Robust Point Set Registration Using Gaussian Mixture Models

The point set registration problem can be described as: for two finite size point sets ${M, S}$, where $M$ represents the moving “model” set and $S$ represents the fixed “scene” set, and both $M$ and $S$ are assumed to be subsets of a finite dimensional real vector space $\mathbb{R}^d$, and they can be of different sizes. The registration task involves estimating a mapping from $\mathbb{R}^d$ to $\mathbb{R}^d$ which yields the best alignment between the two sets $M$ and $S$. An important consideration here is that apart from the set of points being treated as a collection of isolated unstructured points, there is no assumption of additional information about the points (such as mesh structure, labels, features, etc.).

The authors propose to use continuous density functions to represent the aforementioned discrete point sets. The authors provide two arguments to support this reasoning:

• When using a continuous density function, the given point set (the one on which the registration has to be performed) can be interpreted as a statistical sample drawn from a continuous probability distribution of random point locations.

• By using a continuous density function to model these point sets, discrete optimization problems can be potentially converted to continuous optimization problems, and since the latter is easier to solve than the former, this is helpful.

It should be noted, however, that an inherent assumption while using this density based registration approach is that both the point sets have similar sampling rates, and the registration results degrade when this is not the case.

One such continuous function that be used for this task is the Gaussian Mixture Model (GMM). The probability density function of a general GMM is defined as

$$ p(\boldsymbol{x}) = \sum_{i=1}^k w_i \phi(\boldsymbol{x}|\mu_i,\Sigma_i) $$ where $w_i$ represents the weight of the $i^{th}$ Gaussian component and $\mu_i$ and $\Sigma_i$ represent the corresponding mean and the covariance matrix. The covariance matrix is given by $$ \phi(\boldsymbol{x}|\mu_i,\Sigma_i) = \frac{\text{exp}\big[-\frac{1}{2}(\boldsymbol{x}-\mu_i)^T\Sigma_i^{-1}(\boldsymbol{x} - \mu_i)\big]}{\sqrt{(2\pi)^d |\text{det}(\Sigma_i)|}} $$

For the chosen GMM, the following 3 settings were chosen:

• The number of Gaussian components is the number of the points in the point set, and all the components are weighted equally.

• For each component, the spatial location of each point denotes the mean vector.

• All components share the same spherical covariance matrix.

For estimating the similarity between two Gaussian mixtures, the $L2$ distance was chosen by the authors because there exists a closed-form expression for it betweeen Gaussian mixtures, meaning an efficient implementation of the proposed registration algorithm is possible. The $L2$ solution is also strongly related to the inherently robust estimator $L_2E$. Given two sets of points ${M, S}$, the goal of the registration method is to find the parameter $\theta$ of a parametrized spatial transformation family $T$ to minimize the following cost function.

$$ d_{L_{2}} (S, M, \theta) = \int \big(\text{gmm}(S) - \text{gmm}(T(M, \theta))\big)^2 dx $$

The authors demonstrate in the paper how the $L2$ distance similarity metric is robust to noise and outliers, and that it outperforms other metrics such as the maximum likelihood estimator.