Image Denoising via Improved Dictionary Learning with Global Structure and Local Similarity Preservations
Abstract
:1. Introduction
- We developed an image denoising approach that processes advantages of both reconstruction-based and learning-based methods. A practical two-stage optimization solution is proposed for the implementation.
- We introduced a sparse term to reduce the multiplicative noise approximately to additive noise. Consequently, our method is capable of removing both additive and multiplicative noise from a noisy image.
- We used the Laplacian Schatten norm to capture the edge information and preserve small details that may be potentially ignored by learning based methods. Hence, both global and local information can be preserved in our model for image denoising.
- We established a new method that combines Method of Optimal Directions (MOD) with Approximate K-SVD (AK-SVD) for dictionary learning.
2. Proposed Method
2.1. Formulation
- Global Structure Reconstruction: High pass filter emphasizes fine details of an image by effectively enhancing contents that are of high intensity gradient in the image. After high pass filtering, clean image contains the high frequency contents that represent global structures while low frequency contents are eliminated, making the filtered image of low rank. However, since noise usually has high-frequency components too, it may still remain together with the structural information after high-pass filtering. For each pixel, noise usually does not depend on neighboring pixels while the pixels on the global structure such as edges and textures have correlations with their neighbouring pixels. To differentiate noise and structural pixels, we consider minimizing the rank of high-pass filtered image. As Schatten norm can effectively approximate the rank [24], we use the Schatten norm of high-pass filtered image to capture the underlying structures.Let be a matrix with singular value decomoposition (SVD) where and are unitary matrices consisting of singular vectors of X, and is a rectangular diagonal matrix consisting of singular values of X. Then, the Schatten p-norm ( norm) of X is defined asIn this paper, to high-pass filter the image, we adopt an 8-neighborhoods Laplacian operator defined asThis Laplacian filter captures 8-directional connectedness of each pixel and thus the structures of the image as well. By filtering the image with such Laplacian filter, we can obtain a low-rank filtered image containing the global structures of the image. Hence, it is desireable to minimize the rank of to ensure the low-rankness of the global structures. To achieve this goal, we propose to adopt the above defined Hessian Schatten-p norm as rank approximation, and by minimizing , the global structures of the image can be well preserved.Because multiplicative noise is image content dependent, it may remain mixed with the clean image after minimizing Laplacian Schatten norm of the noisy image. To alleviate the effect of multiplicative noise, we introduce a sparse matrix S that may as well capture the outliers in the case of additive noise. In a now-standard way, we minimize the 1-norm of S to obtain the sparsity. In summary, our model is as follows:
- Local Similarity Preservation: We define the local similarity of an image using its patches with a size of pixels. We define an operator that extracts the ith patch from X and orders it as a column vector, i.e., . To preserve the local similarity of image patches we exploit the dictionary learning. Define a dictionary , where is the number of dictionary basis. Each column of is a basis, i.e., and the dictionary is redundant. The local similarity suggests that every patch in the clean image may be sparsely represented over this dictionary. The sparse representation vector is obtained by solving the following constrained minimization problem:
2.2. Practical Solution for Optimization
2.2.1. Global Structure Reconstruction Stage
2.2.2. Dictionary Learning Stage
Algorithm 1 The Laplacian Schatten p-norm and Learning Algorithm (LSLA-p). |
Require: Noisy Image: Y; Penalty parameter: ; Smoothing parameter: ; Stopping tolerence: ; Clearn Image X;
|
3. Experiments
3.1. Parameter Setting
3.2. Performance and Analysis
3.3. Parameter Sensitivity
3.4. Time Comparison and Analysis
3.5. Discussion
4. Conclusions
Author Contributions
Acknowledgments
Conflicts of Interest
Abbreviations
MDPI | Multidisciplinary Digital Publishing Institute |
DOAJ | Directory of open access journals |
TLA | Three Letter Acronym |
LD | Linear Dichroism |
Appendix A
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Parameter | Symbol | Empirical Value for | Empirical Value for |
---|---|---|---|
Penalty parameter | 10 | 10 | |
Penalty parameter | 0.1 | 0.1 | |
Penalty parameter | |||
Smoothing parameter | 0.12 | 0 | |
Stopping tolerence | 0.001 | 0.001 | |
Stopping tolerence | 0.001 | 0.001 |
Image | Face | Kids | ||||||
Method | TV | PSNR | 21.47 | 20.44 | 19.02 | 23.06 | 21.03 | 19.49 |
SSIM | 0.6915 | 0.6435 | 0.6131 | 0.6790 | 0.5932 | 0.5667 | ||
HS | PSNR | 22.13 | 20.92 | 19.42 | 24.03 | 21.60 | 19.96 | |
SSIM | 0.7417 | 0.6992 | 0.6616 | 0.7409 | 0.6706 | 0.6333 | ||
EPLL | PSNR | 22.02 | 20.85 | 19.30 | 24.02 | 21.62 | 19.39 | |
SSIM | 0.7320 | 0.7030 | 0.6636 | 0.7531 | 0.6817 | 0.6366 | ||
BM3D | PSNR | 22.80 | 20.76 | 20.05 | 24.52 | 22.08 | 20.40 | |
SSIM | 0.7536 | 0.6679 | 0.6765 | 0.7603 | 0.6882 | 0.6458 | ||
LSLA-2 | PSNR | 23.25 | 22.50 | 20.95 | 24.69 | 23.03 | 23.68 | |
SSIM | 0.7679 | 0.7396 | 0.6851 | 0.7555 | 0.7052 | 0.6578 | ||
LSLA-1 | PSNR | 23.48 | 22.05 | 21.19 | 24.59 | 23.29 | 22.45 | |
SSIM | 0.7694 | 0.7217 | 0.6912 | 0.7423 | 0.7063 | 0.6825 | ||
Image | Wall | Abdomen | ||||||
Method | TV | PSNR | 20.70 | 18.19 | 16.80 | 22.57 | 20.06 | 18.50 |
SSIM | 0.6521 | 0.5601 | 0.4978 | 0.5579 | 0.4940 | 0.4697 | ||
HS | PSNR | 21.33 | 18.54 | 17.03 | 23.29 | 20.52 | 18.77 | |
SSIM | 0.7043 | 0.5975 | 0.5460 | 0.6384 | 0.5592 | 0.5300 | ||
EPLL | PSNR | 21.36 | 18.38 | 16.76 | 23.51 | 20.64 | 18.84 | |
SSIM | 0.7254 | 0.6254 | 0.5698 | 0.6517 | 0.5915 | 0.5440 | ||
BM3D | PSNR | 21.97 | 19.04 | 17.42 | 24.14 | 21.26 | 19.50 | |
SSIM | 0.7421 | 0.6410 | 0.5838 | 0.6700 | 0.6026 | 0.5603 | ||
LSLA-2 | PSNR | 22.28 | 20.11 | 19.22 | 25.06 | 22.68 | 21.47 | |
SSIM | 0.7598 | 0.6730 | 0.6477 | 0.7530 | 0.6680 | 0.6237 | ||
LSLA-1 | PSNR | 22.51 | 20.31 | 19.12 | 24.97 | 22.72 | 21.37 | |
SSIM | 0.7675 | 0.6736 | 0.6311 | 0.7462 | 0.6663 | 0.6096 |
Image | Noise Level | Method | |||||
---|---|---|---|---|---|---|---|
AA | MIDAL | LSLA-2 | |||||
PSNR | SSIM | PSNR | SSIM | PSNR | SSIM | ||
Nimes | 22.40 | 0.5378 | 22.68 | 0.5041 | 23.64 | 0.5942 | |
25.59 | 0.7572 | 25.36 | 0.7537 | 26.50 | 0.7757 | ||
27.53 | 0.8511 | 27.88 | 0.8910 | 28.51 | 0.8625 | ||
Fields | 24.38 | 0.3369 | 25.13 | 0.3380 | 25.27 | 0.3505 | |
26.43 | 0.4230 | 27.40 | 0.4024 | 27.46 | 0.4622 | ||
26.77 | 0.4464 | 28.27 | 0.5371 | 28.64 | 0.5421 |
Algorithm | Time (s) | |||
---|---|---|---|---|
Face | Kids | Wall | Abdomen | |
BM3D | 1.0284 | 1.0336 | 1.1008 | 3.7333 |
HS1 | 16.9454 | 18.0842 | 17.5324 | 37.0296 |
EPLL | 146.2443 | 78.3126 | 146.561 | 502.4728 |
TV | 0.6696 | 0.6841 | 0.6538 | 1.3947 |
LSLA2 | 124.036 | 185.6624 | 122.9288 | 404.6401 |
LSLA1 | 169.4185 | 139.1726 | 178.3549 | 438.5715 |
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Cai, S.; Kang, Z.; Yang, M.; Xiong, X.; Peng, C.; Xiao, M. Image Denoising via Improved Dictionary Learning with Global Structure and Local Similarity Preservations. Symmetry 2018, 10, 167. https://doi.org/10.3390/sym10050167
Cai S, Kang Z, Yang M, Xiong X, Peng C, Xiao M. Image Denoising via Improved Dictionary Learning with Global Structure and Local Similarity Preservations. Symmetry. 2018; 10(5):167. https://doi.org/10.3390/sym10050167
Chicago/Turabian StyleCai, Shuting, Zhao Kang, Ming Yang, Xiaoming Xiong, Chong Peng, and Mingqing Xiao. 2018. "Image Denoising via Improved Dictionary Learning with Global Structure and Local Similarity Preservations" Symmetry 10, no. 5: 167. https://doi.org/10.3390/sym10050167