# American Institute of Mathematical Sciences

December  2019, 13(6): 1161-1188. doi: 10.3934/ipi.2019052

## Nonlinear fractional diffusion model for deblurring images with textures

 School of Mathematics, Harbin Institute of Technology, Harbin, Heilongjiang 150001, China

* Corresponding author: Jiebao Sun

Received  June 2018 Revised  July 2019 Published  October 2019

Fund Project: The work of the authors was partially supported by the National Natural Science Foundation of China (11971131, U1637208, 61873071, 51476047, 11871133) and the Natural Science Foundation of Heilongjiang Province (LC2018001, A2016003)

It is a long-standing problem to preserve fine scale features such as texture in the process of deblurring. In order to deal with this challenging but imperative issue, we establish a framework of nonlinear fractional diffusion equations, which performs well in deblurring images with textures. In the new model, a fractional gradient is used for regularization of the diffusion process to preserve texture features and a source term with blurring kernel is used for deblurring. This source term ensures that the model can handle various blurring kernels. The relation between the regularization parameter and the deblurring performance is investigated theoretically, which ensures a satisfactory recovery when the blur type is known. Moreover, we derive a digital fractional diffusion filter that lives on images. Experimental results and comparisons show the effectiveness of the proposed model for texture-preserving deblurring.

Citation: Zhichang Guo, Wenjuan Yao, Jiebao Sun, Boying Wu. Nonlinear fractional diffusion model for deblurring images with textures. Inverse Problems & Imaging, 2019, 13 (6) : 1161-1188. doi: 10.3934/ipi.2019052
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##### References:
Blurry image and the deblurred results
Comparison results: (a) Original signal (b) Result of fractional model (c) Result of integer model (d)-(f) Results in [0.44, 0.56] from (a)-(c)
Test images: (a) Barbara ("$300 \times 300$"), (b) Boat ("$512\times 512$"), (c) Aerial ("$512\times 512$"), (d) Chimpanzee ("$256 \times 256$"), (e) Synthetic ("$256 \times 256$"), (f) Cameraman ("$256 \times 256$")
The images in the first row are blurred images with different kernel: (a) Scenario Ⅰ, (b) Scenario Ⅱ, (c) Scenario Ⅲ, (d) Scenario Ⅳ, (e) Scenario Ⅴ. The images in the second row are the corresponding results using our method
The images in the first row are blurred images with different kernel: (a) Scenario Ⅰ, (b) Scenario Ⅱ, (c) Scenario Ⅲ, (d) Scenario Ⅳ, (e) Scenario Ⅴ. The images in the second row are the corresponding results using our method
The images in the first row are blurred images with different kernel: (a) Scenario Ⅰ, (b) Scenario Ⅱ, (c) Scenario Ⅲ, (d) Scenario Ⅳ, (e) Scenario Ⅴ. The images in the second row are the corresponding results using our method
The images in the first row are blurred images with different kernel: (a) Scenario Ⅰ, (b) Scenario Ⅱ, (c) Scenario Ⅲ, (d) Scenario Ⅳ, (e) Scenario Ⅴ. The images in the second row are the corresponding results using our method
The images in the first row are recovered by the proposed model with different $k_1$: (a) $k_1 = 5$, (b) $k_1 = 10$, (c) $k_1 = 50$, (d) $k_1 = 100$. The images in the second row are the corresponding results using variational method with: (e) $k_1 = 5$, (f) $k_1 = 10$, (g) $k_1 = 50$, (h) $k_1 = 100$
Deblurring of the Barbara image, scenario Ⅱ. From left to right: blurred, reconstructed by FastTVMM [24], WFBM [5], our method
Deblurring of the Barbara image, scenario Ⅲ. From left to right: blurred, reconstructed by FastTVMM [24], WFBM [5], our method
Deblurring of the Aerial image, scenario Ⅱ. From left to right: blurred, reconstructed by FastTVMM [24], WFBM [5], our method
Deblurring of the Aerial image, scenario Ⅲ. From left to right: blurred, reconstructed by FastTVMM [24], WFBM [5], our method
The original Barbara image and Aerial image, the marked rectangles are marked for zooming
Zooming regions
Deblurring of the Barbara image, scenario Ⅱ. From left to right, zoomed fragments of the following images are presented: blurred, reconstructed by FastTVMM [24], WFBM [5], our method
Deblurring of the Barbara image, scenario Ⅱ. From left to right, zoomed fragments of the following images are presented: blurred, reconstructed by FastTVMM [24], WFBM [5], our method
Deblurring of the Barbara image, scenario Ⅲ. From left to right, zoomed fragments of the following images are presented: blurred, reconstructed by FastTVMM [24], WFBM [5], our method
Deblurring of the Barbara image, scenario Ⅲ. From left to right, zoomed fragments of the following images are presented: blurred, reconstructed by FastTVMM [24], WFBM [5], our method
Deblurring of the Aerial image, scenario Ⅱ. From left to right, zoomed fragments of the following images are presented: blurred, reconstructed by FastTVMM [24], WFBM [5], our method
Deblurring of the Aerial image, scenario Ⅲ. From left to right, zoomed fragments of the following images are presented: blurred, reconstructed by FastTVMM [24], WFBM [5], our method
Top row: Noisy blurred images. Bottom row: Recovered images corresponding to the above images. Left two columns: scenario Ⅴ and noise levels $\sigma$ = 2, 5, respectively. Right two columns: scenario Ⅱ and noise levels $\sigma$ = 2, 5, respectively
Top row: Noisy blurred images with scenario Ⅰ and noise levels $\sigma = 2$ (left two columns), $\sigma = 5$ (right two columns). Bottom row: Recovered images corresponding to the above images
Left to right: Noisy blurred image (scenario Ⅰ and $\sigma = 2$), recovered image from FastTVMM [24], MSPB [31], WFBM [5], BM3D [11], and recovered image with proposed model
Deblurring results for the Synthetic blurred with scenario Ⅳ and corrupted by noise of standard deviation $\sigma = 3$. From left to right: original image; noisy blurred image; MSPB [31] (PSNR = 28.41, elapsed time = 202 seconds) and proposed model (PSNR = 30.39, elapsed time = 6 seconds)
The five different types of blurring kernels used in the experiments
 Scenario PSF Ⅰ [1 4 6 4 1]$^T$[1 4 6 4 1]/256 Ⅱ $15\times 15$ Gaussian PSF with standard deviation 1.5 Ⅲ fspecial('motion', 20, 45) is used in Matlab Ⅳ fspecial('disk', 3) is used in Matlab Ⅴ [1 1 1 1 1]$^T$[1 1 1 1 1]/25
 Scenario PSF Ⅰ [1 4 6 4 1]$^T$[1 4 6 4 1]/256 Ⅱ $15\times 15$ Gaussian PSF with standard deviation 1.5 Ⅲ fspecial('motion', 20, 45) is used in Matlab Ⅳ fspecial('disk', 3) is used in Matlab Ⅴ [1 1 1 1 1]$^T$[1 1 1 1 1]/25
Values of PSNR for the restoration by using the proposed model with $\alpha = 0.9$, $k_1 = 1$, $\beta = 1$, $\lambda = 5$
 Images $r=0$ $r=0.5$ $r=1$ $r=1.5$ $r=2$ Barbara 40.24 40.24 40.22 40.20 40.18 Boat 45.95 46.26 46.41 46.50 46.57 Aerial 44.15 44.32 44.38 44.42 44.45 Chimpanzee 44.46 44.46 44.48 44.49 44.51 Synthetic 27.11 27.08 27.06 27.05 27.03 Cameraman 42.45 42.57 42.58 42.58 42.58
 Images $r=0$ $r=0.5$ $r=1$ $r=1.5$ $r=2$ Barbara 40.24 40.24 40.22 40.20 40.18 Boat 45.95 46.26 46.41 46.50 46.57 Aerial 44.15 44.32 44.38 44.42 44.45 Chimpanzee 44.46 44.46 44.48 44.49 44.51 Synthetic 27.11 27.08 27.06 27.05 27.03 Cameraman 42.45 42.57 42.58 42.58 42.58
PSNR values for the experiments shown in Figures 2122
 PSNR (noisy) PSNR(recovered) Figure 21 Left two columns $\sigma=2$ 38.63 42.55 $\sigma=5$ 38.53 41.23 Right two columns $\sigma=2$ 39.81 41.74 $\sigma=5$ 39.67 40.55 Figure 22 Left two columns $\sigma=2$ 41.27 44.51 $\sigma=2$ 42.81 46.73 Right two columns $\sigma=5$ 41.09 42.63 $\sigma=5$ 42.67 45.03
 PSNR (noisy) PSNR(recovered) Figure 21 Left two columns $\sigma=2$ 38.63 42.55 $\sigma=5$ 38.53 41.23 Right two columns $\sigma=2$ 39.81 41.74 $\sigma=5$ 39.67 40.55 Figure 22 Left two columns $\sigma=2$ 41.27 44.51 $\sigma=2$ 42.81 46.73 Right two columns $\sigma=5$ 41.09 42.63 $\sigma=5$ 42.67 45.03
Recovery experiment (scenario Ⅰ and additive noise $\sigma = 2$). Figure 23 was obtained using different models
 Methods PSNR Time(s) FastTVMM in [24] 42.91 0.44 MSPB in [31] 43.21 220.12 WFBM in [5] 42.32 19.72 BM3D in [11] 42.98 60.85 Proposed model 42.57 1.45
 Methods PSNR Time(s) FastTVMM in [24] 42.91 0.44 MSPB in [31] 43.21 220.12 WFBM in [5] 42.32 19.72 BM3D in [11] 42.98 60.85 Proposed model 42.57 1.45
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