A dual EM algorithm for TV regularized Gaussian mixture model in image segmentation

Shi Yan; Jun Liu; Haiyang Huang; Xue-Cheng Tai

doi:10.3934/ipi.2019030

Article Contents

2019, Volume 13, Issue 3: 653-677. Doi: 10.3934/ipi.2019030

This issue Previous Article Inverse random source problem for biharmonic equation in two dimensions Next Article Momentum ray transforms

A dual EM algorithm for TV regularized Gaussian mixture model in image segmentation

1.
Laboratory of Mathematics and Complex Systems (Ministry of Education of China), School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China
2.
Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong, China
3.
Depart of Mathematics, University of Bergen, P. O. Box 7800, N-5020, Bergen, Norway

^* Corresponding author: Jun Liu
^* Corresponding author: Jun Liu

Received: April 30, 2018

Revised: September 30, 2018

Published: March 2019

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Abstract

A dual expectation-maximization (EM) algorithm for total variation (TV) regularized Gaussian mixture model (GMM) is proposed in this paper. The algorithm is built upon the EM algorithm with TV regularization (EM-TV) model which combines the statistical and variational methods together for image segmentation. Inspired by the projection algorithm proposed by Chambolle, we give a dual algorithm for the EM-TV model. The related dual problem is smooth and can be easily solved by a projection gradient method, which is stable and fast. Given the parameters of GMM, the proposed algorithm can be seen as a forward-backward splitting method which converges. This method can be easily extended to many other applications. Numerical results show that our algorithm can provide high quality segmentation results with fast computation speed. Compared with the well-known statistics based methods such as hidden Markov random field with EM method (HMRF-EM), the proposed algorithm has a better performance. The proposed method could also be applied to MRI segmentation such as SPM8 software and improve the segmentation results.

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Mathematics Subject Classification: Primary: 68U10.

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Figure 1. Comparison of Dual EM-TV, Original EM-TV [26] and HMRF-EM methods for 2 classes segmentation

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Figure 2. Enlarged red square regions in Figure 1

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Figure 3. Comparison of Dual EM-TV, Original EM-TV [26] and HMRF-EM methods for 4 classes segmentation

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Figure 4. Enlarged red square regions in Figure 3

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Figure 5. An image with two regions, which are the same mean but different variance, and the histogram

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Figure 6. Comparison of segmentation results of GMM-EM, Dual EM-TV and CV model on region with same mean but different variances image

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Figure 7. Comparison between Dual EM-TV and HMRF-EM method

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Figure 8. Segmentation results by Dual EM-TV in color images for two-region images

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Figure 9. Segmentation results by Dual EM-TV in color images for multi-region images

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Figure 10. Segmentation comparison under different noise levels. Noise standard deviation: A: 120/255, B: 240/255, C: 300/255, D: 390/255

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Figure 11. Segmentation comparison of the same multi-region image by different noise levels. Noise standard deviation: A: 50/255, B: 60/255, C: 70/255, D: 80/255

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Figure 12. Results of SPM8 with one slice of MR Images. First row: left image, MRI with 0% RF and 0% noise; right: MRI with 40% RF and 9% noise). Second and third rows: the first and third columns are the newsegment method for left and right images, respectively; the second and fourth columns are the proposed method for left and right images, respectively

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Table 1. Comparison of 2 classes segmentation for Dual EM-TV, Original EM-TV [26] and HMRF-EM method (Heart image with size $ 615\times 615 $)

Methods	CPU time (seconds)	Accuracy (SAI)
Dual EM-TV	1.233	99.70%
Original EM-TV [26]	9.482	98.02%
HMRF-EM 4n	4.052	91.84%
HMRF-EM 8n	3.052	97.38%
%HMRF-EM 12n	2.653	98.69%
HMRF-EM 20n	1.528	99.24%

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Table 2. Comparison of 4 classes segmentation for Dual EM-TV, Original EM-TV [26] and HMRF-EM methods (4-color image with size $ 600\times 600 $)

Methods	CPU time (seconds)	Accuracy (SAI)
Dual EM-TV	1.699	99.90%
Original EM-TV [26]	18.502	99.02%
HMRF-EM 4n	3.480	97.39%
HMRF-EM 8n	3.202	99.81%
%HMRF-EM 12n	2.696	99.87%
HMRF-EM 20n	2.388	99.83%

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Table 3. Comparison of DM in SPM8 on brain images

noise level	5%	5%	7%	7%	9%	9%
brain part	WM	GM	WM	GM	WM	GM
Dual EM-TV	0.9380	0.9122	0.9218	0.8970	0.9035	0.8818
New Segment	0.9317	0.9099	0.9035	0.8822	0.8759	0.8536

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