*3.4. Diffeomorphism Supported by Hyper-Elasticity Regularization*

Although hierarchical cage deformation recursively represents shape deformation to avoid local minima, dense cages possibly lead to more cage degeneration. Therefore, an appropriate regularization process is required when applying hierarchical transformations. For plausible deformation, we used hyper-elastic regularization, which prevents unexpected partial deformation. We utilized and modified the study of Burger et al. [20], which can be easily extended to the cage deformation setting. As shown in Figure 5, the 24 sub-regions of the cage were defined using corner points *p<sup>i</sup>* and seven auxiliary points, which are the volume points *p<sup>V</sup>* and face points *pF*. Tetrahedral sub-regions are defined by the span of a volume point and corresponding face points. Regularization ensures that the transformation is a diffeomorphism; that is, it is reversible and smooth. Hyper-elastic regularization, as defined by Burger et al. [20], is given by

$$S^{hypr}(\mathbf{x}) = \int a\_1 \eta\_{\rm vol}(\mathbf{x}) + a\_2 \eta\_{\rm sur}(\mathbf{x}) + a\_3 \eta\_{\rm len}(\mathbf{x}) d\Omega. \tag{21}$$

where *α<sup>i</sup>* are balancing parameters. The functions *ηvol* , *ηsur* , and *ηlen* penalize changes of volume, surface, and length, respectively. Here, we set the balancing parameter as 10.0 for all experiments. Burger et al. [20] utilized the average points to delineate the volume point *p<sup>V</sup>* and six face points *pF*. However, if the cage is concave (i.e., due to large deformations), the face and volume points are not maintained inside the cage, as shown in Figure 5. As a result, the functions *ηvol* and *ηsur* may have negative values, which can lead to the failure of gradient descent.

**Figure 5.** Cage sub-division with hyper-elastic regularization: (**a**) A tetrahedral sub-division of the 3D cage volume, which is the span of face (blue) and volume (green) points; (**b**) sub-division of cage face; (**c**) the average points (red) located outside of the cage and their negative areas (red triangles); and (**d**) the equal-area points (blue) located inside of cages despite large deformations and their positive areas (yellow triangles).

To achieve robust regularization, we define the face and volume vertices of each cage to have the same sub-area and sub-volume inside of the cage. Assuming that the face point *p<sup>F</sup>* = (*pF<sup>x</sup>* , *pF<sup>y</sup>* , *pF<sup>z</sup>* ) is located inside the quadrilateral, the position *p<sup>F</sup>* of the points dividing the areas △*p*<sup>0</sup> *p*<sup>1</sup> *pF*, △*p*<sup>1</sup> *p*<sup>2</sup> *pF*, △*p*<sup>2</sup> *p*<sup>3</sup> *pF*, and △*p*<sup>3</sup> *p*<sup>0</sup> *p<sup>F</sup>* is defined as follows;

$$
\triangle p\_i p\_{i+1} p\_F = (p\_{i+1} - p\_i) \times (p\_F - p\_i) / 2 = [p\_{i+1} - p\_i]\_\times (p\_F - p\_i) / 2,\tag{22}
$$

where *i* = {0, 1, 2, 3}. The least-squares solution of the above conditions for all triangles is

$$
\begin{bmatrix} [p\_1 - p\_0]\_\times \\ [p\_2 - p\_1]\_\times \\ [p\_3 - p\_2]\_\times \\ [p\_0 - p\_3]\_\times \end{bmatrix} \begin{bmatrix} p\_{F\_x} \\ p\_{F\_y} \\ p\_{F\_z} \end{bmatrix} = \begin{bmatrix} [p\_1]\_\times p\_0 \\ [p\_2]\_\times p\_1 \\ [p\_3]\_\times p\_2 \\ [p\_1 0]\_\times p\_3 \end{bmatrix} . \tag{23}
$$

Similar to the face point, we assume that the volume point is located inside the hexahedron. The volume point *p<sup>V</sup>* = (*pV<sup>x</sup>* , *pV<sup>y</sup>* , *pV<sup>z</sup>* ) partitions 24 sub-tetrahedra of the cage. The volume of a single tetrahedron is given as

$$V\_{p\_{i,j}p\_{i+1,j}p\_{F\_j}} = (p\_{i+1,j} - p\_{i,j}) \times (p\_{F\_j} - p\_{i,j}) \cdot (p\_V - p\_{i,j}) / 6,\tag{24}$$

where *pF<sup>j</sup>* is *j*th face point of the hexahedron and *pi*,*<sup>j</sup>* is the *i*th corner point of the *j*th face. The volume point *p<sup>V</sup>* is obtained by solving the following least-squares problem:

$$
\begin{bmatrix}
[p\_{1,0}-p\_{0,0}] \times p\_{F\_0}-[p\_{1,0}] \times p\_{0,0} \\
[p\_{2,0}-p\_{1,0}] \times p\_{F\_0}-[p\_{2,0}] \times p\_{1,0} \\
[p\_{3,0}-p\_{2,0}] \times p\_{F\_0}-[p\_{3,0}] \times p\_{2,0} \\
[p\_{0,0}-p\_{3,0}] \times p\_{F\_0}-[p\_{0,0}] \times p\_{3,0} \\
\vdots \\
[p\_{0.5}-p\_{3.5}] \times p\_{F\_0}-[p\_{0.5}] \times p\_{3.5} \\
\end{bmatrix}
\begin{bmatrix}
p\_{V\_x} \\
p\_{V\_y} \\
p\_{V\_z}
\end{bmatrix} = 
\begin{bmatrix}
\vdots \\
\end{bmatrix}.
\tag{25}
$$

The robust face/volume points improve the numerical stability of the cage deformation. The cost function is a combination of the dissimilarity measurement and regularization functions.
