[Paper Review] Fast Phase Retrieval from Local Correlation Measurements

Mark Iwen, Aditya Viswanathan+1 more|arXiv (Cornell University)|Jan 10, 2015

Advanced X-ray Imaging Techniques48 references3 citations

TL;DR

This paper presents a fast, deterministic phase retrieval algorithm that recovers a complex signal ${\bf x} \in \mathbb{C}^d$ up to global phase from local correlation-based magnitude measurements in $\mathcal{O}(d\log^4 d)$ time. It uses a lifted linear system for local phase differences and angular synchronization to recover full signal phases, with a sublinear-time extension for sparse signals using $\mathcal{O}(s\log^4 s \cdot \log d)$ measurements.

ABSTRACT

We develop a fast phase retrieval method which can utilize a large class of local phaseless correlation-based measurements in order to recover a given signal ${\bf x} \in \mathbb{C}^d$ (up to an unknown global phase) in near-linear $\mathcal{O} \left( d \log^4 d ight)$-time. Accompanying theoretical analysis proves that the proposed algorithm is guaranteed to deterministically recover all signals ${\bf x}$ satisfying a natural flatness (i.e., non-sparsity) condition for a particular choice of deterministic correlation-based measurements. A randomized version of these same measurements is then shown to provide nonuniform probabilistic recovery guarantees for arbitrary signals ${\bf x} \in \mathbb{C}^d$. Numerical experiments demonstrate the method's speed, accuracy, and robustness in practice -- all code is made publicly available. Finally, we conclude by developing an extension of the proposed method to the sparse phase retrieval problem; specifically, we demonstrate a sublinear-time compressive phase retrieval algorithm which is guaranteed to recover a given $s$-sparse vector ${\bf x} \in \mathbb{C}^d$ with high probability in just $\mathcal{O}(s \log^5 s \cdot \log d)$-time using only $\mathcal{O}(s \log^4 s \cdot \log d)$ magnitude measurements. In doing so we demonstrate the existence of compressive phase retrieval algorithms with near-optimal linear-in-sparsity runtime complexities.

Motivation & Objective

• Develop a computationally efficient phase retrieval method for signals from local correlation measurements.
• Provide deterministic recovery guarantees for non-sparse (flat) signals using a novel lifting and angular synchronization framework.
• Extend the method to compressive phase retrieval for $s$-sparse signals with sublinear runtime complexity.
• Ensure robustness and accuracy under noise through theoretical analysis and numerical validation.
• Enable practical deployment via publicly available code and efficient algorithms.

Proposed method

• Lifts quadratic magnitude measurements into a linear system involving $x_i \overline{x_j}$ for $|j-i| < \delta$, forming a block-structured system of size $(2\delta-1)d$.
• Uses $\mathcal{O}(\delta d)$ local correlation measurements to solve the lifted system efficiently via fast solvers.
• Applies angular synchronization to propagate local phase differences across the signal vector to estimate relative phases.
• Combines recovered magnitudes $|x_j|$ and phases to reconstruct the original signal $\bf x$.
• Extends the method to sparse signals by combining compressive sensing with the proposed phase retrieval framework.
• Employs a randomized measurement design to achieve probabilistic recovery guarantees for arbitrary signals.

Experimental results

Research questions

• RQ1 Can phase retrieval be performed in near-linear time using only local correlation measurements?
• RQ2 What theoretical guarantees can be established for deterministic recovery of flat signals from local correlation data?
• RQ3 Can the method be extended to compressive phase retrieval with sublinear runtime for sparse signals?
• RQ4 How robust is the algorithm to noise, and what error bounds can be derived?
• RQ5 Can angular synchronization be effectively used to recover global phases from local phase differences in a stable and efficient manner?

Key findings

• The algorithm achieves deterministic recovery of all $d$-dimensional signals satisfying a flatness condition in $\mathcal{O}(d\log^4 d)$ time.
• A randomized variant provides non-uniform probabilistic recovery guarantees for arbitrary signals ${\bf x} \in \mathbb{C}^d$.
• For $s$-sparse signals, the method achieves recovery in $\mathcal{O}(s\log^5 s \cdot \log d)$ time using $\mathcal{O}(s\log^4 s \cdot \log d)$ magnitude measurements.
• The method is robust to noise and demonstrates high accuracy and speed in numerical experiments.
• Theoretical analysis confirms that the lifted system matrix is well-conditioned under periodic boundary assumptions.
• The framework enables the first known global robust recovery guarantees for ptychographic phase retrieval problems.