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19
Compressive Sensing
, 2010
"... Compressive sensing is a new type of sampling theory, which predicts that sparse signals and images can be reconstructed from what was previously believed to be incomplete information. As a main feature, efficient algorithms such as ℓ1minimization can be used for recovery. The theory has many poten ..."
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Cited by 50 (12 self)
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Compressive sensing is a new type of sampling theory, which predicts that sparse signals and images can be reconstructed from what was previously believed to be incomplete information. As a main feature, efficient algorithms such as ℓ1minimization can be used for recovery. The theory has many potential applications in signal processing and imaging. This chapter gives an introduction and overview on both theoretical and numerical aspects of compressive sensing.
Domain decomposition methods for linear inverse problems with sparsity constraints
, 2007
"... Quantities of interest appearing in concrete applications often possess sparse expansions with respect to a preassigned frame. Recently, there were introduced sparsity measures which are typically constructed on the basis of weighted ℓ1 norms of frame coefficients. One can model the reconstruction o ..."
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Cited by 25 (6 self)
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Quantities of interest appearing in concrete applications often possess sparse expansions with respect to a preassigned frame. Recently, there were introduced sparsity measures which are typically constructed on the basis of weighted ℓ1 norms of frame coefficients. One can model the reconstruction of a sparse vector from noisy linear measurements as the minimization of the functional defined by the sum of the discrepancy with respect to the data and the weighted ℓ1norm of suitable frame coefficients. Thresholded Landweber iterations were proposed for the solution of the variational problem. Despite of its simplicity which makes it very attractive to users, this algorithm converges slowly. In this paper we investigate methods to accelerate significantly the convergence. We introduce and analyze sequential and parallel iterative algorithms based on alternating subspace corrections for the solution of the linear inverse problem with sparsity constraints. We prove their norm convergence to minimizers of the functional. We compare the computational cost and the behavior of these new algorithms with respect to the thresholded Landweber iterations.
Variational models for image colorization via Chromaticity and Brightness decomposition
 IEEE TRANS. IMAGE PROC
, 2006
"... Colorization refers to an image processing task which recovers color of gray scale images when only small regions with color are given. We propose a couple of variational models using chromaticity color component to colorize black and white images. We first consider Total Variation minimizing (TV) ..."
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Cited by 19 (1 self)
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Colorization refers to an image processing task which recovers color of gray scale images when only small regions with color are given. We propose a couple of variational models using chromaticity color component to colorize black and white images. We first consider Total Variation minimizing (TV) colorization which is an extension from TV inpainting to color using chromaticity model. Secondly, we further modify our model to weighted harmonic maps for colorization. This model adds edge information from the brightness data, while it reconstructs smooth color values for each homogeneous region. We introduce penalized versions of the variational models, we analyze their convergence properties, and we present numerical results including extension to texture colorization.
An iterative algorithm for nonlinear inverse problems with joint sparsity constraints in vectorvalued regimes and an application to color image inpainting
 Inverse Problems
"... This paper is concerned with nonlinear inverse problems where data and solution are vector valued and, moreover, where the solution is assumed to have a sparse expansion with respect to a preassigned frame. We especially focus on such problems where the different components of the solution exhibit a ..."
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Cited by 12 (3 self)
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This paper is concerned with nonlinear inverse problems where data and solution are vector valued and, moreover, where the solution is assumed to have a sparse expansion with respect to a preassigned frame. We especially focus on such problems where the different components of the solution exhibit a common or so–called joint sparsity pattern. Joint sparsity means here that the measure (typically constructed as weighted ℓ1 norms of componentwise ℓq norms of the frame coefficients) promotes a coupling of non–vanishing components. Quite recently, an iterative strategy for linear inverse problems with such joint sparsity constraints was presented. Here we develop an iterative concept for nonlinear inverse problems with joint sparsity constraints for which we show convergence and regularization properties. Moreover, we demonstrate the capabilities of the proposed algorithm in the field of color image inpainting.
Image and video colorization using vectorvalued reproducing kernel Hilbert spaces
, 2010
"... Motivated by the setting of reproducing kernel Hilbert space (RKHS) and its extensions considered in machine learning, we propose an RKHS framework for image and video colorization. We review and study RKHS especially in vectorial cases and provide various extensions for colorization problems. Theor ..."
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Cited by 11 (2 self)
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Motivated by the setting of reproducing kernel Hilbert space (RKHS) and its extensions considered in machine learning, we propose an RKHS framework for image and video colorization. We review and study RKHS especially in vectorial cases and provide various extensions for colorization problems. Theory as well as a practical algorithm is proposed with a number of numerical experiments.
Iteratively reweighted least squares minimization: Proof of faster than linear rate for sparse recovery
 in Proc. 42nd Annu. Conf. Inf. Sci. Syst
"... Abstract — Given an m × N matrix Φ, with m < N, the system of equations Φx = y is typically underdetermined and has infinitely many solutions. Various forms of optimization can extract a “best ” solution. One of the oldest is to select the one with minimal ℓ2 norm. It has been shown that in many ..."
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Cited by 10 (0 self)
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Abstract — Given an m × N matrix Φ, with m < N, the system of equations Φx = y is typically underdetermined and has infinitely many solutions. Various forms of optimization can extract a “best ” solution. One of the oldest is to select the one with minimal ℓ2 norm. It has been shown that in many applications a better choice is the minimal ℓ1 norm solution. This is the case in Compressive Sensing, when sparse solutions are sought. The minimal ℓ1 norm solution can be found by using linear programming; an alternative method is Iterative Reweighted Least Squares (IRLS), which in some cases is numerically faster. The main step of IRLS finds, for a given weight w, the solution with smallest ℓ2(w) norm; this weight is updated at every iteration step: if x (n) is the solution at step n, then w (n) is defined by w (n)
The application of joint sparsity and total variation minimization algorithms to a reallife art restoration problem
 Advances in Computational Mathematics
, 2009
"... art restoration problem ..."
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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.
A MODIFIED TVSTOKES MODEL FOR IMAGE PROCESSING
"... We introduce and investigate the modified TVStokes model for two classical image processing tasks, i.e., image restoration and image inpainting. The modified TVStokes model is a twostep model based on a total variation (TV) minimization in each step and the use of geometric information of the i ..."
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Cited by 2 (0 self)
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We introduce and investigate the modified TVStokes model for two classical image processing tasks, i.e., image restoration and image inpainting. The modified TVStokes model is a twostep model based on a total variation (TV) minimization in each step and the use of geometric information of the image. In the first step, a smoothed and divergence free tangential field of the given image is recovered, and in the second step, the image is reconstructed from the corresponding normals. The existence and the uniqueness of the solution to the minimization problems are established for both steps of the model. Numerical examples and comparisons are presented to illustrate the effectiveness of the model.
EXACT RECONSTRUCTION OF DAMAGED COLOR IMAGES USING A TOTAL VARIATION MODEL
"... Abstract. In this paper the reconstruction of damaged piecewice constant color images is studied using a RGB total variation based model for colorization/inpainting. In particular, it is shown that when color is known in a uniformly distributed region, then reconstruction is possible with maximal fi ..."
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Abstract. In this paper the reconstruction of damaged piecewice constant color images is studied using a RGB total variation based model for colorization/inpainting. In particular, it is shown that when color is known in a uniformly distributed region, then reconstruction is possible with maximal fidelity. Contents