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Unformatted text preview: Compressed Sensing: A Tutorial IEEE Statistical Signal Processing Workshop Madison, Wisconsin August 26, 2007 Justin Romberg Michael Wakin School of ECE EECS Department Georgia Tech University of Michigan Download at: http://users.ece.gatech.edu/ ∼ justin/ssp2007 Data Acquisition • ShannonNyquist sampling theorem: no information loss if we sample at 2x the bandwidth ing pressure on DSP hardware, algo aster sampling and processing rates • DSP revolution: sample then process • Trends (demands): – faster sampling – larger dynamic range – higherdimensional data – lower energy consumption – new sensing modalities Nyquist Sampling • RF applications: to acquire an EM pulse containing frequencies at f m ax , we need to sample at rate ∼ f m ax • Pixel imaging: to get npixel resolution, we need n sensors Fourier imaging (MRI): need dense sampling out to freqs ∼ n ! 2 ! 1 • Resolution determines the measurement complexity • Makes sense, but we know many times signals are much simpler ... Signal and Image Representations • Fundamental concept in DSP: Transformdomain processing • Decompose f as superposition of atoms (orthobasis or tight frame) f ( t ) = X i α i ψ i ( t ) or f = Ψ α e.g. sinusoids, wavelets, curvelets, Gabor functions, ... • Process the coefficient sequence α α i = h f,ψ i i , or α = Ψ T f • Why do this? If we choose Ψ wisely, { α i } will be “simpler” than f ( t ) Classical Image Representation: DCT • Discrete Cosine Transform (DCT) Basically a realvalued Fourier transform (sinusoids) • Model: most of the energy is at low frequencies • Basis for JPEG image compression standard • DCT approximations: smooth regions great, edges blurred/ringing Modern Image Representation: 2D Wavelets • Sparse structure: few large coeffs, many small coeffs • Basis for JPEG2000 image compression standard • Wavelet approximations: smooths regions great, edges much sharper • Fundamentally better than DCT for images with edges Wavelets and Images 1 megapixel image wavelet coeffs (sorted) 2 4 6 8 10 12 x 10 540002000 2000 4000 6000 8000 10000 12000 14000 16000 2 4 6 8 10 12 x 10 5 100 200 300 400 500 600 700 800 900 1000 ⇓ 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 x 10 4300020001000 1000 2000 3000 2 4 6 8 10 12 x 10 543.532.521.510.5 zoom in ( log 10 sorted) Wavelet Approximation 2 4 6 8 10 x 10 554321 1 megapixel image 25k term approx Bterm approx error • Within 2 digits (in MSE) with ≈ 2 . 5% of coeffs • Original image = f , Kterm approximation = f K k f f K k 2 ≈ . 01 · k f k 2 Computational Harmonic Analysis • Sparsity plays a fundamental role in how well we can: – Estimate signals in the presence of noise (shrinkage, softthresholding) – Compress (transform coding) – Solve inverse problems (restoration and imaging) • Dimensionality reduction facilitates modeling: simple models/algorithms are effective • This talk: Sparsity also determines how quickly we can acquire signals nonadaptively Sample then Compress...
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 Spring '10
 RunyiYu
 Gate, Signal Processing

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