Stable and Robust Sampling Strategies for Compressive Imaging.

Area of Science:

• Signal Processing
• Image Reconstruction
• Applied Mathematics

Background:

• Image reconstruction often requires sparse representations in transform domains like wavelets or finite differences.
• Variable density sampling, concentrating on lower frequencies, empirically enhances reconstruction quality.
• Standard compressive sensing theory faces challenges due to correlations between frequency and wavelet domains.

Purpose of the Study:

• To develop a theoretical framework for variable density sampling in image reconstruction.
• To analyze the relationship between sampling strategies and sparsity basis coherence.
• To provide guarantees for stable and noise-robust image reconstruction.

Main Methods:

• Introduced the concept of local coherence to measure sensing vector correlation with the sparsity basis.
• Analyzed local coherence for Fourier measurements and Haar wavelet sparsity.
• Proved the restricted isometry property for sampling matrices derived from an inverse square power-law density.
• Applied the framework to both l1-minimization and total variation minimization reconstruction methods.

Main Results:

• Demonstrated that local coherence can be explicitly controlled and bounded for specific measurement and sparsity bases.
• Established that an inverse square power-law sampling density enables near-optimal embedding dimensions.
• Showcased that the proposed variable-density sampling strategy yields reconstructions stable to sparsity defects and robust to noise.
• Validated the findings for both l1 and total variation minimization techniques.

Conclusions:

• The local coherence framework provides a more refined understanding of optimal sampling for sparse recovery.
• Bounded average coherence, coupled with adapted sampling, is sufficient for optimal sparse recovery, unlike bounded maximal coherence.
• The developed sampling strategy and theoretical guarantees advance the field of compressive sensing for image acquisition.