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We show how certain nonconvex optimization problems that arise in image processing and computer vision can be restated as convex minimization problems. This allows, in particular, the finding of global minimizers via standard convex minimization schemes. 1. Introduction. Image denoising and segmentation are two related, fundamental problems of computer… (More)

This paper proposes a two-phase scheme for removing salt-and-pepper impulse noise. In the first phase, an adaptive median filter is used to identify pixels which are likely to be contaminated by noise (noise candidates). In the second phase, the image is restored using a specialized regularization method that applies only to those selected noise candidates.… (More)

We address the minimization of regularized convex cost functions which are customarily used for edge-preserving restoration and reconstruction of signals and images. In order to accelerate computation, the multiplicative and the additive half-quadratic reformulation of the original cost-function have been pioneered in Geman & Reynolds (1992) and Geman &… (More)

We present a theoretical study of the recovery of an unknown vector x ∈ I R p (a signal, an image) from noisy data y ∈ I R q by minimizing with respect to x a regularized cost-function F (x, y) = Ψ(x, y) + αΦ(x), where Ψ is a data-fidelity term, Φ is a smooth regularization term and α > 0 is a parameter. Typically, Ψ(x, y) = Ax − y 2 where A is a linear… (More)

We focus on the question of how the shape of a cost-function determines the features manifested by its local (and hence global) minimizers. Our goal is to check the possibility that the local minimizers of an unconstrained cost-function satisfy different subsets of affine constraints dependent on the data, hence the word " weak ". A typical example is the… (More)

We consider the restoration of piecewise constant images where the number of the regions and their values are not fixed in advance, with a good difference of piecewise constant values between neighboring regions, from noisy data obtained at the output of a linear operator (e.g., a blurring kernel or a Radon transform). Thus we also address the generic… (More)

Image deblurring [4] from noisy data is a fundamental problem in image processing. Let the true image x belong to a proper function space S(Ω) on Ω = [0, 1] 2 , and the observed digital image y be a vector in R m×m indexed by A = {1, 2, · · · , m} × {1, 2, · · · , m}. The image degradation can be modeled as y = N (Hx), where H : S(Ω) → R m×m is a linear… (More)

We address the denoising of images contaminated with multiplicative noise, e.g. speckle noise. Classical ways to solve such problems are filtering, statistical (Bayesian) methods, variational methods , and methods that convert the multiplicative noise into additive noise (using a logarithmic function), apply a variational method on the log data or shrink… (More)