Compressed Sensing
9 months ago
- #sparsity
- #optimization
- #signal-processing
- Compressed sensing is a signal processing technique for efficiently acquiring and reconstructing signals by solving underdetermined linear systems.
- It exploits signal sparsity through optimization, allowing recovery from fewer samples than required by the Nyquist-Shannon sampling theorem.
- Key conditions for recovery are sparsity (signal must be sparse in some domain) and incoherence (applied via isometric property).
- Applications include MRI, where incoherence is typically satisfied, enabling faster and lower-dose imaging.
- Historical roots trace back to statistics (least squares, L1-norm), robust statistics, and signal processing (seismology, matching pursuit).
- Compressed sensing does not violate the Nyquist-Shannon theorem but offers an alternative for sparse signals with high-frequency components.
- Methods involve solving underdetermined systems with sparsity constraints, using L1-minimization or basis pursuit denoising for noise robustness.
- Total variation (TV) regularization is used in image reconstruction to preserve edges while reducing noise and artifacts.
- Iterative reweighted L1 minimization and edge-preserving TV improve reconstruction by adaptively penalizing coefficients and gradients.
- Directional TV refinement enhances accuracy by estimating and refining orientation fields, preserving texture and edges.
- Applications span photography (single-pixel cameras), holography, facial recognition, network tomography, and astronomy (aperture synthesis).
- In MRI, compressed sensing reduces scan time while maintaining image quality, and in CT, it enables low-dose imaging with fewer projections.