参考文献Chatzis S. P., Varvarigou T. A. A Fuzzy Clustering Approach Toward Hidden Markov Random Field Models for Enhanced Spatially Constrained Image Segmentation. IEEE Transactions on Fuzzy Systems 16(5) (2008) 1351-1361
随机场是什么？打一个很简单的比喻:现在有一个农民，家里有几块地，开春了，要开始种庄稼，每一块地里种什么庄稼，都具有一个概率，所有位置的概率构成了随机场。
下面给出数学表达。对于图像分割，实际上就是给每个像素打标签，标签表明了每个像素被分进了哪个类别。假设想要将10个像素分成3个类，每个像素只能分进1类中。
Q
Q
Q
Q
=
{
1
,
2
,
3
}
Q=\{1,2,3\}
Q = { 1 , 2 , 3 }
S
=
{
1
,
2
,
.
.
.
,
10
}
S=\{1,2,...,10\}
S = { 1 , 2 , . . . , 1 0 }
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j ∈ S
x
j
∈
Q
x_j\in Q
x j ∈ Q
x
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x j
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p
(
x
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>
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p(x_j)>0
p ( x j ) > 0
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j
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∂
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∂
p
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x
j
)
=
p
(
x
j
∣
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∂
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)
p(x_j)=p(x_j|X_{\partial_j})
p ( x j ) = p ( x j ∣ X ∂ j )
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∂
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X_{\partial_j}
X ∂ j
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j
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p
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p
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exp
(
−
U
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x
j
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)
Z
p(x_j)=p(x_j|X_{\partial_j})=\frac{\exp(-U(x_j))}{Z}
p ( x j ) = p ( x j ∣ X ∂ j ) = Z exp ( − U ( x j ) )
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Z
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p
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p(x_j)
p ( x j )
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(
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U(x_j)
U ( x j )
U
(
x
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∑
C
V
C
(
x
j
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U(x_j)=\sum_CV_C(x_j)
U ( x j ) = C ∑ V C ( x j )
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C
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x
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V_C(x_j)
V C ( x j )
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−
∑
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V
C
(
x
j
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p(x_j)=p(x_j|X_{\partial_j})=\frac{\exp(-\sum_{j\in C}V_C(x_j))}{\sum_{x_j=1}^3\exp(-\sum_{j\in C}V_C(x_j))}
p ( x j ) = p ( x j ∣ X ∂ j ) = ∑ x j = 1 3 exp ( − ∑ j ∈ C V C ( x j ) ) exp ( − ∑ j ∈ C V C ( x j ) )
然而，实际生活经验告诉我们，将所有像素一视同仁随便分类显然不科学，不同的像素需要根据其灰度值来进行分类，OK，现在引入另一个条件，像素灰度值。
R
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{
0
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2
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3
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4
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5
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.
.
.
,
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R = { 0 , 1 , 2 , 3 , 4 , 5 , . . . , 2 5 5 }
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p(y_j,x_j)
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p(y_j,x_j)=p(y_j|x_j)p(x_j)
p ( y j , x j ) = p ( y j ∣ x j ) p ( x j )
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x
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μ
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σ
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p(y_j|x_j)=p(y_j|x_j;\theta_{x_j})=N_{x_j}(\mu_{x_j},\sigma_{x_j})
p ( y j ∣ x j ) = p ( y j ∣ x j ; θ x j ) = N x j ( μ x j , σ x j )
μ
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和
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μ x j 和 σ x j
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OK，现在将隐Markov随机场引入FCM中
首先，该算法利用了Ichihashi等人提出的一种基于KL信息惩罚的FCM方法，目标函数如下
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ln
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J=\sum_{i=1}^3\sum_{j=1}^{10}u_{ij}d_{ij}+\lambda\sum_{i=1}^3\sum_{j=1}^{10}u_{ij}\ln(\frac{u_{ij}}{\pi_{ij}})
J = i = 1 ∑ 3 j = 1 ∑ 1 0 u i j d i j + λ i = 1 ∑ 3 j = 1 ∑ 1 0 u i j ln ( π i j u i j )
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\pi_{ij}=p(x_j)=p(x_j=i|X_{\partial_j})=\frac{\exp(-\sum_{j\in C}V_C(a_{ij}))}{\sum_{h=j}^3\exp(-\sum_{j\in C}V_C(a_{hj}))}
π i j = p ( x j ) = p ( x j = i ∣ X ∂ j ) = ∑ h = j 3 exp ( − ∑ j ∈ C V C ( a h j ) ) exp ( − ∑ j ∈ C V C ( a i j ) )
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U(x_j)=-\beta\sum_{j=1}^{10}\sum_{l=1}^8\delta(x_j-x_l)
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\pi_{ij}=p(x_j)=p(x_j=i|X_{\partial_j})=\frac{\beta\exp(-\sum_{l\in \partial_j}\delta(i-x_l))}{\sum_{h=1}^3\beta\exp(-\sum_{l\in \partial_j}\delta(h-x_l))}
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\delta(x_j,x_l)= \begin{cases} 1,x_j=x_l \\ 0, otherwise \end{cases}
δ ( x j , x l ) = { 1 , x j = x l 0 , o t h e r w i s e
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r_{ij}=\frac{\pi_{ij}\exp(-\frac{1}{\lambda}d_{ij})}{\sum_{h=1}^3\pi_{ij}\exp(-\frac{1}{\lambda}d_{ij})}，d_{ij}=\frac{w}{2}\ln(2\pi)+\frac{1}{2}\ln(|\mu_i|)+\frac{(y_j-\mu_i)^2}{2\mu_i}
r i j = ∑ h = 1 3 π i j exp ( − λ 1 d i j ) π i j exp ( − λ 1 d i j ) ， d i j = 2 w ln ( 2 π ) + 2 1 ln ( ∣ μ i ∣ ) + 2 μ i ( y j − μ i ) 2
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i
=
∑
j
=
1
10
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i
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y
j
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10
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\mu_i=\frac{\sum_{j=1}^{10}r_{ij}y_j}{\sum_{j=1}^{10}y_j}
μ i = ∑ j = 1 1 0 y j ∑ j = 1 1 0 r i j y j
σ
i
=
∑
j
=
1
10
r
i
j
(
y
j
−
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=
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\sigma_i=\frac{\sum_{j=1}^{10}r_{ij}(y_j-\mu_i)^2}{\sum_{j=1}^{10}r_{ij}}
σ i = ∑ j = 1 1 0 r i j ∑ j = 1 1 0 r i j ( y j − μ i ) 2
r
i
j
r_{ij}
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x
j
=
arg max
i
{
r
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j
}
x_j=\argmax_i\{r_{ij}\}
x j = i a r g m a x { r i j }
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r
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x
j
=
arg max
i
{
r
i
j
}
x_j=\argmax_i\{r_{ij}\}
x j = i a r g m a x { r i j }
