Stable image reconstruction using total variation minimization

This article presents near-optimal guarantees for accurate and robust image recovery from under-sampled noisy measurements using total variation minimization. In particular, we show that from O(slog(N)) nonadaptive linear measurements, an image can be reconstructed to within the best s-term approximation of its gradient up to a logarithmic factor, and this factor can be removed by taking slightly more measurements. Along the way, we prove a strengthened Sobolev inequality for functions lying in the null space of suitably incoherent matrices.Comment: 25 page

Fast Solvers for Cahn-Hilliard Inpainting

We consider the efficient solution of the modified Cahn-Hilliard equation for binary image inpainting using convexity splitting, which allows an unconditionally gradient stable time-discretization scheme. We look at a double-well as well as a double obstacle potential. For the latter we get a nonlinear system for which we apply a semi-smooth Newton method combined with a Moreau-Yosida regularization technique. At the heart of both methods lies the solution of large and sparse linear systems. We introduce and study block-triangular preconditioners using an efficient and easy to apply Schur complement approximation. Numerical results indicate that our preconditioners work very well for both problems and show that qualitatively better results can be obtained using the double obstacle potential

Split-Bregman iteration for framelet based image inpainting

AbstractImage inpainting plays a significant role in image processing and has many applications. Framelet based inpainting methods were introduced recently by Cai et al. (2007, 2009) [6,7,9] under an assumption that images can be sparsely approximated in the framelet domain. By analyzing these methods, we present a framelet based inpainting model in which the cost functional is the weighted ℓ1 norm of the framelet coefficients of the underlying image. The split-Bregman iteration is exploited to derive an iterative algorithm for the model. The resulting algorithm assimilates advantages while avoiding limitations of the framelet based inpainting approaches in Cai et al. (2007, 2009) [6,7,9]. The convergence analysis of the proposed algorithm is presented. Our numerical experiments show that the algorithm proposed here performs favorably

A New Multiscale Representation for Shapes and Its Application to Blood Vessel Recovery

In this paper, we will first introduce a novel multiscale representation (MSR) for shapes. Based on the MSR, we will then design a surface inpainting algorithm to recover 3D geometry of blood vessels. Because of the nature of irregular morphology in vessels and organs, both phantom and real inpainting scenarios were tested using our new algorithm. Successful vessel recoveries are demonstrated with numerical estimation of the degree of arteriosclerosis and vessel occlusion.Comment: 12 pages, 3 figure