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The problem of phase retrieval is to determine a signal $fin mathcal{H}$, with $mathcal{H}$ a Hilbert space, from intensity measurements $|F(omega)|$, where $F(omega):=langle f , varphi_omegarangle$ are measurements of $f$ with respect to a measurement system $(varphi_omega)_{omegain Omega}subset mathcal{H}$. 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 [8, 4]; moreover the stability deteriorates severely in the dimension of the problem [8]. 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 $D_jsubset Omega$, 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.
In recent work [P. Grohs and M. Rathmair. Stable Gabor Phase Retrieval and Spectral Clustering. Communications on Pure and Applied Mathematics (2018)] the instabilities of the Gabor phase retrieval problem, i.e., the problem of reconstructing a funct
We study the phase reconstruction of signals $f$ belonging to complex Gaussian shift-invariant spaces $V^infty(varphi)$ from spectrogram measurements $|mathcal{G}f(X)|$ where $mathcal{G}$ is the Gabor transform and $X subseteq mathbb{R}^2$. An explic
The problem of reconstructing a function from the magnitudes of its frame coefficients has recently been shown to be never uniformly stable in infinite-dimensional spaces [5]. This result also holds for frames that are possibly continuous [2]. On the
We consider the phase retrieval problem of reconstructing a $n$-dimensional real or complex signal $mathbf{X}^{star}$ from $m$ (possibly noisy) observations $Y_mu = | sum_{i=1}^n Phi_{mu i} X^{star}_i/sqrt{n}|$, for a large class of correlated real a
This paper is concerned with stable phase retrieval for a family of phase retrieval models we name locally stable and conditionally connected (LSCC) measurement schemes. For every signal $f$, we associate a corresponding weighted graph $G_f$, defined