Article

# Experimental compressive phase space tomography

Department of Mechanical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA.
(Impact Factor: 3.49). 04/2012; 20(8):8296-308. DOI: 10.1364/OE.20.008296
Source: PubMed

ABSTRACT

Phase space tomography estimates correlation functions entirely from snapshots in the evolution of the wave function along a time or space variable. In contrast, traditional interferometric methods require measurement of multiple two-point correlations. However, as in every tomographic formulation, undersampling poses a severe limitation. Here we present the first, to our knowledge, experimental demonstration of compressive reconstruction of the classical optical correlation function, i.e. the mutual intensity function. Our compressive algorithm makes explicit use of the physically justifiable assumption of a low-entropy source (or state.) Since the source was directly accessible in our classical experiment, we were able to compare the compressive estimate of the mutual intensity to an independent ground-truth estimate from the van Cittert-Zernike theorem and verify substantial quantitative improvements in the reconstruction.

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Available from: Lei Tian,
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• "However, tomography becomes challenging when the dimensionality of the correlation matrix becomes large. Recently, it was proposed experimentally in [9] to recover an approximately low-rank correlation matrix, which often holds in physics, by only taking a small number of measurements in the form of (1). • Phase Retrieval: Due to the physical constraints, one can only measure amplitudes of the Fourier coefficients of an optical object. "
##### Article: Exact and Stable Covariance Estimation From Quadratic Sampling via Convex Programming
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ABSTRACT: Statistical inference and information processing of high-dimensional data streams and random processes often require efficient and accurate estimation of their second-order statistics. With rapidly changing data and limited storage, it is desirable to extract the covariance structure from a single pass over the data stream. In this paper, we explore a quadratic random sampling model which imposes minimal memory requirement and low computational complexity during the sampling process, and are shown to be optimal in preserving low-dimensional covariance structures. Specifically, two popular structural assumptions of covariance matrices, namely sparsity and low rank, are investigated. We show that a covariance matrix with either structure can be perfectly recovered from a minimal number of sub-Gaussian quadratic measurements, via efficient convex relaxation for respective structure. The proposed convex optimization algorithm has a variety of potential applications in large-scale data stream processing, high-frequency wireless communication, phase space tomography in optics, non-coherent subspace detection, etc. By introducing a novel notion of mixed-norm restricted isometry property (RIP-$\ell_{2}/\ell_{1}$), we show that our method admits accurate and universal recovery in the absence of noise, as soon as the number of measurements exceeds the theoretic sampling limits. We also show this approach is robust to noise and imperfect structural assumptions, i.e. it admits high-accuracy recovery even when the covariance matrix is only approximately low-rank or sparse. Our methods are inspired by the recent breakthroughs in phase retrieval, and the analysis framework herein recovers and improves upon best-known phase retrieval guarantees with simpler proofs.
IEEE Transactions on Information Theory 10/2013; 61(7). DOI:10.1109/TIT.2015.2429594 · 2.33 Impact Factor
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