OSI-flex: Optimization-Based Shearing Interferometry for Joint Phase and Shear Estimation Using a Flexible Open-Source Framework

Shearing interferometry is a common-path quantitative phase imaging technique in which an object beam interferes with a laterally shifted replica of itself, providing high temporal stability, reduced sensitivity to environmental noise, compact design, and compatibility with partially coherent illumination that suppresses coherence-related artifacts. Its principal limitation, however, is that it yields only sheared phase-difference measurements rather than the absolute phase, thereby requiring additional reconstruction step. In this work, we introduce OSI-flex, a flexible, open-source computational framework for quantitative phase reconstruction from sheared phase-difference measurements. The method leverages modern machine learning tools, namely automatic differentiation and the advanced ADAM (Adaptive Moment Estimation) optimizer. The method simultaneously estimates the phase and shear values, enabling it to adapt to experimental conditions where the shear cannot be precisely determined. Because defining shear value is inherently difficult in most systems, yet crucial for effective phase reconstruction, this joint optimization leads to robust and reliable phase retrieval. OSI-flex is highly versatile, supporting arbitrary numbers, magnitudes, and orientations of shear vectors. While optimal reconstruction is achieved with two orthogonal shears, the inclusion of regularization - specifically total variation minimization and sign constraint - enables OSI-flex to remain effective with nonorthogonal or even single-shear measurements. Moreover, OSI-flex accommodates a wide range of shear magnitudes, from subpixel (differential configuration) to several dozen pixels (semi-total shear configuration). Validation with simulations and experimental data confirms quantitative accuracy on calibrated phase objects and demonstrates robustness with 3D-printed cell phantom and follicular thyroid cells.