Shape-controllable geometry completion for point cloud models

Abstract

Geometry completion is an important operation for generating a complete model. In this paper, we present a novel geometry completion algorithm for point cloud models, which is capable of filling holes on either smooth models or surfaces with sharp features. Our method is built on the physical diffusion pattern. We first decompose each pass hole-boundary contraction into two steps, namely normal propagation and position sampling. Then the normal dissimilarity constraint is incorporated into these two steps to fill holes with sharp features. Our algorithm implements these two steps alternately and terminates until generating no new hole boundary. Experimental results demonstrate its feasibility and validity of recovering the potential geometry shapes.

Appendix: Computing the equilibrium position

To compute the equilibrium position $O_{b_i^{'}}$ for a former pass hole-boundary point $b_i^{'}$, as shown in Fig. 3, we introduce a point $q_0$ whose position just exactly locates in $O_{b_i^{'}}$, as depicted in Fig. 18. The equilibrium position of $q_0$ on the direction of vector $\overrightarrow{O_{b_i^{'}}b_i^{'}}$ must have the same position with point $b_i^{'}$, that is to say $O_{q_0}$ coincides with the position of $b_i^{'}$. Taking the elastic force received by $O_{q_0}$ from $b_i^{'}$ into account, its value should be the positive maximum (equals 1, corresponding to the maximum repulsive force) according to the definition of the elastic force in Eq. (5). The overlap positions can be seen as the extremely close distance between $O_{q_0}$ and $b_i^{'}$. Without loss of generality, we assign this repulsive force along the vector $\overrightarrow{b_i^{'}O_{b_i^{'}}}$. Therefore, we have $r_{b_i^{'}}(O_{q_0})=1$, specifically $1.0-\exp \left( {\big | O_{q_0} - O_{b_i^{'}} \big |}_{\overrightarrow{b_i^{'}O_{b_i^{'}}}}{/}{ \sigma _r^2} \right) =1$. By substituting $O_{q_0}$ with $b_i^{'},$ we have $\exp \left( {\big | b_i^{'} - O_{b_i^{'}} \big |}_{\overrightarrow{b_i^{'}O_{b_i^{'}}}}{/}{ \sigma _r^2} \right) =0$.

Our purpose is to compute the equilibrium position $O_{b_i^{'}}$ for point ${b_i^{'}}$. So, we need to take the logarithm for the above equation. However, the right side of this equation equals zero and cannot enforce a logarithm operation immediately. For the sake of numerical computing, we use a small constant $10^{-4}$ to approximate instead of zero and make our computation feasible. Finally, we can use the following equation to compute $O_{b_i^{'}}$ for $b_i^{'}$ if the parameter $\sigma _r$ is assigned,

$$\begin{aligned} \exp \left( {\big | b_i^{'} - O_{b_i^{'}} \big |}_{\overrightarrow{b_i^{'}O_{b_i^{'}}}}{/}{ \sigma _r^2} \right) =10^{-4}. \end{aligned}$$

Fig. 18

Computing the equilibrium position $O_{b_i^{'}}$ for a point $b_i^{'}$