poisson denoising using sparse representations and ... · the noise power is measured by the peak...

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Motivation and Goals Poisson noise appears in many imaging applications such as fluorescence microscopy, computed tomography (CT), night vision, etc. We aim at recovering such noisy images. We propose an algorithm that relies on the Poisson noise statistics and sparsity-based image modeling, leading to state-of-the-art results {raja, elad}@cs.technion.ac.il Sparsity Based Reconstruction By maximizing the log-likelihood of the Poisson distribution we get the following minimization problem A regular sparsity prior leads to a non-negative optimization Instead, similar to [2], we use that leads to Note that [2] uses a GMM, which is different from our scheme D is a “given“ dictionary and counts the non-zero elements This problem is NP-hard and thus we seek an approximation The Poisson Denoising Problem x 0 and y - clean and noisy images The goal is to recover x 0 from y Each element y[i] in y is (an integer) i.i.d. Poisson distributed with mean and variance x 0 [i]: large x 0 [i] large y[i], small x 0 [i] small y[i] The noise power is measured by the peak value: maximal value in x 0 In scenarios of low noise/large peak, the Poisson noise looks like an additive i.i.d. Gaussian noise: x 0 y The Anscombe transform can be used to convert the Poisson noise into a Gaussian one, and then standard denoising methods can be applied In this work we deal with the case of high noise/small peak value (peak<4) x 0 y The Anscombe transform is no more effective and the Poisson noise has to be treated directly Image Peak/PSNR [dB] Peak = 0.1 Peak =0.2 Peak = 1 Peak =2 Saturn Ours 19.1 23.58 27.38 29.46 NLSPCA [2] 20.56 23 27 28.44 BM3D with Anscombe [3] 18.98 21.71 25.95 27.7 Flag Ours 14.04 16.28 22.38 25.18 NLSPCA [2] 14.45 16.44 20.37 21.03 BM3D with Anscombe [3] 12.88 14.23 18.51 20.95 House Ours 16.84 18.91 22.6 24.85 NLSPCA [2] 17.76 18.99 21.94 23.23 BM3D with Anscombe [3] 16.13 18.24 22.35 24.06 Swoosh Ours 21.84 23.82 31.18 33.59 NLSPCA [2] 19.16 23.81 30.71 32.9 BM3D with Anscombe [3] 19.66 21.09 26.35 30.17 0 0 0 [] | [] ! 0 m e P i m i m m y x 100 peak 0.1 peak Poisson Greedy Algorithm for Sparse Coding Input: Group of noisy patches such that their original clean versions have the same support in their representation with k non-zeros We find these support by a greedy procedure: Initialization: the support T={ }, t=0 For t=1:1:k Find new support element and representations: min log T T x 1x y x 0 min exp s.t. T T k α 1 D α y Dα α 0 min log s.t. , 0 T T k α 1D α y α 0 , k x D αα 0 exp , k x D α α 0 Our Sparse Poisson Denoising Algorithm Improvements Dictionary learning stage after we have representations for all the patches we do a Newton step for updating D Error based stopping criterion: add elements to the support till the error between the reconstructed patches and the patches of the reconstructed image stop decreasing (2) Gaussian filtering used only for the task of clustering the patches (1) Divide the image into set of overlapping patches (3) Cluster (using the Gaussian filtered image) the noisy patches into large number of small groups (4) Apply the Poisson greedy algorithm for each group assuming that its patches have the same non-zero locations (support) in their representations (5) Form the final image from the reconstructed patches by averaging ..… ..… ..… ..… ..… Recovery Performance References [1] R. Giryes and M Elad, Sparsity Based Poisson Denoising, in Proc. IEEEI’12, Eilat, Israel, Nov. 2012 [2] J. Salmon, Z. T. Harmany, C.-A. Deledalle, and R. Willett, Poisson noise reduction with non-local PCA, CoRR, vol. abs/1206.0338, 2012 [3] M. Makitalo and A. Foi, Optimal inversion of the Anscombe transformation in low-count Poisson image denoising, IEEE Transactions on Image Processing, vol. 20, no. 1, pp. 99109, Jan. 2011 1 ,..., l y y 1 1 1 1 ,..., , 1 ˆ ˆ ,..., , arg min 1 exp , t t t t l t t t T T l i i i T j T j j il j y D D exp ,1 i i l D α ..… ..… ..… ..… ..… ..… ..… ..… ..… ..… ..… A global dictionary D is used for all groups Noisy, peak =0.2 Recovered Original Noisy, peak =1 Recovered Poisson Denoising using Sparse Representations and Dictionary Learning Raja Giryes and Michael Elad The Computer Science Department, Technion Israel Institute of Technology

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Page 1: Poisson Denoising using Sparse Representations and ... · The noise power is measured by the peak value: maximal value in x 0 In scenarios of low noise/large peak, the Poisson noise

Motivation and Goals Poisson noise appears in many imaging applications

such as fluorescence microscopy, computed tomography (CT), night vision, etc.

We aim at recovering such noisy images.

We propose an algorithm that relies on the Poisson noise statistics and sparsity-based image modeling, leading to state-of-the-art results

{raja, elad}@cs.technion.ac.il

Sparsity Based Reconstruction By maximizing the log-likelihood of the Poisson distribution we get the

following minimization problem

A regular sparsity prior leads to a non-negative optimization

Instead, similar to [2], we use that leads to

Note that [2] uses a GMM, which is different from our scheme

D is a “given“ dictionary and counts the non-zero elements

This problem is NP-hard and thus we seek an approximation

The Poisson Denoising Problem

x0 and y - clean and noisy images

The goal is to recover x0 from y

Each element y[i] in y is (an integer) i.i.d. Poisson distributed with mean and variance x0 [i]:

large x0 [i] large y[i], small x0 [i] small y[i]

The noise power is measured by the peak value: maximal value in x0

In scenarios of low noise/large peak, the Poisson noise looks like an additive i.i.d. Gaussian noise:

x0 y

The Anscombe transform can be used to convert the Poisson noise into a Gaussian one, and then standard denoising methods can be applied

In this work we deal with the case of high noise/small peak value (peak<4)

x0 y

The Anscombe transform is no more effective and the Poisson noise has to be treated directly

Image Peak/PSNR [dB]

Peak = 0.1 Peak =0.2 Peak = 1 Peak =2

Saturn Ours 19.1 23.58 27.38 29.46

NLSPCA [2] 20.56 23 27 28.44

BM3D with Anscombe [3] 18.98 21.71 25.95 27.7

Flag Ours 14.04 16.28 22.38 25.18

NLSPCA [2] 14.45 16.44 20.37 21.03

BM3D with Anscombe [3] 12.88 14.23 18.51 20.95

House Ours 16.84 18.91 22.6 24.85

NLSPCA [2] 17.76 18.99 21.94 23.23

BM3D with Anscombe [3] 16.13 18.24 22.35 24.06

Swoosh Ours 21.84 23.82 31.18 33.59

NLSPCA [2] 19.16 23.81 30.71 32.9

BM3D with Anscombe [3] 19.66 21.09 26.35 30.17

0

0

0[ ] | [ ] !

0

m

eP i m i m

m

y x

100peak

0.1peak

Poisson Greedy Algorithm for Sparse Coding Input: Group of noisy patches such that their original clean versions have

the same support in their representation with k non-zeros

We find these support by a greedy procedure:

Initialization: the support T={ }, t=0

For t=1:1:k

Find new support element and representations:

min logT Tx

1 x y x

0

min exp s.t. T T k α

1 Dα y Dα α

0

min log s.t. , 0T T k α

1 Dα y Dα α Dα

0, k x Dα α

0

exp , k x Dα α

0

Our Sparse Poisson Denoising Algorithm

Improvements Dictionary learning stage – after we have representations for

all the patches we do a Newton step for updating D

Error based stopping criterion: add elements to the support till the error between the reconstructed patches and the patches of the reconstructed image stop decreasing

(2) Gaussian filtering –

used only for the

task of clustering

the patches

(1) Divide the

image into

set of

overlapping

patches

(3) Cluster (using the Gaussian

filtered image) the noisy

patches into large number

of small groups

(4) Apply the

Poisson greedy

algorithm for

each group

assuming that its

patches have the

same non-zero

locations

(support)

in their

representations (5) Form the final

image from the

reconstructed

patches by averaging

..… ..… ..…

..… ..…

Recovery Performance

References [1] R. Giryes and M Elad, Sparsity Based Poisson Denoising, in Proc. IEEEI’12, Eilat, Israel, Nov. 2012 [2] J. Salmon, Z. T. Harmany, C.-A. Deledalle, and R. Willett, Poisson noise reduction with non-local PCA,

CoRR, vol. abs/1206.0338, 2012 [3] M. Makitalo and A. Foi, Optimal inversion of the Anscombe transformation in low-count Poisson image

denoising, IEEE Transactions on Image Processing, vol. 20, no. 1, pp. 99–109, Jan. 2011

1,..., ly y

1 1

1

1,..., , 1

ˆ ˆ,..., , arg min 1 exp ,t tt t

l

t t t T T

l i i iT j T jj i l

j y

D D

exp ,1i i l Dα

..…

..…

..… ..…

..…

..…

..…

..… ..…

..…

..…

A global dictionary D

is used for all groups

Nois

y, p

eak

=0.2

R

eco

vere

d

Ori

ginal

N

ois

y, p

eak

=1

Reco

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d

Poisson Denoising using Sparse Representations and Dictionary Learning

Raja Giryes and Michael Elad

The Computer Science Department, Technion – Israel Institute of Technology