ nssag,ipi 1 ,…, ∑ nssag,i

Mxi Mx p 1 p Mx

q

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Outline

▪ Introduction

▪ Background

▪ Proposed method

▪ Experimental results

▪ Conclusion

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Data preparation

SubjectClass #1

Elder (non-dementia)Class #2

Elder (very mild AD)Class #3

Elder (mild and moderate AD)

Number 100 70 30

Gender (male/female)

27/73 31/39 10/20

Age (year) ±SD 75.6 ± 9.2 76.2 ± 7.2 78 ± 6.9

CDR 0 0.5 1 or 2

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Data preparation

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Data set Class number ratio

Set #1 #1 and #2 170 100:70

Set #2 #1 and #3 130 100:30

Set #3 #1, #2 and #3 200 100:100

Class #1 Elder (non-dementia)Class #2 Elder (very mild AD)Class #3 Elder (mild and moderate AD)

Experimental setup

▪ In order to evaluate the performance of the proposed method, two experiments are set up.

1. t-test statistical analysis of FD features between 3D-to-slice image with BC method (3D-BC) and 3D-to-slice image with ITBC method (3D-ITBC).

2. Classification by k-NN, SVM and HLP methods with 10k fold cross-validation. The FD features are extracted from MRI with q=5; 15 values (5FDsag,q, 5FDcor,q and 5FDaxi,q).

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Experimental results 1

DatasetAxial view Sagittal view Coronal view

3D-BC 3D-ITBC 3D-BC 3D-ITBC 3D-BC 3D-ITBC

Set #1 .373 .093 .395 .016* .403 .042*

Set #2 .590 .000* .510 .001* .484 .008*

Set #3 .673 .004* .725 .001* .747 .006*

* Significant difference (2-tailed) at p < 0.05

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Experimental results 2

Datasetk-NN SVM (Linear) SVM (RBF) HLP (Polynomial)

Accuracy SD Accuracy SD Accuracy SD Accuracy SD

Set #1 51.76 12.34 45.88 12.02 52.35 12.54 55.29 10.07

Set #2 64.62 11.00 61.54 9.59 66.92 15.41 78.46 3.24

Set #3 57.00 13.98 54.00 8.43 60.00 12.25 56.00 8.43

Datasetk-NN SVM (Linear) SVM (RBF) HLP (Polynomial)

Accuracy SD Accuracy SD Accuracy SD Accuracy SD

Set #1 61.76 8.43 59.41 13.71 64.71 7.34 65.29 15.55

Set #2 72.31 12.14 75.38 12.46 76.15 2.43 86.15 7.94

Set #3 59.00 10.49 64.00 10.49 68.00 11.83 67.50 9.20

BC3D

ITBC3D

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Outline

▪ Introduction

▪ Background

▪ Proposed method

▪ Experimental results

▪ Conclusion

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Conclusion

▪ The proposed method uses equivalent property of FD to estimate the FD values in 3D gray-level images.

▪ The FD of gray-level images more efficiency than the FD of binary images.

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Thank you for your attentionQ&A