Celebratio Mathematica — Daubechies

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R. Al-Aifari, I. Daubechies, P. Grohs, and G. Thakur: “Reconstructing real-valued functions from unsigned coefficients with respect to wavelet and other frames,” J. Fourier Anal. Appl. 23 : 6 (December 2017), pp. 1480–1494. MR 3735589 ArXiv 1601.07579 article

In this paper we consider the following problem of phase retrieval: given a collection of real-valued band-limited functions \[ \{\psi_{\lambda}\}_{\lambda\in\Lambda} \subset L^2(\mathbb{R}^d) \] that constitutes a semi-discrete frame, we ask whether any real-valued function \( f\in L^2(\mathbb{R}^d) \) can be uniquely recovered from its unsigned convolutions \[ \{|f*\psi_{\lambda}|\}_{\lambda\in\Lambda} .\] We find that under some mild assumptions on the semi-discrete frame and if \( f \) has exponential decay at \( \infty \), it suffices to know \( |f*\psi_{\lambda}| \) on suitably fine lattices to uniquely determine \( f \) (up to a global sign factor). We further establish a local stability property of our reconstruction problem. Finally, for two concrete examples of a (discrete) frame of \( L^2(\mathbb{R}^d) \), \( d=1,2 \) we show that through sufficient oversampling one obtains a frame such that any real-valued function with exponential decay can be uniquely recovered from its unsigned frame coefficients.

Rima Alaifari	Related
Philipp Grohs	Related
Gaurav S. Thakur	Related

R. Alaifari, I. Daubechies, P. Grohs, and R. Yin: “Stable phase retrieval in infinite dimensions,” Found. Comput. Math. 19 : 4 (August 2019), pp. 869–900. MR 3989716 Zbl 07091714 ArXiv 1609.00034 article

The problem of phase retrieval is to determine a signal \( f\in \mathcal{H} \), with \( \mathcal{H} \) a Hilbert space, from intensity measurements \( |F(\omega)| \), where \[ F(\omega):=\langle f,\phi_{\omega}\rangle \] are measurements of \( f \) with respect to a measurement system \[ (\phi_{\omega})_{\omega\in \Omega}\subset \mathcal{H} .\] Although phase retrieval is always stable in the finite dimensional setting whenever it is possible (i.e. injectivity implies stability for the inverse problem), the situation is drastically different if \( \mathcal{H} \) is infinite-dimensional: in that case phase retrieval is never uniformly stable [Cahill et al. 2016; Alaifari and Grohs 2017]; moreover the stability deteriorates severely in the dimension of the problem [Cahill et al. 2016]. On the other hand, all empirically observed instabilities are of a certain type: they occur whenever the function \( |F| \) of intensity measurements is concentrated on disjoint sets \( D_j\subset \Omega \), i.e., when \[ F = \sum_{j=1}^k F_j \] where each \( F_j \) is concentrated on \( D_j \) (and \( k \geq 2 \)). Motivated by these considerations we propose a new paradigm for stable phase retrieval by considering the problem of reconstructing \( F \) up to a phase factor that is not global, but that can be different for each of the subsets \( D_j \), i.e., recovering \( F \) up to the equivalence \[ F \sim \sum_{j=1}^k e^{i \alpha_j} F_j. \] We present concrete applications (for example in audio processing) where this new notion of stability is natural and meaningful and show that in this setting stable phase retrieval can actually be achieved, for instance if the measurement system is a Gabor frame or a frame of Cauchy wavelets.

Rima Alaifari	Related
Philipp Grohs	Related
Rujie (Rachel) Yin	Related