139 research outputs found

    A Convex Variational Model for Restoring Blurred Images with Multiplicative Noise

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    Abstract. In this paper, a new variational model for restoring blurred images with multiplicative noise is proposed. Based on the statistical property of the noise, a quadratic penalty function technique is utilized in order to obtain a strictly convex model under a mild condition, which guarantees the uniqueness of the solution and the stabilization of the algorithm. For solving the new convex variational model, a primal-dual algorithm is proposed and its convergence is studied. The paper ends with a report on numerical tests for the simultaneous deblurring and denoising of images subject to multiplicative noise. A comparison with other methods is provided as well. Key words. Convexity, deblurring, multiplicative noise, primal-dual algorithm, total variation regularization, variational model. AMS subject classifications. 52A41, 65K10, 65K15, 90C30, 90C4

    Image reconstruction under non-Gaussian noise

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    Bregman Cost for Non-Gaussian Noise

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    One of the tasks of the Bayesian inverse problem is to find a good estimate based on the posterior probability density. The most common point estimators are the conditional mean (CM) and maximum a posteriori (MAP) estimates, which correspond to the mean and the mode of the posterior, respectively. From a theoretical point of view it has been argued that the MAP estimate is only in an asymptotic sense a Bayes estimator for the uniform cost function, while the CM estimate is a Bayes estimator for the means squared cost function. Recently, it has been proven that the MAP estimate is a proper Bayes estimator for the Bregman cost if the image is corrupted by Gaussian noise. In this work we extend this result to other noise models with log-concave likelihood density, by introducing two related Bregman cost functions for which the CM and the MAP estimates are proper Bayes estimators. Moreover, we also prove that the CM estimate outperforms the MAP estimate, when the error is measured in a certain Bregman distance, a result previously unknown also in the case of additive Gaussian noise

    ๋น„๊ฐ€์šฐ์‹œ์•ˆ ์žก์Œ ์˜์ƒ ๋ณต์›์„ ์œ„ํ•œ ๊ทธ๋ฃน ํฌ์†Œ ํ‘œํ˜„

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    ํ•™์œ„๋…ผ๋ฌธ(๋ฐ•์‚ฌ)--์„œ์šธ๋Œ€ํ•™๊ต ๋Œ€ํ•™์› :์ž์—ฐ๊ณผํ•™๋Œ€ํ•™ ์ˆ˜๋ฆฌ๊ณผํ•™๋ถ€,2020. 2. ๊ฐ•๋ช…์ฃผ.For the image restoration problem, recent variational approaches exploiting nonlocal information of an image have demonstrated significant improvements compared with traditional methods utilizing local features. Hence, we propose two variational models based on the sparse representation of image groups, to recover images with non-Gaussian noise. The proposed models are designed to restore image with Cauchy noise and speckle noise, respectively. To achieve efficient and stable performance, an alternating optimization scheme with a novel initialization technique is used. Experimental results suggest that the proposed methods outperform other methods in terms of both visual perception and numerical indexes.์˜์ƒ ๋ณต์› ๋ฌธ์ œ์—์„œ, ์˜์ƒ์˜ ๋น„๊ตญ์ง€์ ์ธ ์ •๋ณด๋ฅผ ํ™œ์šฉํ•˜๋Š” ์ตœ๊ทผ์˜ ๋‹ค์–‘ํ•œ ์ ‘๊ทผ ๋ฐฉ์‹์€ ๊ตญ์ง€์ ์ธ ํŠน์„ฑ์„ ํ™œ์šฉํ•˜๋Š” ๊ธฐ์กด ๋ฐฉ๋ฒ•๊ณผ ๋น„๊ตํ•˜์—ฌ ํฌ๊ฒŒ ๊ฐœ์„ ๋˜์—ˆ๋‹ค. ๋”ฐ๋ผ์„œ, ์šฐ๋ฆฌ๋Š” ๋น„๊ฐ€์šฐ์‹œ์•ˆ ์žก์Œ ์˜์ƒ์„ ๋ณต์›ํ•˜๊ธฐ ์œ„ํ•ด ์˜์ƒ ๊ทธ๋ฃน ํฌ์†Œ ํ‘œํ˜„์— ๊ธฐ๋ฐ˜ํ•œ ๋‘ ๊ฐ€์ง€ ๋ณ€๋ถ„๋ฒ•์  ๋ชจ๋ธ์„ ์ œ์•ˆํ•œ๋‹ค. ์ œ์•ˆ๋œ ๋ชจ๋ธ์€ ๊ฐ๊ฐ ์ฝ”์‹œ ์žก์Œ๊ณผ ์ŠคํŽ™ํด ์žก์Œ ์˜์ƒ์„ ๋ณต์›ํ•˜๋„๋ก ์„ค๊ณ„๋˜์—ˆ๋‹ค. ํšจ์œจ์ ์ด๊ณ  ์•ˆ์ •์ ์ธ ์„ฑ๋Šฅ์„ ๋‹ฌ์„ฑํ•˜๊ธฐ ์œ„ํ•ด, ๊ต๋Œ€ ๋ฐฉํ–ฅ ์Šน์ˆ˜๋ฒ•๊ณผ ์ƒˆ๋กœ์šด ์ดˆ๊ธฐํ™” ๊ธฐ์ˆ ์ด ์‚ฌ์šฉ๋œ๋‹ค. ์‹คํ—˜ ๊ฒฐ๊ณผ๋Š” ์ œ์•ˆ๋œ ๋ฐฉ๋ฒ•์ด ์‹œ๊ฐ์ ์ธ ์ธ์‹๊ณผ ์ˆ˜์น˜์ ์ธ ์ง€ํ‘œ ๋ชจ๋‘์—์„œ ๋‹ค๋ฅธ ๋ฐฉ๋ฒ•๋ณด๋‹ค ์šฐ์ˆ˜ํ•จ์„ ๋‚˜ํƒ€๋‚ธ๋‹ค.1 Introduction 1 2 Preliminaries 5 2.1 Cauchy Noise 5 2.1.1 Introduction 6 2.1.2 Literature Review 7 2.2 Speckle Noise 9 2.2.1 Introduction 10 2.2.2 Literature Review 13 2.3 GSR 15 2.3.1 Group Construction 15 2.3.2 GSR Modeling 16 2.4 ADMM 17 3 Proposed Models 19 3.1 Proposed Model 1: GSRC 19 3.1.1 GSRC Modeling via MAP Estimator 20 3.1.2 Patch Distance for Cauchy Noise 22 3.1.3 The ADMM Algorithm for Solving (3.7) 22 3.1.4 Numerical Experiments 28 3.1.5 Discussion 45 3.2 Proposed Model 2: GSRS 48 3.2.1 GSRS Modeling via MAP Estimator 50 3.2.2 Patch Distance for Speckle Noise 52 3.2.3 The ADMM Algorithm for Solving (3.42) 53 3.2.4 Numerical Experiments 56 3.2.5 Discussion 69 4 Conclusion 74 Abstract (in Korean) 84Docto

    Multiscale hierarchical decomposition methods for images corrupted by multiplicative noise

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    Recovering images corrupted by multiplicative noise is a well known challenging task. Motivated by the success of multiscale hierarchical decomposition methods (MHDM) in image processing, we adapt a variety of both classical and new multiplicative noise removing models to the MHDM form. On the basis of previous work, we further present a tight and a refined version of the corresponding multiplicative MHDM. We discuss existence and uniqueness of solutions for the proposed models, and additionally, provide convergence properties. Moreover, we present a discrepancy principle stopping criterion which prevents recovering excess noise in the multiscale reconstruction. Through comprehensive numerical experiments and comparisons, we qualitatively and quantitatively evaluate the validity of all proposed models for denoising and deblurring images degraded by multiplicative noise. By construction, these multiplicative multiscale hierarchical decomposition methods have the added benefit of recovering many scales of an image, which can provide features of interest beyond image denoising

    Variational approach for restoring blurred images with cauchy noise

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