1 Micha Feigin, Danny Feldman, Nir Sochen

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13 Defined as a set {D,X,Y} such that DY t X Figure courtesy Michael Elad

14 Given a D and y i, how to find x i Constraint : x i is sufficiently sparse Finding exact solution difficult Approximate solution good enough ?

15 Select d k with max projection on residue x k = arg min ||y-D k x k || Update residue r = y - D k x k Check terminating condition D, yx

16 Greedy algorithm Can find approximate solution Close solution if T is small enough Simplistic in nature Tends to be unstable for large T

17 Select atoms from input Atoms can be patches from the image Patches are overlapping Initialize Dictionary Sparse Coding (OMP) Update Dictionary One atom at a time

18 Use OMP or any other fast method Output gives sparse code for all signals Minimize error in representation Initialize Dictionary Sparse Coding (OMP) Update Dictionary One atom at a time

19 Replace unused atom with minimally represented signal Identify signals that use k-th atom (non zero entries in rows of X) Initialize Dictionary Sparse Coding (OMP) Update Dictionary One atom at a time

20 Deselect k-th atom from dictionary Find coding error matrix of these signals Minimize this error matrix with rank-1 approx from SVD Initialize Dictionary Sparse Coding (OMP) Update Dictionary One atom at a time

21 Initialize Dictionary Sparse Coding (OMP) Update Dictionary One atom at a time

22 A cost function for : Y = Z + n Solve for Prior term

23 Break problem into smaller problems Aim at minimization at the patch level Select i-th patch of Z accounted for implicitly by OMP

24 Solution : Denoising by normalized weighted averaging Initialize Dictionary Sparse Coding (OMP) Update Dictionary One atom at a time Averaging of patches

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26 Replace input set Y with a weighted set C Construct an optimal dictionary for C (instead of Y) Compute coefficients for Y based on this dictionary Advantages: The coreset C is much smaller than Y Constructing C is much cheaper than solving the original problem

x4000 pixels

28 K-SVD Denoiser for LD images [Michael Elad et al.]

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32 Uniform random sampling generally does not produce a good coreset o For a mostly uniform image (such as line drawings), patches with features will be chosen with low probability o Essential to choose them to reconstruct the image To construct a good dictionary, we need to o Make sure outliers are represented o Give them low priority for affecting the final outcome

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35 Tree Computation p1p1 p2p2 p3p3 p4p4 p5p5 p7p7 p6p6 p8p8 p9p9 p 10 p 11 p 12 p 13 p 15 p 14 p 16 From a presentation by Piotr Indyk

36 Coresets greatly speed up calculation Allow handling much larger inputs (at lower running times) Can stabilize random seeded and greedy algorithms due to smaller input Allow multiple runs on random seeded methods to allow majority vote methods (at a lower total run time)

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