參考文獻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
隨機場是什麼？打一個很簡單的比喻:現在有一個農民，家裏有幾塊地，開春了，要開始種莊稼，每一塊地裏種什麼莊稼，都具有一個概率，所有位置的概率構成了隨機場。
下面給出數學表達。對於圖像分割，實際上就是給每個像素打標籤，標籤表明了每個像素被分進了哪個類別。假設想要將10個像素分成3個類，每個像素只能分進1類中。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 } j ∈ S j\in S j ∈ S x j ∈ Q x_j\in Q x j ∈ Q x j x_j x j j j j p ( x j ) > 0 p(x_j)>0 p ( x j ) > 0 j j j ∂ \partial ∂ p ( x j ) = p ( x j ∣ X ∂ j ) p(x_j)=p(x_j|X_{\partial_j}) p ( x j ) = p ( x j ∣ X ∂ j ) X ∂ j X_{\partial_j} X ∂ j j j j j j j p ( x j ) = p ( x j ∣ X ∂ j ) = exp ( − U ( x j ) ) 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 ) ) Z Z Z p ( x j ) p(x_j) p ( x j ) U ( x j ) U(x_j) U ( x j ) U ( x j ) = ∑ C V C ( x j ) U(x_j)=\sum_CV_C(x_j) U ( x j ) = C ∑ V C ( x j ) V C ( x j ) V_C(x_j) V C ( x j ) j j j Z Z Z p ( x j ) = p ( x j ∣ X ∂ j ) = exp ( − ∑ j ∈ C V C ( x j ) ) ∑ x j = 1 3 exp ( − ∑ j ∈ C V C ( x j ) ) 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，實際生活經驗告訴我們，將所有像素一視同仁隨便分類顯然不科學，不同的像素需要根據其灰度值來進行分類，OK，現在引入另一個條件，像素灰度值。R R R R = { 0 , 1 , 2 , 3 , 4 , 5 , . . . , 255 } R=\{0,1,2,3,4,5,...,255\} R = { 0 , 1 , 2 , 3 , 4 , 5 , . . . , 2 5 5 } j ∈ S j\in S j ∈ S y j ∈ R y_j\in R y j ∈ R y j y_j y j j j j y j y_j y j x j x_j x j p ( y j , x j ) p(y_j,x_j) p ( y j , x j ) p ( y j , x j ) p(y_j,x_j) p ( y j , x j ) 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) p ( y j , x j ) = p ( y j ∣ x j ) p ( x j ) p ( x j ) p(x_j) p ( x j ) p ( y j ) p(y_j) p ( y j ) p ( y j ∣ x j ) p(y_j|x_j) p ( y j ∣ x j ) j j j p ( y j ∣ x j ) p(y_j|x_j) p ( y j ∣ x j ) θ \theta θ p ( y j ∣ x j ) = p ( y j ∣ x j ; θ x j ) = N x j ( μ x j , σ x j ) 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 x j 和 σ x j \mu_{x_j}和\sigma_{x_j} μ x j 和 σ x j x j x_j x j x j x_j x j p ( y j ∣ x j ) p(y_j|x_j) p ( y j ∣ x j )
OK，現在將隱Markov隨機場引入FCM中
首先，該算法利用了Ichihashi等人提出的一種基於KL信息懲罰的FCM方法，目標函數如下J = ∑ i = 1 3 ∑ j = 1 10 u i j d i j + λ ∑ i = 1 3 ∑ j = 1 10 u i j ln ( u i j π i j ) 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 ) π i j = p ( x j ) = p ( x j = i ∣ X ∂ j ) = exp ( − ∑ j ∈ C V C ( a i j ) ) ∑ h = j 3 exp ( − ∑ j ∈ C V C ( a h j ) ) \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 ) ) a i j = ( x j = i , X θ j ) a_{ij}=(x_j=i,X_{\theta_j}) a i j = ( x j = i , X θ j ) X θ j X_{\theta_j} X θ j X θ j X_{\theta_j} X θ j x θ j ′ x'_{\theta_j} x θ j ′ a i j = ( x j = i , x θ j ′ ) a_{ij}=(x_j=i,x'_{\theta_j}) a i j = ( x j = i , x θ j ′ ) U ( x j ) U(x_j) U ( x j ) U ( x j ) = − β ∑ j = 1 10 ∑ l = 1 8 δ ( x j − x l ) U(x_j)=-\beta\sum_{j=1}^{10}\sum_{l=1}^8\delta(x_j-x_l) U ( x j ) = − β j = 1 ∑ 1 0 l = 1 ∑ 8 δ ( x j − x l ) x l x_l x l x j x_j x j ∂ j \partial_j ∂ j π i j = p ( x j ) = p ( x j = i ∣ X ∂ j ) = β exp ( − ∑ l ∈ ∂ j δ ( i − x l ) ) ∑ h = 1 3 β exp ( − ∑ l ∈ ∂ j δ ( h − x l ) ) \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))} π i j = p ( x j ) = p ( x j = i ∣ X ∂ j ) = ∑ h = 1 3 β exp ( − ∑ l ∈ ∂ j δ ( h − x l ) ) β exp ( − ∑ l ∈ ∂ j δ ( i − x l ) ) δ ( x j − x l ) \delta(x_j-x_l) δ ( x j − x l ) δ ( x j , x l ) = { 1 , x j = x l 0 , o t h e r w i s e
\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 J J J r i j r_{ij} r i j μ i \mu_i μ i σ i \sigma_i σ i r i j = π i j exp ( − 1 λ d i j ) ∑ h = 1 3 π i j exp ( − 1 λ d i j ) ， d i j = w 2 ln ( 2 π ) + 1 2 ln ( ∣ μ i ∣ ) + ( y j − μ i ) 2 2 μ i 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 μ i = ∑ j = 1 10 r i j y j ∑ j = 1 10 y j \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 − μ i ) 2 ∑ j = 1 10 r i j \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} r i j 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 } π i j \pi_{ij} π i j μ i \mu_i μ i σ i \sigma_i σ i r i j r_{ij} r i j r i j r_{ij} r i j 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 }
