Branch Points in One-Dimensional Gaussian Scale Space

Abstract

Scale space analysis combines global and local analysis in a single methodology by simplifying a signal. The simplification is indexed using a continuously varying parameter denoted scale. Different analyses can then be performed at their proper scale. We consider evolution of a polynomial by the parabolic partial differential heat equation. We first study a basis for the solution space, the heat polynomials, and subsequently the local geometry around a branch point in scale space. By a branch point of a polynomium we mean a scale and a location where two zeros of the polynomial merge. We prove that the number of branch points for a solution is \lfloor\frac{n}{2}\rfloor for an initial polynomial of degree n. Then we prove that the branch points uniquely determine a polynomial up to a constant factor. Algorithms are presented for conversion between the polynomial's coefficients and its branch points.