No abstract

In compressive digital holography, we reconstruct sparse object wavefields from undersampled holograms by solving an ℓ-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 counter-parts, 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.

Inverse problem approaches for image reconstruction can improve resolution recovery over spatial filtering methods while reducing interference artifacts in digital off-axis holography. Prior works implemented explicit regularization operators in the image space and were only able to match intensity measurements approximatively. As a consequence, convergence to a strictly compatible solution was not possible. In this paper, we replace the non-convex image reconstruction problem for a sequence of surrogate convex problems. An iterative numerical solver is designed using a simple projection operator in the data domain and a Nesterov acceleration of the simultaneous Kaczmarz method. For regularization, the complex-valued object wavefield image is represented in the multiresolution CDF 9/7 wavelet domain and an energy-weighted preconditioning promotes minimum-norm solutions. Experiments demonstrate improved resolution recovery and reduced spurious artifacts in reconstructed images. Furthermore, the method is resilient to additive Gaussian noise and subsampling of intensity measurements.

Compressive sensing is a mathematical framework, which seeks to capture the information of an object using as few measurements as possible. Recently, it has been applied to holography, where the most frequently used reconstruction method is l 1 -norm minimization with the Haar wavelet as the sparsifying operator. In this work, we promote the CDF 9/7 wavelet as the sparsifying operator. We demonstrate that the CDF 9/7 wavelet performs better than the Haar wavelet.