Hough Transform and Laguerre Geometry for the Recognition and Reconstruction of Special 3D Shapes

We put the Hough transform, a method from Image Processing, into relation to Laguerre geometry, a concept of classical geometry, and study both concepts in the 3D case. It is shown how Laguerre geometry, which works in the set of oriented planes, is used in the detection of special shapes such as planes, spheres, rotational cones and cylinders, general cones and cylinders, and general developable surfaces. We perform shape recognition tasks by principal component analysis on a set of points in the so-called Blaschke model of Laguerre geometry. These points are Blaschke image points of estimated tangent planes at the given data points. Finally we present examples and show how the implementation also takes advantage of mathematical morphology on images, which are defined on meshes.