An iterative algorithm for phase retrieval with sparsity constraints: application to frequency domain optical coherence tomography

Abstract

We address the problem of phase retrieval, which is frequently encountered in optical imaging. The measured quantity is the magnitude of the Fourier spectrum of a function (in optics, the function is also referred to as an object). The goal is to recover the object based on the magnitude measurements. In doing so, the standard assumptions are that the object is compactly supported and positive. In this paper, we consider objects that admit a sparse representation in some orthonormal basis. We develop a variant of the Fienup algorithm to incorporate the condition of sparsity and to successively estimate and refine the phase starting from the magnitude measurements. We show that the proposed iterative algorithm possesses Cauchy convergence properties. As far as the modality is concerned, we work with measurements obtained using a frequency-domain optical-coherence tomography experimental setup. The experimental results on real measured data show that the proposed technique exhibits good reconstruction performance even with fewer coefficients taken into account for reconstruction. It also suppresses the autocorrelation artifacts to a significant extent since it estimates the phase accurately.

Analysis-form sparse phase retrieval using variable-splitting

“…For instance, a wide range of recent methods introduce image sparsity as a prior model. These methods include Fienupstyle alternating projections [7], [8], [9], semidefinite programming via "matrix lifting" [10], [11], [12], [13], [14], [15], generalized approximate message passing [16], greedy pursuit [17], and variable splitting [18]. However, with the exception of semidefinite programming, these methods are restricted to synthesis-form or canonical sparsity.…”

Phase Recovery Based on the Sparse Measurement

“…Since Gerchberg and Saxton presented a classical algorithm-GS [6], which applied two images to the iteration between the time domain and frequency domain back and forth, in 1972, various greedy algorithms of phase recovery have been proposed such as the Fienup-type algorithm [7,8]. Another famous algorithm for phase recovery is based on the transport of intensity equation (TIE) [9] which revealed the relationship between phase and the first derivative of intensity with respect to the optical axis.…”

Iterative projection approach for phase retrieval of semi-sparse wave field

“…Interested authors are encouraged to read literatures [19][20][21][22]. Motivated by sparse signal processing communities, many phase retrieval approaches by incorporating sparsity property of wave field are discussed [23][24][25][26][27][28][29][30]. However, it is just assumed that the underling wave field is sparse in these literatures.…”

“…This is the reason why we address the issue in the paper. Different from existing sparse phase retrieval approaches in [23][24][25][26][27][28][29][30], we propose an iterative projection approach with phase sparse constraint for semi-sparse http://asp.eurasipjournals.com/content/2014/1/24 wave field. As phase sparse constraint is exploited, the proposed approach has superior performances than the existing iterative projection approaches in terms of rapid convergence, smaller residual error, noise stability, and suitability in large-scale phase retrieval problems.…”