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 \( \left\lfloor {\frac{n}{2}} \right\rfloor \) 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.
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Johansen, P., Nielsen, M. & Olsen, O.F. Branch Points in One-Dimensional Gaussian Scale Space. Journal of Mathematical Imaging and Vision 13, 193–203 (2000). https://doi.org/10.1023/A:1011241531216
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DOI: https://doi.org/10.1023/A:1011241531216