=Paper= {{Paper |id=Vol-2485/paper33 |storemode=property |title=Automatic Choice of Denoising Parameter in Perona-Malik Model |pdfUrl=https://ceur-ws.org/Vol-2485/paper33.pdf |volume=Vol-2485 |authors=Andrey Nasonov,Nikolay Mamaev,Olga Volodina,Andrey Krylov }} ==Automatic Choice of Denoising Parameter in Perona-Malik Model== https://ceur-ws.org/Vol-2485/paper33.pdf

Automatic Choice of Denoising Parameter in PeronaMalik Model A.V. Nasonov1 , N.V. Mamaev1 , O.S. Volodina1 , A.S. Krylov1 nasonov@cs.msu.ru | mamaev.nikolay93@mail.ru | olya.volodina@gmail.com | kryl@cs.msu.ru 1 Faculty of Computational Mathematics and Cybernetics Lomonosov Moscow State University Moscow, Russia In this work, we propose a no-reference method for automatic choice of the parameters of Perona-Malik image diffusion algorithm for the problem of image denoising. The idea of the approach it to analyze and quantify the presence of structures in the difference image between the noisy image and the processed image as the mutual information value. We apply the proposed method to photographic images and to retinal images with modeled Gaussian noise with different parameters and analyze the effects of no-reference parameter choice compared to the optimal results. The proposed algorithm shows the effectiveness of no-reference parameter choice for the problem of image denoising. Keywords: Image denoising, non-linear diffusion, mutual information, automatic parameter choice 1. Introduction 2. PeronaMalik image diffusion One of the main challenges in image processing is One of the methods for image denoising is based denoising, as images are often corrupted by noise dur- on non-linear diffusion that considers the cleaned im- ing acquisition, transmission or storage. The goal is to age as the solution of the heat conduction. The dif- restore the original image by removing all noise while fusion coeﬀicient is chosen to reduce the diffusivity preserving the contents. Image denoising is usually in locations, which have more likelihood to be edges. needed as a preparation step in other image process- Such methods allow to preserve edges while denoising ing methods. There has been a great research effort due to the right choice of coeﬀicient. Koenderink [6] in that field, yet the problem remains unsolved. In and Hummel [7] pointed out that an imaged convolved this paper, we will use non-linear diffusion method with Gaussian kernel can be viewed as the solution of proposed in [1] by Perona and Malik, which repre- the heat conduction equation with original image as sents a filtered image as a solution of nonlinear diffu- initial condition. sion equation with the original image as initial state ∂u = div(c∇u), (x, t) ∈ Ω × [0, T ], and homogeneous Neumann boundary conditions. By ∂t choosing the diffusion parameter, one can manage to u(x, 0) = l0 , x ∈ Ω, clean flat areas and preserve edges. Non-linear diffu- ∂u sion is an iterative process so there is a problem of = 0, (x, t) ∈ ∂Ω × [0, T ], ∂⃗n stop mechanism. where l0 is the input image defined in spatial domain Most algorithms depend on noise level and thus Ω, c is the diffusion coeﬀicient, u(x, T ) is the result of must be controlled by parameters entered by a user heat distribution at moment T . or estimated automatically. A common approach for In linear diffusion, the coeﬀicient c is considered automatic choice of the parameters is to estimate the to be constant and independent of the image. In non- noise level and then choose the parameters according linear diffusion, the coeﬀicient c is a function of image to this noise level [2]. gradient magnitude c = c(|∇u|), which controls the A less common approach is to analyze the preser- blurring effect. Setting c to 1 in interior of each re- vation of image contents after image restoration and gion and 0 at the boundaries will encourage smoothing to pose the stopping criterion of anisotropic diffu- within a region and stop it on the edge, so that the sion. For example, the work [3] analyzes the edge boundaries remain sharp. In [1] Perona and Malik characteristics, the work [4] calculates image statis- proposed two functions as ( edge-estimator: ( s )2 ) tics for speckle noise reduction. In [5], a analysis of c1 (s) = exp − (1) the contents in the difference image between the origi- K nal noisy image and the processed image is performed. and 1 c2 (s) = ( s )2 , (2) Its idea comes from an assumption, that in the ideal 1+ case the difference image must contain just random K values without any structures from the original image. where K is the parameter of the method. If there are structures from the original noisy image, The diffusion equation can be solved numerically then we have wiped out the important information as by simple step algorithm: well as the noise. un+1 = un + tn · c(|∇u|)∆u, In this work, we investigate the automatic choice u0 = u(x, 0) = I0 , ∑ of the parameters for Perona-Malik image diffusion for tn = T. Gaussian noise for photographic and retinal images. n Copyright © 2019 for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0). In our work, we use the model (2). We have also noticed that for each noise level the ratio K/T can be fixed, and the parameter optimiza- 3. Target images tion becomes one-dimensional, but for different noise We have analyzed the automatic choice of the pa- level the optimal ratio K/T set is different. rameters for the Perona-Malik image diffusion algo- In order to go from two-dimensional to one- rithm for images of the following two classes: dimensional parameter optimization for any noise level, we have analyzed the behavior of optimal (K, T ) • Photographic images from TID database [8]; values and √ have found out that a set of optimal points • Retinal images from DRIVE database [9]. (log K, T ) lies along a line. Therefore, we introduce single-argument parameterization for (K, T ) values: An example of those images is shown in Fig. 1. K = q1 q2p , In order to model noisy images, we have added (3) white Gaussian noise with different levels σ in [1, 32] T = p2 , range to the reference images. where the coeﬀicients q1 and q2 are chosen experimen- tally by optimizing the full-reference metrics values. For both TID and DRIVE images, we have fixed q1 = 0.1 and optimized q2 value. The ranges of op- timal values for TID images and for DRIVE images are different, but they intersects. We have chosen q2 = 4600 from the intersection. 5. Noreference parameter choice We use the algorithm [5] for non-reference parame- TID database [8] DRIVE database [9]. ter choice. The algorithm is based on the assumption that the difference between input noisy and denoised Fig. 1. An example of reference images used for the images should not have features belonging to original analysis in the paper. image. In order to detect the presence of these fea- tures, the algorithm analyses the eigenvalues of Hes- 4. Fullreference parameter analysis sian matrix for scale and direction evaluation of ridges For each noisy image, we have obtained a pair and edges. The outcome of the algorithm is value µ of (K, T ) parameters that maximizes PSNR and — the mutual information that can be expressed as SSIM [10] metric values. We have found that for each the structure-to-noise ratio for the difference image. image there is a set of (K, T ) values producing the The lower the value µ is, the less details are corrupted results that are almost indistinguishable from the op- compared to noise removal. timal result. The set is banana-shaped and lies per- We use the following scenario: an image denoising pendicular to the line passing though the zero point. algorithm is executed with different parameters, then Fig. 2 shows an example of optimal (K, T ) values for the mutual information µ value is calculated between one of the images for different noise levels. the input image and each denoising result, and the image that minimizes the mutual information is cho- sen as the optimal result. In practice, there can be several local minima, and a special analysis should be performed in order to choose the optimal result. After replacing the two-parameter model with the single-parameter model (3), we find the optimal p value using both full-reference and no-reference ap- proach based on calculating the mutual information coeﬀicient. Noise = 3 Noise = 8 It has been found that mutual information corre- lates well with PSNR and SSIM values for noise level Fig. 2. A visualization of optimal (K, T ) parameters σ > 2. An example is shown in Fig. 3. A argument for an image with different noise levels in terms of where PSNR and/or SSIM reaches its maximum is PSNR. The horizontal axis represents K value in close to a local minimum of µ(p) function. In the case logarithmic scale. The vertical axis represents T of several local minima points, we find the one that value. Top-left corner is (0, 0) point. White regions maximizes the drop: corresponds to (K, T ) values that produce images with PSNR values close to the optimal value. Black popt = argp max ′ µ(p′ ) − µ(p). (4) p

2. Despite the fact that the proposed no-reference al- gorithm has worse PSNR and SSIM values than the optimal ones, the difference between the results of the proposed algorithm and the optimal results is practi- cally indistinguishable, and the effectiveness of image denoising is clearly visible. The individual results are shown in Fig. 4, 5, 6. 7. Conclusion The paper has shown that the parameters of the Perona-Malik image denoising algorithm can be auto- matically and effectively chosen by the algorithm that analyzes the presence of structures from the input im- age in the difference image. The work was supported by Russian Science Foun- Fig. 5. Denoising by the proposed method. TID dation grant 17-11-01279. image I08, noise σ = 32. TID DRIVE Optimization method PSNR SSIM PSNR SSIM Input noisy images 30.76 0.8083 30.76 0.6174 Full-reference, double-parameter, by PSNR 34.25 0.9252 39.21 0.9293 Full-reference, double-parameter, by SSIM 34.10 0.9286 38.87 0.9338 Full-reference, single-parameter, by PSNR 34.22 0.9236 38.99 0.9224 Full-reference, single-parameter, by SSIM 34.03 0.9267 38.40 0.9297 No-reference, single-parameter, by MU (proposed) 33.77 0.9135 38.97 0.9218 Table 1. PSNR and SSIM results for different scenarios of denoising parameter choice for TID and DRIVE images. [5] Nikolay Mamaev, Dmitry Yurin, and Andrey Krylov. Choice of the parameter for bm3d de- noising algorithm using no-reference metric. In 2018 7th European Workshop on Visual Informa- tion Processing (EUVIP), pages 1–6. IEEE, 2018. [6] Jan J Koenderink. The structure of images. Bi- ological cybernetics, 50(5):363–370, 1984. [7] Robert A Hummel. Representations based on zero-crossings in scale-space. In Readings in Computer Vision, pages 753–758. Elsevier, 1987. [8] N. Ponomarenko, L. Jin, O. Ieremeiev, V. Lukin, K. Egiazarian, J. Astola, B. Vozel, K. Chehdi, M. Carli, F. Battisti, and C.-C. Jay Kuo. Im- age database tid2013: Peculiarities, results and perspectives. Signal Processing: Image Commu- nication, 30:57–77, 2015. [9] M.D. Abramoff J.J. Staa and, M. Niemeijer, M.A. Viergever, and B. van Ginneken. Ridge based vessel segmentation in color images of the retina. IEEE Transactions on Medical Imaging, 23:501–509, 2004. [10] Z. Wang, A. Bovik, H. Sheikh, and E. Simoncelli. Image quality assessment: from error visibility to Fig. 6. Denoising by the proposed method. structural similarity. IEEE Transactions on Im- DRIVE image I02, noise σ = 13. age Processing, 13(4):600–612, 2004. 8. References [1] Pietro Perona and Jitendra Malik. Scale-space and edge detection using anisotropic diffusion. IEEE Transactions on pattern analysis and ma- chine intelligence, 12(7):629–639, 1990. [2] Karl Krissian and Santiago Aja-Fernández. Noise-driven anisotropic diffusion filtering of mri. IEEE transactions on image processing, 18(10): 2265–2274, 2009. [3] Chourmouzios Tsiotsios and Maria Petrou. On the choice of the parameters for anisotropic dif- fusion in image processing. Pattern recognition, 46(5):1369–1381, 2013. [4] Santiago Aja-Fernández and Carlos Alberola- López. On the estimation of the coeﬀicient of variation for anisotropic diffusion speckle filter- ing. IEEE Transactions on Image Processing, 15 (9):2694–2701, 2006.