## Multidimensional MRFBased Segmentation

In this section, we provide a multispectral image-segmentation solution based on mMRF model.

### 8.4.2.1 mMRF Model

Similar to MRF model discussed in section 8.2.1, our introduction of mMRF model is also based on MAP criterion. Assume the input images I1; I2,..., Id

Id yd

Figure 8.14: Illustration of d-dimensional image space.

are observed in d channels as illustrated in Fig. 8.14 and the label matrix of segmentation result is X. Then a posterior probability can be expressed in Eq. (8.39).

Another way to express the input data is to view them as d-dimensional vector Ys = [ys1, ys2,..., ysd]T, where the value of ith dimension, ySii, represents the intensity at site s in image Ii. There is relation that p(X | Ii, I2,...,Id ) = p( X | Yi, Y2,

where S is the total number of pixels in each image. Based on Bayes' theorem, a posterior probability is with form p(X | Y ) a p(Yi, Y2,...,Ys | X) p( X)

By assuming the conditional independence of each dimension given the segmentation result, the conditional probability can be expressed as p(Yu Y2,...,Ys | X) a [] p( ysA, ys^ ...,ys,d | X)

= n [p(ysi IX)p(ys,21X) ■■■ p(ySdd | X)] (8.42)

For prior probability, the neighborhood is defined as a block in mMRF model. Figure 8.15 is an illustration of 3 by 3 neighborhood in d-dimensional scenario. The black nodes are the pixels at location s in each channel and the gray nodes Figure 8.15: Illustration of a 3 by 3 by d neighborhood in mMRF model. are the pixels in neighborhood. The prior probability of the whole image is p(X) = Z exp - £ Fs(x) = 1 exp - Y,[Vsn(s) + Vse(s)] • (8.43)

Figure 8.15: Illustration of a 3 by 3 by d neighborhood in mMRF model. are the pixels in neighborhood. The prior probability of the whole image is p(X) = Z exp - £ Fs(x) = 1 exp - Y,[Vsn(s) + Vse(s)] • (8.43)

Similar to the energy definition of monochrome image, the clique energy Vs(x) for mMRF model at location s is a summation of spatial constraint energy VsN(x) and edge constraint energy VsE(s) in all channels. Their definitions are given in the following equations, respectively:

where the VsN i(s) and VsE i(s) represent the component in the ith dimension. Compared to the energy function in traditional mMRF models [46, 82, 83], an additional edge constraint VsE is added, which preserves the details of each dimension in the probability description and provides an even more accurate description of the energy function. This can makes the label updating process more sensitive at the regions boundaries.

Based on the above definition, we can find the a posteriori probability as p(Y | X) a p(Y | X)p(X) a exp j -

i=i 2s2i

and the energy function of the whole image is expressed as

Based on MAP, the optimal segmentation is

which is also equivalent to the minimization of image energy E(X).