Total Variation as a local filter

International audienceIn the Rudin-Osher-Fatemi (ROF) image denoising model, Total Variation (TV) is used as a global regularization term. However, as we observe, the local interactions induced by Total Variation do not propagate much at long distances in practice, so that the ROF model is not far from being a local filter. In this paper, we propose to build a purely local filter by considering the ROF model in a given neighborhood of each pixel. We show that appropriate weights are required to avoid aliasing-like effects, and we provide an explicit convergence criterion for an associated dual minimization algorithm based on Chambolle's work. We study theoretical properties of the obtained local filter, and show that this localization of the ROF model brings an interesting optimization of the bias-variance trade-off, and a strong reduction a ROF drawback called "staircasing effect". We finally present a new denoising algorithm, TV-means, that efficiently combines the idea of local TV-filtering with the non-local means patch-based method

On a fast bilateral filtering formulation using functional rearrangements

We introduce an exact reformulation of a broad class of neighborhood filters, among which the bilateral filters, in terms of two functional rearrangements: the decreasing and the relative rearrangements. Independently of the image spatial dimension (one-dimensional signal, image, volume of images, etc.), we reformulate these filters as integral operators defined in a one-dimensional space corresponding to the level sets measures. We prove the equivalence between the usual pixel-based version and the rearranged version of the filter. When restricted to the discrete setting, our reformulation of bilateral filters extends previous results for the so-called fast bilateral filtering. We, in addition, prove that the solution of the discrete setting, understood as constant-wise interpolators, converges to the solution of the continuous setting. Finally, we numerically illustrate computational aspects concerning quality approximation and execution time provided by the rearranged formulation.Comment: 29 pages, Journal of Mathematical Imaging and Vision, 2015. arXiv admin note: substantial text overlap with arXiv:1406.712

Total variation regularization for manifold-valued data

We consider total variation minimization for manifold valued data. We propose a cyclic proximal point algorithm and a parallel proximal point algorithm to minimize TV functionals with $\ell^p$-type data terms in the manifold case. These algorithms are based on iterative geodesic averaging which makes them easily applicable to a large class of data manifolds. As an application, we consider denoising images which take their values in a manifold. We apply our algorithms to diffusion tensor images, interferometric SAR images as well as sphere and cylinder valued images. For the class of Cartan-Hadamard manifolds (which includes the data space in diffusion tensor imaging) we show the convergence of the proposed TV minimizing algorithms to a global minimizer

Exact discrete minimization for TV+L0 image decomposition models

International audiencePenalized maximum likelihood denoising approaches seek a solution that fulfills a compromise between data fidelity and agreement with a prior model. Penalization terms are generally chosen to enforce smoothness of the solution and to reject noise. The design of a proper penalization term is a difficult task as it has to capture image variability. Image decomposition into two components of different nature, each given a different penalty, is a way to enrich the modeling. We consider the decomposition of an image into a component with bounded variations and a sparse component. The corresponding penalization is the sum of the total variation of the first component and the L0 pseudo-norm of the second component. The minimization problem is highly non-convex, but can still be globally minimized by a minimum s-t-cut computation on a graph. The decomposition model is applied to synthetic aperture radar image denoising

Split-Bregman-based sparse-view CT reconstruction

Total variation minimization has been extensively researched for image denoising and sparse view reconstruction. These methods show superior denoising performance for simple images with little texture, but result in texture information loss when applied to more complex images. It could thus be beneficial to use other regularizers within medical imaging. We propose a general regularization method, based on a split-Bregman approach. We show results for this framework combined with a total variation denoising operator, in comparison to ASD-POCS. We show that sparse-view reconstruction and noise regularization is possible. This general method will allow us to investigate other regularizers in the context of regularized CT reconstruction, and decrease the acquisition times in µCT

Multiscale bilateral filtering for improving image quality in digital breast tomosynthesis

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/135115/1/mp3283.pd