D ebruitage non-local Adaptation au type de bruit et aux structures - - PowerPoint PPT Presentation

Congr` es SMAI 2011, 5e Biennale Fran¸ caise des Math´ ematiques Appliqu´ ees

Guidel, Bretagne, 23-27 mai 2011

D´ ebruitage non-local Adaptation au type de bruit et aux structures de l’image

Charles Deledalle Collaborateurs: Lo¨ ıc Denis, Vincent Duval, Jospeh Salmon, Florence Tupin Institut Telecom, Telecom ParisTech, CNRS LTCI, Paris, France May 25, 2011

• C. Deledalle (Telecom ParisTech)

D´ ebruitage NL (SMAI 2011) May 25, 2011 1 / 19

Motivation

Noise: fluctuations which corrupt a signal or an image, Examples of noise in imagery:

Gaussian noise:

ex: optical imagery.

Poisson noise: due to low flux,

ex: optical imagery, microscopy, astronomy.

Speckle noise: due to coherent summation of random phasors

ex: SAR imagery, SONAR imagery, ultrasound imagery.

Signal dependent noise

Noise variance is a function of the true image, Generally modeled by non-Gaussian distributions.

Poisson distributions

• C. Deledalle (Telecom ParisTech)

D´ ebruitage NL (SMAI 2011) May 25, 2011 2 / 19

Overview of denoising approaches

Image denoising: find an estimation of the true image from the noisy image. How to denoise an image?

Three main approaches, Lots of hybrid methods.

Problems of non-local approaches

Designed for Gaussian noise, Adaptation to the local structures.

Sparsifying transforms (wavelets, dictionnaries) Variational / Markovian Approaches Non-local methods Noisy image with Poisson noise

• C. Deledalle (Telecom ParisTech)

D´ ebruitage NL (SMAI 2011) May 25, 2011 3 / 19

Overview of denoising approaches

Image denoising: find an estimation of the true image from the noisy image. How to denoise an image?

Three main approaches, Lots of hybrid methods.

Problems of non-local approaches

Designed for Gaussian noise, Adaptation to the local structures.

Sparsifying transforms (wavelets, dictionnaries) Variational / Markovian Approaches Non-local methods Noisy image with Poisson noise BM3D Non-local Total Variation

• C. Deledalle (Telecom ParisTech)

D´ ebruitage NL (SMAI 2011) May 25, 2011 3 / 19

Overview of denoising approaches

Image denoising: find an estimation of the true image from the noisy image. How to denoise an image?

Three main approaches, Lots of hybrid methods.

Problems of non-local approaches

Designed for Gaussian noise, Adaptation to the local structures.

Sparsifying transforms (wavelets, dictionnaries) Variational / Markovian Approaches Non-local methods Noisy image with Poisson noise BM3D Non-local Total Variation

• C. Deledalle (Telecom ParisTech)

D´ ebruitage NL (SMAI 2011) May 25, 2011 3 / 19

Table of contents

1

Limits of non-local filtering

2

Adaptation to the noise model

3

Adaptation to local image structures

• C. Deledalle (Telecom ParisTech)

D´ ebruitage NL (SMAI 2011) May 25, 2011 4 / 19

Table of contents

1

Limits of non-local filtering

2

Adaptation to the noise model

3

Adaptation to local image structures

• C. Deledalle (Telecom ParisTech)

D´ ebruitage NL (SMAI 2011) May 25, 2011 5 / 19

Non-local filtering

Non-local approach [Buades et al., 2005]

Local filters: loss of resolution, Non-local filers: data-driven adaptive weights, Weights are based on patch similarity.

• C. Deledalle (Telecom ParisTech)

D´ ebruitage NL (SMAI 2011) May 25, 2011 5 / 19

Non-local filtering

Non-local approach [Buades et al., 2005]

Local filters: loss of resolution, Non-local filers: data-driven adaptive weights, Weights are based on patch similarity.

• C. Deledalle (Telecom ParisTech)

D´ ebruitage NL (SMAI 2011) May 25, 2011 5 / 19

Non-local filtering

Non-local approach [Buades et al., 2005]

Local filters: loss of resolution, Non-local filers: data-driven adaptive weights, Weights are based on patch similarity.

• C. Deledalle (Telecom ParisTech)

D´ ebruitage NL (SMAI 2011) May 25, 2011 5 / 19

Non-local filtering – Limits of the squared differences

Non-local means [Buades et al., 2005]

Define weights from the squared differencess between patches 1 and 2: with s+b and t+b the b-th respective pixels in Bs and Bt.

• C. Deledalle (Telecom ParisTech)

D´ ebruitage NL (SMAI 2011) May 25, 2011 6 / 19

Non-local filtering – Limits of the squared differences

Non-local means [Buades et al., 2005]

Define weights from the squared differencess between patches 1 and 2: with s+b and t+b the b-th respective pixels in Bs and Bt. Beyond Gaussian noise? squared differences: adapted for Gaussian noise, Which criterion for non-Gaussian noise? How to choose the “optimal” parameters? Lo¨ ıc Denis Florence Tupin

• C. Deledalle (Telecom ParisTech)

D´ ebruitage NL (SMAI 2011) May 25, 2011 6 / 19

Non-local filtering – Limits of the squared differences

The rare patch effect

Around edges with high contrast, almost all weights can be zeros: with s+b and t+b the b-th respective pixels in Bs and Bt. The rare patch effect leads to a noise halo. Beyond the rare patch effect? Square patches non-adapted to heterogeneous area. How to use efficiently non square patches? How to choose the best patch shape? Vincent Duval Joseph Salmon

• C. Deledalle (Telecom ParisTech)

D´ ebruitage NL (SMAI 2011) May 25, 2011 7 / 19

Table of contents

1

Limits of non-local filtering

2

Adaptation to the noise model

3

Adaptation to local image structures

• C. Deledalle (Telecom ParisTech)

D´ ebruitage NL (SMAI 2011) May 25, 2011 8 / 19

Non-local estimation under non-Gaussian noise

Patch-similarities: how to replace the squared differences? [Deledalle et al., 2009]

Weights have to select pixels with close true values, Compare patches ⇔ test the hypotheses that patches have: H0 : same true values , H1 : independent true values .

[Deledalle et al., 2009] Deledalle, C., Denis, L., and Tupin, F. (2009). Iterative Weighted Maximum Likelihood Denoising with Probabilistic Patch-Based Weights. IEEE Transactions on Image Processing, 18(12):2661–2672.

• C. Deledalle (Telecom ParisTech)

D´ ebruitage NL (SMAI 2011) May 25, 2011 8 / 19

Non-local estimation under non-Gaussian noise

Patch-similarities: how to replace the squared differences? [Deledalle et al., 2009]

Weights have to select pixels with close true values, Compare patches ⇔ test the hypotheses that patches have: H0 : same true values , H1 : independent true values .

• C. Deledalle (Telecom ParisTech)

D´ ebruitage NL (SMAI 2011) May 25, 2011 8 / 19

Non-local estimation under non-Gaussian noise

Patch-similarities: how to replace the squared differences? [Deledalle et al., 2009]

Weights have to select pixels with close true values, Compare patches ⇔ test the hypotheses that patches have: H0 : same true values , H1 : independent true values . Next, how should we set the parameters α and β?

• C. Deledalle (Telecom ParisTech)

D´ ebruitage NL (SMAI 2011) May 25, 2011 8 / 19

Non-local estimation – Influence of the parameters

How to choose the parameters?

(trade-off noisy/pre-filtered)

• C. Deledalle (Telecom ParisTech)

D´ ebruitage NL (SMAI 2011) May 25, 2011 9 / 19

Non-local estimation – Influence of the parameters

Noisy Noisy Noisy Blurry Blurry Blurry Artifacts Artifacts

How to choose the parameters?

(trade-off noisy/pre-filtered)

Visually?

• C. Deledalle (Telecom ParisTech)

D´ ebruitage NL (SMAI 2011) May 25, 2011 9 / 19

Non-local estimation – Influence of the parameters

How to choose the parameters?

(trade-off noisy/pre-filtered)

Visually? Mean squared error (MSE)?

• C. Deledalle (Telecom ParisTech)

D´ ebruitage NL (SMAI 2011) May 25, 2011 9 / 19

Non-local estimation – Influence of the parameters

How to choose the parameters?

(trade-off noisy/pre-filtered)

Visually? Mean squared error (MSE)? How to estimate the MSE?

• C. Deledalle (Telecom ParisTech)

D´ ebruitage NL (SMAI 2011) May 25, 2011 9 / 19

Automatic setting of the denoising parameters

Unsupervised filtering

MSE estimators: unbiased risk estimators

Estimator Gaussian Poisson General SURE PURE

[Stein, 1981] [Chen, 1975]

Wavelet SUREshrink

[Donoho et al., 1995]

SURE-LET PURE-LET

[Blu et al., 2007] [Luisier et al., 2010]

NL means SURE based NL means Poisson NL means

[Van De Ville et al., 2009] [Deledalle et al., 2010a]

Local-SURE NL means

[Duval et al., 2010]

SURE: Stein’s Unbiased Risk Estimator PURE: Poisson Unbiased Risk Estimator [Deledalle et al., 2010a] Deledalle, C., Tupin, F., and Denis, L. (2010a). Poisson NL means: Unsupervised non local means for Poisson noise. In Image Processing (ICIP), 2010 17th IEEE International Conference on, pages 801–804. IEEE. Best student paper award

• C. Deledalle (Telecom ParisTech)

D´ ebruitage NL (SMAI 2011) May 25, 2011 10 / 19

Results on simulations

Noisy image NL Means Our method

(a) Gaussien +0.87 dB (b) Poisson +1.13 dB (c) Speckle +4.00 dB (d) Impuls. +3.82 dB

• C. Deledalle (Telecom ParisTech)

D´ ebruitage NL (SMAI 2011) May 25, 2011 11 / 19

Comparisons with Poisson noise on true data [Deledalle et al., 2010a]

(a) Noisy image

[Buades et al., 2005]

(b) NL means

[Le et al., 2007]

(c) Poisson-TV Cardiac mitochondrion, Confocal fluorescence microscopy, Image courtesy of Y. Tourneur.

[Luisier et al., 2010]

(e) PURE-LET

Our approach

(f) Poisson NL means

• C. Deledalle (Telecom ParisTech)

D´ ebruitage NL (SMAI 2011) May 25, 2011 12 / 19

Results on polarimetric SAR data [Deledalle et al., 2010b]

(Complex Wishart distributions)

(a) High-resolution noisy image

c DLR

(b) Our estimation (4096 × 4096: 2 min 10) [Deledalle et al., 2010b] Deledalle, C., Tupin, F., and Denis, L. (2010b). Polarimetric SAR estimation based on non-local means. In the proceedings of IGARSS, Honolulu, Hawaii, USA, July 2010.

• C. Deledalle (Telecom ParisTech)

D´ ebruitage NL (SMAI 2011) May 25, 2011 13 / 19

Table of contents

1

Limits of non-local filtering

2

Adaptation to the noise model

3

Adaptation to local image structures

• C. Deledalle (Telecom ParisTech)

D´ ebruitage NL (SMAI 2011) May 25, 2011 14 / 19

Adaptation to local image structures

The rare patch effect depends on the patch size/shape:

Choose different size and shapes, Calculate efficiently each associated estimate, Combine properly the different estimates.

• C. Deledalle (Telecom ParisTech)

D´ ebruitage NL (SMAI 2011) May 25, 2011 14 / 19

Efficient calculation based on FFT

Squared differences for general patch shapes

If candidates x′ = x + δ, then the squared differences between the two general patches at x and x′ is given by:

• τ∈Ω

S(τ)(v(x + τ) − v(x + δ + τ))2

• Discrete convolution product

For all displacements δ

Calculate the squared differences: ∀x, ∆δ(x) = (v(x) − v(x + δ))2 Convolve ∆δ by the shape S(−τ) using Fast Fourier Transform (FFT) Deduce the weights associated to all candidates x′ = x + δ.

Complexity: search window size × image size × log(image size)

[Deledalle et al., 2011b] Deledalle, C., Duval, V., and Salmon, J. (2011b). Non-local methods with shape-adaptive patches (NLM-SAP). Journal of Mathematical Imaging and Vision.

• C. Deledalle (Telecom ParisTech)

D´ ebruitage NL (SMAI 2011) May 25, 2011 15 / 19

Proper combination

Estimate the local MSE associated to all estimators ⇒ SURE has too large variance

Noisy SURE

[Deledalle et al., 2011b] Deledalle, C., Duval, V., and Salmon, J. (2011b). Non-local methods with shape-adaptive patches (NLM-SAP). Journal of Mathematical Imaging and Vision.

• C. Deledalle (Telecom ParisTech)

D´ ebruitage NL (SMAI 2011) May 25, 2011 16 / 19

Proper combination

Estimate the local MSE associated to all estimators ⇒ SURE has too large variance

Noisy SURE

[Deledalle et al., 2011b] Deledalle, C., Duval, V., and Salmon, J. (2011b). Non-local methods with shape-adaptive patches (NLM-SAP). Journal of Mathematical Imaging and Vision.

• C. Deledalle (Telecom ParisTech)

D´ ebruitage NL (SMAI 2011) May 25, 2011 16 / 19

Proper combination

Estimate the local MSE associated to all estimators ⇒ SURE has too large variance

Noisy SURE

[Deledalle et al., 2011b] Deledalle, C., Duval, V., and Salmon, J. (2011b). Non-local methods with shape-adaptive patches (NLM-SAP). Journal of Mathematical Imaging and Vision.

• C. Deledalle (Telecom ParisTech)

D´ ebruitage NL (SMAI 2011) May 25, 2011 16 / 19

Proper combination

Estimate the local MSE associated to all estimators ⇒ SURE has too large variance

Noisy SURE

[Deledalle et al., 2011b] Deledalle, C., Duval, V., and Salmon, J. (2011b). Non-local methods with shape-adaptive patches (NLM-SAP). Journal of Mathematical Imaging and Vision.

• C. Deledalle (Telecom ParisTech)

D´ ebruitage NL (SMAI 2011) May 25, 2011 16 / 19

Proper combination

Estimate the local MSE associated to all estimators ⇒ SURE has too large variance Regularize the estimation (e.g. with Yaroslavsky filter)

Regularized SURE

[Deledalle et al., 2011b] Deledalle, C., Duval, V., and Salmon, J. (2011b). Non-local methods with shape-adaptive patches (NLM-SAP). Journal of Mathematical Imaging and Vision.

• C. Deledalle (Telecom ParisTech)

D´ ebruitage NL (SMAI 2011) May 25, 2011 16 / 19

Proper combination

Estimate the local MSE associated to all estimators ⇒ SURE has too large variance Regularize the estimation (e.g. with Yaroslavsky filter) Combine the estimates using a convex aggregation

Regularized SURE

e.g. the Exponential Weighted Aggregation [Leung and Barron, 2006]

[Deledalle et al., 2011b] Deledalle, C., Duval, V., and Salmon, J. (2011b). Non-local methods with shape-adaptive patches (NLM-SAP). Journal of Mathematical Imaging and Vision.

• C. Deledalle (Telecom ParisTech)

D´ ebruitage NL (SMAI 2011) May 25, 2011 16 / 19

Analysis of local choices [Deledalle et al., 2011b]

Yaroslavsky

(a) 15 pie sizes/shapes

Anisotropic diffusion

(b) Regularized SURE (c) Patch orientations

• C. Deledalle (Telecom ParisTech)

D´ ebruitage NL (SMAI 2011) May 25, 2011 17 / 19

Results and comparisons [Deledalle et al., 2011b]

(a) NL Means (b) BM3D [Dabov et al., 2007] (c) BM3D [Dabov et al., 2007] (d) [Goossens et al., 2008] (e) Our approach (f) Our approach

30 seconds on 256 × 256 images with the online Matlab implementation

• C. Deledalle (Telecom ParisTech)

D´ ebruitage NL (SMAI 2011) May 25, 2011 18 / 19

Conclusion and perspectives

Conclusions

Noise adaptation: patch similarity is linked to detection theory, Choose global parameters: use risk estimation (SURE, PURE...). Adaptation to local structures: patch shape/size varies inside images Choose local best shape/size: use regularized risk map (SURE, PURE...)

Perspectives

Mix both methods, Extend to other problems using patches:

Change detection, Stereo vision, Object tracking.

0.2 0.4 0.6 0.8 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Probability of false alarm Probability of detection Poisson LB LG QB QG KB S N

Parties craquelées 40 cm/jour 20 Parties craquelées Parties craquelées Chutes de sérac de Lognan

• C. Deledalle (Telecom ParisTech)

D´ ebruitage NL (SMAI 2011) May 25, 2011 19 / 19

Conclusion and perspectives

Questions?

deledalle@telecom-paristech.fr

http://perso.telecom-paristech.fr/~deledall/

→ More details, articles and pieces of software available.

• C. Deledalle (Telecom ParisTech)

D´ ebruitage NL (SMAI 2011) May 25, 2011 19 / 19

[Buades et al., 2005] Buades, A., Coll, B., and Morel, J. (2005). A Non-Local Algorithm for Image Denoising. Computer Vision and Pattern Recognition, 2005. CVPR 2005. IEEE Computer Society Conference on, 2. [Chen, 1975] Chen, L. (1975). Poisson approximation for dependent trials. The Annals of Probability, 3(3):534–545. [Dabov et al., 2007] Dabov, K., Foi, A., Katkovnik, V., and Egiazarian, K. (2007). Image denoising by sparse 3-D transform-domain collaborative filtering. IEEE Transactions on image processing, 16(8):2080. [Deledalle et al., 2009] Deledalle, C., Denis, L., and Tupin, F. (2009). Iterative Weighted Maximum Likelihood Denoising with Probabilistic Patch-Based Weights. IEEE Transactions on Image Processing, 18(12):2661–2672. [Deledalle et al., 2011a] Deledalle, C., Denis, L., and Tupin, F. (2011a). NL-InSAR : Non-Local Interferogram Estimation. IEEE Transactions on Geoscience and Remote Sensing, 49(4):1441–1452. [Deledalle et al., 2011b] Deledalle, C., Duval, V., and Salmon, J. (2011b). Non-local methods with shape-adaptive patches (nlm-sap). Journal of Mathematical Imaging and Vision. [Deledalle et al., 2010a] Deledalle, C., Tupin, F., and Denis, L. (2010a). Poisson NL means: Unsupervised non local means for Poisson noise. In Image Processing (ICIP), 2010 17th IEEE International Conference on, pages 801–804. IEEE.

• C. Deledalle (Telecom ParisTech)

D´ ebruitage NL (SMAI 2011) May 25, 2011 20 / 19

[Deledalle et al., 2010b] Deledalle, C., Tupin, F., and Denis, L. (2010b). Polarimetric SAR estimation based on non-local means. In the proceedings of IGARSS, Honolulu, Hawaii, USA, July 2010 (accepted for publication). [Goossens et al., 2008] Goossens, B., Luong, H., Piˇ zurica, A., and Philips, W. (2008). An improved non-local denoising algorithm. In Proc. Int. Workshop on Local and Non-Local Approximation in Image Processing (LNLA’2008), Lausanne, Switzerland. [Kay, 1998] Kay, S. (1998). Fundamentals of Statistical signal processing, Volume 2: Detection theory. Prentice Hall PTR. [Le et al., 2007] Le, T., Chartrand, R., and Asaki, T. (2007). A variational approach to reconstructing images corrupted by Poisson noise.

• J. of Math. Imaging and Vision, 27(3):257–263.

[Leung and Barron, 2006] Leung, G. and Barron, A. R. (2006). Information theory and mixing least-squares regressions. 52(8):3396–3410. [Luisier et al., 2010] Luisier, F., Vonesch, C., Blu, T., and Unser, M. (2010). Fast interscale wavelet denoising of Poisson-corrupted images. Signal Processing, 90(2):415–427. [Stein, 1981] Stein, C. (1981). Estimation of the mean of a multivariate normal distribution. The Annals of Statistics, pages 1135–1151.

• C. Deledalle (Telecom ParisTech)

D´ ebruitage NL (SMAI 2011) May 25, 2011 21 / 19

[Van De Ville and Kocher, 2009] Van De Ville, D. and Kocher, M. (2009). SURE-Based Non-Local Means. IEEE Signal Processing Letters, 16(11):973–976.

• C. Deledalle (Telecom ParisTech)

D´ ebruitage NL (SMAI 2011) May 25, 2011 22 / 19

Results on interferometric SAR data [Deledalle et al., 2011a]

(Circular complex Gaussian distribution)

Noisy channels Our estimation

c ONERA c CNES

[Deledalle et al., 2011a] Deledalle, C., Denis, L., and Tupin, F. (2011a). NL-InSAR : Non-Local Interferogram Estimation. IEEE Transactions on Geoscience and Remote Sensing, 49(4):1441–1452.

• C. Deledalle (Telecom ParisTech)

D´ ebruitage NL (SMAI 2011) May 25, 2011 23 / 19

Results on polarimetric SAR data [Deledalle et al., 2010b]

(Wishart distributions)

(a) Low-resolution noisy image

c NASA

(b) Our estimation [Deledalle et al., 2010b] Deledalle, C., Tupin, F., and Denis, L. (2010b). Polarimetric SAR estimation based on non-local means. In the proceedings of IGARSS, Honolulu, Hawaii, USA, July 2010.

• C. Deledalle (Telecom ParisTech)

D´ ebruitage NL (SMAI 2011) May 25, 2011 24 / 19

Non-local estimation under non-Gaussian noise

Patch-similarities: how to replace the squared differences? [Deledalle et al., 2009]

Weights have to select pixels with close true values, Compare patches ⇔ test the hypotheses that patches have: H0 : same true values , H1 : independent true values .

[Deledalle et al., 2009] Deledalle, C., Denis, L., and Tupin, F. (2009). Iterative Weighted Maximum Likelihood Denoising with Probabilistic Patch-Based Weights. IEEE Transactions on Image Processing, 18(12):2661–2672.

• C. Deledalle (Telecom ParisTech)

D´ ebruitage NL (SMAI 2011) May 25, 2011 25 / 19

Non-local estimation under non-Gaussian noise

Patch-similarities: how to replace the squared differences? [Deledalle et al., 2009]

Weights have to select pixels with close true values, Compare patches ⇔ test the hypotheses that patches have: H0 : same true values , H1 : independent true values .

• 1. Similarity between noisy patches

Based on detection theory, we propose to evaluate the generalized likelihood ratio (GLR) of both hypotheses given the noisy patches [Kay, 1998]. → For speckle noise, we obtain the following criterion: − log GLR(v1, v2) = 2 log v1 v2 + v1 v2

• − 2 log 2

→ For Poisson noise, we obtain the following criterion: − log GLR(k1, k2) = k1 log k1 + k2 log k2 − (k1 + k2) log k1 + k2 2

• .
• C. Deledalle (Telecom ParisTech)

D´ ebruitage NL (SMAI 2011) May 25, 2011 25 / 19

Non-local estimation under non-Gaussian noise

Patch-similarities: how to replace the squared differences? [Deledalle et al., 2009]

Weights have to select pixels with close true values, Compare patches ⇔ test the hypotheses that patches have: H0 : same true values , H1 : independent true values .

• 2. Similarity between pre-filtered patches

We propose to refine weights by using the similarity between pre-filtered patches.

Idea motivated by [Polzehl et al., 2006, Brox et al., 2007, Goossens et al., 2008, Louchet et al., 2008]

A statistical test for the hypothesis H0 can be given by the symmetrical Kullback-Leibler divergence: → For Poisson noise, we obtain the following criterion: DKL(ˆ θ1ˆ θ2) =

• ˆ

θ1 − ˆ θ2

• log

ˆ θ1 ˆ θ2 .

• C. Deledalle (Telecom ParisTech)

D´ ebruitage NL (SMAI 2011) May 25, 2011 25 / 19

Non-local estimation under non-Gaussian noise

Patch-similarities: how to replace the squared differences? [Deledalle et al., 2009]

Weights have to select pixels with close true values, Compare patches ⇔ test the hypotheses that patches have: H0 : same true values , H1 : independent true values .

• C. Deledalle (Telecom ParisTech)

D´ ebruitage NL (SMAI 2011) May 25, 2011 25 / 19

Non-local estimation under non-Gaussian noise

Patch-similarities: how to replace the squared differences? [Deledalle et al., 2009]

Weights have to select pixels with close true values, Compare patches ⇔ test the hypotheses that patches have: H0 : same true values , H1 : independent true values . Next, how should we set the parameters α and β?

• C. Deledalle (Telecom ParisTech)

D´ ebruitage NL (SMAI 2011) May 25, 2011 25 / 19

Automatic setting of the denoising parameters

PURE in Poisson NL means

Based on the same ideas as SURE based NL means: PURE is obtained in closed-form for Poisson NL means, with almost same computation time.

Selection of the parameters

Optimum α and β obtained iteratively using Newton’s method:

• αn+1

βn+1

• =
• αn

βn

• − H−1∇

with H the Hessian and ∇ the gradient. The first and second order differentials of PURE are also obtained in closed-forms.

Find the best denoising level using similarities of noisy and pre-filtered patches!

20 25 30 35 40 45 Risk MSE PURE −10 −8 −6 −4 −2 2 Variations of the risk ∂ MSE / ∂ α ∂ PURE / ∂ α Zero 5 10 15 20 2 4 6 α Slope variations of the risk ∂2 MSE / ∂ α2 ∂2 PURE / ∂ α2 Zero MSE PURE ∂ MSE / ∂ β ∂ PURE / ∂ β Zero 5 10 15 20 β ∂2 MSE / ∂ β2 ∂2 PURE / ∂ β2 Zero

MSE and PURE and their first and sec-

• nd order variations with respect to the

parameters α and β

• C. Deledalle (Telecom ParisTech)

D´ ebruitage NL (SMAI 2011) May 25, 2011 26 / 19

Automatic setting of parameters

Risk minimisation

Choose the parameters α et β minimising the mean squared error (MSE): E 1 N λ − ˆ λ2

• = 1

N

• s
• λ2

s + E

• ˆ

λ2

s

• − E
• λsˆ

λs

• s λ2

s idependent of the paramaters,

• s E
• ˆ

λ2

s

• can be estimated from ˆ

λ, How to estimate

s E

• λs ˆ

λs

• ?

Poisson unbiased risk estimator (PURE) [Chen, 1975, Luisier et al., 2010]

If k is damaged by Poisson noise and ˆ λ = h(k) then E

• λsˆ

λs

• = E
• ksλs
• with λ = h(k) and k defined by kt =

kt − 1 if t = s kt

• therwise

PURE is given by: R(ˆ λ) = 1 N

• s
• λ2

s + ˆ

λ2

s − 2ksλs

• .
• C. Deledalle (Telecom ParisTech)

D´ ebruitage NL (SMAI 2011) May 25, 2011 27 / 19

Automatic setting of parameters

Risk minimisation

Choose the parameters α et β minimising the mean squared error (MSE): E 1 N λ − ˆ λ2

• = 1

N

• s
• λ2

s + E

• ˆ

λ2

s

• − E
• λsˆ

λs

• s λ2

s idependent of the paramaters,

• s E
• ˆ

λ2

s

• can be estimated from ˆ

λ, How to estimate

s E

• λs ˆ

λs

• ?

PURE - Proof

Let be k a r.v. following a Poisson distribution and h(.) a function: E [kh(k − 1)] =

∞

• k=1

kh(k − 1)λke−λ k! = λ

∞

• k=1

h(k − 1)λk−1e−λ (k − 1)! = E [λh(k)]

• C. Deledalle (Telecom ParisTech)

D´ ebruitage NL (SMAI 2011) May 25, 2011 27 / 19

Automatic setting of parameters

PURE in Poisson NL means

[Van De Ville and Kocher, 2009] use an estimator of the risk for NL means and Gaussian noise. Based on the same ideas: We obtained PURE in closed-form, with almost same computation time.

Selection of the parameters

We propose to optimize α and β iteratively using Newton’s method:

• αn+1

βn+1

• =
• αn

βn

• − H−1∇

with H the Hessian and ∇ the gradiant. The first and second order differentials of PURE are also obtained in closed-forms.

20 25 30 35 40 45 Risk MSE PURE −10 −8 −6 −4 −2 2 Variations of the risk ∂ MSE / ∂ α ∂ PURE / ∂ α Zero 5 10 15 20 2 4 6 α Slope variations of the risk ∂2 MSE / ∂ α2 ∂2 PURE / ∂ α2 Zero MSE PURE ∂ MSE / ∂ β ∂ PURE / ∂ β Zero 5 10 15 20 β ∂2 MSE / ∂ β2 ∂2 PURE / ∂ β2 Zero

MSE and PURE and their first and sec-

• nd order variations with respect to the

parameters α and β

• C. Deledalle (Telecom ParisTech)

D´ ebruitage NL (SMAI 2011) May 25, 2011 28 / 19

Numerical results - PSNRs

Peppers (256 × 256) Noisy 3.14 13.14 17.91 23.92 MA filter 19.20 20.93 21.11 21.16 PURE-LET [Luisier et al., 2010] 19.33 24.29 27.27 30.79 NL means [Buades et al., 2005] 18.12 23.33 26.98 30.64 Poisson NL means 19.90 25.32 28.07 31.06 αopt (209) (13.6) (10.05) (9.21) βopt (0.72) (1.31) (2.76) (7.64) #iterations (13.5) (8.02) (7.03) (6.90) Cameraman (256 × 256) Noisy 3.28 13.27 18.03 24.05 MA filter 18.71 20.15 20.29 20.33 PURE-LET [Luisier et al., 2010] 19.67 24.32 26.87 30.36 NL means [Buades et al., 2005] 18.17 23.53 26.77 29.39 Poisson NL means 19.89 25.07 27.42 29.47 αopt (62.1) (9.48) (8.81) (7.34) βopt (0.51) (1.19) (3.57) (16.19) #iterations (11.0) (6.80) (7.60) (11.3) PSNR values averaged over ten realisations using different methods on images damaged by Poisson noise with different levels of degradation. The averaged optimal parameters and the averaged number of iterations of the proposed Poisson NL means are given.

• C. Deledalle (Telecom ParisTech)

D´ ebruitage NL (SMAI 2011) May 25, 2011 29 / 19

Conclusion

• C. Deledalle (Telecom ParisTech)

D´ ebruitage NL (SMAI 2011) May 25, 2011 30 / 19

Conclusion

• C. Deledalle (Telecom ParisTech)

D´ ebruitage NL (SMAI 2011) May 25, 2011 30 / 19

Conclusion

• C. Deledalle (Telecom ParisTech)

D´ ebruitage NL (SMAI 2011) May 25, 2011 30 / 19