Phase Retrieval By Projections

By Jameson Cahill, Peter G. Casazza, Jesse Peterson, L. Woodland. 2013

74 citations recorded by the source corpus.

The problem of recovering a vector from the absolute values of its inner products against a family of measurement vectors has been well studied in mathematics and engineering. A generalization of this phase retrieval problem also exists in engineering: recovering a vector from measurements consisting of norms of its orthogonal projections onto a family of subspaces. There exist semidefinite programming algorithms to solve this problem, but much remains unknown for this more general case. Can families of subspaces for which such measurements are injective be completely classified? What is the minimal number of subspaces required to have injectivity? How closely does this problem compare to the usual phase retrieval problem with families of measurement vectors? In this paper, we answer or make incremental steps toward these questions. We provide several characterizations of subspaces which yield injective measurements, and through a concrete construction, we prove the surprising result that phase retrieval can be achieved with $2M-1$ projections of arbitrary rank in $\HH_M$. Finally we present several open problems as we discuss issues unique to the phase retrieval problem with subspaces.

Subfields: Computer Vision and Pattern Recognition, Radiation

View Paper: https://www.semanticscholar.org/paper/b25bd53e966879e93a80c2099b165dd243c370d9

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Citations in the corpus

Phase Retrieval via Wirtinger Flow: Theory and Algorithms (2014, background): “If the loss function is non-negative and vanishes only when x = y, then a solution to the generalized phase retrieval problem (1.1) is any solution to minimize f(z) ∶= 1 2m m ∑ r=1 `(yr, ∣a ∗ rz∣ 2 ), z ∈ Cn. (2.1) Although one could study many loss functions, we shall focus in this paper on the…”
Phase Retrieval from Coded Diffraction Patterns (2013, unknown): “7,42,36,35] and the references therein and also the related paper [43]. We note that some recent articles study fundamental lower bounds on noiseless reconstruction from magnitude of frame coecients [12,19,5,15,39]. Please see the two blog posts [2] and [3] and references there in for a comprehensiv”
Signal recovery from Pooling Representations (2013, methodology): “If the outputs of the nonlinearity are nonnegative (as for the half rectification function), thenp = 1 corresponds to average pooling, and the casep = ∞ is max pooling.”