HMRF-EM Framework for Brain MR Image Segmentation

As has been stated in the previous section, FM model-based segmentation methods do not utilize any spatial information and therefore are not robust and appropriate in certain cases. The MEM algorithm for brain MR image segmentation suffers from the same problem. But as an effective way to remove bias field, the MEM algorithm is worth improving by overcoming this drawback. We present in this section how our HMRF-EM framework can be easily extended to incorporate an additional bias field correction step. More specifically, we seek an EM solution for three dependent unknowns: the bias field, the image classification and the involved model parameters. In the E step, we calculate the MAP estimate of the bias field and the class labels to form the Q-function. In the M step, we calculate the ML estimate of the parameters using the estimated bias field and the class labels in the E step.

• E step:
  

  $$\text{MAP estimation}\quad B^{(t)} \: =\:\arg\max_B p(B\vert\mathbf y, \mathbf x^{(t-1)}, \theta^{(t)})$$ (40)

  $$\text{MAP estimation}\quad{\mathbf x}^{(t)} =\: \arg\min_{\mathbf x \in \mathcal X} U({\mathbf x\vert y},B^{(t)},\theta^{(t)})$$ (41)

  
  

• M step:
  

  $$\text{ML estimation}\quad \theta^{(t+1)}\: =\:\arg\max_\theta P({\mathbf y}\vert\theta,{\mathbf x^{(t)}},B^{(t)})$$ (42)

  
  

The complete algorithm is described in Figure 5.

  

Figure 5: HMRF-EM algorithm for brain MR image segmentation and bias field correction