Module compressed_sensing 

Compressed sensing and sparse signal recovery algorithms.

Provides Discrete Cosine Transform (DCT) bases, random measurement matrices, Basis Pursuit (ISTA/LASSO/FISTA), Orthogonal Matching Pursuit (OMP), sparsity metrics, dictionary learning (K-SVD), sparse coding, FISTA, restricted isometry property (RIP) analysis, Bernoulli measurement matrices, MRI-like signal reconstruction, and theoretical recovery guarantees for compressed sensing problems.

Overview

Compressed sensing (CS) exploits the sparsity of natural signals to allow faithful reconstruction from far fewer measurements than the Nyquist rate. The key ingredients are:

• A sparsifying basis (DCT, Wavelet, etc.) in which the signal has few non-zero coefficients.
• A measurement matrix (random Gaussian or Bernoulli) that is incoherent with the sparsifying basis.
• A recovery algorithm (Basis Pursuit / ISTA / FISTA / OMP) that finds the sparsest signal consistent with the measurements.

The BasisPursuit struct implements ISTA (slow) and FISTA (fast, with Nesterov momentum) for L1-regularised least-squares (LASSO) recovery. OrthogonalMatchingPursuit provides a greedy alternative. KSvd implements the K-SVD dictionary learning algorithm.

Structs

BasisPursuit

Basis Pursuit via iterative shrinkage-thresholding (ISTA and FISTA).

DctBasis

Discrete Cosine Transform (DCT-II) basis for sparse representation.

KSvd

K-SVD dictionary learning algorithm.

MriCompressedSensing

Compressed sensing reconstruction for MRI-like k-space data.

OrthogonalMatchingPursuit

Orthogonal Matching Pursuit (OMP) for sparse signal recovery.

RandomMeasurementMatrix

A random Gaussian measurement matrix for compressed sensing.

RecoveryGuarantee

Theoretical guarantees for exact sparse recovery.

SparseSignal

Utility for generating and manipulating synthetic sparse signals.

SparsityMetrics

Metrics for quantifying signal sparsity and dictionary coherence.

Functions

compression_ratio

Compute the compression ratio m / n.

l2_norm

Compute the ℓ₂ norm of a slice.

mat_transpose_vec

Compute the transposed matrix-vector product y = A^T x.

mat_vec

Compute the matrix-vector product y = A x.

normalise

Normalise a vector to unit ℓ₂ norm in place.

nyquist_rate

Compute the Nyquist sampling rate for a band-limited signal.

soft_threshold

Apply element-wise soft-thresholding: sign(x) * max(|x| - lambda, 0).

spectral_norm

Estimate the spectral norm (largest singular value) of a via power iteration.