Sparse Signal Models

• Outline
  • Sparsity
• Sparse solutions for under-determined linear systems
  • Regularization
  • \(l_2\) Regularization
  • Convexity
  • \(l_1\) Regularization
  • \(l_1\) norm minimization problem as a linear programming problem
• Dictionary based representations
  • Dictionaries
  • Redundant dictionaries and sparse signals
  • Sparse approximation problem
  • Synthesis and analysis
• p-norms and sparse signals
  • l1 , l2 and max norms
  • Sparse signals
• Compressible signals
  • K-term approximation of general signals
  • Compressible signals
• Tools for dictionary analysis
• Spark
• Coherence
  • Lower bounds for spark
  • Uniqueness-Coherence
  • Singular values of sub-dictionaries
  • Embeddings using sub-dictionaries
• Babel function
  • Computation of Babel function
  • Babel function and spark
  • Babel function and singular values
  • Babel function and gram matrix
  • Quasi incoherent dictionaries
  • Implementing the Babel function
• Dirac-DCT dictionary
  • Hands-on with Dirac DCT dictionaries
• Dirac-Hadamard dictionary
  • Hands-on with Dirac Hadamard dictionaries
• Dictionaries with Wavelet Toolbox
  • Constructing Dictionaries