[Paper Review] Phase Retrieval from 4N-4 Measurements

Friedrich Philipp|arXiv (Cornell University)|Mar 19, 2014

Advanced X-ray Imaging Techniques13 references3 citations

TL;DR

This paper proves that phase retrieval of complex polynomials of degree less than $N$ is possible using only $4N-4$ phaseless Fourier measurements—specifically, $2N-1$ intensity measurements of the polynomial and $2N-3$ of its derivative on the unit circle. It presents a constructive algorithm that recovers the polynomial up to a global phase factor, offering a minimal and structured measurement system that achieves the conjectured lower bound of $4N-4$ measurements.

ABSTRACT

We prove by means of elementary methods that phase retrieval of complex polynomials p of degree less than N is possible with 4N-4 phaseless Fourier measurements of p and p'. In addition, we provide an associated algorithm and prove that it recovers p up to global phase.

Motivation & Objective

• To establish that phase retrieval is possible with $4N-4$ phaseless Fourier measurements for complex polynomials of degree less than $N$, achieving the conjectured minimal measurement count.
• To provide a self-contained, elementary proof of phase retrieval using only evaluations on the unit circle, avoiding complex algebraic geometry.
• To present a constructive recovery algorithm that uniquely reconstructs the polynomial up to a global phase factor from $4N-4$ intensity measurements.
• To demonstrate that the measurement system is structured and practical, using only point evaluations on the unit circle for both the polynomial and its derivative.

Proposed method

• The method relies on representing the polynomial and its derivative as trigonometric polynomials on the unit circle, enabling the use of point evaluations as linear measurements.
• It uses the fact that the magnitude of a polynomial and its derivative at $2N-1$ and $2N-3$ distinct points on the unit circle, respectively, contain sufficient information for phase retrieval.
• The core technique involves reconstructing the autocorrelation coefficients $\delta_{i,j} = \overline{\alpha_i}\alpha_j$ from intensity measurements and their derivatives.
• A recursive algorithm is developed to recover the coefficients $\alpha_k$ of the polynomial by solving a system of linear equations derived from the measured intensities and their derivatives.
• The algorithm uses normalization via $r = \sqrt{\delta_{m,m}}$ to fix the global phase, ensuring the output polynomial is a unimodular multiple of the original.
• The recovery process is proven via induction on the degree of the polynomial, showing that $\delta_{i,j} = \overline{\alpha_i}\alpha_j$ is correctly reconstructed at each step.

Experimental results

Research questions

• RQ1 Can phase retrieval be achieved with exactly $4N-4$ intensity measurements for complex polynomials of degree less than $N$?
• RQ2 Is there a constructive and elementary method to recover the polynomial up to global phase from $4N-4$ phaseless Fourier measurements on the unit circle?
• RQ3 Can the measurement system be structured using only evaluations on the unit circle, without requiring measurements on other circles as in prior work?
• RQ4 Does the proposed algorithm correctly reconstruct the polynomial from $4N-4$ intensity measurements of the polynomial and its derivative?
• RQ5 Is the minimal measurement bound of $4N-4$ achievable with a practical and implementable measurement scheme?

Key findings

• The paper proves that $4N-4$ phaseless Fourier measurements—$2N-1$ from the polynomial and $2N-3$ from its derivative on the unit circle—suffice for phase retrieval of complex polynomials of degree less than $N$.
• The proof is elementary and self-contained, avoiding advanced algebraic geometry, unlike prior results that relied on such tools.
• A recovery algorithm is constructed that reconstructs the polynomial up to a global phase factor from the $4N-4$ intensity measurements.
• The algorithm correctly recovers the autocorrelation coefficients $\delta_{i,j} = \overline{\alpha_i}\alpha_j$ through recursive computation and normalization.
• The method achieves the conjectured minimal measurement count of $4N-4$, confirming the $(4N-4)$-Conjecture for structured measurement systems.
• The measurement system is practical and uses only evaluations on the unit circle, making it suitable for applications in optics and signal processing.