Medial-axis-driven shape deformation with volume preservation

Abstract

The medial axis is a natural skeleton for shapes. However, it is rarely used in the existing skeleton-based shape deformation techniques. In this paper, we propose a novel medial-axis-driven skin surface deformation algorithm with volume preservation property. Specifically, an as-rigid-as-possible deformation scheme is used to deform the medial axis so that its local transform is as close as possible to a rigid transform. We maintain surface features of the deformed shape based on an implicit skinning method. Our experiments show that the proposed algorithm effectively preserves the volume of deformed shape and addresses the bending and twisting problems associated with traditional skeleton-based shape deformation techniques.

Appendices

Appendix: Footprint sphere on medial slab

For the medial slab ${C}_{l}$ of a triangle face ${f}_{ijk}$, the footprint sphere of point p on the slab is the sphere ${m}_{n}=\{{{c}}_{n}, {r}_{n}\}$ with minimal scalar function ${E}_{m}({m}_{n})$ defined as:

$$\begin{aligned} {E}_{m}({m}_{n})=||{p}-{{c}}_{n}||^{2} - {r}_{n}^{2}. \end{aligned}$$

(16)

Suppose $\{{{c}}_{i}, {r}_{i}\}$, $\{{{c}}_{j}, {r}_{j}\}$, and $\{{{c}}_{k}, {r}_{k}\}$ are the three spheres defining this medial slab, then we have ${{c}}_{n}={\beta }_{i}{{c}}_{i}+{\beta }_{j}{{c}}_{j}+(1-{\beta }_{i}-{\beta }_{j}){{c}}_{k}$, and ${r}_{n}={\beta }_{i} {r}_{i} + {\beta }_{j} {r}_{j} + (1-{\beta }_{i}-{\beta }_{j}) {r}_{k}$. The scalar function ${E}_{m}({m}_{n})$ can be written as:

$$\begin{aligned} {E}_{m}({\beta }_{i},{\beta }_{j})= & {} ||{p} - ({\beta }_{i}{{c}}_{i} +{\beta }_{j}{{c}}_{j}+(1-{\beta }_{i}-{\beta }_{j}){{c}}_{k}) ||^{2} \nonumber \\&- ({\beta }_{i} {r}_{i}+ {\beta }_{j} {r}_{j} + (1-{\beta }_{i}-{\beta }_{j}) {r}_{k})^{2}\nonumber \\= & {} {A}_{i} {\beta }_{i}^{2} + {A}_{j} {\beta }_{j}^{2} + {A}_{k} (1-{\beta }_{i}-{\beta }_{j})^{2} \nonumber \\&+ 2{B}_{ij} {\beta }_{i} {\beta }_{j}+ 2{B}_{ik} {\beta }_{i} (1-{\beta }_{i}-{\beta }_{j}) \nonumber \\&+2{B}_{jk} {\beta }_{j}(1-{\beta }_{i}-{\beta }_{j}), \end{aligned}$$

(17)

where

$$\begin{aligned} {A}_{i}= & {} ({p} - {{c}}_{i})^{T}({p}-{{c}}_{i}) - {r}_{i}^{2}, \nonumber \\ {A}_{j}= & {} ({p} - {{c}}_{j})^{T}({p}-{{c}}_{j}) - {r}_{j}^{2}, \nonumber \\ {A}_{k}= & {} ({p} - {{c}}_{k})^{T}({p}-{{c}}_{k}) - {r}_{k}^{2}, \nonumber \\ {B}_{ij}= & {} ({p} - {{c}}_{i})^{T}({p}-{{c}}_{j}) - {r}_{i} {r}_{j}, \nonumber \\ {B}_{ik}= & {} ({p} - {{c}}_{i})^{T}({p}-{{c}}_{k}) - {r}_{i} {r}_{k}, \nonumber \\ {B}_{jk}= & {} ({p} - {{c}}_{j})^{T}({p}-{{c}}_{k}) - {r}_{j} {r}_{k}. \end{aligned}$$

(18)

The Hessian matrix of ${E}_{m}({\beta }_{i},{\beta }_{j})$ is:

$$\begin{aligned}&{\mathrm {H}}({E}_{m}) = \left( \begin{array}{cc} \frac{\partial ^{2} {E}_{m}}{\partial {\beta }_{i}^{2}} &{} \frac{\partial ^{2} {E}_{m}}{\partial {\beta }_{i} \partial {\beta }_{j}}\\ \frac{\partial ^{2} {E}_{m}}{\partial {\beta }_{j} \partial {\beta }_{i}} &{} \frac{\partial ^{2} {E}_{m}}{\partial {\beta }_{j}^{2}} \end{array}\right) \nonumber \\&\quad =2\left( \begin{array}{cc} {A}_{i} \,+\, {A}_{k} \,-\, 2{B}_{ik} &{} {A}_{k} \,+\, {B}_{ij} \,-\, {B}_{ik} \,-\, {B}_{jk}\\ {A}_{k} \,+\, {B}_{ij} \,-\, {B}_{ik} \,-\, {B}_{jk} &{} {A}_{j} \,+\, {A}_{k} \,-\, 2{B}_{jk} \end{array}\right) .\nonumber \\ \end{aligned}$$

(19)

Let us denote:

$$\begin{aligned} {H}_{11}= & {} {A}_{i} + {A}_{k} - 2{B}_{ik}, \nonumber \\ {H}_{12}= & {} {A}_{k} + {B}_{ij} - {B}_{ik} - {B}_{jk}, \nonumber \\ {H}_{22}= & {} {A}_{j} + {A}_{k} - 2{B}_{jk}. \end{aligned}$$

(20)

since two Euclidean vectors v and ${\mathrm {\textit{w}}}$ satisfy the law of cosines:

$$\begin{aligned} ({{v}}-{\mathrm {\textit{w}}})^{T}({{v}}-{\mathrm {\textit{w}}})={{v}}^{T}{{v}} +{\mathrm {w}}^{T}{\mathrm {\textit{w}}}-2{{v}}^{T}{\mathrm {\textit{w}}}, \end{aligned}$$

(21)

we can rewrite ${H}_{11}$, ${H}_{12}$, and ${H}_{22}$ as:

$$\begin{aligned} {H}_{11}= & {} ({{c}}_{i}-{{c}}_{k})^{T}({{c}}_{i}-{{c}}_{k}) -({r}_{i}-{r}_{k})^{2}, \nonumber \\ {H}_{12}= & {} ({{c}}_{i}-{{c}}_{k})^{T}({{c}}_{j}-{{c}}_{k}) - ({r}_{i}-{r}_{k})({r}_{j}-{r}_{k}), \nonumber \\ {H}_{22}= & {} ({{c}}_{j}-{{c}}_{k})^{T}({{c}}_{j}-{{c}}_{k}) -({r}_{j}-{r}_{k})^{2}. \end{aligned}$$

(22)

If we denote the 4-dimensional vectors ${{v}}_{i}$ and ${{v}}_{j}$ in Minkowski space as:

$$\begin{aligned} {{v}}_{i}= & {} \left[ ({{c}}_{i}-{{c}}_{k})^{T}, ({r}_{i}-{r}_{k})\right] ^{T}, \nonumber \\ {{v}}_{j}= & {} \left[ ({{c}}_{j}-{{c}}_{k})^{T}, ({r}_{j}-{r}_{k})\right] ^{T}, \end{aligned}$$

(23)

then ${H}_{11}$, ${H}_{12}$, and ${H}_{22}$ can be written using Minkowski inner product $g(\cdot ,\cdot )$ as:

$$\begin{aligned} {H}_{11}= & {} g({{v}}_{i},{{v}}_{i}), \nonumber \\ {H}_{12}= & {} g({{v}}_{i},{{v}}_{j}), \nonumber \\ {H}_{22}= & {} g({{v}}_{j},{{v}}_{j}). \end{aligned}$$

(24)

As shown in Fig. 3 of the paper, as long as the two spheres ${m}_{i}$ and ${m}_{k}$ are not arranged as “one inside another”, then we can guarantee ${H}_{11}>0$. Similarly ${H}_{22}>0$ also holds for general valid configurations of ${m}_{j}$ and ${m}_{k}$.

Since both $g({{v}}_{i},{{v}}_{i}) >0$ and $g({{v}}_{j},{{v}}_{j}) >0$, i.e., ${{v}}_{i}$ and ${{v}}_{j}$ are spacelike vectors in Minkowski space, they satisfy the usual Cauchy-Schwarz inequality (see Formula 3 of [14]):

$$\begin{aligned} g({{v}}_{i},{{v}}_{i}) g({{v}}_{j},{{v}}_{j}) \ge g ({{v}}_{i},{{v}}_{j})^{2}, \end{aligned}$$

(25)

with equality holds when ${{v}}_{i}$ and ${{v}}_{j}$ are co-linear. For a general medial slab, ${{v}}_{i}$ and ${{v}}_{j}$ will not be co-linear; thus, we have $g({{v}}_{i},{{v}}_{i}) g({{v}}_{j},{{v}}_{j}) - g({{v}}_{i},{{v}}_{j})^{2} > 0$.

Thus the determinant of Hessian ${\mathrm {H}}({E}_{m})$ is positive:

$$\begin{aligned} \left| {\mathrm {H}}({E}_{m})\right| = 2({H}_{11}{H}_{22}-{H}_{12}^{2}) > 0, \end{aligned}$$

(26)

the scalar function ${E}_{m}$ will have a unique global minimum, and thus the footprint sphere can be solved from minimizing Eq. (17) with $\left[ \frac{\partial {E}_{m}}{\partial {\beta }_{i}}, \frac{\partial {E}_{m}}{\partial {\beta }_{j}}\right] =[0,0]$.

Minimizing ARAP deformation energy

For our medial mesh ${M}_{s}$, the ARAP deformation energy is defined as:

$$\begin{aligned} {E}_{d} (\{{{R}}_{j},{\mathrm {t}}_{j}\}, \{\tilde{{{c}}}_{i}\}) = \sum _{i} \sum _{j \in \mathscr {N}(i)}^{n} ||{{R}}_{j} {{c}}^{0}_{ij} + {\mathrm {t}}_{j} - \tilde{{{c}}}_{i} ||^{2}, \end{aligned}$$

(27)

where $\mathscr {N}(i)$ denotes the set of indices of medial primitives $\{{C}_{j}|j\in \mathscr {N}(i)\}$ which are connected with the medial sphere ${m}_{i}$. ${{R}}_{j}$ and ${\mathrm {t}}_{j}$ are the rotation matrix and translation vector for medial primitive ${C}_{j}$. $\tilde{{{c}}}_{i}$ is the deformed position of the medial sphere ${m}_{i}$, and ${{c}}^{0}_{ij}$ is its corresponding center position in ${C}_{j}$ at the rest pose.

Each medial primitive has its local coordinate system with origin on the center of the primitive. So the translation vector can be simply: ${\mathrm {t}}_{j}=\frac{1}{3}(\tilde{{{c}}}_{i}+\tilde{{{c}}}_{j}+\tilde{{{c}}}_{k})$ for triangle ${f}_{ijk}$, and ${\mathrm {t}}_{j}=\frac{1}{2}(\tilde{{{c}}}_{i}+\tilde{{{c}}}_{j})$ for edge ${e}_{ij}$ on the medial mesh. To minimize ${E}_{d}$ in Eq. (27), the rotation matrix ${{R}}_{j}$ of all primitives and the medial sphere central positions $\tilde{{{c}}}_{i}$ need to be solved in turn iteratively.

In each iteration, if we first fix the value of $\tilde{{{c}}}_{i}$, we can minimize ${E}_{d}$ and solve the rotation matrices ${{R}}_{j}$ as follows. Let us denote $\tilde{{{c}}}_{ij} = \tilde{{{c}}}_{i} - {\mathrm {t}}_{j}$, for each primitive $j \in \mathscr {N}(i)$. Then:

$$\begin{aligned} {E}_{d}= & {} \sum _{i} \sum _{j \in \mathscr {N}(i)}^{n} ||{{R}}_{j} {{c}}^{0}_{ij} + {\mathrm {t}}_{j} - \tilde{{{c}}}_{i} ||^{2} \nonumber \\= & {} \sum _{j} \sum _{i \in \mathscr {V}(j)} ||{{R}}_{j} {{c}}^{0}_{ij} - \tilde{{{c}}}_{ij} ||^{2} \nonumber \\= & {} \sum _{j} \sum _{i \in \mathscr {V}(j)} \left( {{{c}}^{0}_{ij}}^{T}{{c}}^{0}_{ij} - 2{{{c}}^{0}_{ij}}^{T} {{R}}_{j}^{T} \tilde{{{c}}}_{ij} + \tilde{{{c}}}_{ij}^{T}\tilde{{{c}}}_{ij}\right) , \end{aligned}$$

(28)

where $\mathscr {V}(j)$ is the set of vertices for primitive j. Since ${{{c}}^{0}_{ij}}^{T}{{c}}^{0}_{ij}$ and $\tilde{{{c}}}_{ij}^{T}\tilde{{{c}}}_{ij}$ are fixed for now, minimizing ${E}_{d}$ is equivalent to maximizing the following ${F}_{d}$:

$$\begin{aligned} {F}_{d}= & {} \sum _{j} \sum _{i \in \mathscr {V}(j)} \left( {{{c}}^{0}_{ij}}^{T} {{R}}_{j}^{T} \tilde{{{c}}}_{ij}\right) \nonumber \\= & {} \hbox {trace}\left( \sum _{j} \sum _{i \in \mathscr {V}(j)} \left( {{{c}}^{0}_{ij}}^{T} {{R}}_{j}^{T} \tilde{{{c}}}_{ij}\right) \right) \nonumber \\= & {} \sum _{j} \sum _{i \in \mathscr {V}(j)} \hbox {trace}\left( {{{c}}^{0}_{ij}}^{T} {{R}}_{j}^{T} \tilde{{{c}}}_{ij}\right) \nonumber \\= & {} \sum _{j} \sum _{i \in \mathscr {V}(j)} \hbox {trace}\left( {{R}}_{j}^{T} \tilde{{{c}}}_{ij}{{{c}}^{0}_{ij}}^{T}\right) \nonumber \\= & {} \sum _{j} \hbox {trace}\left( {{R}}_{j}^{T} \sum _{i \in \mathscr {V}(j)} \tilde{{{c}}}_{ij}{{{c}}^{0}_{ij}}^{T}\right) \nonumber \\= & {} \sum _{j} \hbox {trace}\left( {{R}}_{j}^{T} {\mathrm {A}}_{j}\right) \nonumber \\= & {} \sum _{j} {F}_{d}^{j}, \end{aligned}$$

(29)

where the matrix ${\mathrm {A}}_{j} \,=\, \sum \nolimits _{i \in \mathscr {V}(j)} \tilde{{{c}}}_{ij}{{{c}}^{0}_{ij}}^{T}$, and ${F}_{d}^{j} \,=\, \hbox {trace}({{R}}_{j}^{T} {\mathrm {A}}_{j})$. Since each ${{R}}_{j}$ is independent of each other, we can maximize each ${F}_{d}^{j}$ individually. We decompose ${\mathrm {A}}_{j}$ using singular value decomposition (SVD): ${\mathrm {A}}_{j} = {{U}}_{j} {{D}}_{j} {{V}}_{j}^{T}$. Then ${F}_{d}^{j} = \hbox {trace}({{R}}_{j}^{T} {{U}}_{j} {{D}}_{j} {{V}}_{j}^{T}) = \hbox {trace}({{V}}_{j}^{T}{{R}}_{j}^{T} {{U}}_{j} {{D}}_{j})$. Since ${{D}}_{j}$ is a diagonal matrix, the trace achieves maximum when ${{V}}_{j}^{T}{{R}}_{j}^{T} {{U}}_{j}$ is an identity matrix. So ${{R}}_{j}$ can be solved as:

$$\begin{aligned} {{R}}_{j} = {{U}}_{j} {{V}}_{j}^{T}. \end{aligned}$$

(30)

After getting the rotation matrices for primitives, we can assume ${{R}}_{j}$ to be fixed, and minimize ${E}_{d}$ by solving for medial sphere center positions ${\tilde{{{c}}}}_{i}$. They can be simply solved as:

$$\begin{aligned} \tilde{{{c}}}_{i} = \frac{1}{|{\mathscr {N}}(i)|} \sum _{j \in {\mathscr {N}}(i)} \left( {{R}}_{j} {{c}}^{0}_{ij} + {\mathrm {t}}_{j}\right) , \end{aligned}$$

(31)

where $|\mathscr {N}(i)|$ is the number of primitives that are connected to medial sphere i.

It should be noted that in each iteration, we compute the optimal ${{R}}_{j}$ and $\tilde{{{c}}}_{i}$ in turn, and each of these steps will decrease the energy ${E}_{d}$ until converged.