1. Improving K-SVD Denoising by Post-Processing its Method-Noise
Yaniv Romano
The Electrical Engineering Department
The Technion – Israel Institute of technology
Haifa 32000, Israel
Michael Elad
The Computer Science Department
The Technion – Israel Institute of technology
Haifa 32000, Israel
The research leading to these results has received funding from the European Research Council under European Union's Seventh Framework Programme, ERC Grant agreement no , and by the Intel Collaborative Research Institute for Computational Intelligence.

2. Background Improving K-SVD Denoising By
Post-Processing Its Method-Noise
By: Michael Elad and Yaniv Romano

3. ? Denoising Remove Additive Noise Noisy Image Original Image
Gaussian Noise
Improving K-SVD Denoising By
Post-Processing Its Method-Noise
By: Michael Elad and Yaniv Romano

4. D D Sparsity Model – The Basics … z = 
We assume the existence of a dictionary DIR dn whose columns are the atom signals.
Signals are modeled as sparse linear combinations of the dictionary atoms:
where  is sparse, meaning that it is assumed to contain mostly zeros.
The computation of  from z (or its noisy version) is called sparse-coding. D … z = D 
Improving K-SVD Denoising By
Post-Processing Its Method-Noise
By: Michael Elad and Yaniv Romano

5. K-SVD Denoising [Elad & Aharon (`06)]
Initial Dictionary
Noisy Image
Reconstructed Image
Update the Dictionary
Denoising each patch
Using OMP
This method (and variants of it) is very popular and lead to
state-of-the-art results in various applications.
The problem: we model small patches of the image while disregarding their inter-relations.
Improving K-SVD Denoising By
Post-Processing Its Method-Noise
By: Michael Elad and Yaniv Romano

6. Residual Content Noisy image Denoised image Method Noise
The residual image contains “stolen”
original image information
The observed problem:
Improving K-SVD Denoising By
Post-Processing Its Method-Noise
By: Michael Elad and Yaniv Romano

7. Local Vs. Global
The OMP ensures that the patch representation is orthogonal to its residual.
After patch-averaging, orthogonality of the overall residual to the chosen atoms is lost. α1 α2 +
Improving K-SVD Denoising By
Post-Processing Its Method-Noise
By: Michael Elad and Yaniv Romano

8. Our Assumption
The method-noise contains residual image content that can be represented as a linear combination of the very same atoms used in the initial denoising stage. + ? α1 α2 +
Improving K-SVD Denoising By
Post-Processing Its Method-Noise
By: Michael Elad and Yaniv Romano

9. The Proposed Algorithm
Improving K-SVD Denoising By
Post-Processing Its Method-Noise
By: Michael Elad and Yaniv Romano

10. Dij Starting Point K-SVD Denoise D, Supp Improving K-SVD Denoising By
Post-Processing Its Method-Noise
By: Michael Elad and Yaniv Romano

11. The Core Idea Given: Supp ,D, , Output:
residual patches
pre-chosen subspaces
1. Project the onto the
Perform iterations of the form:
, and then
2. Average the projected patches.
Output:
Activity mask
Improving K-SVD Denoising By
Post-Processing Its Method-Noise
By: Michael Elad and Yaniv Romano

12. Projection D Given Sij = Supp and D, solve: But
We found that allowing a slight modification to these supports may lead to further denoising improvement. So
Sparse-coding additional (e.g. 1) atoms using the given supports as initial solution for the OMP.
Improving K-SVD Denoising By
Post-Processing Its Method-Noise
By: Michael Elad and Yaniv Romano

13. Stopping Criterion is met
Repeat:
Projection of the residual patches.
Averaging the results.
Until ?
Stopping Criterion is met
Improving K-SVD Denoising By
Post-Processing Its Method-Noise
By: Michael Elad and Yaniv Romano

14. Stopping Criterion
Apply “Pearson’s correlation test” between the active regions
of the k-iteration denoised and method-noise images
Improving K-SVD Denoising By
Post-Processing Its Method-Noise
By: Michael Elad and Yaniv Romano

15. Stopping Criterion
Apply “Pearson’s correlation test” between the active regions
of the k-iteration denoised and method-noise images
A correlation closer to zero implies less dependence between these two images.
The iteration that obtains the minimum absolute value of the correlation is the best one.
minimum
|correlation|
Improving K-SVD Denoising By
Post-Processing Its Method-Noise
By: Michael Elad and Yaniv Romano

16. Experiments [dB] Regular K-SVD results Boosting results
Barbara
Couple
Fingerprint
House
Boats
Average
Boost
Est. Err.
15/24.61
32.49
32.61
31.52
31.61
30.04
30.16
34.37
31.80
31.89
32.04
32.13
0.08
0.02
20/22.11
30.88
31.12
30.08
30.22
28.45
28.66
33.22
33.21
30.43
30.57
30.61
30.76
0.15
0.01
25/20.18
29.57
29.87
28.90
29.12
27.24
27.53
32.19
32.20
29.34
29.52
29.45
29.65
0.20
50/14.16
25.40
25.87
25.27
25.69
23.21
23.88
28.02
28.23
25.92
26.17
25.56
25.97
0.41
0.03
75/10.63
22.92
23.13
23.56
23.79
19.99
21.65
25.06
25.42
24.01
24.20
23.11
23.64
0.53
0.06
100/8.14
21.85
21.89
22.61
22.68
18.31
20.02
23.77
22.85
22.26
0.04
Boosting the K-SVD consistently.
The stopping criterion estimation error is almost negligible.
Improving K-SVD Denoising By
Post-Processing Its Method-Noise
By: Michael Elad and Yaniv Romano

17. Experiments Regular K-SVD results Boosting results
PSNR = dB PSNR = dB
Improving K-SVD Denoising By
Post-Processing Its Method-Noise
By: Michael Elad and Yaniv Romano

18. Experiments Regular K-SVD results Boosting results
PSNR = dB PSNR = dB
Improving K-SVD Denoising By
Post-Processing Its Method-Noise
By: Michael Elad and Yaniv Romano

19. Experiments Regular K-SVD results Boosting results
PSNR = dB PSNR = dB
Improving K-SVD Denoising By
Post-Processing Its Method-Noise
By: Michael Elad and Yaniv Romano

20. Experiments Regular K-SVD results Boosting results
PSNR = dB PSNR = dB
Improving K-SVD Denoising By
Post-Processing Its Method-Noise
By: Michael Elad and Yaniv Romano

21. Summary
It suffers from the gap between the local approach and the global need
Its method-noise contains “stolen” image content
K-SVD denoising is a patch-based technique
What about
other patch-based
techniques?
Extract the residual information based on the initial supports
We tried and the proposed solution is effective for them as well !!
The support of
each patch
is the set of its
similar patches
Any examples?
NonLocal-Means, BM3D*
*Boosting its first stage
Improving K-SVD Denoising By
Post-Processing Its Method-Noise
By: Michael Elad and Yaniv Romano

22. Thank You Questions ? We Are Done Improving K-SVD Denoising By
Post-Processing Its Method-Noise
By: Michael Elad and Yaniv Romano