1. Stein Unbiased Risk Estimator Michael Elad

2. The Objective We have a denoising algorithm of some sort, and we want to set its parameters so as to extract the best out of it Algorithm +

3. Derivation – 1 Lets open the norm into its ingredients: Therefore, we will proceed with the second term and show that in fact it can be computed Easy Impossible? No matter

4. Derivation – 2 Using the fact that we get Again, the first term is fine for us to compute, while the second seems hard (we do not know the noise vector!) Easy Impossible?

5. Derivation – 3 Using the definition of expectation This may look ugly BUT …..

6. Derivation – 4 We notice that the same integral can be written as which should remind us of integration by parts:

7. Derivation – 5 Using this to our expression leads to Assuming that the function h is finite for all y, this term is zero The derivative w.r.t. v can be replaced by a derivative w.r.t. y

8. Derivation – 6 One last step – the expression we got is in fact an expectation …

9. Wrap Up (1) We got the following expression after all the above steps The squared norm of the estimated image An inner product between the noisy and the denoised images Sum over the “sensitivity” of our algorithm to perturbations in the input vector Our estimator is true up to an unknown constant

10. Wrap Up (2) Since we cannot compute the expectation, we will simply drop it with the hope that the summation over all the image pixels is sufficient to provide the desired accuracy  If you want to set the parameters, , do this while minimizing the above expression  This implies that the algorithm should be differentiable w.r.t. the input.

11. Example – Thresholding Algorithm + Lets come back to the global image denoising scheme by thresholding

12. Example – Smoothing Lets make sure that our estimator is differentiable by smoothing it (assume k is even)

13. Example - SURE Lets make sure that our estimator is differentiable by smoothing it (assume k is even)

14. Example - SURE We can simplify the last term Some Properties:

15. Example - SURE Bottom line: Does this work?