Phase retrieval by designed Hadamard complementary coded apertures

Phase retrieval is a challenging inverse problem where amplitude and phase are estimated from diffracted intensities, with applications ranging from microscopy to astronomy. Current computational imaging techniques employ random complementary coded apertures to recover complex optical fields, but require at least 20 masks for effective reconstruction, limiting real-time applications. We propose a novel approach using eight binary Hadamard complementary coded apertures designed to minimize the condition number, thereby ensuring a well-conditioned inverse problem. Our method significantly reduces acquisition time while enhancing reconstruction quality. Using the Fresnel propagation regime and the hybrid input-output algorithm, we validate our approach through extensive simulations with 23 Kodak dataset images across various noise levels. Results demonstrate that our Hadamard approach outperforms conventional random coded methods in reducing the required number of masks. Furthermore, experimental results confirm our technique successfully recovers both simple phase objects like lenses and complex arbitrary phases displayed on spatial light modulators, achieving superior visual quality measured by naturalness image quality evaluation metrics compared to conventional patterns.