3D Hough transform for sphere recognition on point clouds

Abstract

Three-dimensional object recognition on range data and 3D point clouds is becoming more important nowadays. Since many real objects have a shape that could be approximated by simple primitives, robust pattern recognition can be used to search for primitive models. For example, the Hough transform is a well-known technique which is largely adopted in 2D image space. In this paper, we systematically analyze different probabilistic/randomized Hough transform algorithms for spherical object detection in dense point clouds. In particular, we study and compare four variants which are characterized by the number of points drawn together for surface computation into the parametric space and we formally discuss their models. We also propose a new method that combines the advantages of both single-point and multi-point approaches for a faster and more accurate detection. The methods are tested on synthetic and real datasets.

Appendix A: Computing inlier threshold T to compare 1-Point and 4-Point HT methods

Given a point cloud with a certain inlier rate, we want to compute the inlier rate threshold T to identify whether it is more convenient to use a single-point approach or a four point for sphere detection.

According to [22], the error function for $H_1$ is:

$$\begin{aligned} \varepsilon _{H1} = \frac{1}{24}\cdot c^4 \mathrm {e}^{-c} \end{aligned}$$

(21)

where $c = s_{H1}\cdot N_\text {in}/N$. Similarly, we have:

$$\begin{aligned} \varepsilon _{H4} = b\cdot \mathrm {e}^{-b} \end{aligned}$$

(22)

where $b = s_{H4}\cdot (N_\text {in}/N)^4$. From Eqs. 21 and 22, we have:

$$\begin{aligned} s_{H1} = s_{H4}\cdot (N_\text {in}/N)^3\cdot \frac{c}{b} \approx s_{H4}\cdot (N_\text {in}/N)^3 \end{aligned}$$

(23)

Total costs of $H_1$ and $H_4$ are, respectively: $c_{H1}\cdot \overline{S}^3\cdot s_{H4} \cdot (N_\text {in}/N)^3$ and $c_{H4}\cdot s_{H4}$ thus, if we make those costs equal we can compute the threshold $T = N_\text {in}/N$ as:

$$\begin{aligned} T = \root 3 \of {\frac{c_{H4}}{c_{H1}\cdot \overline{S}^3}} \end{aligned}$$

(24)

Equation 24 introduces a criterion for choosing the best between $H_1$ and $H_4$ which is based on the estimated inlier rate $N_\text {in}/N$ of the input cloud. It also shows that the threshold $T$ depends on the medium size of the cells $\overline{S}$, a parameter that can be chosen a priori. For small values of $\overline{S}$ and very noisy clouds (e.g., with $N_\text {in}/N < 0.05$), the criterion would indicate $H_4$ as best choice ($N_\text {in}/N > T$), but with a high number of quadruples as input and thus with a long execution time. In contrast, for relatively big values of $\overline{S}$ one should prefer $H_1$, but there is a risk of voting for big spheres that are tangent to the target. CMPHT leverages this property by reducing the search area using $H_1$ with coarse quantization, and then refining the results with $H_4$.