Demisting the Hough Transform for 3D Shape Recognition and Registration

Abstract

In applying the Hough transform to the problem of 3D shape recognition and registration, we develop two new and powerful improvements to this popular inference method. The first, intrinsic Hough, solves the problem of exponential memory requirements of the standard Hough transform by exploiting the sparsity of the Hough space. The second, minimum-entropy Hough, explains away incorrect votes, substantially reducing the number of modes in the posterior distribution of class and pose, and improving precision. Our experiments demonstrate that these contributions make the Hough transform not only tractable but also highly accurate for our example application. Both contributions can be applied to other tasks that already use the standard Hough transform.

Appendix Proof of the Integer Nature of Vote Weights

Theorem 1

Given Eq. (3), an integer set of optimal values of $\varvec{\uptheta }$ exists, i.e. for which $\theta _{ij}\in \{0,1\}~\forall i,j$.

Proof

Let $\varvec{\uptheta }^{\prime }$ denote a globally optimal value of $\varvec{\uptheta }$, i.e. one that minimizes Eq. (3). Let us consider only the vote weights of the $i^\mathrm th$ feature, and assume the other vote weights are fixed at their optimal value, i.e. $\varvec{\uptheta }_j = \varvec{\uptheta }_j^{\prime } ~\forall j\ne i$. The objective function can then be written as

$$\begin{aligned} f(\varvec{\uptheta }_i)&= -\int _\mathcal H p(\mathbf y |\varvec{\uptheta }_i) \ln p(\mathbf y |\varvec{\uptheta }_i^{\prime }) ~\mathrm d \mathbf y ,\end{aligned}$$

(14)

$$\begin{aligned} p(\mathbf y |\varvec{\uptheta }_i)&= C(\mathbf y ) + \omega _i\sum _{j=1}^{J_i}\theta _{ij} K(\mathbf x _{ij},\mathbf y ), \end{aligned}$$

(15)

where $C(\mathbf y )$ is a function which is independent of $\varvec{\uptheta }_i$. Note that we have further assumed that the instance of $\varvec{\uptheta }_i$ in the $\ln p(\mathbf y |\varvec{\uptheta }_i)$ term in Eq. (14) is also at its optimal value, $\varvec{\uptheta }_i^{\prime }$. This allows us to rewrite the objective function as follows:

$$\begin{aligned} f(\varvec{\uptheta }_i)&= D - \sum _{j=1}^{J_i}\theta _{ij} a_{ij}\end{aligned}$$

(16)

$$\begin{aligned} a_{ij}&= \int _\mathcal H \omega _iK(\mathbf x _{ij},\mathbf y ) \ln p(\mathbf y |\varvec{\uptheta }_i^{\prime }) ~\mathrm d \mathbf y \end{aligned}$$

(17)

where $D$ is a constant, as are the values $a_{ij}$. Given the constraints of Eq. (2), minimizing Eq. (16) with respect to $\varvec{\uptheta }_i$, can always be achieved by setting $\theta _{ij} = 1$ for one $j$ for which $a_{ij}$ is largest, and setting all other weights to 0. In addition, Gibbs’ inequality Falk (1970) implies that Eq. (14) is minimized when $\varvec{\uptheta }_i= \varvec{\uptheta }_i^{\prime }$ (as we require them to be). Therefore the $i^\mathrm th$ feature must have an integer set of optimal weights. This argument can be applied to each feature independently. $\square$