Annals of Mathematical Sciences and Applications

Volume 3 (2018)

Number 2

The Marr conjecture and uniqueness of wavelet transforms

Pages: 473 – 528

DOI: https://dx.doi.org/10.4310/AMSA.2018.v3.n2.a4

Authors

Benjamin Allen (Dept. of Mathematics, Emmanuel College, Boston, Ma., U.S.A.; Program for Evolutionary Dynamics, Harvard University, Cambridge, Ma., U.S.A.; and Center for Mathematical Sciences and Applications, Harvard University, Cambridge, Ma., U.S.A.)

Mark A. Kon (Department of Mathematics and Statistics, Boston University, Boston, Massachusetts, U.S.A.; and Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Ma., U.S.A.)

Abstract

The inverse question of identifying a function from the nodes (zeroes) of its wavelet transform arises in a number of fields. These include whether the nodes of a heat or hypoelliptic equation solution determine its initial conditions, and in mathematical vision theory the Marr conjecture, on whether an image is mathematically determined by its edge information. We prove a general version of this conjecture by reducing it to the moment problem, using a basis dual to the monomial basis $x^{\alpha}$ on $\mathbb{R}^n$.

Keywords

edge detection, wavelet, Gaussian blurring, Hermite polynomial, moment expansion

2010 Mathematics Subject Classification

Primary 41A27. Secondary 41A58.

Full Text (PDF format)

Received 15 October 2015

Published 9 August 2018