Total variation denoising in l1l^1 anisotropy

We aim at constructing solutions to the minimizing problem for the variant of Rudin-Osher-Fatemi denoising model with rectilinear anisotropy and to the gradient flow of its underlying anisotropic total variation functional. We consider a naturally defined class of functions piecewise constant on rectangles (PCR). This class forms a strictly dense subset of the space of functions of bounded variation with an anisotropic norm. The main result shows that if the given noisy image is a PCR function, then solutions to both considered problems also have this property. For PCR data the problem of finding the solution is reduced to a finite algorithm. We discuss some implications of this result, for instance we use it to prove that continuity is preserved by both considered problems.Comment: 34 pages, 9 figure

영상 복원 문제의 변분법적 접근

학위논문 (박사)-- 서울대학교 대학원 : 수리과학부, 2013. 2. 강명주.Image restoration has been an active research area in image processing and computer vision during the past several decades. We explore variational partial differential equations (PDE) models in image restoration problem. We start our discussion by reviewing classical models, by which the works of this dissertation are highly motivated. The content of the dissertation is divided into two main subjects. First topic is on image denoising, where we propose non-convex hybrid total variation model, and then we apply iterative reweighted algorithm to solve the proposed model. Second topic is on image decomposition, in which we separate an image into structural component and oscillatory component using local gradient constraint.Abstract i 1 Introduction 1 1.1 Image restoration 2 1.2 Brief overview of the dissertation 3 2 Previous works 4 2.1 Image denoising 4 2.1.1 Fundamental model 4 2.1.2 Higher order model 7 2.1.3 Hybrid model 9 2.1.4 Non-convex model 12 2.2 Image decomposition 22 2.2.1 Meyers model 23 2.2.2 Nonlinear filter 24 3 Non-convex hybrid TV for image denoising 28 3.1 Variational model with non-convex hybrid TV 29 3.1.1 Non-convex TV model and non-convex HOTV model 29 3.1.2 The Proposed model: Non-convex hybrid TV model 31 3.2 Iterative reweighted hybrid Total Variation algorithm 33 3.3 Numerical experiments 35 3.3.1 Parameter values 37 3.3.2 Comparison between the non-convex TV model and the non-convex HOTV model 38 3.3.3 Comparison with other non-convex higher order regularizers 40 3.3.4 Comparison between two non-convex hybrid TV models 42 3.3.5 Comparison with Krishnan et al. [39] 43 3.3.6 Comparison with state-of-the-art 44 4 Image decomposition 59 4.1 Local gradient constraint 61 4.1.1 Texture estimator 62 4.2 The proposed model 65 4.2.1 Algorithm : Anisotropic TV-L2 67 4.2.2 Algorithm : Isotropic TV-L2 69 4.2.3 Algorithm : Isotropic TV-L1 71 4.3 Numerical experiments and discussion 72 5 Conclusion and future works 80 Abstract (in Korean) 92Docto

A regularization approach to blind deblurring and denoising of QR barcodes

QR bar codes are prototypical images for which part of the image is a priori known (required patterns). Open source bar code readers, such as ZBar, are readily available. We exploit both these facts to provide and assess purely regularization-based methods for blind deblurring of QR bar codes in the presence of noise