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.
Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
