Three-Dimensional Laplacian Spatial Filter of a Field of Vectors for Geometrical Edges Magnitude and Direction Detection in Point Cloud Surfaces

Abstract

Detecting geometrical features and edges magnitudes and directions in 3-D surfaces remains an important problem that has a wide range of application such as 3-D object recognition and detection. The objective of the current research is to detect geometrical edges magnitude and direction of the surface of the three-dimensional images using the high-pass spatial filters of the second-order (the Laplacian) derivative that handle the vectors as one solid quantity, instead of separate layers of axes, in a way similar to using the second order derivative to detect intensity edges in the gray-scale images. The problem of the current research is that the state of the art high-pass spatial filter of the Laplacian operators are not applicable to the vector quantities because they use the component-wise Cartesian vector subtraction. In the current research, based on Solid Vector Subtraction operation, we: (1) propose a novel definition for the second-order (the Laplacian) derivative of a field of vectors geometrical edges magnitude and direction detector, where the field is the group of the pixels of the 3-D image and the vectors are the normal vectors to the surface of that field; (2) do behavioral analyses on the geometrical Step, Plane, and Ramp areas; and (3) performance analyses on TUM data set, and comparison study on NYUD data set that shows our Laplacian edge detector is efficient.