Affine phase retrieval for sparse signals via ℓ1\ell_1 minimization

Affine phase retrieval is the problem of recovering signals from the magnitude-only measurements with a priori information. In this paper, we use the $\ell_1$ minimization to exploit the sparsity of signals for affine phase retrieval, showing that $O(k\log(en/k))$ Gaussian random measurements are sufficient to recover all $k$-sparse signals by solving a natural $\ell_1$ minimization program, where $n$ is the dimension of signals. For the case where measurements are corrupted by noises, the reconstruction error bounds are given for both real-valued and complex-valued signals. Our results demonstrate that the natural $\ell_1$ minimization program for affine phase retrieval is stable.Comment: 22 page

Signal Recovery From Product of Two Vandermonde Matrices

In this work, we present some new results for compressed sensing and phase retrieval. For compressed sensing, it is shown that if the unknown $n$-dimensional vector can be expressed as a linear combination of $s$ unknown Vandermonde vectors (with Fourier vectors as a special case) and the measurement matrix is a Vandermonde matrix, exact recovery of the vector with $2s$ measurements and $O(\mathrm{poly}(s))$ complexity is possible when $n \geq 2s$. Based on this result, a new class of measurement matrices is presented from which it is possible to recover $s$-sparse $n$-dimensional vectors for $n \geq 2s$ with as few as $2s$ measurements and with a recovery algorithm of $O(\mathrm{poly}(s))$ complexity. In the second part of the work, these results are extended to the challenging problem of phase retrieval. The most significant discovery in this direction is that if the unknown $n$-dimensional vector is composed of $s$ frequencies with at least one being non-harmonic, $n \geq 4s - 1$ and we take at least $8s-3$ Fourier measurements, there are, remarkably, only two possible vectors producing the observed measurement values and they are easily obtainable from each other. The two vectors can be found by an algorithm with only $O(\mathrm{poly}(s))$ complexity. An immediate application of the new result is construction of a measurement matrix from which it is possible to recover almost all $s$-sparse $n$-dimensional signals (up to a global phase) from $O(s)$ magnitude-only measurements and $O(\mathrm{poly}(s))$ recovery complexity when $n \geq 4s - 1$

Single-shot phase retrieval: a holography-driven problem in Sobolev space

The phase-shifting digital holography (PSDH) is a widely used approach for recovering signals by their interference (with reference waves) intensity measurements. Such measurements are traditionally from multiple shots (corresponding to multiple reference waves). However, the imaging of dynamic signals requires a single-shot PSDH approach, namely, such an approach depends only on the intensity measurements from the interference with a single reference wave. In this paper, based on the uniform admissibility of plane (or spherical) reference wave and the interference intensity-based approximation to quasi-interference intensity, the nonnegative refinable function is applied to establish the single-shot PSDH in Sobolev space. Our approach is conducted by the intensity measurements from the interference of the signal with a single reference wave. The main results imply that the approximation version from such a single-shot approach converges exponentially to the signal as the level increases. Moreover, like the transport of intensity equation (TIE), our results can be interpreted from the perspective of intensity difference.Comment: 37page