Bounds and Conditions for Compressive Digital Holography Using Wavelet Sparsifying Bases

Abstract

In compressive digital holography, we reconstruct sparse object wavefields from undersampled holograms by solving an 1 -minimization problem. Applying wavelet transformations to the object wavefields produces the necessary sparse representations, but prior work clings to transformations with too few vanishing moments. We put several wavelet transformations belonging to different wavelet families to the test by evaluating their sparsifying properties, the number of hologram samples that are required to reconstruct the sparse wavefields perfectly, and the robustness of the reconstructions to additive noise and sparsity defects. In particular, we recommend the CDF 9/7 and 17/11 wavelet transformations, as well as their reverse counterparts, because they yield sufficiently sparse representations for most accustomed wavefields in combination with robust reconstructions. These and other recommendations are procured from simulations and are validated using biased, noisy holograms.

Unitary Transforms Using Time-Frequency Warping for Digital Holograms of Deep Scenes

“…For the sensing of holograms with sparse/incomplete data, it could benefit applications such as viewpoint inference and despeckling of holograms [12], scene reconstruction from subsampled holograms using compressed sensing [28], or for compressive holographic tomography [29]. The transform bounds also indicate what types of object surface shapes are possible to (fully) recover from a hologram.…”