9.2 Compressed sensing and its principles

Compressed sensing revolutionizes signal processing by efficiently acquiring and reconstructing sparse signals using fewer measurements. It exploits signal sparsity and incoherence principles, enabling applications in medical imaging, radar systems, and wireless communication.

This technique relies on the Restricted Isometry Property and low coherence between measurement matrices and sparsifying bases. Recovery algorithms promote sparsity, with L1-norm minimization and greedy methods being common approaches for solving the underdetermined system of equations.

Fundamental Concepts of Compressed Sensing

Core Principles and Signal Representation

Top images from around the web for Core Principles and Signal Representation

• 

  Frontiers | An Intelligent EEG Classification Methodology Based on Sparse Representation ... View original

  Is this image relevant?

• 

  Frontiers | An Intelligent EEG Classification Methodology Based on Sparse Representation ... View original

  Is this image relevant?

1 of 1

Top images from around the web for Core Principles and Signal Representation

• 

  Frontiers | An Intelligent EEG Classification Methodology Based on Sparse Representation ... View original

  Is this image relevant?

• 

  Frontiers | An Intelligent EEG Classification Methodology Based on Sparse Representation ... View original

  Is this image relevant?

1 of 1

• Compressed sensing enables efficient acquisition and reconstruction of sparse signals from fewer measurements than traditional sampling methods
• Many natural signals can be represented sparsely in some basis or domain
• Exploits signal sparsity to reduce the number of measurements needed for accurate reconstruction
• Measurement process involves taking random linear projections of the signal
• Relies on two key principles:
  • Sparsity refers to the signal having few non-zero coefficients in some representation
  • Incoherence ensures that the sampling/sensing modality is sufficiently different from the sparsifying basis
• Reconstruction process typically involves solving an optimization problem to recover the original signal from the compressed measurements

Examples and Applications

• Medical imaging (MRI scans)
• Radar systems (target detection)
• Wireless communication (signal compression)
• Seismic data processing (oil exploration)
• Astronomical imaging (radio telescope data)

Conditions for Signal Recovery

Measurement Matrix Properties

• Restricted Isometry Property (RIP) crucial condition for the measurement matrix
• RIP approximately preserves the Euclidean length of sparse vectors
• Low coherence between the measurement matrix and the sparsifying basis required for effective compressed sensing
• Number of measurements required related to the sparsity level of the signal
• Signal must be compressible, well-approximated by a sparse representation in some basis
• Choice of measurement matrix critical, random matrices often satisfy necessary conditions
  • Examples: Gaussian random matrices, Bernoulli matrices, partial Fourier matrices

Recovery Algorithms and Requirements

• Recovery algorithm must promote sparsity in the solution
• Typical approaches:
  • L1-norm minimization (convex optimization)
  • Greedy algorithms (iterative methods)
• Examples of recovery algorithms:
  • Basis Pursuit (BP)
  • Orthogonal Matching Pursuit (OMP)
  • Compressive Sampling Matching Pursuit (CoSaMP)
• Algorithm choice depends on:
  • Computational complexity
  • Recovery guarantees
  • Noise tolerance

Compressed Sensing as Sparse Recovery

Problem Formulation

• Expressed as $y = Ax + e$, where:
  • y represents measurement vector
  • A denotes measurement matrix
  • x signifies sparse signal
  • e indicates noise
• Goal recovers sparse signal x from underdetermined system of equations
• Typically formulated as optimization problem:
  • Minimize $\|x\|_0$ subject to $\|y - Ax\|_2 \leq \varepsilon$
  • $\|x\|_0$ represents L0 norm (number of non-zero entries)
  • $\varepsilon$ denotes noise tolerance

Relaxation and Algorithmic Approaches

• L0 minimization NP-hard, often relaxed to L1 minimization:
  • Minimize $\|x\|_1$ subject to $\|y - Ax\|_2 \leq \varepsilon$
• Greedy algorithms like Orthogonal Matching Pursuit (OMP) iteratively build sparse solution
• Basis Pursuit (BP) algorithm solves L1 minimization problem when no noise present
• Examples of other algorithms:
  • Iterative Hard Thresholding (IHT)
  • Gradient Projection for Sparse Reconstruction (GPSR)
  • Least Absolute Shrinkage and Selection Operator (LASSO)

Advantages vs Limitations of Compressed Sensing

Benefits and Applications

• Reduced data acquisition time and storage requirements
  • Example: Faster MRI scans with fewer measurements
• Potential for improved signal-to-noise ratio in certain applications
  • Example: Enhanced radar detection in noisy environments
• Ability to recover signals from incomplete or corrupted measurements
  • Example: Reconstructing images from partial data in astronomy
• Applicability to various domains (medical imaging, radar, communications)
• Enables novel sensing architectures (single-pixel camera in computational imaging)
• Efficient sampling of high-dimensional signals with sparse representation
  • Example: Hyperspectral imaging with reduced data collection

Challenges and Constraints

• Increased computational complexity in signal reconstruction compared to traditional sampling
  • Example: Longer processing times for large-scale problems
• Sensitivity to noise and model mismatch in practical applications
  • Example: Degraded performance in high-noise environments
• Requirement for signals to be sparse or compressible in some known basis
  • Example: Limited effectiveness for truly dense signals
• Potential difficulties in designing hardware that implements random sampling efficiently
  • Example: Challenges in creating truly random measurement matrices in physical systems
• Performance degrades as signal becomes less sparse or noise level increases
• Critical choice of sparsifying basis and measurement matrix impacts effectiveness in specific applications
  • Example: Selecting appropriate wavelet basis for image compression