Generalized sparse Hadamard single-pixel Imaging

Scattering presents significant challenges for practical imaging applications. Single-pixel imaging offers advantages over conventional approaches, yet further advances are still required. In this work, we introduce sparse Hadamard speckle patterns with tunable sparsity and propose sparse Hadamard single-pixel imaging (SHSI), a new framework built on these patterns. By integrating orthogonality, sparsity, and pseudo-randomness, the proposed patterns provide clear benefits over traditional Hadamard speckle patterns and their variants. SHSI improves noise robustness across a wide range of sampling rates by encoding sparse but spatially correlated information within each illumination pattern. It also achieves high-resolution imaging with low-order Hadamard matrices, which substantially reduces memory requirements. Moreover, SHSI operates without prior knowledge, training, or post-processing, thereby simplifying implementation. To further enhance performance, we introduce two operational modes and develop a complete theoretical framework that is validated through simulations and experiments. SHSI has the potential to advance computational imaging platforms based on sequential correlation measurements and to enable real-time reconstruction of occluded or embedded targets within scattering media.