Surface Approximation by Means of Gaussian Process Latent Variable Models and Line Element Geometry

Abstract

The close relation between spatial kinematics and line geometry has been proven to be fruitful in surface detection and reconstruction. However, methods based on this approach are limited to simple geometric shapes that can be formulated as a linear subspace of line or line element space. The core of this approach is a principal component formulation to find a best-fit approximant to a possibly noisy or impartial surface given as an unordered set of points or point cloud. We expand on this by introducing the Gaussian process latent variable model, a probabilistic non-linear non-parametric dimensionality reduction approach following the Bayesian paradigm. This allows us to find structure in a lower dimensional latent space for the surfaces of interest. We show how this can be applied in surface approximation and unsupervised segmentation to the surfaces mentioned above and demonstrate its benefits on surfaces that deviate from these. Experiments are conducted on synthetic and real-world objects.

Special Issue “Statistical Data Modeling and Machine Learning with Applications II”

“…Article [9] presents new theoretical and applied results in stochastic processes in spatial kinematics and line geometry for modeling some characteristics of 3D surfaces. The authors introduced theoretical principles on line-element geometry, kinematic surfaces, and the Gaussian process latent variable model (GPLVM).…”

Gaussian processes for 3D measurements

“…Experiments are conducted on synthetic and real world objects. We published the findings of this chapter in [32].…”