with nodes for the spinal structure as shown in Fig. 1. Each node represents a connected disc-vertebra-disc component of the spinal structure, in which both the discs and the vertebral body are modelled as rectangular shapes. We assign a parameter set to to describe the positions, sizes and orientations of the vertebral body and the upper/lower intervertebral discs of as shown in Fig. 2. define a connection matrix of the graph . On this graphical model, the observation model of a single component is defined as and the potential among neighboring components and with is defined as . From a probabilistic point of view, represents the probability that the configuration of the node match the observed image(s) I and the potential encodes the geometrical constraint between components. The identification of the spinal structure is then to find the configuration , that maximizes.
2.2 Component Observation Model
Intensity observation model : The intensity observation model represents the probability that the parameterized model of with the correspondent parameter set fits the appearance of the observed image(s) I. Each determines a disc-vertebra-disc template as shown in Fig. 2. We assume that the interior area of the vertebral body has a homogeneous intensity distribution modeled as a Gaussian distribution . While the border region, which is defined as a small neighborhood outside the vertebral body as shown in Fig. 2, is assumed to obey a different intensity distribution from the interior area of the vertebral body. For each pixel that falls in the interior and the border region of the template with an intensity value , the image appearance value of s is computed as
We further define , where is the cross-correlation between the image appearance values and a binary template which sets value 1 to the interior area of the template and 0 to the border region. can be learned from the observed image(s) I once is given, i.e., to fit a Gaussian distribution with the intensity values of the interior region of the vertebral body determined by .
Gradient observation model : Similar to , we can define , where is the cross-correlation between the gradient image values of the observed image(s) in the template area and a binary gradient template, which sets 0 in the interior area and 1 in the border region. This means that the interior region of the vertebral body is homogeneous and high gradient values should only happen on the border of the vertebral template.
Local variance observation model : We define the local variance image of a pixel in the image(s) I as the intensity variance in a small window centered at this pixel. We set , where is the cross-correlation between the local variance values and a binary template identical to the gradient template. Similar to the gradient observation model, this item is used to model the observation that intensities of the interior area of a vertebral body should be more homogeneous than those of the border region.
2.3 Potentials Between Components
Size constraint : is used to set constraint on the sizes (radius and height of the vertebral body) of the neighboring components and is defined as
which means that neighboring components should have similar sizes and that the strength of the constraint should decay with the order distance between these two components.