Phase Retrieval for the Cauchy Wavelet Transform

Authors: Stéphane Mallat Irène Waldspurger

Publish Date: 2015/04/08

Volume: 21, Issue: 6, Pages: 1251-1309

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Abstract

We consider the phase retrieval problem in which one tries to reconstruct a function from the modulus of its wavelet transform We study the uniqueness and stability of the reconstruction In the case where the wavelets are Cauchy wavelets we prove that the modulus of the wavelet transform uniquely determines the function up to a global phase We show that the reconstruction operator is continuous but not uniformly continuous We describe how to construct pairs of functions which are far away in L2norm but whose wavelet transforms are very close in modulus The principle is to modulate the wavelet transform of a fixed initial function by a phase which varies slowly in both time and frequency This construction seems to cover all the instabilities that we observe in practice we give a partial formal justification to this fact Finally we describe an exact reconstruction algorithm and use it to numerically confirm our analysis of the stability questionWe first remark that tildeB F and tildeB G admit meromorphic extensions to mathbb C Indeed if the z k k are the zeros of F+ialpha in mathbb H this set has no accumulation point in overlineH if z infty was an accumulation point z infty +ialpha in mathbb H would be an accumulation point of the zeros of F and as F is holomorphic it would be the null function From the classical properties of Blaschke products tildeB F converge over mathbb C and so does tildeB GLet z be such that 0mathrmImzle alpha The zeros of B F are the zeros of F in mathbb H counted with multiplicity Thus zialpha is a zero of B F+ialpha with multiplicity mu Fz It is a zero of B G+ialpha with multiplicity mu GzBecause mathrmImzialpha le 0 it is not a zero of tildeB F resp tildeB G but may be a pole As a pole its multiplicity is the multiplicity of overlinezialpha =overlinez+ialpha as a zero of F+ialpha resp G+ialpha it is mu Foverlinez+2ialpha resp mu Goverlinez+2ialpha

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