GPU-based radial view-based culling for continuous self-collision detection of deformable surfaces

Abstract

We propose a graphics processing unit-based approach to accelerate the radial view-based culling method for continuous self-collision detection of deformable surfaces. The deformable surfaces may have small round-shaped holes and ghost triangles are used to fill the holes. We identify the key processes of the radial view-based culling method, including triangle classification, traversal of bounding volume hierarchies and handling violated triangles (i.e., the triangles intersecting with ghost triangles). We propose efficient parallel processing techniques to perform these key processes on a programmable graphics unit. We have evaluated our proposed approach on several examples. Experimental results show that our approach significantly cuts down the cost of the key processes of the radial-based culling method, compared with the serial implementation on CPU.

Appendices

Appendix A. The pseudo-code of algorithms

Algorithm 1 is the entire process of CSCD. Algorithm 2 is the pseudo-code for the boundary handling of ghost triangles using a stack.

Appendix B. Characteristic functions

In sake of completeness, we list the characteristic functions which are defined in [32]. Denote a triangle of deformable mesh $M$ as $T(\mathbf {p}_0(t), \mathbf {p}_1(t), \mathbf {p}_2(t))$.

Let $\mathbf {q}$ be an observer point. The characteristic function $\phi (T(t), \mathbf {q}(t))$ is defined as

$$\begin{aligned} \phi (T(t), \mathbf {q}(t)) = \mathbf {n}_T(t) \cdot (\mathbf {p}_0(t) - \mathbf {q}(t)), \end{aligned}$$

(1)

where $\mathbf {n}_T(t)$ is the continuous normal vector of triangle $T$, i.e., $\mathbf {n}_T(t) = (\mathbf {p}_1(t)-\mathbf {p}_0(t))\times (\mathbf {p}_2(t)-\mathbf {p}_0(t))$. The normal vector of a triangle points outside of the deformable mesh.

Let $\overline{q_0q_1}$ be an observer line segment. A vertex, e.g., $\mathbf {p}_0(t)$, of triangle $T$ is projected onto $\overline{q_0q_1}$. We compute the projected component $u(t)$ of $\mathbf {p}_0(t)$ as follows:

$$\begin{aligned} u(t) = \frac{(\mathbf {p}_0(t) - \mathbf {q}_0(t))\cdot (\mathbf {q}_1(t)-\mathbf {q}_0(t))}{||\mathbf {q}_1(t)-\mathbf {q}_0(t)||^2}. \end{aligned}$$

The range of $u(t)$ is evaluated as follows:

• 1. Case I. If $\exists t \in [0, \Delta t]\ \mathrm{s.t.}\ u(t) \ge 1$, set $\mathbf {p}'(t) = \mathbf {q}_1(t)$.

• 2. Case II. If $\exists t \in [0, \Delta t]\ \mathrm{s.t.}\ u(t) \le 0$, set $\mathbf {p}'(t) = \mathbf {q}_0(t)$.

• 3. Case III. If $\exists t \in [0, \Delta t]\ \mathrm{s.t.}\ 0<u(t)<1$, set $\mathbf {p}'(t) = \mathbf {q}_0(t)+u(t)(\mathbf {q}_1(t)-\mathbf {q}_0(t))$.

The characteristic function of triangle $T$ is defined as:

$$\begin{aligned} \phi (T(t), \overline{q_0q_1}(t)) = \mathbf {n}_T(t) \cdot (\mathbf {p}_0(t) - \mathbf {p}'(t)). \end{aligned}$$

(2)

To evaluate the sign of the characteristic functions, readers are referred to [32] for details.