The document proposes a fast global image denoising algorithm based on a nonstationary gamma-normal statistical model. The algorithm effectively removes Gaussian and Poisson noise while satisfying constraints on computational cost to process large datasets with minimal user input. It develops a probabilistic data model and defines the joint prior distribution, leading to a Bayesian estimate of the hidden image field. The algorithm uses a Gauss-Seidel procedure on a trellis of neighborhood graphs to iteratively find optimal hidden variable values. Experimental results show the algorithm achieves similar denoising performance to other techniques but with significantly less computation time.

1. Fast Global Image Denoising Algorithm
on the Basis of Nonstationary Gamma-
Normal Statistical Model
Gracheva Inessa, gia1509@mail.ru,
Kopylov Andrey, and.kopylov@gmail.com,
Krasotkina Olga, O.V.Krasotkina@yandex.ru
Tula State University, Tula, Russia
AIST Conference, April 2015, Ekaterinburg

2. Formulation of the problem
What do we want?
 the method should effectively remove Gaussian noise as well as
Poissonian noise;
 the method should satisfy strict constraints in terms of computational
cost, so as to be able to process large data sets;
the algorithm required as less user input as possible in order to facilitate
its application and to enhance the reproducibility of its results.
To achieve these purpose a nonstationary gamma-normal noise model
has been proposed in the framework of Bayesian approach to the problem of
image processing. This model allows us to develop a fast global algorithm
on the basis of Gauss-Seudel procedure and Kalman filter-interpolator.

3. Image denoising
( , )tY y t T= ∈ ( , )tX x t T= ∈
1 2 1 1 2 2{ ( , ): 1,..., , 1,..., }T t t t t N t N= = = =
x X∈y Y∈
The analyzed
image
The processing
result

4. Probabilistic data model
The joint conditional probability density:
where is the variance of the observation noise.
The a priori joint distribution:
where is the proportionality coefficients to the variance
of the observation noise ; V is the neighborhood
graphs of image elements having the form of a lattice.
2
/2 /2
1 1
( | , ) exp( ( ) )
(2 ) 2
t tN N
t T
Y X y xδ
δ π δ ∈
Φ = − −∑
2
1/2
,( 1)/2
1 1 1
( | , ) exp ( )
2
(2 )
t t
t t V tn N
t
t T
X x xδ
δλ
δλ π
′ ′′
′ ′′∈−
∈
 
Ψ Λ µ × − − ÷
   
 ÷
 
∑
∏
δ=)( 2
teE
tλ
t tr λδ=
1 N1t1
1
N2
t2

5. Gamma - distribution
))/1(exp()/1(),|/1( 1
ttt λϑλϑαλγ α
−∝ −
2 1
2
(1/ | , , ) (1/ ) exp (1/ )
2
t t t
µ
δµ λ
γ λ δ λ µ λ λ
δµ
+
 
µ − ÷
 
2
1)1(
2)/1(,
1)1(
)/1(
λ
µδ
δµλ
λ
µδ
λ
++
=
++
= tt VarE2
(1/ ) , (1/ )t tE Var
α α
λ λ
ϑ ϑ
= =
1
1
( | , ) ( | , )
1 1
exp exp ( 1) ln
t
t T t T t
t
t T t T t Tt t
G
α
α ϑ γ λ α ϑ
λ
ϑ α λ ϑ
λ λ
−
∈ ∈
∈ ∈ ∈
 
Λ = µ × ÷
 
   
× − = − − − ÷  
   
∏ ∏
∑ ∑ ∑
( | , , )
1 1 1
exp ln ,
2
t
t T t
G δ λ µ
λ λ
δµ λ λ∈
Λ =
  
= − +  ÷
  
∑
where
1 1 1
1 1 ,
2 2
λ
α ϑ
δ µ δµ
  
= + + =  ÷
  
Gamma-distributed of the inverse factors :tλ/1
with mathematical expectations and variances:
We come to the a priori distribution density:

6. Normal gamma-distribution
So, have completely defined the joint prior normal gamma-distribution
of both hidden fields and
Coupled with the conditional density of the observable field, it makes
basis for Bayesian estimation of the field .
The joint a posteriori distribution of hidden elements, namely, those of
field and its instantaneous factors
( , )tX x t T= ∈ ( , )t t TλΛ = ∈
( , | , , ) ( | , ) ( | , , )H X X Gδ λ µ δ δ λ µΛ = Ψ Λ Λ
( , )tX x t T= ∈
( | , ) ( | , ) ( | , )
( , | , , , )
( '| ', ) ( '| , ) ( | ', ) ' '
X G Y X
P X Y
X G Y X dX d
δ α ϑ δ
δ α ϑ
δ α ϑ δ
Ψ Λ Λ Φ
Λ =
Ψ Λ Λ Φ Λ∫∫
( , )tX x t T= ∈ ( , )t t TλΛ = ∈

7. The Bayesian estimate of the
hidden random field
The Bayesian estimate of is the maximum point of the numerator
The conditionally optimal factors are
defined independently of each other:
The zero conditions for the derivatives, excluding the trivial solutions
lead to the equalities
( , )X Λ
,
2
2
' ''
', ''
( , | , ) arg min ( , | , , ),
( , | , , ) ( )
1
( ) / (1 1/ )ln .
X
t t
t T
t t t
t t V t
X J X Y
J X Y y x
x x
λ µ λ µ
λ µ
λ µ µ λ
λ
Λ
∈
∈

 Λ = Λ


Λ = − +

  
 + − + + +  
  
∑
∑
) )
,
2
1
2
' ''
', ''
( , | , ) arg min ( , | , , ),
( , | , , ) ( )
(1/ )( ) 1/
(1 1/ )ln .
1 1/
X
N
t t
t
t t
t t V
X J X Y
J X Y y x
x x
λ µ λ µ
λ µ
λ µ
µ
µ
Λ
=
∈

 Λ = Λ


Λ = − +

  − +
 + 
+  
∑
∑
) )
( , , ) [ ( , , ), ]tX X t Tλ µ λ λ µΛ = ∈
))
2
' ''
1
( , , ) arg min ( | , , ) : [( ) / ] (1 1/ )ln 0t t t
t t
X J X x xλ µ λ µ λ µ µ λ
λ λΛ
 ∂
Λ = Λ − + + + = 
∂  
)
,∞→tλ
2
' ''(1/ )( ) 1/
( , , )
1 1/
t t
t
x x
X
λ µ
λ λ µ λ
µ
− +
=
+
)

8. “The saturation effect”
“The saturation effect” of the unsmoothness penalty for sufficiently large
values of the parameter µ and fixed value λ =1.

9. The Gauss - Seudel procedure
Minimize of the objective function gives the approximation to the
estimate of the mutually agreed field starting with the initial values
, and new value of the hidden component gives the
solution of the conditional optimization problem
( , )k k
tX x t T= ∈
) )
2 2
' ''
', '' '
( , 1,..., ) arg min ( , | , , )
1
arg min ( ) ( ) .
k k k
X
t t t t
X t T t t V t
X x t N J X Y
y x x x
λ µ
λ∈ ∈
= = = Λ =
 
− + − 
 
∑ ∑
) )
1
2 2
' ''
', '' '
arg min ( , | , , )
1
arg min ( ) ( ) .
k k
k k
t t t t
t T t t V t
J X Y
y x x x
λ µ
λ
+
Λ
Λ ∈ ∈
Λ = Λ =
 
= − + − 
 
∑ ∑
) )
) ) )
0 0
( , )t t Tλ λΛ = = ∈
))

10. Approximation trellis
neighborhood graph sequence trees
For finding the values of the hidden field at each vertical row of the picture,
we use a separate pixel neighborhood tree which is defined, on the whole
pixel set and has the same horizontal branches as the others.
The resulting image processing procedure is aimed at finding optimal
values only for the hidden variables at the stem nodes in each tree.
The simplicity of such a succession of elementary trees allows for a
significant simplification of the optimization procedure.
1

11. Comparison of Poissonian and
Gaussian noise removal algorithms
Image
Peak intensity
Input PSNR
Peppers 256x256
120 60 30
23.92 20.92 17.91
Haar-Fisz 29.27 27.30 24.54
Platelet 29.07 27.44 25.73
PURE-LET 30.28 28.51 26.72
Our algorithm 29.53 28.65 25.77
Image
Peak intensity
Input PSNR
Einstein 512x512
120 60 30
8.34 14.35 19.23
FBF 24.86 26.21 29.63
SURE-LET 24.56 26.30 29.43
BM3D 25.03 26.59 29.99
Our algorithm 24.76 26.44 29.88
Gaussian noisePoissonian noise

12. Relative computation time of
various denoising techniques (sec)
Methods Image size
256x256 512x512
Platelet 856 1112
PURE-LET 4.6 8.2
BM3D 4.1 7.2
Haar-Fisz 1.3 3.2
FBF 0.5 1.8
SURE-LET 0.4 1.6
Our
algorithm
0.16 0.53
A plot of the time of the algorithm (s) on
the image size.

13. Experimental results
a) b)
c) d)
a)The original Einstein image. b)Noisy version with simulated Gaussian noise.
c)Denoised with our algorithm. d)Denoised with BM3D.

14. Experimental results
a)
b) c)
a)Part of the original MRI slice from the IGBMC Imaging Center data sets.
b)Denoised with our algorithm. c)Denoised with PURE-LET

15. THANKS FOR YOUR ATTENTION!
Acknowledgements
This research is funded by RFBR grant,13-07-00529.