Robust Segmentation Framework for Spine Trauma Diagnosis

(the signed distance function) as a level set $\phi =0$ [16]. The evolution of the level set function $\phi (t)$ is governed by $\frac{\partial \phi }{\partial t}+ F|\nabla \phi |=0$ , where is the speed function. Based on the variational framework, an energy function $E(\phi )$ is defined in relation to the the speed function. The minimization of such energy generates the Euler-Lagrange equation, and the evolution of the equation is through calculus of variation:

$\begin{aligned} \frac{\partial \phi }{\partial t} = -\frac{\partial E(\phi )}{\partial \phi }. \end{aligned}$

In this work, the fusion of energies whereby a shape prior distribution estimator

and an edge-mounted Willmore energy $E_{\textit{w}_0}$ is employed:

$E(\phi )= \lambda E_s + E_{\textit{w}_0},$

where $\lambda$ $(0<\lambda \le 1)$ is the weight parameter, which is tuned to suit the segmentation of normal and abnormal spinal vertebrae.

2.1 Computing Prior Shape Energy via Kernel Density Estimation

Kernel density estimation (KDE) is a nonparametric approach for estimating the probability density function of a random variable. Without assuming the prior shapes are Gaussian distributed, KDE presents advantage in estimating the shape distribution even with a small number of training set, in addition to modeling shapes with high complexity and structure. In this study, we adopted the prior shape energy formulation discussed by Cremers et al. [2].

The density estimation is formulated as a sum of Gaussian of shape dissimilarity measures $d^2(\phi , \phi _i)$ , $i=1, 2, \dots , N$ :

$\begin{aligned} P(\phi )\propto \frac{1}{N} \sum _{i=1}^Ne^{-\frac{d^2(\phi ,\phi _i)}{2\sigma ^2}}, \end{aligned}$

where the shape dissimilarity measure $d^2(\phi , \phi _i)$ is defined as

$\begin{aligned} d^2(\phi ,\phi _i)&=\int _\Omega \frac{1}{2}\left( H(\phi ) - H(\phi _m)\right) ^2 dx,\\ \sigma ^2&= \frac{1}{N} \sum _{i=1}^N\min _{j\ne i}d^2(\phi _i,\phi _j), \end{aligned}$

and $H(\phi )$ is the Heaviside function. By maximizing the conditional probability

$\begin{aligned} P(\phi |I) = \frac{P(I|\phi )P(\phi )}{P(I)}, \end{aligned}$

and considering the shape energy as

$\begin{aligned} E_s(\phi ) = -\log P(\phi |I), \end{aligned}$

the variational with respect to $\phi$ becomes

$\begin{aligned} \frac{\partial E_s}{\partial \phi }=&\frac{\sum _{i=1}^N \alpha _i \frac{\partial }{\partial \phi } d^2(\phi , \phi _i)}{2\sigma ^2 \sum _{i=1}^N \alpha _i}\\ =&\sum _{i=1}^N \frac{e^{-\frac{d^2(\phi , \phi _i)}{2\sigma ^2}}}{2\sigma ^2 \sum _{i=1}^N \alpha _i} \Bigg ( 2\delta (\phi ) \bigg [H(\phi )-H(\phi _i(x-\mu _\phi ))\bigg ]\\&\quad + \int \bigg [H(\phi (\xi )-H(\phi _i(\xi -\mu _\phi ))\bigg ] \delta \phi (\xi ) \frac{(x-\mu _\phi )^T\nabla \phi (\xi ) }{\int H\phi dx} d\xi \Bigg ), \end{aligned}$

where $\mu _{\phi }$ is the centroid of $\phi$ and $\alpha _i=\exp \left( -\frac{1}{2\sigma ^2}d^2(\phi ,\phi _i)\right)$ is the weight factor for $i = 1,2,\dots ,N$ .

2.2 Computing Local Geometry Energy via Willmore Flow

Willmore energy is a function of mean curvature, which is a quantitative measure of how much a given surface deviates from a sphere. It is formulated as

$E_\textit{w} = \frac{1}{2}\int _M h^2 dA,$

where

is a

-dimensional surface embedded in $\mathbb R^{d+1}$ and

the mean curvature on

[18]. For image segmentation, the Willmore energy provides an internal energy that gives a useful description of a region, where the effect of edge indicator is not significant. In these regions, smoothness of the shape of the curve should be maintained and extended, which can be regarded as a weak form of inpainting [3].

As a geometric functional, the Willmore energy is defined on the geometric representation of a collection of level sets. Its gradient flow can be well represented by defining a suitable metric, the Frobenius norm, on the space of the level sets. Frobenius norm is a convenient choice as it is equivalent to the

-norm of a matrix and more importantly it is computationally attainable. As Frobenius norm is an inner-product norm, the optimization in the variational method comes naturally.

Based on the formulation by Droske and Rumpf [3], the Willmore flow or the variational form for the Willmore energy with respect to $\phi$ is

$\frac{\partial E_{\textit{w}}}{\partial \phi } = -\Vert \nabla \phi \Vert \left( \Delta _M h +h(t) \left( \Vert S(t)\Vert _2^2-\frac{1}{2}h(t)^2\right) \right) ,$

where $\Delta _M h = \Delta h -h\frac{\partial h}{\partial n} - \frac{\partial ^2h}{\partial n^2}$ is the Laplacian Beltrami operator on

with $n=\frac{\nabla \phi }{\Vert \nabla \phi \Vert }$ , $S = (I-n\otimes n)(\nabla \times \nabla )\phi$ is the shape operator on $\phi$ and $\Vert S\Vert _2$ is the Frobenius norm of