模型参数的快速估计
小波域HMT模型参数的快速 估计及其在图像降噪中的应用（六）
小波域HMT模型参数的快速估计
小波系数的分类
模型参数的快速估计
模型参数的快速估计
• 节点的状态概率 • 节点的状态转移概率 • 节点的方差
小波域HMT模型参数的快速 估计及其在图像降噪中的应用（七）
Gibbs效应的消除
小波域HMT模型参数的快速 估计及其在图像降噪中的应用（十一）
实验结果（PSNR：20.0107 28.2667）
小波域HMT模型参数的快速 估计及其在图像降噪中的应用（十二）
实验结果分析（PSNR：19.9862 25.3904）
小波域HMT模型参数的快速 估计及其在图像降噪中的应用（十三）
基于HMT模型的降噪算法
wˆ ( yi ) E wi yi , θ
m
P(Si
m
yi
,
θ
)
2 i,m
y
i
2 n
2 i,m
2 n
i,m
小波域HMT模型参数的快速 估计及其在图像降噪中的应用（六）
小波域HMT模型参数的快速估计
小波系数的分类
模型参数的快速估计
• 阈值选取
小波系数的分类
• 阈值分类
Wavelet domain method
y xn take wavelet transform both sides: Y X N, where Y , X , and N denote the wavelet transform of y, x, n respectively. Generally, people assume that the elements in Y and X are independent. Yi Xi Ni , where index i is the position of coefficient. coefficients of y, x, and n respectively. Bayesian framework in wavelet domain:
(Y ) arg min{ log P(Y | X ) log P( X )}
X
(Y )i arg min{ log P(Yi | Xi ) log P( Xi )}
Xi
Wavelet domain Wiener filtering
Assume that P( X i ) exp( X i2 / 2 )
自然图象小波变换的两个性质
非高斯分布特性：小波系数的边缘分 布呈现出非高斯特性，即呈现出“尖 峰长尾”的状态。
保持性：在空间同一位置，小波系数 “大”或“小”的状态具有在尺度间 传递的特性。
小波域HMT模型参数的快速 估计及其在图像降噪中的应用（二）
Crouse提出的小波域HMT模型
子带系数的高斯混合模型
Let
(
y)
sgn(
y)(| 0
y
|
)
| y |
otherwise
Theorem: for any estimates rule which satisfies | ( y) || y |,
then we have
E( ( y), x) E( ( y), x)
Adaptive wavelet domain
n2 .
Some results about uniform
shrinkage
y xu
where u is a nuisanse term which only satisfies
| u | 1.
E(x, x) sup || x x ||2
|u|1
where x satisfies: |x || x |
2 1 L
Xi2
1 L
Yi2 n2
For different scale, is different, compute the separately.
n is the variance of noise.
n
median(| HH 0.6745
|)
Wavelet domain uniform
x'
Given x ', y x ' n, P( y | x ') P(n)
( y) arg min{ log P(n) log P(x ')}
x'
Finding appropriate probability distribution expressions
P(n), P(x ') are the key points.
x ( y) x.
MAP framework
A MAP(maximize a posterior ) method to
define a estimator as follows:
( y) arg max P(x ' | y) such that y x ' n.
x'
From Bayesian formula: P(x ' | y) P( y | x ')P(x ') P( y)
Wavelet based image denoising (小波域图像去噪)
Peng Silong 2009.04.24
Outline
• Mathematical model of denoising and MAP framework
• Wavelet domain Wiener method • Wavelet domain uniform shrinkage
each position i, we use a adaptive threshold parameter .
i
n2 . (i)
(i)
max( 1 L
Yj2
jB(i )
n2, 0),
where
B(i)
is
a
neighbor
of i which have L points close to Yi.
shrinkage (Donoho)
Threshold method: Hard threshold:
(Y )i Y0i
| Yi |
otherwise
Soft threshold:
(Y )i
sgn(Yi
)(| 0
Yi
| )
(Y )i
Yi
0
Yi
Yi | Yi | Yi
| Yi |
P(Y | X ) P(N )
P(Ni ) exp(Ni2 / n2 )
Then
(Y )i arg min{ log P(Y | X ) log P( X )}
X
= arg min{(Yi X i )2 / n2 X i2 / 2 )
Xi
F(Xi
)
(Yi
Xi noising • Adaptive wavelet domain shrinkage
denoising • HMT based denoising method
Mathematical model
x is a ideal image or signal, y is the observation of x added by a noise n. y xn Denosing problem: find a estimator to give a estimate of x from y,
2
0
F X i
2( X i
Yi ) / n2
2Xi
/ 2
Xi
2 2 n2
Yi
Parameter estimating
In the Wiener filtering method, there are two parameters:
and n, for , we can see that it is a proper result that
otherwise
Parameter estimating
is a big problem, in Donoho's work, it is a global
parameter:
n 2 log M , where M is the number of points
considered. Another better choice:( by Chang, Yu, Vetterli)
shrinkage (Chang, Yu, Vetterli)
n2 , to avoid the global effect ( one part affects another
part), it will be better to use localized parameter, that is, in
Stationary wavelet transform A 3 level SWT filter bank
SWT filters
• 高斯尺度混合模型
其中：U为零均值高斯向量，Z是一个独立的正的乘量，X是高
斯向量的无穷混合。
尺度间模型
• 隐马尔可夫树（HMT）模型（Crouse1998） • 双变量模型（Sendur2002）
结合尺度内与尺度间的混合模型
• 条件高斯模型
小波域HMT模型参数的快速 估计及其在图像降噪中的应用（一）
since y is the observasion, P( y) is a constant,
( y) arg max P( y | x ')P(x ')
x'
P( y)