Signal/image registration via a polynomial system solution method and the Pascal Triangle of a discrete image

Shanshan Huang, Purdue University

Abstract

The first part of this thesis is concerned with the problem of signal and image registration. We first introduce an existing method for solving systems of polynomial equations proposed by Ji Zhang and our motivation to apply this method to the registration of 1D and 2D discrete time functions, such as discrete images. In this part, we focus on the problem of finding the translation that best maps a piece of signal (represented by sample points) onto a template. For simplicity, we restrict ourselves to discrete translations. We then propose a method to represent discrete time functions in 1D and 2D case so to be able to formulate the registration problem as a polynomial equation system. We modify Zhang's solution method so to take advantage of the specific structure of our problem. Moreover, we show that in the case where all the Fourier coefficients of the 1D template signal are nonzero and the query points are given in consecutive times, the singular value decomposition step in Zhang's method can be replaced by explicitly solving for a complete solution set of the polynomial system. Furthermore, we show how to solve the image registration problem following Zhang's solution method of multivariate polynomial systems. Finally, we test our method on some simple signal and image registration problems. The second part of this thesis is concerned with the problem of shape analysis. We define the Pascal Triangle of a discrete (gray scale) image as a pyramidal arrangement of complex-valued moments and we explore its geometric significance. In particular, we show that the entries of row k of this triangle correspond to the Fourier series coefficients of the moment of order k of the Radon transform of the image. Group actions on the plane can be naturally prolonged onto the entries of the Pascal Triangle. We study the prolongation of some common group actions, such as rotations and reflections, and we propose simple tests for detecting equivalences and self-equivalences under these group actions. The application of this work is the problem of characterizing the geometry of objects on images, for example by detecting approximate symmetries.

Degree

Ph.D.

Advisors

Boutin, Purdue University.

Subject Area

Applied Mathematics

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