Introduction to Compressive Sensing Richard Baraniuk, Compressive sensing. IEEE Signal Processing Magazine, 24(4), pp. 118-121, July 2007) Emmanuel Candès and Michael Wakin, An introduction to compressive sampling. IEEE Signal Processing Magazine, 25(2), pp. 21 - 30, March 2008 A course on compressive sensing, http://w3.impa.br/~aschulz/CS/course.html
Outline Introduction to compressive sensing (CS) First CS theory Concepts and applications Theory Compression Reconstruction
Introduction Compressive sensing First CS theory Compressed sensing Compressive sampling First CS theory E. Cand`es, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inform. Theory, vol. 52, no. 2, pp. 489–509, Feb. 2006. Cand`es Romberg Tao
Compressive Sensing: concept and applications
Compression/Reconstruction Transmit X RNx1 CS sampling yRMx1 Quantization human coding RMxN Measurement matrix CS Reconstruction Optimization Inverse transform (e.g., IDCT) X’ Inverse Quantization human coding y’ s : transform basis (e.g., DCT basis)
Theory and Core Technology compression K-sparse most of the energy is at low frequencies K non-zero wavelet (DCT) coefficients
Compression Measurement matrix
Compression transform basis coefficient
Compression transform basis coefficient
Reconstruction
Reconstruction: optimization (1) NP-hard problem (2) Minimum energy ≠ k-sparse (3) Linear programming [1][2] Orthogonal matching pursuit (OMP) (4) Greedy algorithm [3]
Compressive sensing: significant parameters What measurement matrix should we use? How many measurements? (M=?) K-sparse?
Measurement Matrix Incoherence (1) Correlation between and
Examples = noiselet, = Haar wavelet (,)=2 = noiselet, = Daubechies D4 (,)=2.2 = noiselet, = Daubechies D8 (,)=2.9 Noiselets are also maximally incoherent with spikes and incoherent with the Fourier basis = White noise (random Gaussian)
Restricted Isometry Property (RIP) preserving length For each integer k = 1, 2, …, define the isometry constant k of a matrix A as the smallest number such that A approximately preserves the Euclidean length of k-sparse signals (2) Imply that k-sparse vectors cannot be in the nullspace of A (3) All subsets of s columns taken from A are in fact nearly orthogonal To design a sensing matrix , so that any subset of columns of size k be approximately orthogonal.
How many measurements ?
Single-Pixel CS Camera [Baraniuk and Kelly, et al.]
On the Interplay Between Routing and Signal Representation for Compressive Sensing in Wireless Sensor Networks G. Quer, R. Masiero, D. Munaretto, M. Rossi, J. Widmer and M. Zorzi University of Padova, Italy. DoCoMo Euro-Labs, Germany Information Theory and Applications Workshop (ITA 2009)
Network Scenario Setting X x11 x12 x13 x14 x21 x22 x23 x24 … .. Irregular network setting [4] Graph wavelet Diffusion wavelet Example of the considered multi-hop topology.
Measurement matrix Built on routing path …………………… ……………… …………………… ……………………
Measurement matrix R1: is built according to routing protocol, randomly selected from {+1, -1} R2: is built according to routing protocol randomly selected from (0, 1] R3: has all coefficients in randomly selected from {+1, -1} R4: has all coefficients in randomly selected from(0, 1]
Transform basis T1: DCT T2: Haar Wavelet T3: Horizontal difference T4: Vertical difference + Horizontal difference
Degree of sparsity H-diff VH-diff Haar DCT
Incoherence DCT Haar H-diff VH-diff
Performance Comparison Random sampling (RS) each node sends its data with probability P = M/N, the data packets are not processed at internal nodes but simply forwarded. RS-CS the data values are combined with that of any other node encountered along the path. Routing path
Reconstruction Error
Reconstruction Error pre-distribution for T3 and T4 [5]
Research issues when applying CS in Sensor Networks How to construct measurement matrix Incoherent with transform basis Distributed M=? How to choose transformation basis Sparsity Incoherent with measurement matrix Irregular sensor deployment Graph wavelet Diffusion wavelet
References [1] Bloomfield, P., Steiger, W., Least Absolute Deviations: Theory, Applications, and Algorithms. Progr. Probab. Statist. 6, Birkhäuser, Boston, MA, 1983. [2] Chen, S. S., Donoho, D. L., Saunders, M. A, Atomic decomposition by basis pursuit. SIAM J. Sci. Comput. 20 (1999), 33–61. [3] J. Tropp and A. C. Gilbert, “Signal recovery from partial information via orthogonal matching pursuit,” Apr. 2005, Preprint. [4] J. Haupt, W.U. Bajwa, M. Rabbat, and R. Nowak, “Compressed sensing for networked data,” IEEE Signal Processing Mag., vol. 25, no. 2, pp. 92-101, Mar. 2008. [5] M. Rabbat, J. Haupt, A. Singh, and R. Novak, “Decentralized Compression and Predistribution via Randomized Gossiping,” in IPSN, 2006.