We consider the problem of efficiently approximating and encoding high-dimensional data sampled from a probability distribution ρ in R, that is nearly supported on a d-dimensional setM for example supported on a d-dimensional Riemannian manifold. Geometric MultiResolution Analysis (GMRA) provides a robust and computationally efficient procedure to construct low-dimensional geometric approximati...

Abstract. This paper presents the multiscale analysis and numerical algorithms for the convectiondiffusion equations with rapidly oscillating periodic discontinuous coefficients. The multiscale asymptotic expansions are developed and an explicit rate of convergence is derived for the convex domains. An efficient multiscale hybrid FEM-FVM algorithm is constructed, and numerical experiments are r...

In the framework of heavy mid-level processing for high speed imaging, a nonlinear bi-dimensional network is proposed, allowing the implementation of active curve algorithms. Usually this efficient type of algorithm is prohibitive for real-time image processing due to its calculus charge and the inadequate structure for the use of serial or parallel architectures. Another kind of implementation...

Several empirical results appeared in the literature during the last decade have shown that it is often possible to separate images and other multidimensional data into geometrically distinct constituents. A rigorous mathematical analysis of the geometric separation problem in the two-dimensional setting was recently introduced by Donoho and Kutyniok [6], who proposed a mathematical framework t...

The analysis to detect arrhythmias and life-threatening conditions are highly essential in today world and this analysis can be accomplished by advanced non-linear processing methods for accurate analysis of the complex signals of heartbeat dynamics. In this perspective, recent developments in the field of multiscale information content have lead to the Microcanonical Multiscale Formalism (MMF)...

Proof. If E ≡ +∞ onX, then there is nothing to prove. So, assume E 6≡ +∞. Denote α = infu∈X E(u) ≥ −∞ and choose a strictly decreasing sequence (αm) such that αm ↘ α as m → ∞. Let Kαm = {u ∈ X : E(u) ≤ αm}, then by assumption Kαm is compact. Moreover, it’s clear that Kαm+1 ⊂ Kαm for all m ∈ N, i.e. we have a decreasing nesting of compact sets. By Cantor’s Intersection Theorem, we know that K = ∞ ⋂

The restoration and recovery of a defective skull can be performed through operative techniques to implant a customized prosthesis. Recently, image processing, surface reconstruction and geometric methods have been used for digital prosthesis design. In this chapter we review state-of-the-art approaches in this field and discuss related issues. The field of prosthesis modeling may include metho...

In this paper we present a multiscale non-parametric shape model for the segmentation of 3D branching structures such as vasculature or biological neuronal networks. The proposed technique models these structures as surfaces formed by varying-sized interconnected cylinders using higher order active contours. A geometric shape model is built using this approach and incorporated into a surface es...

Climate modeling encompasses an enormous range of spatial and temporal scales. One of its great remaining challenges is bridging the scale gaps between global climate processes, basinscale and regional impacts, and smaller-scale ecosystem dynamics. Improving and accessing enhanced computational resources will help bridge this gap. Nonetheless, enhanced computer power alone is insufficient witho...