1. Image Denoising with K-SVD Priyam Chatterjee EE 264 – Image Processing & Reconstruction Instructor : Prof. Peyman Milanfar Spring 2007

2. 2 Sparseland Model  Defined as a set {D,X,Y} such that DY t X Figure courtesy Michael Elad

3. 3 Sparse Coding  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 ?

4. 4 Orthogonal Matching Pursuit 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

5. 5 OMP : features  Greedy algorithm  Can find approximate solution  Close solution if T is small enough  Simplistic in nature

6. 6 Dictionary Selection  What D to use ?  A fixed overcomplete set of basis :  Steerable wavelet  Contourlet  DCT Basis  ….  Data Adaptive Dictionary – learn from data

7. 7 K-SVD Algorithm  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

8. 8 K-SVD Algorithm  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

9. 9 K-SVD Algorithm  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

10. 10 K-SVD Algorithm  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

11. 11 K-SVD Algorithm  [U,S,V] = svd(E k )  Replace coeff of atom d k in X with entries of s 1 v 1  d k = u 1 /||u 1 || 2 Initialize Dictionary Sparse Coding (OMP) Update Dictionary One atom at a time

12. 12 Denoising framework  A cost function for : Y = Z + n  Solve for Prior term

13. 13 Denoising Framework  Break problem into smaller problems  Aim at minimization at the patch level Select i-th patch of Z accounted for implicitly by OMP

14. 14 Denoising Framework  Solution :  Denoising by normalized weighted averaging Initialize Dictionary Sparse Coding (OMP) Update Dictionary One atom at a time Averaging of patches

15. 15 Proof of the pudding – low noise Denoising under presence of AWGN of std. dev 10 PSNR 28.12 dBPSNR 34.16 dB

16. 16 High noise case – std dev 50 PSNR 14.75 dB PSNR 24.93 dB

17. 17 Outside the math :  Similar atoms in dictionary should be replaced with signals that are least represented  Atoms which are least used should be replaced by signals that are least represented