Studies on the sparsifying operator in compressive digital holography.

Area of Science:

• Optics and Photonics
• Digital Imaging
• Signal Processing

Background:

• Compressive digital holography reconstructs object wavefields from undersampled holograms.
• Sparse representations are crucial for reconstruction, typically achieved via wavelet transformations.
• Previous methods used wavelets with insufficient vanishing moments, limiting reconstruction quality.

Purpose of the Study:

• To evaluate various wavelet transformations for their sparsifying properties in digital holography.
• To determine the minimum number of hologram samples needed for accurate wavefield reconstruction.
• To assess the robustness of reconstructions against noise and sparsity defects.

Main Methods:

• Solving an ℓ1-minimization problem for wavefield reconstruction.
• Applying and comparing multiple wavelet transformations from different families.
• Conducting simulations and validating results with biased, noisy holograms.

Main Results:

• Identified CDF 9/7 and 17/11 wavelet transformations (and their reverses) as optimal.
• These transformations provide sufficient sparsity for common wavefields.
• Achieved robust reconstructions even with noisy and incomplete holographic data.

Conclusions:

• CDF 9/7 and 17/11 wavelets offer superior performance in compressive digital holography.
• The choice of wavelet transformation significantly impacts reconstruction accuracy and robustness.
• The study provides practical recommendations for selecting effective wavelet transformations for sparse wavefield reconstruction.