Parameter learning and fractional differential operators: Applications in regularized image denoising and decomposition problems

Figure 1.  Decomposition of the original image (left) in structural component $ u $ (middle) and textural component $ v $ (right)

Figure 2.  Fractional Laplacian (1) of image $ u_d $. A higher exponent $ s $ in (1) results in stronger smoothing of high oscillations

Figure 3.  Riesz potential (2) of image $ u_d $. A lower exponent $ s $ in (2) smooths the image $ u_d $

Figure 4.  The denoising via solving (3) strongly depends on the choice of the parameters $ s $ and $ \alpha $

Figure 5.  Energy functional $ j_n $ with function $ \varphi_1 $ and different choices for $ s $ and $ \alpha $

Figure 6.  Results for the image "boat". As clearly seen, the model eliminates the fog based on the normally distributed noise

Figure 7.  In the image "peppers" there is a significant reduction of noise while maintaining the edges

Figure 8.  With a higher resolution($ n = 1566 $) than in the sample images before ($ n = 512 $), the noise can be better suppressed optically. The denoised image is visually almost identical to the image in [1]. The small deviation of the optimal parameters is due to the influence of the function $ \varphi_1(s,\alpha) $

Figure 9.  On the synthetic image we get a good denoising performance, which is reflected in the high SSIM value. The denoised image is visually almost identical to the image in [1]

Figure 10.  We study the influence of the noise on the parameter $ s $. The numerical results coincide with theoretical considerations that more noise implies a stronger smoothing of the image, i.e a higher value of the parameter $ s $

Figure 11.  In the case of the parameter $ \alpha $ the numerical results coincide with theoretical considerations. A higher noise has the result that the denoised image is more far away from the noisy image, i.e a lower value of the parameter $ \alpha $

Figure 12.  In comparison to the fractional Laplacian model, the ROF model produces smoother edges. Moreover the ROF model has a better SSIM-value

Figure 13.  Runtime comparison between the fractional Laplacian model and the ROF model. Both models show empirically a linear runtime, but the fractional Laplacian model has a reduced computing time by factors 16-22. We used MATLAB R2015a with CPU i3-3240 and 8 GB RAM

Figure 14.  The PSNR(peak signal-to-noise ratio) of the ROF model outperforms the Laplacian model, but the discrepancy depends on the choice of the image. We observe a decreasing PSNR for a larger variance, implying a stronger noise. A higher PSNR value implies a better image quality

Figure 15.  The SSIM(structural similarity index measure) of the ROF model outperforms the Laplacian model, again the performance gap highly depends on the image. We observe a decreasing SSIM for stronger noise. A higher SSIM value implies a better image quality

Figure 16.  Detail of the image "Baboon". The results of the Fractional Laplacian and ROF model are similar.

Figure 17.  Image components with fixed $ s_1 = 0.2, s_2 = -1, \beta = 1 $ and different choices for parameter $ \alpha $ for the noise-free image. The choice of the parameter $ \alpha $ has a strong impact on the decomposition

Figure 18.  Decomposition of the image "kentaur" with optimal parameters $ \bar{s_1} = 0.172,\; \bar{\alpha} = 9999.7,\; \bar{s_2} = -0.926 $ and $ \bar{\beta} = 10410 $. We see a significant improvement of the SSIM-value and the desired image decomposition

Figure 19.  The component $ u $ of (26) is obtained with optimal parameters $ \bar{s_1} = 0.351,\; \bar{\alpha} = 9999.4,\; \bar{s_2} = -0.918 $ and $ \bar{\beta} = 9997.2 $. The fractional image decomposition model achieves a significantly better result than the fractional image denoising model with an improvement of the SSIM value by 0.4. The result is less blurred and has a better contrast

Figure 20.  Detail of the image "pepper" ($ \sigma = 0.1 $). The model (29) can not sufficiently distinguish between noise and component $ v $

Figure 21.  Detail of the image "Baboon" ($ \sigma = 0.1 $)