# Gauss-Newton Optimization for Phase Recovery from the Bispectrum

## TL;DR

This paper introduces Gauss-Newton optimization methods for phase recovery from the bispectrum in speckle interferometry, demonstrating improved image reconstruction quality over traditional first-order methods with comparable computational effort.

## Contribution

It develops and implements Gauss-Newton schemes for phase recovery, including projected variants for bounded image intensities, advancing optimization techniques in this domain.

## Key findings

- Gauss-Newton methods outperform gradient descent in image quality.
- Projected Gauss-Newton enforces pixel bounds effectively.
- Methods are computationally efficient and publicly available.

## Abstract

Phase recovery from the bispectrum is a central problem in speckle interferometry which can be posed as an optimization problem minimizing a weighted nonlinear least-squares objective function. We look at two different formulations of the phase recovery problem from the literature, both of which can be minimized with respect to either the recovered phase or the recovered image. Previously, strategies for solving these formulations have been limited to first-order optimization methods such as gradient descent or quasi-Newton methods. This paper explores Gauss-Newton optimization schemes for the problem of phase recovery from the bispectrum. We implement efficient Gauss-Newton optimization schemes for all the formulations. For the two of these formulations which optimize with respect to the recovered image, we also extend to projected Gauss-Newton to enforce element-wise lower and upper bounds on the pixel intensities of the recovered image. We show that our efficient Gauss-Newton schemes result in better image reconstructions with no or limited additional computational cost compared to previously implemented first-order optimization schemes for phase recovery from the bispectrum. MATLAB implementations of all methods and simulations are made publicly available in the BiBox repository on Github.