Edge detection with mixed noise based on maximum a posteriori approach

Yuying Shi; Zijin Liu; Xiaoying Wang; Jinping Zhang

doi:10.3934/ipi.2021035

Article Contents

2021, Volume 15, Issue 5: 1223-1245. Doi: 10.3934/ipi.2021035

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Edge detection with mixed noise based on maximum a posteriori approach

School of Mathematics and Physics, North China Electric Power University, Beijing 102206, China

^* Corresponding author: wangxiaoying@ncepu.edu.cn
^* Corresponding author: wangxiaoying@ncepu.edu.cn

Received: September 1, 2020

Revised: February 1, 2021

Early access: April 27, 2021

Published online: October 1, 2021

The first author is supported by Beijing Natural Science Foundation (No. Z200001).

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Abstract

Edge detection is an important problem in image processing, especially for mixed noise. In this work, we propose a variational edge detection model with mixed noise by using Maximum A-Posteriori (MAP) approach. The novel model is formed with the regularization terms and the data fidelity terms that feature different mixed noise. Furthermore, we adopt the alternating direction method of multipliers (ADMM) to solve the proposed model. Numerical experiments on a variety of gray and color images demonstrate the efficiency of the proposed model.

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Mathematics Subject Classification: Primary: 68U10.

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Figure 1. Comparisons of three different clean images without noise. Column 1: the original clean images; Column 2: detected edges; Column 3: restored images.

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Figure 2. Comparisons of two images with Gaussian noise and salt & pepper noise. Column 1: noisy images; Column 2: detected edges; Column 3: restored images.

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Figure 3. Comparisons of two images with Laplace noise and Gaussian noise. Column 1: noisy images; Column 2: detected edges; Column 3: restored images.

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Figure 4. Comparisons of two images with RVIN and Gaussian noise. Column 1: noisy images; Column 2: detected edges; Column 3: restored images.

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Figure 5. Comparisons of color image with Gaussian noise and salt & pepper noise. Column 1: noisy images; Column 2: detected edges; Column 3: restored images.

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Figure 6. Comparisons of color image with Laplace noise and Gaussian noise. Column 1: noisy images; Column 2: detected edges; Column 3: restored images.

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Figure 7. Comparisons of color images with RVIN and Gaussian noise. Column 1: noisy images; Column 2: detected edges; Column 3: restored images.

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Figure 8. Comparisons of the two models with G1 and SP1. Column 1: noisy images; Column 2: detected edges; Column 3: restored images.

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Figure 11. Results with different λ and λ₁.

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Figure 12. Results with different α and β.

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Figure 13. Results with different r.

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Figure 9. Comparisons of the two models with G1 and L1. Column 1: noisy images; Column 2: detected edges; Column 3: restored images.

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Figure 10. Comparisons of the two models with G1 and R1. Column 1: noisy images; Column 2: detected edges; Column 3: restored images.

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Figure 14. ROC curves using the two models (28) and (29) with three different mixed noise.

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Table 1. AUC of two models with different mixed noise

noise	G1 and S1	G1 and L1	G1 and R1	G2 and S2	G2 and L2	G2 and R2
model (28)	0.8095	0.8872	0.8025	0.8415	0.8503	0.8305
model (29)	0.8254	0.8897	0.8273	0.8844	0.8801	0.8683

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Table 2. Running times of two models with different mixed noise

noise	G1 and S1	G1 and L1	G1 and R1	G2 and S2	G2 and L2	G2 and R2
model (28)	2.83	2.87	2.88	3.02	2.85	2.87
model (29)	2.61	2.37	2.40	2.27	2.25	2.32

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Table 3. Iteration numbers of two models with different mixed noise

noise	G1 and S1	G1 and L1	G1 and R1	G2 and S2	G2 and L2	G2 and R2
model (28)	88	70	93	75	73	91
model (29)	74	65	81	68	60	80

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