Localized Bases and Band-Limited Methods for Image Modeling and Analysis

University Of California-Davis, Davis CA

Investigators

Abstract

The objective of this project is to develop a mathematical framework for the modeling and analysis of images in two and three dimensions based on certain basis functions. The foundation of this framework is a class of band-limited functions (that is, functions that have compactly supported Fourier transforms) which are also locally concentrated in the spatial domain. Although images are seldom truly band-limited, they are generally considered piecewise smooth, and are frequently modeled so locally (or piecewisely) by smooth functions such as polynomials and trigonometric functions. Since common images contain many features such as edges and textures that are usually locally compact in the Fourier domain and simultaneously limited in the spatial (or image) domain, functions of this type are an ideal choice of basis for image representations. The PI will develop numerical methods for the construction of the band-limited basis functions, as well as efficient methods for the analysis and synthesis of images in such bases. Furthermore, algorithms for targeted image analysis problems such as restoration and texture analysis will be implemented. The PI will also investigate the adaptive construction of optimal (or near-optimal) representations of images, and test the effectiveness of such techniques on a realistic collection of 2D and 3D images. Potential applications of the mathematical theory and image analysis algorithms developed in this project include satellite imagery, remote sensing, and image-based diagnoses in fields such as medicine, and material science. For example, in satellite imagery where data are acquired with limited resolution, one central task is to magnify or interpolate the images so that the faintest details embedded can be reliably enhanced. Since the methods proposed in this project provide highly adaptive, localized image representation mechanisms, and therefore are likely to produce advanced image enhancement techniques with fewer artifacts than traditional methods, information may be more readily and reliably extracted. Research proposed in this project also comprises methods and techniques that have been studied in several disciplines: the theory of band-limited functions in mathematics, fast algorithms in numerical computation, edge detection and texture analysis in computer vision. Thus, the project affords opportunities for cross-pollination between researchers and students in mathematics, computer science, electrical engineering and statistics, and will particular benefit the participating students by providing a fertile ground for them to make connections between mathematics and engineering.