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# Nonlinear fractional diffusion model for deblurring images with textures

• * Corresponding author: Jiebao Sun

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.

Mathematics Subject Classification: Primary: 26A33, 68U10; Secondary: 65M06.

 Citation:

• Figure 1.  Blurry image and the deblurred results

Figure 2.  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)

Figure 3.  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 ("$256 \times 256$")

Figure 4.  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

Figure 5.  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

Figure 6.  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

Figure 7.  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

Figure 8.  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) $k_1 = 100$

Figure 9.  Deblurring of the Barbara image, scenario Ⅱ. From left to right: blurred, reconstructed by FastTVMM [24], WFBM [5], our method

Figure 10.  Deblurring of the Barbara image, scenario Ⅲ. From left to right: blurred, reconstructed by FastTVMM [24], WFBM [5], our method

Figure 11.  Deblurring of the Aerial image, scenario Ⅱ. From left to right: blurred, reconstructed by FastTVMM [24], WFBM [5], our method

Figure 12.  Deblurring of the Aerial image, scenario Ⅲ. From left to right: blurred, reconstructed by FastTVMM [24], WFBM [5], our method

Figure 13.  The original Barbara image and Aerial image, the marked rectangles are marked for zooming

Figure 14.  Zooming regions

Figure 15.  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

Figure 16.  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

Figure 17.  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

Figure 18.  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

Figure 19.  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

Figure 20.  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

Figure 21.  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

Figure 22.  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

Figure 23.  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

Figure 24.  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)

Table 1.  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

Table 2.  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

Table 3.  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

Table 4.  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
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