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A frameletbased image inpainting algorithm
 Applied and Computational Harmonic Analysis
"... Abstract. Image inpainting is a fundamental problem in image processing and has many applications. Motivated by the recent tight frame based methods on image restoration in either the image or the transform domain, we propose an iterative tight frame algorithm for image inpainting. We consider the c ..."
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Abstract. Image inpainting is a fundamental problem in image processing and has many applications. Motivated by the recent tight frame based methods on image restoration in either the image or the transform domain, we propose an iterative tight frame algorithm for image inpainting. We consider the convergence of this frameletbased algorithm by interpreting it as an iteration for minimizing a special functional. The proof of the convergence is under the framework of convex analysis and optimization theory. We also discuss the relationship of our method with other waveletbased methods. Numerical experiments are given to illustrate the performance of the proposed algorithm. Key words. Tight frame, inpainting, convex analysis 1. Introduction. The problem of inpainting [2] occurs when part of the pixel data in a picture is missing or overwritten by other means. This arises for example in restoring ancient drawings, where a portion of the picture is missing or damaged due to aging or scratch; or when an image is transmitted through a noisy channel. The task of inpainting is to recover the missing region from the incomplete data observed. Ideally, the restored image should possess shapes and patterns consistent
Restoration of color images by vector valued BV functions and variational calculus
 SIAM J. Appl. Math
, 2006
"... Abstract. We analyze a variational problem for the recovery of vector valued functions and we compute its numerical solution. The data of the problem are a small set of complete samples of the vector valued function and a significant incomplete information where the former are missing. The incomplet ..."
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Cited by 20 (12 self)
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Abstract. We analyze a variational problem for the recovery of vector valued functions and we compute its numerical solution. The data of the problem are a small set of complete samples of the vector valued function and a significant incomplete information where the former are missing. The incomplete information is assumed as the result of a distortion, with values in a lower dimensional manifold. For the recovery of the function we minimize a functional which is formed by the discrepancy with respect to the data and total variation regularization constraints. We show existence of minimizers in the space of vector valued BV functions. For the computation of minimizers we provide a stable and efficient method. First we approximate the functional by coercive functionals on W 1,2 in terms of Γconvergence. Then we realize approximations of minimizers of the latter functionals by an iterative procedure to solve the PDE system of the corresponding EulerLagrange equations. The numerical implementation comes naturally by finite element discretization. We apply the algorithm to the restoration of color images from a limited color information and gray levels where the colors are missing. The numerical experiments show that this scheme is very fast and robust. The reconstruction capabilities of the model are shown, also from very limited (randomly distributed) color data. Several examples are included from the real restoration problem of the A. Mantegna’s art frescoes in Italy.
The application of joint sparsity and total variation minimization algorithms to a reallife art restoration problem
 Advances in Computational Mathematics
, 2009
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Theory and Computation of Variational Image Deblurring
, 2005
"... To recover a sharp image from its blurry observation is the problem known as image deblurring. It frequently arises in imaging sciences and technologies, including optical, medical, and astronomical applications, and is crucial for allowing to detect important features and patterns such as those of ..."
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Cited by 10 (1 self)
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To recover a sharp image from its blurry observation is the problem known as image deblurring. It frequently arises in imaging sciences and technologies, including optical, medical, and astronomical applications, and is crucial for allowing to detect important features and patterns such as those of a distant planet or some microscopic tissue. Mathematically, image deblurring is intimately connected to backward diffusion processes (e.g., inverting the heat equation), which are notoriously unstable. As inverse problem solvers, deblurring models therefore crucially depend upon proper regularizers or conditioners that help secure stability, often at the necessary cost of losing certain highfrequency details in the original images. Such regularization techniques can ensure the existence, uniqueness, or stability of deblurred images. The present work follows closely the general framework described in our recent monograph [18], but also contains more updated views and approaches to image deblurring, including, e.g., more discussion on stochastic signals, the Bayesian/Tikhonov approach to Wiener filtering, and the iteratedshrinkage algorithm of Daubechies et al. [30,31] for waveletbased deblurring. The work thus contributes to the development of generic, systematic, and unified frameworks in contemporary image processing.
UNCONDITIONALLY STABLE SCHEMES FOR HIGHER ORDER INPAINTING
"... Abstract. Inpainting methods with third and fourth order equations have certain advantages in comparison with equations of second order such as the smooth interpolation of image information even over large distances. Because of this such methods became very popular in the last couple of years. Solvi ..."
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Cited by 9 (7 self)
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Abstract. Inpainting methods with third and fourth order equations have certain advantages in comparison with equations of second order such as the smooth interpolation of image information even over large distances. Because of this such methods became very popular in the last couple of years. Solving higher order equations numerically can be a computational demanding task though. Discretizing a fourth order evolution equation with a brute force method may restrict the time steps to a size up to order ∆x 4 where ∆x denotes the step size of the spatial grid. In this work we will present a more educated way of discretization, namely efficient semiimplicit schemes that are guaranteed to be unconditionally stable. We will explain the main idea of these schemes and present applications in image processing for inpainting with the CahnHilliard equation, TVH −1 inpainting, and inpainting with LCIS (low curvature image simplifiers). 1.
Spatiotemporal texture synthesis and image inpainting for video applications
 In IEEE International Conference on Image processing Vol.2
, 2005
"... In this paper we investigate the application of texture synthesis and image inpainting techniques for video applications. Working in the nonparametric framework, we use 3D patches for matching and copying. This ensures temporal continuity to some extent which is not possible to obtain by working ..."
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In this paper we investigate the application of texture synthesis and image inpainting techniques for video applications. Working in the nonparametric framework, we use 3D patches for matching and copying. This ensures temporal continuity to some extent which is not possible to obtain by working with individual frames. Since, in present application, patches might contain arbitrary shaped and multiple disconnected holes, fast fourier transform (FFT) and summed area table based sum of squared difference (SSD) calculation [1] cannot be used. We propose a modification of above scheme which allows its use in present application. This results in significant gain of efficiency since search space is typically huge for video applications. 1.
RESTORATION OF IMAGES CORRUPTED BY IMPULSE NOISE AND MIXED GAUSSIAN IMPULSE NOISE USING BLIND INPAINTING
, 1304
"... Abstract. This article studies the problem of image restoration of observed images corrupted by impulse noise and mixed Gaussian impulse noise. Since the pixels damaged by impulse noise contain no information about the true image, how to find this set correctly is a very important problem. We propos ..."
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Cited by 4 (1 self)
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Abstract. This article studies the problem of image restoration of observed images corrupted by impulse noise and mixed Gaussian impulse noise. Since the pixels damaged by impulse noise contain no information about the true image, how to find this set correctly is a very important problem. We propose two methods based on blind inpainting and ℓ0 minimization that can simultaneously find the damaged pixels and restore the image. By iteratively restoring the image and updating the set of damaged pixels, these methods have better performance than other methods, as shown in the experiments. In addition, we provide convergence analysis for these methods, these algorithms will converge to coordinatewise minimum points. In addition, they will converge to local minimum points (or with probability one) with some modifications in the algorithms.
Faithful recovery of vector valued functions from incomplete data. Recolorization and art restoration
 in Proceedings of the First International Conference on Scale Space and Variational Methods in Computer Vision, Lecture Notes in Comput. Sci. 4485
, 2007
"... Abstract. On March 11, 1944, the famous Eremitani Church in Padua (Italy) was destroyed in an Allied bombing along with the inestimable frescoes by Andrea Mantegna et al. contained in the Ovetari Chapel. In the last 60 years, several attempts have been made to restore the fresco fragments by traditi ..."
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Cited by 3 (2 self)
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Abstract. On March 11, 1944, the famous Eremitani Church in Padua (Italy) was destroyed in an Allied bombing along with the inestimable frescoes by Andrea Mantegna et al. contained in the Ovetari Chapel. In the last 60 years, several attempts have been made to restore the fresco fragments by traditional methods, but without much success. We have developed an efficient pattern recognition algorithm to map the original position and orientation of the fragments, based on comparisons with an old gray level image of the fresco prior to the damage. This innovative technique allowed for the partial reconstruction of the frescoes. Unfortunately, the surface covered by the fragments is only 77 m 2, while the original area was of several hundreds. This means that we can reconstruct only a fraction (less than 8%) of this inestimable artwork. In particular the original color of the blanks is not known. This begs the question of whether it is possible to estimate mathematically the original colors of the frescoes by making use of the potential information given by the available fragments and the gray level of the pictures taken before the damage. Can one estimate how faithful such restoration is? In this paper we retrace the development of the recovery of the frescoes as an inspiring and challenging reallife problem for the development of new mathematical methods. We introduce two models for the recovery of vector valued functions from incomplete data, with applications to the fresco recolorization problem. The models are based on the minimization of a functional which is formed by the discrepancy with respect to the data and additional regularization constraints. The latter refer to joint sparsity measures with respect to frame expansions for the first functional and functional total variation for the second. We establish the relations between these two models. As a byproduct we develop the basis of a theory of fidelity in color recovery, which is a crucial issue in art restoration and compression.
Geometric and stochastic analysis of reactiondiffusion patterns
 Int. J. Pure Appl. Math
"... After Turing’s ingenious work on the chemical basis of morphogenesis fifty years ago, reactiondiffusion patterns have been extensively studied in terms of modelling and analysis of pattern formations (in both chemistry and biology), pattern growing in complex laboratory environments, and novel appli ..."
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After Turing’s ingenious work on the chemical basis of morphogenesis fifty years ago, reactiondiffusion patterns have been extensively studied in terms of modelling and analysis of pattern formations (in both chemistry and biology), pattern growing in complex laboratory environments, and novel applications in computer graphics. A fundamental question that remains unanswered in the literature is what one precisely means by (reactiondiffusion) patterns. Most patterns have only been discovered, identified, or explained by human vision and human intelligence. Inspired by the recent advancement in mathematical image and vision analysis (Miva), the current paper develops both geometric and stochastic tools and frameworks for identifying, classifying, and characterizing common reactiondiffusion patterns and pattern formations. In essence, it presents a data mining theory for the scientific simulations of reactiondiffusion patterns, or various analytical tools for the automatic characterization of generic complex patterns by artificial intelligence.