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Elaine Hale, Wotao Yin, Yin Zhang Computational and Applied Mathematics Rice University 2007-08-13, McMaster University, ICCOPT II TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAAAAAA

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2 The FPC (fixed-point continuation) algorithm A short introduction to Compressed Sensing An imaging perspective Image compression Scene Picture 10 Mega Pixels Why do we compress images?

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3 The FPC (fixed-point continuation) algorithm Introduction to Compressed Sensing Image are compressible Image compression –Take an input image u –Pick a good dictionary © –Find a sparse representation x of u such that || ©x - u || 2 is small –Save x Because Only certain part of information is important (e.g. objects and their edges) Some information is unwanted (e.g. noise) This is traditional compression.

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4 The FPC (fixed-point continuation) algorithm Introduction to Compressed Sensing An imaging perspective This is traditional compression. 10 Mega Pixels 100 Kilobytes

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5 The FPC (fixed-point continuation) algorithm Introduction to Compressed Sensing An imaging perspective This is traditional compression. n : 10 Mega Pixels k:100 Kilobytes

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6 The FPC (fixed-point continuation) algorithm Introduction to Compressed Sensing If only 100 kilobytes are saved, why do we need a 10- megapixel camera in the first place? Answer: a traditional compression algorithm needs the complete image to compute © and x Can we do better than this?

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7 The FPC (fixed-point continuation) algorithm Introduction to Compressed Sensing Let k =|| x || 0, n =dim( x )=dim( u ). In compressed sensing based on l 1 minimization, the number of measurements is m =O( k log( n / k )) (Donoho, Candés-Tao)

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8 The FPC (fixed-point continuation) algorithm Introduction to Compressed Sensing Input Linear encoding Signal acquisition Signal reconstruction Signal representation

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9 The FPC (fixed-point continuation) algorithm Introduction to Compressed Sensing Input Linear encoding Signal acquisition Signal reconstruction Signal representation

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10 The FPC (fixed-point continuation) algorithm Introduction to Compressed Sensing Input: b = Bu = B©x, A = B© Output: x In compressed sensing, m =dim( b )<

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11 The FPC (fixed-point continuation) algorithm We solve the following problems

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12 The FPC (fixed-point continuation) algorithm Existing approaches Methods exactly solving the l1-problem –GPSR (Figueiredo, Nowak, Wright) –Iterative Shresholding (Daubechies, Defrise, and De Mol) –l1_ls (Kim, Koh, Lustig, Boyd, Gorinvesky) –l1_magic (Candes, Romberg) –Lasso, LARS –…… Methods not solving the l1-problem per se –OMP, StOMP, Random Projections, Sparse Tree Rep., Belief Propogation, Gradient Pursuits, Sparsify, Randomized Algorithms, Chaining Pursuit, HHS Pursuit.

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13 The FPC (fixed-point continuation) algorithm Difficulties Large scales Completely dense data: A However Solutions x are expected to be sparse The matrices A are often fast transforms

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14 The FPC (fixed-point continuation) algorithm Original problem: Linearize f and solve iteratively: Must keep x close to x k : Linearization Combine

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15 The FPC (fixed-point continuation) algorithm Shrinkage (Soft-thresholding)

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16 The FPC (fixed-point continuation) algorithm Original problem: First-order Taylor series: Must keep x close to x k : Linearization Combine

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17 The FPC (fixed-point continuation) algorithm An (well-known) optimality theorem Theorem: Let ¿ >0. x solves the convex problem if and only if

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18 The FPC (fixed-point continuation) algorithm The sketch of the algorithm In our report, we used 1/º instead of º.

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19 The FPC (fixed-point continuation) algorithm The sketch of the algorithm Identical to iterative thresholding by Daubechies et al (2004) for convex quadratic f.

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20 The FPC (fixed-point continuation) algorithm The sketch of the algorithm

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21 The FPC (fixed-point continuation) algorithm Summary of properties Simple –Easy to implement –Free of matrix factorization (Low memory requirement) –Takes advantage of fast transforms Flexible Parallelizable Convergence speed? Practical performance?

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22 The FPC (fixed-point continuation) algorithm Convergence Under some conditions on f ( x ) (e.g., bounded max/min Hessian eigenvalues): Strong convergence (Combettes and Vajs) q -linearly convergent in 2-norm Finite convergent on certain quantities –Zeros in x * –The signs of non-zero components in x *

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23 The FPC (fixed-point continuation) algorithm Computations At each iteration Shrinkage (soft thresholding): Min TV: non-trivial in 2 and higher dimensional spaces

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24 The FPC (fixed-point continuation) algorithm

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25 The FPC (fixed-point continuation) algorithm Computations At each iteration Shrinkage: TV minimization: parametric max-flow (see our Poster)

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26 The FPC (fixed-point continuation) algorithm Continuation Convergence requires Support and signs can be obtained earlier than convergence Faster convergence is achieved by All of the above suggests using an increasing sequence of

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27 The FPC (fixed-point continuation) algorithm Some References I.Daubechies, M.Defrise, C.De Mol. Iterative Thresholding, CPAM, 2004. D.Donoho, Y.Tsaig, I.Drori, J.-C.Starck. StOMP, preprint, 2006 M.Figueiredo, R.Nowak, S.Wright. GPSR, preprint, 2007. S.Kim, K.Koh, M.Lustig, S.Boyd, D.Gorinevsky. l1_ls, preprint, 2007. E.Candes, J.Romberg. l1-magic. 2005 A report on FPC is available on my homepage.

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