Hauptseminar Bildanalyse - Topic: Single-Image Noise Level ... · Zhu, Xiang and Peyman Milanfar (2010). “Automatic parameter selection for denoising algorithms using a no-reference

Introduction Noise level estimation based on PCA Patch selection Optimal noise parameter Results References

Hauptseminar BildanalyseTopic: Single-Image Noise Level Estimation for Blind Denoising

Michael Jobst

June 8, 2016

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The paperSingle-Image Noise Level Estimation for Blind Denoising

Tokyo Institute of Technologyin IEEE Transactions on Image Processing, Vol. 22, No.12,

December 2013

Xinhao Liu Masayuki Tanaka Masatoshi Okutomi



Questions

Single-Image Noise Level Estimation for Blind Denoising

▶ Single-Image:▶ Why?▶ Problems?

▶ Noise▶ What is noise (in the paper)?

▶ Level Estimation▶ What is the noise level?

▶ Blind Denoising▶ Blind vs. non-blind▶ Why?

▶ How to achieve the desired result?



BM3D denoising (Dabov et al., 2006)



Data representation

▶ input: grayscale image▶ M square patches of 49 pixels▶ image zi , i ∈ M: ground truth (noise free) image patches▶ image yi , i ∈ M: noisy image patches▶ gaussian noise vector ni , variance σ2n, zero-mean▶ yi = zi + ni



Principal Component Analysis▶ Covariance Matrix

Σy =1M

M∑i=1

yiyTi

Comparision: principal components of an image (a) and noise (b)



Principal Component Analysis▶ noise even in all dimensions and added to the image▶ idea: image data does not span the whole space▶ eigenvector of covariance matrix with minimum eigenvalue

represents noise levelλmin(Σy ) = λmin(Σz) + σ2n ∧ λmin(Σz) = 0 ⇔ σ2n = λmin(Σy )



Problem solved



Problem solved?



Patch energy

▶ gradient matrix of patch yi

Gyi =[Dhyi Dv yi

]▶ joint gradient matrix of patch yi

Cyi =GTyi Gyi

▶ trace xii (= sum of eigenvalues) of joint gradient matrix Cyi isthe patch energy

ξi = tr(Cyi )



Patch energy

▶ trace xii (= sum of eigenvalues) of joint gradient matrix Cyi isthe patch energy

ξi = tr(Cyi )



Threshold for patch selection▶ flat patch zf , added noise: yf = zf + n▶ gradient matrix Gyf only depends on the noise n, thus the

noise parameter σ2n▶ possible to derive texture strength of the bare noise ξ(n)▶ approximate ξ(n) by a gamma distribution▶ threshold τ is the 99.9999-th percentile (→ inverse Gamma

cumulative distribution)



Threshold for patch selection

τ = F −1(

δ,N22 ,

2N2 σ

2ntr(DTh Dh + DTv Dv )

)

▶ Problem: we use (the true) σ2n which we do not know▶ chicken-and-egg problem cries for an iterative approach



Iterative threshold calculation



Tests



Regression model for optimal noise parameter

▶ regression model with unknown parameter vector θ

σ̂′n = R(σ̂0n, σ̂n; θ)

▶ quadratic regression model

σ̂′n = a0 + a1σ̂n + a2σ̂0n + a3σ̂nσ̂0n + a4(σ̂0n)2 + a5(σ̂n)2 + ε

▶ generation of samples by brute force optimization▶ solve with least-squares



Experimental results



References I

Liu, X., M. Tanaka, and M. Okutomi (2013). “Single-Image NoiseLevel Estimation for Blind Denoising”. In: IEEE Transactions onImage Processing 22.12, pp. 5226–5237. issn: 1057-7149. doi:10.1109/TIP.2013.2283400.

Zhu, Xiang and Peyman Milanfar (2010). “Automatic parameterselection for denoising algorithms using a no-reference measureof image content”. In: Image Processing, IEEE Transactions on19.12, pp. 3116–3132.

Feng, XiaoGuang and Peyman Milanfar (2002). “Multiscaleprincipal components analysis for image local orientationestimation”. In: Signals, Systems and Computers, 2002.Conference Record of the Thirty-Sixth Asilomar Conference on.Vol. 1. IEEE, pp. 478–482.


http://dx.doi.org/10.1109/TIP.2013.2283400


References II

Dabov, Kostadin et al. (2006). “Image denoising withblock-matching and 3D filtering”. In: Electronic Imaging 2006.International Society for Optics and Photonics,pp. 606414–606414.

Burger, H. C., C. J. Schuler, and S. Harmeling (2012). “Imagedenoising: Can plain neural networks compete with BM3D?” In:Computer Vision and Pattern Recognition (CVPR), 2012 IEEEConference on, pp. 2392–2399. doi:10.1109/CVPR.2012.6247952.


http://dx.doi.org/10.1109/CVPR.2012.6247952


Thank you for your attention!Questions?


Gradient covariance matrix▶ gradient at point k = (xk , yk) (Zhu and Milanfar, 2010):

∇f (k) = ∇f (xk , yk) =[

δf (xk , yk)δx ,

δf (xk , yk)δy

]T=: [px (k), py (k)]T

▶ gradient matrix at point k (Feng and Milanfar, 2002):

Gwi =

...

∇f (k)T...

, k ∈ wi▶ joint gradient matrix:

Cwi = GTwi Gwi =[ ∑

k∈wi p2x (k)

∑k∈wi px (k)py (k)∑

k∈wi px (k)py (k)∑

k∈wi p2y (k)

]

Appendix I Michael Jobst Hauptseminar Bildanalyse

Threshold from Gamma distribution

▶ gamma distribution of ξ over the noise free patches:

ξ(n) ∼ Gamma(

N22 ,

2N2 σ

2ntr(DTh Dh + DTv Dv )

)

▶ confidence interval of δ = 1 − 10−6, threshold τ

P(0 < ξ(n) < τ) = δ

Appendix II Michael Jobst Hauptseminar Bildanalyse

IntroductionNoise level estimation based on PCAPatch selectionOptimal noise parameterResultsAppendix

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Hauptseminar Bildanalyse - Topic: Single-Image Noise Level ... · Zhu, Xiang and Peyman Milanfar (2010). “Automatic parameter selection for denoising algorithms using a no-reference