  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 [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 $D_j\subset \Omega$,
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.

Functional Analysis