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

Detecting geometrical edges magnitude of the surface of the point clouds is still an important problem that has a wide range of applications such as object recognition and simultaneous localization and mapping. The objective is to use the first-order (the Gradient) derivative of a field of vectors to detect the 3-D geometrical edges that is analogous to the Gradient of a field of scalars that is typically used for gray-scale edge detection. The problem is that the state of the art high-pass spatial filter of the Gradient operators are not applicable to the vector quantities because they are based on the componentwise Cartesian vector subtraction. The contributions are that: (1) we propose a novel mathematical Solid Vector subtraction operation that handle the vectors as one solid quantity, instead of separate layers of axes; (2) we propose a novel definition of the geometrical Step edge, Plane and Ramp geometrical areas; (3) based on our proposed Absolute Solid Vector subtraction operation, we propose a novel algorithm for the first-order (the Gradient) derivative of a field of vectors to detect geometrical edges magnitude, and to classify Plane segment, and Step and Ramp segment. (4) We do behavioral analyses on the geometrical Step, Plane, and Ramp areas; and (5) performance analyses on TUM data set; and comparison study on NYUD data set that shows our less complex Gradient geometrical edge detector is efficient.