Towards the Mathematic Formalization of Parametric Generalized Cylinders and Initial Results in Modeling 3D Image Data

Generalized cylinders (GCs) have been proposed in computer vision for object modeling in two dimensional images. However approaches so far have not investigated closed form solutions, based on a formal generic definition that covers every possible family of surfaces. In this work we derive the parametric definition of the torus-like surface that is generated when moving a planar closed curve along a three dimensional curve and provide the analytical solution for the distance of an arbitrary point from the surface. As an application, a family of surfaces that approximate a long bone is selected and analytical parametric equations are derived. The measure of match (MoM) between the parametric surface of GC and a three-dimensional (3D) image is defined, using the previously derived analytic parametric equation. The MoM is sampled along pairs of surface shape controlling parameters and its unimodality is evaluated for synthetic 3D image, under various levels of Gaussian noise. Finally, a global optimization technique is applied to determine shape controlling parameter values to fit the synthetic 3D image data.