*3.3. Multi-Resolution Cage Deformation Representation*

In this section, we present a cage deformation method using multi-resolution to represent a gradual deformation. In the registration process, the resolution of the cage determines the degree of freedom of shape deformation. With increasing resolution of the cage, the deformation model can represent a more detailed shape change. However, a dense cage has the disadvantage that it can lose the overall shape. A method for maintaining local shape features through multi-resolution or hierarchical data structures is, thus, used as a complementary method.

We assumed the generation of the cage based on a regular lattice grid. The primitive shape of the cage obtained from the lattice structure is a cube with eight vertices and six quadrilateral faces, where the points inside the cage can be represented as linear combinations of cage control vertices. As shown in Figure 4, the cage can be partitioned into the inner sub-regions, where the control points of this sub-region can be created using the control points of the outer region. We denote the vertex *v* using cage control points *P <sup>n</sup>* at the deformation depth of *n* as follows:

$$\upsilon = F(\upsilon; P^n) = \sum\_{i=0}^7 \varphi\_i(\upsilon) p\_i^n,\tag{15}$$

where *p n i* is *i*th cage control point at the deformation depth *n*. If we recursively acquire the sub-region of the region Ω that surrounds the source model, we can denote the higher-level control points using the lower level control points. The generalized formula presents the corner points of the sub-division, which are recursively described in the multi-resolution process below:

$$P^m = F(P^m; P^n) = \sum\_{i=0}^7 \varphi\_i(P^m) p\_{i\cdot\cdot\cdot}^n \tag{16}$$

where *<sup>m</sup>* <sup>&</sup>gt; *<sup>n</sup>* and *<sup>m</sup>*, *<sup>n</sup>* <sup>∈</sup> <sup>N</sup>. Thus, if we represent the shape using a chain of cage deformations, the deformed vertex *v*¯, with respect to the level *n* deformation, is

$$\mathfrak{v} = F(v; P^n, \partial P^n) = \sum\_{i=0}^7 \varphi\_i(v) (p\_i^n + \partial p\_i^n). \tag{17}$$

Similarly, the deformed vertex *v*¯ by level *n* deformation after level *n* − 1 deformation is

$$\begin{split} \bar{v} &= F(v; P^n, P^{n-1}) \\ &= \sum\_{i=0}^{7} \varphi\_i(v) \left( \sum\_{j=0}^{7} \varphi\_j(p\_i) (p\_j^{n-1} + \partial p\_j^{n-1}) + \partial p\_i^n \right) . \end{split} \tag{18}$$

To cooperate with the gradient descent, we reformulate *<sup>∂</sup> ∂p<sup>i</sup> d*(*v<sup>s</sup>* , *v<sup>t</sup>* , *P*) as a multi-resolution process. The partial derivative of the given cost function at the (*n* − 1) th level is given as follows:

$$\begin{split} \frac{\partial}{\partial p\_i^{n-1}} d(v\_s, v\_t, P) &= \frac{\partial}{\partial p\_i^{n-1}} \| \sum\_{i=0}^7 \varphi\_i(v\_s) p\_i^n - v\_t \|^2 \\ &= \frac{\partial}{\partial p\_i^{n-1}} \| \sum\_{i=0}^7 \varphi\_i(v\_s) \sum\_{j=0}^7 \varphi\_j(p\_i^n) p\_j^{n-1} - v\_t \|^2 \\ &= 2 \| v\_s - v\_t \| \cdot \| \sum\_{i=0}^7 \varphi\_i(v\_s) \varphi\_j(p\_i^n) \| . \end{split} \tag{19}$$

Modification of the control point *∂p<sup>i</sup>* in the multi-resolution cage sub-division is carried out by

$$
\partial p\_i = \partial p\_i^n + \partial p\_i^{n-1} + \dots + \partial p\_i^1. \tag{20}
$$
