Model-Based Vertebra Identification from X-Ray Image(s)

with nodes for the spinal structure as shown in Fig. 1. Each node $V_{i} ,i = 0,1, \ldots ,N - 1$ 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 ${\mathbf{X}}_{i} = \{ x_{i} ,y_{i} ,r_{i} ,h_{i} ,a_{i} (\theta_{i} ),h_{i}^{u} ,\theta_{i}^{u} ,h_{i}^{l} ,\theta_{i}^{l} \}$ to $V_{i}$ to describe the positions, sizes and orientations of the vertebral body and the upper/lower intervertebral discs of $V_{i}$ as shown in Fig. 2. $E = \left\{ {e_{i,j} } \right\},i,j = 0,1,2, \ldots ,N - 1$ define a connection matrix of the graph . On this graphical model, the observation model of a single component $V_{i}$ is defined as $p\left( {{\mathbf{I}} |{\mathbf{X}}_{i} } \right),i = 0,1, \ldots ,N - 1$ and the potential among neighboring components $V_{i}$ and $V_{j}$ with $e_{i,j} = 1$ is defined as $p\left( {{\mathbf{X}}_{i} ,{\mathbf{X}}_{j} } \right), i,j = 0,1, \ldots ,N - 1,e_{i,j} = 1$ . From a probabilistic point of view, $p\left( {{\mathbf{I}} |{\mathbf{X}}_{i} } \right)$ represents the probability that the configuration ${\mathbf{X}}_{i}$ of the node ${\mathbf{V}}_{i}$ match the observed image(s) I and the potential $p\left( {{\mathbf{X}}_{i} ,{\mathbf{X}}_{j} } \right)$ encodes the geometrical constraint between components. The identification of the spinal structure is then to find the configuration ${\mathbf{X}} = \{ {\mathbf{X}}_{0} ,{\mathbf{X}}_{1} , \ldots ,{\mathbf{X}}_{i} , \ldots ,{\mathbf{X}}_{N - 1} \}$ , that maximizes.

Fig. 1

A schematic view of the graphical model based representation of a spinal structure

Fig. 2

A schematic view of the vertebral body template for the component observation model

$p({\mathbf{X}}|{\mathbf{I}}) \propto \prod\limits_{i} {p\left( {{\mathbf{I}} |{\mathbf{X}}_{i} } \right)} \prod\limits_{{e_{i,j} = 1}} {p\left( {{\mathbf{X}}_{i} ,{\mathbf{X}}_{j} } \right)} .$

(1)

2.2 Component Observation Model

The component observation model $p\left( {{\mathbf{I}} |{\mathbf{X}}_{i} } \right)$ is to match a template, which is determine by ${\mathbf{X}}_{i}$ , with the observed image(s) I. We define our component observation model as:

$p\left( {{\mathbf{I}} |{\mathbf{X}}_{i} } \right) = p_{I} \left( {{\mathbf{I}} |{\mathbf{X}}_{i} } \right)p_{G} \left( {{\mathbf{I}} |{\mathbf{X}}_{i} } \right)p_{V} \left( {{\mathbf{I}} |{\mathbf{X}}_{i} } \right) .$

(2)

The three items in Eq. (2) come from the intensity, gradient and local variance of the template as detailed below:

Intensity observation model $p_{I} \left( {{\mathbf{I}} |{\mathbf{X}}_{i} } \right)$ : The intensity observation model represents the probability that the parameterized model of $V_{i}$ with the correspondent parameter set $X_{i}$ fits the appearance of the observed image(s) I. Each ${\mathbf{X}}_{i}$ 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 ${\mathcal{N}}(\mu_{i} ,\sigma_{i} )$ . 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 ${\mathbf{I}}(s)$ , the image appearance value of s is computed as

$p\left( {s |{\mathbf{X}}_{i} } \right) = e^{{ - \frac{{({\mathbf{I}}\left( s \right) - \mu_{i} )^{2} }}{{2\sigma_{i}^{2} }}}} .$

(3)

We further define $p_{I} \left( {{\mathbf{I}} |{\mathbf{X}}_{i} } \right) = e^{{\omega_{I} c_{I}^{i} }}$ , where $c_{I}^{i}$ is the cross-correlation between the image appearance values $p\left( {s |{\mathbf{X}}_{i} } \right)$ and a binary template which sets value 1 to the interior area of the template and 0 to the border region. $$$ \omega_{I} > 0 $$” src=”/wp-content/uploads/2016/12/A312884_1_En_12_Chapter_IEq32.gif”></SPAN> is a weighting factor. Intuitively this means that we assume that the interior region of the template should obey the Gaussian distribution and the border area should have a different intensity distribution. The Gaussian model <SPAN id=IEq33 class=InlineEquation><IMG alt=$ can be learned from the observed image(s) I once ${\mathbf{X}}_{i}$ is given, i.e., to fit a Gaussian distribution with the intensity values of the interior region of the vertebral body determined by ${\mathbf{X}}_{i}$ .
Gradient observation model $p_{G} \left( {{\mathbf{I}} |{\mathbf{X}}_{i} } \right)$ : Similar to $p_{I} \left( {{\mathbf{I}} |{\mathbf{X}}_{i} } \right)$ , we can define $p_{G} \left( {{\mathbf{I}} |{\mathbf{X}}_{i} } \right) = e^{{\omega_{G} c_{G}^{i} }}$ , where $c_{G}^{i}$ 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 $p_{V} \left( {{\mathbf{I}} |{\mathbf{X}}_{i} } \right)$ : We define the local variance image $I_{V}$ of a pixel in the image(s) I as the intensity variance in a small window centered at this pixel. We set $p_{V} \left( {{\mathbf{I}} |{\mathbf{X}}_{i} } \right) = e^{{\omega_{V} c_{V}^{i} }}$ , where $c_{V}^{i}$ 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.

We only consider the image observation model of the vertebral bodies but ignore the observation model of the discs. This is due to the fact that for X-ray image(s) with different view direction(s), the above mentioned observation model is more reliable for the vertebral bodies than for the discs. A unified observation model for the discs is more difficult to be designed.

It can also be observed that the three components in the observation model do not need to be trained with training data as done in [1, 2, 5]. Instead their parameters can be directly learned from the target X-ray image(s) I.

2.3 Potentials Between Components

We define inter-node potentials to apply geometric constraints between neighboring nodes such that all the nodes will be assembled to a meaningful spinal structure. More specifically, we have:

$p\left( {{\mathbf{X}}_{i} ,{\mathbf{X}}_{j} } \right) = p_{S} \left( {{\mathbf{X}}_{i} ,{\mathbf{X}}_{j} } \right)p_{O} \left( {{\mathbf{X}}_{i} ,{\mathbf{X}}_{j} } \right)p_{D} \left( {{\mathbf{X}}_{i} ,{\mathbf{X}}_{j} } \right)$

(4)

The three items in (4) specify the size, the orientation and the distance constraints as detailed below:

Size constraint $p_{S} \left( {{\mathbf{X}}_{i} ,{\mathbf{X}}_{j} } \right)$ : $p_{S} \left( {{\mathbf{X}}_{i} ,{\mathbf{X}}_{j} } \right)$ is used to set constraint on the sizes (radius and height of the vertebral body) of the neighboring components and is defined as

$p_{S} \left( {{\mathbf{X}}_{i} ,{\mathbf{X}}_{j} } \right) = e^{{ - \frac{{(\omega_{r} \frac{{\left| {r_{i} - r_{j} } \right|}}{{\left| {r_{i} + r_{j} } \right|}} + \omega_{h} \frac{{\left| {h_{i} - h_{j} } \right|}}{{\left| {h_{i} + h_{j} } \right|}})}}{|i - j|}}}$

(5)
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.
Orientation constraint $p_{O} \left( {{\mathbf{X}}_{i} ,{\mathbf{X}}_{j} } \right)$ : we define
$p_{O} \left( {{\mathbf{X}}_{i} ,{\mathbf{X}}_{j} } \right) = e^{{ - \frac{{\omega_{{0(a_{i} (\theta_{i} ) \cdot a_{j} (\theta_{j} ))}} }}{|i - j|}}}$

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