Phase Retrieval via Matrix Completion

By E.J. Candes, Yonina C. Eldar, Thomas Strohmer, Vladislav Voroninski. SIAM Review, 2011

1,083 citations recorded by the source corpus.

This paper develops a novel framework for phase retrieval, a problem which arises in X-ray crystallography, diffraction imaging, astronomical imaging, and many other applications. Our approach, called PhaseLift, combines multiple structured illuminations together with ideas from convex programming to recover the phase from intensity measurements, typically from the modulus of the diffracted wave. We demonstrate empirically that a complex-valued object can be recovered from the knowledge of the magnitude of just a few diffracted patterns by solving a simple convex optimization problem inspired by the recent literature on matrix completion. More importantly, we also demonstrate that our noise-aware algorithms are stable in the sense that the reconstruction degrades gracefully as the signal-to-noise ratio decreases. Finally, we introduce some theory showing that one can design very simple structured illumination patterns such that three diffracted figures uniquely determine the phase of the object we wish to...

Subfields: Computer Vision and Pattern Recognition, Electrical and Electronic Engineering, Radiation

View Paper: https://www.semanticscholar.org/paper/f07bf34f49af25c82cb305e493815ea27a775fde

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Citations in the corpus

Dropping Convexity for Faster Semi-definite Optimization (2015, background): “…includes matrix completion [21, 42, 49, 22], aﬃne rank minimization [65, 41, 8], covariance / inverse covariance selection [38, 50], phase retrieval [64, 18, 77, 68], Euclidean distance matrix completion [56], ﬁnding the square root of a PSD matrix [40], and sparse PCA [26], just to name a few.”
Solving Systems of Random Quadratic Equations via Truncated Amplitude Flow (2016, background): “In many fields of physical sciences and engineering, the problem of recovering the phase from intensity/magnitudeonly measurements is commonly referred to as phase retrieval [7], [8], [9].”
Phase Retrieval Using Alternating Minimization (2013, background): “Inspired by the recent literature on trace norm relaxation of the rank constraint, they design SDPs to solve this problem.”
How hard is the tensor rank (2016, background): “As it turns out, this simply looking problem takes one of the central places in modern applied mathematics [1, 13, 15, 14, 16, 17, 39, 59, 84], and the current paper is the first one that gives a full description of its algorithmic complexity.”
Phase Retrieval with Application to Optical Imaging: A contemporary overview (2015, methodology): “( ) Tr X This approach is referred to as PhaseLift [39].”
Phase recovery, MaxCut and complex semidefinite programming (2012, methodology): “which is a SDP (called PhaseLift by [13]) in the variable X ∈ Hp.”
Implicit Regularization in Nonconvex Statistical Estimation: Gradient Descent Converges Linearly for Phase Retrieval, Matrix Completion, and Blind Deconvolution (2017, background): “This arises, for example, in X-ray crystallography [15], and in latent-variable models where the hidden variables are captured by the missing signs [33].”
Deep Scattering Spectrum (2013, background): “Recent convex relaxation approaches [35], [36] are able to compute exact solutions, but they require too much computation and memory for audio applications.”
The non-convex geometry of low-rank matrix optimization (2016, background): “A good initialization can lead to global convergence if the initial iterate lies in the attraction basin of the global optima [2, 4, 12, 51].”
Phase Retrieval via Wirtinger Flow: Theory and Algorithms (2014, background): “Indeed, for a large number of unknowns in the tens of thousands, say, the memory requirements are far out of reach of desktop computers so that these SDP relaxations are de facto impractical.”
On the Compressive Spectral Method (2014, background): “L(1) optimization and its related models are at the center of many problems in the fields of imaging science and data analysis; see, for example, [2, 17, 18, 16].”
Compressed modes for variational problems in mathematics and physics (2013, neutral): “μ ψ j 1 + D ψ j; Ĥψ j E s:t: ψ j;ψk = δjk; [4]”
Solving Quadratic Equations via PhaseLift When There Are About as Many Equations as Unknowns (2012, positive): “Then approximate this combinatorially hard problem by using a convex surrogate for the nonconvex rank functional: PhaseLift [2, 3] is the relaxation”