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The Marr conjecture and uniqueness of wavelet transforms. (English) Zbl 1404.42062
In \({\mathbb R}^d\), the Richter or Mexican Hat wavelet \(M\) is the Laplacian of a Gaussian. We let \(M_\sigma(x) = \sigma^{-d} M(x/\sigma)\). The continuous wavelet transform of a function \(f\) at scale \(\sigma\) is then \(f * M_\sigma\). The question is whether \(f\) is uniquely determined by the zeroes of its continuous wavelet transform.
The original Marr conjecture states: A locally integrable function \(f\) of sufficiently rapid decay is uniquely determined (up to a constant multiple) by the zero sets of \(f * M_\sigma\) for any sequence of positive scales \(\{ \sigma_j\}_{j=1}^\infty\) tending to infinity. Some special cases of the conjecture have been proved in the past. This paper proves the conjecture in more generality, and also considers wavelets \(\psi\) other than \(M\).
The main result is that it is sufficient that \(f\) is integrable and of exponential decay, and that \(\psi\) is smooth, with all derivatives growing slower than exponentially, and satisfying the “Genericity Condition”. That condition states that the regular zero set of any derivative of order \(m\) does not contain the regular zero set of any derivative of order \(n\), \(m, n \geq 0\). Here “derivative of order \(m\)” means any linear combination of partial derivatives of total order \(m\).
The Mexican Hat wavelet in one dimension satisfies the Genericity Condition. In higher dimensions the condition can be reduced to a condition on the zeroes of Laplace-Hermite polynomials. For some values of \(m\), \(n\) in dimension \(d=2\), the authors have verified the condition numerically.
Other results include counterexamples for the case when conditions are not satisfied; a counterexample to show that the conjecture does not hold for the discrete case (where the zero crossings are only known on a lattice); and a corollary that shows that the zeros of a solution to the heat equation determine the initial conditions uniquely (under some assumptions).

42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
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