Optical sensing and imaging applications often suffer from a combination of low resolution object reconstructions and a large number of sensors (thousands), which depending on the frequency can be quite expensive or bulky. A key objective in optical design is to minimize the number of sensors (which reduces cost) for a given target resolution level (image quality) and permissible total sensor array size (compactness). Equivalently, for a given imaging hardware one seeks to maximize image quality, which in turn means fully exploiting the available sensors as well as all priors about the properties of the sought-after objects such as sparsity properties, and other, which can be incorporated into data processing schemes for object reconstructions. In this paper we propose a compressive-sensing-based method to process through-focus optical field data captured at a sensor array. This method applies to both two-dimensional (2D) and three-dimensional (3D) objects. The proposed approach treats in-focus and out-of-focus data as projective measurements for compressive sensing, and assumes that the objects are sparse under known linear transformations applied to them. This prior allows reconstruction via familiar compressive sensing methods based on 1-norm minimization. The proposed compressive through-focus imaging is illustrated in the reconstruction of canonical 2D and 3D objects, using either coherent or incoherent light. The obtained results illustrate the combined use of through-focus imaging and compressive sensing techniques, and also shed light onto the nature of the information that is present in in-focus and out-of-focus images.
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