Image Denoising via Adaptive SoftThresholding Based on Non-local Samples
Hangfan Liu, Ruiqin Xiong, Jian Zhang, and Wen Gao
Institute of Digital Media, Peking University
CVPR2015
Outline
• This paper proposes a new image denoising method using
adaptive signal modeling and adaptive soft-thresholding:
①Content adaptive model is adopt to address the non-stationarity of the
images instead of global models with adaptive soft-thresholding .
②The distribution model of each patch is estimated individually with nonzero expectation.
③Non-local similarity is applied to estimate the parameters of the
distribution of each patch.
④The proposed approach outperforms BM3D and CSR.
Denoising Formulation
• The general degradation model:
y  xn
n
N  0,  n2 I 
• The denoising model with sparse prior:
x  arg min
x

2
y  x 2   xi
2
i
q
p
From Bayesian Viewpoint
• The model above can be written as below according to MAP
estimation
1
1
2
q
x  arg min 2 y  x 2 
xi p

2
x
2 n
2 i
Under this assumption, different bands are treated equally
with same weight, but different bands vary in statistical
characteristics.
For example, the variance of coefficients in low frequency
band is usually much more significant than that in a high
frequency band.
• The band adaptive scheme is proposed,
x  arg min
x
1
2 n2
xi
1
yx 2  
2 i i
q
2
p
i
is standard deviation vector of xi with the
k-th element being the standard deviation of
the k-th element in patch xi .
PSNR comparison for uniform
denoising and band adaptive
denoising. PCA is used as transform
domain for its signal adaptive and
near optimality for decorrelation.
Model the Coefficients
• Generalized Gaussian distribution is used
1

G  u,   
exp   u 
1


2 1  
 
• 9 images → 65000 groups → 60 similar
patches per group
• Apply PCA transform to each group.
k
k


• For the k-th band of the i-th group, i and i
are the mean and standard deviation of the
coefficients.
• The coefficients of this band are centralized
and variance normalized,
Cik :
Cik  ik
 ik
• After that, the coefficients of this band from
all the groups are gathered as samples and
form a variance normalized distribution.
• KL divergence is used to measure the fitting error
of GGD to determine the parameter .
• The optimal  falls near 1.3 for all the 3 bands,
here  is approximated to 1, which indicates the
optimal distribution of the coefficients is
Laplacian distribution.
AST-NLS
• For the patch yi , its similar patches is
searched non-locally according to,
d  i, j  
yi  y j
2
2
S2
and the M most similar patches is grouped
into Yi .
• For each Yi , a corresponding i is trained
using PCA transformation.
• The proposed objective function is
xi  i
1
2
x  arg min
yx 2 
2
x
i
2 2 n
i
1
i and i are the expectation vector and
standard deviation vector of xi , where the kth entry of them are the expectation and
standard deviation of coefficients in band k
respectively. These parameters are not
available in practice.
• The expectation i  k  is estimated as the
median value of the coefficients in the k-th
band because the coefficients are assumed to
conform to Laplacian distribution. In this way,
the influence of outliers are excluded.
• Further, since the noise is assumed to be i.i.d.
Gaussian distributed, the standard deviation
of coefficients in the k-th band  i  k  can be
estimated by

 i  k   max  yi  k    n2 ,0
2

Where  yi is the standard deviation vector
calculated by coefficients of observed noisy
image y
2
2
1
 yi    j   j
M jLi

Where
 j  i y j

j  Li
M is the number of similar patches in the
group.
Numerical Solutions
y  x 2  c1  yi  xi
• Because
2
2
2
i
 c1   i yi   i xi
2
2
i
 i   i
Hence the objective function is recast into,
  arg min 

i
c2

2
n
i   i  
2
2
i
 i  i
i
1
• Component wise soft-thresholding operation,


n
 i  i  max  i   i 
, 0   sgn i   i
c2   i 



• Then xi can be calculated by
xi  Ti i
• After obtaining all the patches, the estimated
image can be yielded by simply averaging


T
x    Ei Ei 
 i

1
 E x 
T
i i
i
• Besides, iterative regularization is used to
update the observed noisy image,

y : x   y  x

Then the y is used to regenerate x .
Experimental Results
Noisy
BM3D
LPG-PCA
CSR
AST-NLS
Parameter Selection