Bounds and Conditions for Compressive Digital Holography Using Wavelet Sparsifying Bases

Numerous experiments have been conducted with success in the field of compressive digital holography, but the theory to determine optimal measurement conditions is lagging behind. In contrast to a prior study that expects object wavefields to be sparse in the spatial domain, we investigate how the configuration of the interferometer influences the reconstruction of wavefields that are sparse in a multiresolution orthogonal wavelet basis. In particular, we derive expressions for the coherence between the free-space wave propagation operator and the basis functions of a Shannon multiresolution representation as a function of the wavelength, the propagation distance, the image sensor's pixel pitch, and the scale of the basis functions. These expressions reveal that the coherence as a function of the Fresnel number is subject to specific scaling and translating rules as the scale of the basis functions changes. For a multiresolution orthogonal wavelet representation and digital holograms that are recorded in the near field, we deduce subsequently the optimal configuration of the interferometer and we show by means of hypothesis testing that the associated phase transition bound coincides with the weak threshold for block-sparse compressive sensing with a block length of 2, which is an optimal bound for the class of complex-valued compressive sensing problems. By means of experiments with a USAF 1951 resolution target and an angle grid, we validate our findings and demonstrate that the reconstructed object wavefields are resilient to sparsity defects and additive noise.