This thesis describes a new adaptive approximation technique called high resolution pursuit (HRP), and demonstrates how HRP can be used to extract features that are suitable for object recognition. Recently, adaptive approximation techniques have become popular for obtaining representations of large classes of signals. These techniques include method of frames, matching pursuit, and basis pursu...

In this paper a new multiscale modeling technique is proposed. It relies on a recently introduced measure-theoretic approach, which allows one to manage the microscopic and the macroscopic scale under a unique framework. In the resulting coupled model the two scales coexist and share information. This way it is possible to perform numerical simulations in which the trajectories and the density ...

To investigate complex biophysical relationships driving directed cell migration, we developed a biomimetic platform that allows perturbation of microscale geometric constraints with concomitant nanoscale contact guidance architectures. This permits us to elucidate the influence, and parse out the relative contribution, of multiscale features, and define how these physical inputs are jointly pr...

Traditional methods of time-frequency and multiscale analysis, such as wavelets and Gabor frames, have been successfully employed for representing most classes of pseudodifferential operators. However these methods are not equally effective in dealing with Fourier Integral Operators in general. In this paper, we show that the shearlets, recently introduced by the authors and their collaborators...

The analysis of collections of visual data, e.g., their classification, modeling and clustering, has become a problem of high importance in a variety of applications. Meanwhile, image data captured in uncontrolled environments by arbitrary users is very likely to be exposed to geometric transformations. Therefore, efficient methods are needed for analyzing high-dimensional visual data sets that...

In this paper, we provide a framework based upon diffusion processes for finding meaningful geometric descriptions of data sets. We show that eigenfunctions of Markov matrices can be used to construct coordinates called diffusion maps that generate efficient representations of complex geometric structures. The associated family of diffusion distances, obtained by iterating the Markov matrix, de...

We perform a systematic multiscale analysis for the 2-D incompressible Euler equation with rapidly oscillating initial data using a Lagrangian approach. The Lagrangian formulation enables us to capture the propagation of the multiscale solution in a natural way. By making an appropriate multiscale expansion in the vorticity-stream function formulation, we derive a well-posed homogenized equatio...

The computation of the geometric transformation between a reference and a target image, known as image registration or alignment, corresponds to the projection of the target image onto the transformation manifold of the reference image (the set of images generated by its geometric transformations). It often takes a nontrivial form such that exact computation of projections on the manifold is di...

The geometric analysis of point cloud data as well as of triangle meshes is intimately tied to the notion of scale at which the data is examined. It is well known that different features of the data manifest at different scales and the discovery of scales at which prominent features appear in a stable form is a challenging research problem. This question has recently been addressed by both clas...