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 ℓ1-norm 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.

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 us-ing 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 us-ing 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 recon-structs 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.

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 Re-weighted 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)

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 ℓ1-minimization 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.

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 real-life 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.

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

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. 1 Introduction and

Digital inpainting methods provide an important tool in the restoration of images in a wide range of applications. We present mathematical methods with certain higher order partial differential equations for the inpainting of ancient frescoes. In particular we discuss the Cahn-Hilliard equation for the inpainting of binary structure and a higher order total variation approach. As an example for the preformance of our algorithms we consider the recently discovered Neidhart frescoes in Vienna. Keywords: Image interpolation, Cahn-Hilliard equation, total variation minimization, fresco restoration 2000 Mathematics Subject Classification. 65K10, 65N21, 49M30, 68U10 1

Abstract. We introduce and investigate the modified TV-Stokes model for two classical image processing tasks, i.e., image restoration and image inpainting. The modified TV-Stokes model is a two-step 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. Key words. Total Variation, image restoration, image inpainting AMS subject classifications. 65F10, 65N30, 65N55 1. Introduction. Variational

Iteratively Re-weighted Least Squares minimization (IRLS) appears for the first time in the approximation practice in the Ph.D. thesis of C. L. Lawson in 1961 for L∞ minimization. In the 1970s extensions of Lawson’s algorithm for ℓp-minimization were proposed, as reported in the work of M. R. Osborne. IRLS has been proposed for sparse recovery in signal processing in [5] and for total variation minimization in [3]. We would like to present the analysis of IRLS as provided in [1], and an application in color image restoration [4]. We conclude our talk by illustrating recent joint results with H. Rauhut and R. Ward on the use of IRLS for nuclear norm minimization in low-rank matrix completion [2].