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# 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

#### 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.

Received 15 October 2015

Published 9 August 2018