# Direct Fitting of Gaussian Mixture Models

## TL;DR

This paper introduces a method for directly fitting Gaussian Mixture Models to 3D meshes, improving shape representation and registration accuracy over traditional point cloud approaches.

## Contribution

It presents a novel formulation for fitting GMMs directly to triangular meshes, enhancing model quality and registration performance for 3D geometric data.

## Key findings

- Improved 3D registration accuracy for meshes and RGB-D frames.
- Enables fitting higher-quality GMMs with diverse initializations.
- Applicable to various geometric objects and sensor data.

## Abstract

When fitting Gaussian Mixture Models to 3D geometry, the model is typically fit to point clouds, even when the shapes were obtained as 3D meshes. Here we present a formulation for fitting Gaussian Mixture Models (GMMs) directly to a triangular mesh instead of using points sampled from its surface. Part of this work analyzes a general formulation for evaluating likelihood of geometric objects. This modification enables fitting higher-quality GMMs under a wider range of initialization conditions. Additionally, models obtained from this fitting method are shown to produce an improvement in 3D registration for both meshes and RGB-D frames. This result is general and applicable to arbitrary geometric objects, including representing uncertainty from sensor measurements.