Stable phase retrieval in infinite dimensions. (English) Zbl 1440.94010

Alaifari, Rima; Daubechies, Ingrid; Grohs, Philipp; Yin, Rujie

Found. Comput. Math. 19, No. 4, 869-900 (2019).

Summary: 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 \(F(\omega ):=\langle f, \varphi _\omega \rangle \) are measurements of \(f\) with respect to a measurement system \((\varphi _\omega )_{\omega \in \Omega }\subset \mathcal{H}\). 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 (R. Alaifari and P. Grohs [SIAM J. Math. Anal. 49, No. 3, 1895–1911 (2017; Zbl 1368.42028)]; J. Cahill et al. [Trans. Am. Math. Soc., Ser. B 3, 63–76 (2016; Zbl 1380.46015)]); moreover, the stability deteriorates severely in the dimension of the problem (J. Cahill et al. [loc. cit.]). 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 \(k \ge 2\)). 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
\[F \sim \sum_{j=1}^k e^{\mathrm{i}\alpha_j} F_j.\]
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.

MSC:

94A12	Signal theory (characterization, reconstruction, filtering, etc.)
42C40	Nontrigonometric harmonic analysis involving wavelets and other special systems
30H10	Hardy spaces
46C05	Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product)

Keywords:

phase retrieval; Fourier optics; stability; entire functions

Citations:

Zbl 1368.42028; Zbl 1380.46015

Software:

Wirtinger Flow; PhaseLift

References:

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