Subsampling strategies for compressive Hadamard single-pixel imaging

Single-pixel imaging (SPI) based on Hadamard multiplexing enables efficient image acquisition with a reduced number of measurements, but its performance critically depends on the choice of subsampling strategy. In this work, we conduct a comprehensive evaluation of subsampling strategies for compressive Hadamard SPI, aiming to identify the method that selects the best-suited subset of Hadamard basis vectors. Natural images are known to contain most of their information in low spatial frequencies. Hence, the motivation for subsampling low-sequency patterns first, where sequency is the Hadamard analog of spatial frequency and corresponds to the number of sign changes in a Hadamard pattern, is intuitive but lacks theoretical grounding. This motivates the evaluation of multiple strategies which combine known subsampling schemes-deterministic, variable density sampling, and multilevel density sampling-with several reorderings of the Hadamard basis elements. These follow specific metrics defined on the Hadamard basis, including sequency, total variation, cake-cutting, and to our knowledge a novel data-driven variance-based ordering. In addition, spread spectrum and scrambled Hadamard approaches, relying on a randomization of the Hadamard basis, are considered. The performance of these strategies is assessed across different sampling ratios, Gaussian noise levels, and reconstruction methods, including pseudoinverse, ISTA, and a deep-learning-based approach (PinvNet), both on simulations and on an actual SPI setup. Results on a common, large dataset of images (BSDS500) indicate that deterministic selection of informative Hadamard patterns consistently outperforms probabilistic and randomized compressive sensing strategies. The proposed variance-based ordering achieves state-of-the-art performance for linear and sparsity-based reconstructions, while sequency ordering remains competitive for deep-learning-based reconstruction. These findings highlight the importance of selecting the most informative Hadamard patterns and open the way to data-driven approaches to assess their information content.