Unique recovery from edge information

Abstract

We study the inverse problem of recovering a function f from the nodes (zeroes) of its wavelet transform. The solution also provides an answer to a generalization of the Marr conjecture in wavelet and mathematical vision theory, regarding whether an image is uniquely determined by its edge information. The question has also other forms, including whether nodes of heat and related equation solutions determine their initial conditions. The general Marr problem reduces in a natural way to the moment problem for reconstructing f, using the moment basis on Rd (Taylor monomials xα), and its dual basis (derivatives δ(α) of of the Dirac delta distribution), expanding the wavelet transform in moments of f. If f has exponential decay and the wavelet's derivatives satisfy generic positions for their zeroes, then f can be uniquely recovered. We show this is the strongest statement of its type. For the original Gaussian wavelet unique recovery reduces to genericity of zeroes of so-called Laplace-Hermite polynomials, which is proved in one dimension.