An MRF-ICA Based Algorithm for Image Separation

Separation of sources from one-dimensional mixture signals such as speech has been largely explored. However, two-dimensional sources (images) separation problem has only been examined to a limited extent. The reason is that ICA is a very general-purpose statistical technique, and it does not take the spatial information into account while separating mixture images. In this paper, we introduce Markov random field model to incorporate the spatial information into ICA. MRF is considered as a powerful tool to model the joint probability distribution of the image pixels in terms of local spatial interactions. An MRF-ICA based algorithm is proposed for image separation. It is successfully demonstrated on artificial and real images. 1 Motivation Over the past years, Independent component analysis (ICA) [1] has been widely used in many different areas such as audio processing, biomedical signal processing, image processing and econometrics. As the origination of ICA, separation of sources from one-dimensional mixture signals such as speech has been largely explored [2]. But image separation problem has only been examined to a limited extent. The reason is that ICA is a very general-purpose statistical technique, and it does not think of the spatial information while separating mixture images. In this paper, we introduce Markov random field (MRF) to incorporate the spatial information into ICA. MRF is considered as a powerful stochastic tool to model the joint probability distribution of the image pixels in terms of local spatial interactions [3,4]. Using MRF as the representation of image spatial information has the following advantages. First, MRF is the model of context in the image. Second, the MRF model of prior information need not be an accurate model of the image itself. An MRF-ICA algorithm is proposed for image separation. It is successfully demonstrated on artificial and real images. The remainder of the paper is organized as follows. Section 2 describes the MRF model. Section 3 develops the MRF-ICA algorithm. Experiments results of applying the MRF-ICA algorithm are reported in Section 4. Section 5 contains our conclusions. 392 S. Jia and Y. Qian 2 MRF Model for Image Separation MRF models are mainly used in feature extraction and image segmentation. Readers are referred to [3,4] for details. First we introduce some notations. An image specifies the gray levels for all pixels in a M N × lattice. Let F f = be the feature vector extracted from a random image ( ) X x = , where F denotes a random variable and f is an instance of F . The image can be represented by the vector random variable ( ) 1 2 , , , MN X X X X = K . The energy demonstrating MRF can be divided into two components [5,6]: the feature modeling component and the region labeling component. They are described as follows: ( ) ( ) 2 2 1 1 1 : ( ) 1 ln( ) 2 2 , MN MN c F R i i r k k i i i r i i r k i E E J x x x μ σ θ σ = = = + ⎛ ⎞ ⎜ ⎟ ⎡ ⎤ = + = ⎣ ⎦ ⎜ ⎟ ⎝ ⎠ − ∑ ∑∑ (1) where k i μ and k i σ are the mean and standard deviation for the ith pixel in the kth feature component; ( , ) 1 J a b = − if a b = , 0 if a b ≠ , c is the number of computing dependence of pixels, and{ } 1, , c θ θ K are the weighting parameters. Here, we give an assumption that the standard deviations of feature vector are equal. It not only facilitates the subsequent computation of MRF, but also can be basically satisfied in actual situation without loss of generality. For the convenience of following derivation, Table 1 gives several symbols. And we take two-image separation problem as an example. Using the assumption, the MRF energy can be expressed in terms of matrix forms: { } 2 2 1 ln 2 2 2 T T VS VS VS R MN E VS VS VS E σ μ μ μ σ = + ⋅ − ⋅ ⋅ + ⋅ + (2) SubstitutingVS VX w = ⋅ into (2) results in ( ) { } 2 2 1 ln 2 2 2 T T T R VS VS VS MN E w w VX VX w VX w E σ μ μ μ σ = + ⋅ ⋅ ⋅ − ⋅ ⋅ ⋅ + ⋅ + (3) In order to reduce the computational load, we let c be 2 and all parametersθ equal to 1. We only compute the energy of left-right neighbors; the up-down neighbors can be computed in the same way. Here we introduce two matrixes: a M and b M to facilitate the representation of the region labeling component. ( ) ( ) ( ) ( ) ( ) 2 1 : & 1 1 1 , 2 , MN MN i i r a b R Left Right i i i r i J J M VS M VS E w x x + + = = = ⎛ ⎞ ⎡ ⎤ = ⎡ ⎤ ≈ ⋅ ⋅ × ⎜ ⎟ ⎣ ⎦ ⎣ ⎦ ⎝ ⎠ ∑∑ ∑ (4) Because the derivative of function J does not exist, we introduce function 2 2 ( ) x H x e− = − to fit it. And Equation (4) can be written as ( ) ( ) ( ) [ ] 1 & 1 2 MN a b R Left Right i i M M VS E w H + = ⋅ × ⎛ ⎞ = − ⎜ ⎟ ⎝ ⎠ ∑ (5) Differentiating (5) with respect to w ( ) ( ) & R Left Right a b E H M M VX w w ∂ ∂ = ⋅ ⎡ − ⋅ ⎤ ⎣ ⎦ ∂ ∂ (6) An MRF-ICA Based Algorithm for Image Separation 393 where 2 2 i i i w H w we ∂ ∂ = . The result can be considered to be zero regardless of the value of i w . So the result of (6) is zero. Similarly, the derivative of energy of up-down neighbors is also zero. That is, 0 R E w ∂ ∂ = . Now we use gradient descent methods to obtain the extreme value of MRF model ( ) ( ) { } 2 1 ( ) 2 0 2 T T T T VS E VX VX VX VX VX w w w w σ μ ∂ ⎡ ⎤ = ⋅ ⋅ + ⋅ ⋅ − ⋅ = ⎣ ⎦ ∂ (7) Solving it ( ) ( ) 1 T T VS VX VX VX w μ − = ⋅ ⋅ ⋅ (8) This is the solution formula to w . Table 1. The symbols Symbol Explanation Dimension VX The matrix obtained by converting mixtures into vectors 2 MN × w Demixing vector that we want to find 2 1 × VS The estimated image-vector 1 MN × 1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 a b M M = = ⎛ ⎞ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ O O O O L L The two matrixes are used for dealing with the region labeling component ( ) 1 MN MN + ⋅ 3 MRF-ICA Algorithm The algorithm is described as follows: 1. A random vector w is initialized. 2. MRF-ICA step: Transform the image matrixes into vectors, and use FastICA algorithm [7] to estimate w ; normalize the obtained w , and then make use of (8) to compute w and normalize it again. 3. Repeat step 2 until a stopping criterion is satisfied, an estimated independent component vector is acquired. 4. Repeat all above steps until all independent components have been acquired. 4 Experimental Results 4.1 Results on Artificial Images Figure 1(a) shows the source images. Here we take the following mixing matrix: 0.2 0.5 0.6 0.3 A = ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ 394 S. Jia and Y. Qian The mixtures are displayed in Fig. 1(b). The estimates using ICA and MRF-ICA are given in Fig. 1(c) and (d) respectively. It is clear that MRF-ICA algorithm is more effective than FastICA. The color of left figure is reversed, but this has no significance. Now we consider three-image separation problem. Figure 2(a) and (b) illustrate the source images and the mixtures respectively. The corresponding mixing matrix is: 0.5 0.4 0.6 0.6 0.5 0.4 0.4 0.6 0.5 A = ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ The estimates using ICA and MRF-ICA are given in Fig. 2(c) and (d) respectively. Apparently the estimates of MRF-ICA are better than ICA. 4.2 Results on Real Images Figure 3(a) and (b) illustrate the real source images and mixtures respectively. Here we still use the two-dimension mixing matrix A mentioned above. The estimates using ICA are given in Fig. 3(c). And the second figure is almost the same as the right of the mixtures. The MRF-ICA estimates are displayed in Fig. 3(d). As can be seen, they are similar to the original source images. 20 40 60 80 100 120 20 40 60