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Compressed Sensing

This page contains resources about Compressed Sensing, Sparse Sampling and Sparse Signal Processing.

Contents

Subfields and ConceptsEdit

  • Incoherence / Incoherent Sampling / Incoherent bases
    • Canonical/Kroneker basis
    • Fourier basis
    • Random basis
    • Random sequences / codes
    • Inverse Discrete Cosine Transform (IDCT) / Heisenberg
    • Wavelet basis
  • Coherent-based Sampling
  • Coherence / Mutual Coherence
  • Local Coherence
  • Null Space Property
  • Restricted Isometry Property
  • Underdetermined Linear System
  • Uncertainty Principles (between sparsity basis and measurement system)
    • Continuous Uncertainty Principles (Heisenberg)
    • Discrete Uncertainty Principle (Donoho and Stark)
      • Dirac Comb / Picket Fence
    • Quantitative Uncertainty Principle
      • Quantitative Robust Uncertainty Principle
  • Sparse Approximation / Sparse Representation
    • Basis Pursuit
    • Matching Pursuit
  • Sparse Signal Recovery / Sparse Signal Reconstruction
    • Exact Recovery Theorem
    • Stable Recovery / Stability Theorem
  • Sub-Nyquist Sampling
  • Nonlinear Sampling Theorem
  • Iterative Reweighted Least Squares
  • Sparse Principal Component Analysis (PCA)
  • Structure Sparse PCA
  • B-Splines
  • E-Splines
  • Wavelets
  • Bayesian Compressive Sensing
    • Variational Bayesian Compressive Sensing
  • Sparse Bayesian Models
  • Inverse Problems (Optimization)
    • Regularization
      • Regularized least squares
      • L0 penalization / Spike-and-slab prior
      • L1-regularization / LASSO / Laplace prior
      • L2-regularization / Ridge Regression / Gaussian prior 
      • Elastic nets 
      • Total Variation (TV) Regularization (i.e. L1-norm of the gradient)

Online CoursesEdit

Books and Book ChaptersEdit

  • Theodoridis, S. (2015). "Chapter 9: Sparsity-Aware Learning". Machine Learning: A Bayesian and Optimization Perspective. Academic Press.
  • Hastie, T., Tibshirani, R., & Wainwright, M. (2015). "Chapter 10: Signal Approximation and Compressed Sensing". Statistical learning with sparsity: the lasso and generalizations. CRC Press.
  • Eldar, Y. C. (2015). Sampling theory: Beyond bandlimited systems. Cambridge University Press.
  • Carmi, A. Y., L. Mihaylova, & S. J. Godsill (Eds.). (2014). Compressed Sensing and Sparse Filtering. Springer.
  • Rish, I., & Grabarnik, G. (2014). Sparse modeling: theory, algorithms, and applications. CRC Press.
  • Foucart, S., & Rauhut, H. (2013). A mathematical introduction to compressive sensing. Birkhäuser.
  • Murphy, K. P. (2012). "Chapter 13: Sparse linear models". Machine Learning: A Probabilistic Perspective. MIT Press.
  • Baraniuk, R., Davenport, M. A., Duarte, M. F., & Hegde, C. (2011). An introduction to compressive sensing. Connexions e-textbook.
  • Starck, J. L., Murtagh, F., & Fadili, J. M. (2010). "Chapter 11: Compressed Sensing". Sparse image and signal processing: wavelets, curvelets, morphological diversity. Cambridge University Press.
  • Elad, M. (2010). Sparse and Redundant Representations: From Theory to Applications in Signal and Image Processing. Springer.
  • Mallat, S. (2008). A Wavelet Tour of Signal Processing: The Sparse Way. Academic Press.
  • Saad, Y. (2003). Iterative Methods for Sparse Linear Systems. Siam.
  • MacKay, D. J. (2003). "Part VI: Sparse Graph Codes". Information Theory, Inference and Learning Algorithms. Cambridge University Press.

Scholarly ArticlesEdit

  • Chen, Y., Bhojanapalli, S., Sanghavi, S., & Ward, R. (2014). Coherent matrix completion. In Proceedings of the 31st International Conference on Machine Learning (pp. 674-682).
  • Davenport, M. A., Duarte, M. F., Eldar, Y. C., & Kutyniok, G. (2011). Introduction to compressed sensing. Preprint, 93(1), 2.
  • Fornasier, M., & Rauhut, H. (2011). Compressive sensing. In Handbook of mathematical methods in imaging (pp. 187-228). Springer New York.
  • Yang, J., Wright, J., Huang, T. S., & Ma, Y. (2010). Image super-resolution via sparse representation. IEEE Transactions on image processing, 19(11), 2861-2873.
  • Dai, W., & Milenkovic, O. (2009). Subspace pursuit for compressive sensing signal reconstruction. IEEE Transactions on Information Theory, 55(5), 2230-2249.
  • Starck, J. L., & Fadili, M. J. (2009). An overview of inverse problem regularization using sparsity. In Image Processing (ICIP), 16th IEEE International Conference on, 1453-1456.
  • Duarte, M. F., Davenport, M. A., Takhar, D., Laska, J. N., Sun, T., Kelly, K. E., & Baraniuk, R. G. (2008). Single-pixel imaging via compressive sampling.IEEE Signal Processing Magazine, 25(2), 83.
  • Ji, S., Xue, Y., & Carin, L. (2008). Bayesian compressive sensing. IEEE Transactions on Signal Processing, 56(6), 2346-2356.
  • Candès, E. J., & Wakin, M. B. (2008). An introduction to compressive sampling. IEEE Signal Processing Magazine, 25(2), 21-30.
  • Blu, T., Dragotti, P. L., Vetterli, M., Marziliano, P., & Coulot, L. (2008). Sparse sampling of signal innovations. IEEE Signal Processing Magazine, 25(2), 31-40.
  • Lustig, M., Donoho, D. L., Santos, J. M., & Pauly, J. M. (2008). Compressed sensing MRI. IEEE Signal Processing Magazine, 25(2), 72-82.
  • Lustig, M., Donoho, D., & Pauly, J. M. (2007). Sparse MRI: The application of compressed sensing for rapid MR imaging. Magnetic resonance in medicine, 58(6), 1182-1195.
  • Dragotti, P. L., Vetterli, M., & Blu, T. (2007). Sampling moments and reconstructing signals of finite rate of innovation: Shannon meets Strang–Fix.IEEE Transactions on Signal Processing, 55(5), 1741-1757.
  • Baraniuk, R. G. (2007). Compressive sensing. IEEE Signal Processing Magazine, 24(4).
  • Candes, E., & Romberg, J. (2007). Sparsity and incoherence in compressive sampling. Inverse problems, 23(3), 969.
  • Candes, E. J., Romberg, J., & Tao, T. (2006). Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information.IEEE Transactions on information theory, 52(2), 489-509.
  • Candes, E. J., Romberg, J. K., & Tao, T. (2006). Stable signal recovery from incomplete and inaccurate measurements. Communications on pure and applied mathematics, 59(8), 1207-1223.
  • Candse, E. J. (2006, August). Compressive sampling. In Proceedings of the international congress of mathematicians (Vol. 3, pp. 1433-1452).
  • Donoho, D. L. (2006). Compressed sensing. IEEE Transactions on information theory, 52(4), 1289-1306.
  • Castro, R., Haupt, J., & Nowak, R. (2006). Compressed sensing vs. active learning. In IEEE International Conference on Acoustics, Speech and Signal Processing, Proceedings. (Vol. 3, pp. III-III). IEEE.
  • Elad, M., & Bruckstein, A. M. (2002). A generalized uncertainty principle and sparse representation in pairs of bases. IEEE Transactions on Information Theory, 48(9), 2558-2567.
  • Donoho, D. L., & Stark, P. B. (1989). Uncertainty principles and signal recovery. SIAM Journal on Applied Mathematics, 49(3), 906-931.

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