and Angle-Preserving Parameterization for Vertebra Surface Mesh

of a vertebra with one genus is represented by triangular patches. For each vertex $\varvec{v}_i$ $(i = 1, 2, \ldots , N_{\fancyscript{M}})$ in $\fancyscript{M}$ , its 1-ring region consists of $N_R^{(i)}$ patches containing the vertex $\varvec{v}_i$ .

2.1 Modified Self-organizing Deformable Model

The mSDM is a deformable model whose shape is deformed by using a competitive learning and an energy minimization approach. Regrading a tissue model as the mSDM, the model is mapped onto a target surface by fitting the model to the target surface. The overview of the mSDM algorithm is as follows.

Step. M-1

A vertebra model $\fancyscript{M}$ is deformed to fit to the target surface by the original SDM algorithm [9].

Step. M-2

The deformed model may contain foldovers. To realize the one-to-one correspondence, the foldover is removed by moving the vertices to within their 1-ring region.

Step. M-3

After the Step. M-2, the feature vertices move away from their corresponding points. Free-Form Deformation (FFD) [10] is applied to the mapped model to correct the positions of the feature vertices.

Step. M-4

To compare with the models, it is desirable that the mapped model preserves the original geometric properties as far as possible. Such deformation is achieved by an area- and/or angle-preserving mapping $\varPhi$ . Practically, the model obtained after Step. M-3 is deformed by minimizing an objective function $E(\varPhi )$ that is a weighted linear combination of area error term $E_{area}$ and angle error term $E_{angle}$ :

$\begin{aligned} E(\varPhi ) = \mu E_{area} + (1- \mu ) E_{angle} \simeq \sum _{i} e(\varPhi ); \end{aligned}$

(1)

$\begin{aligned} e(\varPhi ) = \mu \psi e_{area}(\varPhi ) + (1- \mu ) e_{angle}(\varPhi ); \end{aligned}$

(2)

$\begin{aligned} e_{area}(\varPhi ) = \sum _{p_k^{(i)} \in R_i} \Bigl | \frac{\varPhi (A_{p_k^{(i)}})}{\sum _{p_k^{(i)}\in R_i} \varPhi (A_{p_k^{(i)}})}- \frac{A_{p_k^{(i)}}}{\sum _{p_k^{(i)}\in R_i} A_{p_k^{(i)}}} \Bigr | ; \end{aligned}$

(3)

$\begin{aligned} e_{angle}(\varPhi ) = \sum _{p_k^{(i)} \in R_i} \sum _{d=0}^3 |\varPhi (\theta _{p_k^{(i)}}^d) - \theta _{p_k^{(i)}}^d| , \end{aligned}$

(4)

where $\psi$ is a scaling factor to adjust the ranges of the two error terms. $A_{i,k}$ and $\theta _{i,k}^d$ are the area and one angle of the patch $p_{i,k} (k = 1, 2, \ldots ,N_R^{(i)})$ included in the 1-ring region $$R_i$$

of the vertex $\varvec{v}_i$ . $\varPhi (A)$ and $\varPhi (\theta )$ is the area and angle of the patch in $\varPhi (\fancyscript{M})$ . The area errors in Eq. (3) is obtained by the total difference between the area ratio of a patch before and after the mapping. Here, the area ratio of the patch $p_k^{(i)}$ is defined as the ratio of the area $A_{p_k^{(i)}}$ of $p_k^{(i)}$ to the whole area of the 1-ring region $$R_i$$

of $p_k^{(i)}$ . In the same way, the angle error in Eq. (4) is obtained by the total difference between the angles of the patch $p_k^{(i)}$ before and after the mapping. Changing the weighting factor $\mu$ from 0 to 1, the mapping becomes from angle- to area-preserving mapping. The setting of $\mu$ determines the kinds of geometrical features preserved after the mapping. The discussion about the setting of $\mu$ will be found in [1]. From Eqs. (1) and (2), the minimization of Eq. (1) is replaced as the optimization problem of positioning the vertices in the 1-ring region by moving them repeatedly. The optimal mapping is found by applying a greedy algorithm with Eq. (2). Then, the vertices are not on the target surface completely. Therefore, they are mapped onto the nearest patch of the target surface after the movement.

Once a tissue model and its target surface are given, our mSDM is performed automatically with no users’ manual intervention. Moreover, the mSDM framework controls the movement of several feature vertices of the model. Practically, when users manually specify the feature vertices and their corresponding locations on the target surface, the SDM algorithm maps the tissue model onto the target surface while moving the feature vertices toward their corresponding locations. This characteristic of the mSDM makes it easy to find the correspondence between tissue models by mapping the anatomical feature vertices of the tissue model onto their specific locations on the target surface.

2.2 Vertebra Parameterization Using mSDM

Our parameterization method uses an intermediate surface $\fancyscript{I}$ whose shape is close to vertebrae shape. Here, the intermediate surface is determined as a surface model which has intermediate shape between the vertebra and the torus. Figure 1 shows the overview of the mapping a vertebra $\fancyscript{M}$ onto a torus $\fancyscript{T}$ . When the vertebra is mapped directly onto the torus whose shape is far from the vertebra, users need to tune the parameters in the mSDM algorithm and the scale of the torus. Especially, the latter tuning is to change the size of the torus and its hole according to the hole size of each individual vertebra. If the user fails the initial settings, we sometimes obtain the incomplete one-to-one mapping between the vertebra and the torus. Figure 2a, b show the examples of the vertebra models obtained by mapping the vertebra onto the torus directly. As shown in Fig. 2, the accuracy of the direct mapping depends on the parameters and the scale tuning heavily. On the other hand, through our experiments, the use of the intermediate surface results in the stable mapping of the vertebra models independent of the tuning for individual models. At first, we find one-to-one mapping ${}^{\fancyscript{M}}\varPhi _{\fancyscript{I}}$ between $\fancyscript{M}$ and $\fancyscript{I}$ . Similarly, one-to-one mapping ${}^{\fancyscript{I}}\varPhi _{\fancyscript{T}}$ from $\fancyscript{I}$ to $\fancyscript{T}$ is found. Combining the two mappings ${}^{\fancyscript{M}}\varPhi _{\fancyscript{I}}$ and ${}^{\fancyscript{I}}\varPhi _{\fancyscript{T}}$ , we obtain the mapping ${}^{\fancyscript{M}}\varPhi _{\fancyscript{T}}$ from $\fancyscript{M}$ to $\fancyscript{T}$ .

Fig. 1

Vertebra parameterization using mSDM

Fig. 2

Mapped vertebra models onto the torus by applying direct mapping a before and b after parameter and scale tuning of the SDM and the torus

Our parameterization method consists of four steps.

Step. P-1

Select an intermediate surface $\fancyscript{I}$ .

Step. P-2

Find a mapping $^{\fancyscript{I}}\varPhi _{\fancyscript{T}}$ by mapping from $\fancyscript{I}$ to $\fancyscript{T}$ with mSDM.

Step. P-3

Find a mapping ${}^{\fancyscript{M}}\varPhi _{\fancyscript{I}}$ by mapping from $\fancyscript{M}$ to $\fancyscript{I}$ with mSDM.

Step. P-4

Obtain the direct mapping ${}^{\fancyscript{M}}\varPhi _{\fancyscript{T}}$ by combining ${}^{\fancyscript{M}}\varPhi _{\fancyscript{I}}$ and $^{\fancyscript{I}}\varPhi _{\fancyscript{T}}$ .

The following describes the detail of Step. P-4. Let us denote as $\fancyscript{M}^{(\fancyscript{I})} = {}^{\fancyscript{M}}\varPhi _{\fancyscript{I}}(\fancyscript{M})$ the model $\fancyscript{M}$ mapped onto the intermediate surface $\fancyscript{I}$ . For each vertex $\varvec{v}_i^{(\fancyscript{I})} = {}^{\fancyscript{M}}\varPhi _{\fancyscript{I}}(\varvec{v}_i)$ in $\fancyscript{M}^{(\fancyscript{I})}$ , we find a closest patch $p_i^{\fancyscript{I}}$ in $\fancyscript{I}$ to the vertex $\varvec{v}_i^{(\fancyscript{I})}$ . When, $p_i^{\fancyscript{I}}$ consists of three vertices $\varvec{r}_{i,1}, \varvec{r}_{i,2}, \varvec{r}_{i,3}$ (Fig. 1), the coordinate of $\varvec{v}_i^{(\fancyscript{I})}$ is represented by

$\begin{aligned} \varvec{v}_i^{(\fancyscript{I})} = \varvec{r}_{i,1} + \alpha _1 (\varvec{r}_{i,2} - \varvec{r}_{i,1}) + \alpha _2 (\varvec{r}_{i,3} - \varvec{r}_{i,1}) , \end{aligned}$

(5)

where $\alpha _1$ and $\alpha _2$ ( $0\le \alpha _1, \alpha _2 \le 1$ ) are real number parameters. Similarly, using the mapping ${}^{\fancyscript{I}}\varPhi _{\fancyscript{T}}$ , the vertices $\varvec{r}_{i,*}$ in $\fancyscript{I}$ are represented by $\varvec{r}_{i,*}^{(\fancyscript{T})} = {}^{\fancyscript{I}}\varPhi _{\fancyscript{T}}(\varvec{r}_{i,*})$ . Therefore, the vertices $\hat{\varvec{v}}_i^{(\fancyscript{T})} = {}^{\fancyscript{M}}\hat{\varPhi }_{\fancyscript{T}}(\varvec{v}_i)$ of the model mapped on the torus are obtained by replacing the coordinates of vertices in $\fancyscript{I}$ with that in $\fancyscript{I}^{(\fancyscript{T})}$ in Eq. (5):

$\begin{aligned} \hat{\varvec{v}}_i^{(\fancyscript{T})}&\simeq {}^{\fancyscript{I}}\varPhi _{\fancyscript{T}} {}^{\fancyscript{M}}\varPhi _{\fancyscript{I}} (\varvec{v}_i) \\&= \varvec{r}_{i,1}^{(\fancyscript{T})} + \alpha _1 (\varvec{r}_{i,2}^{(\fancyscript{T})} - \varvec{r}_{i,1}^{(\fancyscript{T})}) + \alpha _2 (\varvec{r}_{i,3}^{(\fancyscript{T})} - \varvec{r}_{i,1}^{(\fancyscript{T})}) .\nonumber \end{aligned}$