Denoising is the problem of removing the inherent noise from an image. The standard noise model is additive white Gaussian noise, where the observed image f is related to the underlying true image u by the degradation model f = u+ η, and η is supposed to be at each pixel independently and identically distributed as a zero-mean Gaussian random variable. Since this is an ill-posed problem, Rudin,...

We derive a number of methods to solve efficiently simple optimization problems subject to a totalvariation (TV) regularization, under different norms of the TV operator and both for the case of 1-dimensional and 2-dimensional data. In spite of the non-smooth, non-separable nature of the TV terms considered, we show that a dual formulation with strong structure can be derived. Taking advantage ...

Waveform inverse problems are mathematically ill-posed and, therefore, regularization methods are required to obtain stable and unique solutions. The Total Variation (TV) regularization method is used to resolve sharp interfaces obtaining solutions where edges and discontinuities are preserved. TV regularization accomplishes these goals by imposing sparsity on the gradient of the model paramete...

Domain decomposition methods are well-known techniques to address a very large scale problem by splitting it into smaller scale sub-problems. The theory of such methods is fully clarified when the energy minimized by the method is either smooth and strictly convex or splits additively with respect to the decomposition. Otherwise counterexamples to convergence exist. In this talk we present a co...

We show that the classical Kac’s random walk on S starting from the point mass at e1 mixes in O(n log n) steps in total variation distance. This improves a previous bound by Diaconis and Saloff-Coste of O(n).

We propose a denoising algorithm for medical images based on a combination of the total variation minimization scheme and the wavelet scheme. We show that our scheme offers effective noise removal in real noisy medical images while maintaining sharpness of objects. More importantly, this scheme allows us to implement an effective automatic stopping time criterion.

We study the extension of total variation (TV), total deformation (TD), and (second-order) total generalised variation (TGV) to symmetric tensor fields. We show that for a suitable choice of finite-dimensional norm, these variational semi-norms are rotation-invariant in a sense natural and well-suited for application to diffusion tensor imaging (DTI). Combined with a positive definiteness const...

The purpose of the present note is to draw reader’s attention to an analytic approach proposed by the author in [8] and refined in [9]. In contrast to the celebrated Flajolet-Odlyzko method (see [4]), it allows to obtain asymptotic formulas for the ratio of coefficients of two power series when separately the coefficients do not have a regular asymptotic behavior as their index increases. Moreo...

We propose a new definition for the gradient field of a discrete image, defined on a twice finer grid. The differentiation process from the image to its gradient field is viewed as the inverse operation of linear integration, and the proposed mapping is nonlinear. Then, we define the total variation of an image as the `1 norm of its gradient field amplitude. This new definition of the total var...