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
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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