Based on point spread function (PSF) engineering and astigmatism due to a pair of cylindrical lenses, a novel compressed imaging mechanism is proposed to achieve single-shot incoherent 3D imaging. The speckle-like PSF of the imaging system is sensitive to axial shift, which makes it feasible to reconstruct a 3D image by solving an optimization problem with sparsity constraint. With the experimentally calibrated PSFs, the proposed method is demonstrated by a synthetic 3D point object and real 3D object, and the images in different axial slices can be reconstructed faithfully. Moreover, 3D multispectral compressed imaging is explored with the same system, and the result is rather satisfactory with a synthetic point object. Because of the inherent compatibility between the compression in spectral and axial dimensions, the proposed mechanism has the potential to be a unified framework for multi-dimensional compressed imaging.
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