Collaborative Research: Analysis and processing of multidimensional data using sparse directional multiscale representations

Missouri State University, Springfield MO

Investigators

Abstract

Labate, DMS-1008900 Guo, DMS-1008907 Following the spectacular success of wavelets in signal and image processing, several attempts have been made to adapt their optimal efficiency from the one- to the multi-dimensional setting. In fact, in spite of their remarkable properties, wavelets are not very efficient in capturing the intrinsic geometry of multidimensional phenomena. In recent years, directional multiscale methods such as the shearlet representation, introduced by the investigators and their collaborators, have emerged as the most effective extension of the wavelet framework to the multidimensional setting. Indeed, the shearlet representation encompasses the mathematical theory of affine systems and, to date, is the only method able to combine optimal sparsity and fast transforms through the power of multiresolution analysis. The proposed research focuses on applications of the shearlet approach to a number of challenging problems of analysis and processing of multidimensional data. First, the shearlet representation is applied to provide a precise geometric characterization of the discontinuities of multivariate functions and distributions. Combining techniques from harmonic analysis and differential geometry, this provides the groundwork for the development of improved algorithms for edge detection and feature extraction. Second, the shearlet framework is applied to develop a new generation of methods for the regularized inversion of ill-posed problems. Building on the ability of shearlets to provide sparse representations of Fourier integral operators, efficient decompositions for the Radon and Ray transforms are computed. These are used to develop algorithms for the Radon inversion from local and incomplete data and for image deconvolution. Third, a novel mathematical and computational approach for viewpoint-invariant texture retrieval is introduced. This is achieved by jointly designing a framework for feature extraction and similarity measurement in an appropriate statistical setting, and relies on the unique ability of shearlets to capture local geometric information. Over the past several years, there has been a continuously increasing pressure to handle more efficiently the ever larger and higher dimensional data sets generated from a wide range applications such as electronic surveillance, remote sensing, and medical imaging. The challenge is to rapidly, accurately and reliably extract the relevant information, so that it can be efficiently processed, transmitted and stored. The project focuses on the applications of the shearlet representation -- a method introduced by the investigators and their collaborators that provides a unique combination of optimal sparsity and computational efficiency, within an innovative mathematical and computational framework. The notion of optimal sparsity, in particular, implies that this approach has the ability to very effectively and reliably identify the most relevant features contained in the data. Specifically, this project leads to advanced techniques for edge detection, feature extraction, and texture retrieval from medical, industrial and satellite imagery. This results in innovative and improved computational algorithms for the analysis and processing of high-dimensional data and facilitates technological advances in sensitive applications such as remote sensing, medical diagnostics, data classification and electronic surveillance.