Local curve and surface detection in spatial data using Gaussian mixtures

Discovering the intrinsic surfaces in noisy spatial data is widely used in object detection and recognition in geoscience. Typical examples are identification of tectonic faults from seismic catalogs, volume smoothing from laser reliefs and tracking the path of landslides from GPS data. This paper aims to detect curves and surfaces in noisy point clouds represented by Gaussian mixture models (GMM). The number of components of the model is selected with information criteria and the parameters are estimated with likelihood and clustering methods. By using the concept of surface ridges of differential geometry, local curves and surfaces can be identified with the major axes of the GMM system. Next, by applying the mean shift algorithm and projection matrices of the local Hessian, an efficient estimator can be derived. Finally, a sliced 2D approach for 3D point clouds is applied to simulated and seismic data (Data and Matlab software are provided in the supplementary material).