Surface parametrization and shape description

Procedures for the parameterization and description of the surface of simply connected 3-D objects are presented. Critical issues for shape-based categorization and comparison of 3-D objects are addressed, which are generality with respect to object complexity, invariance to standard transformations, and descriptive power in terms of object geometry. Starting from segmented volume data, a relational data structure describing the adjacency of local surface elements is generated. The representation is used to parametrize the surface by defining a continuous, one-to-one mapping from the surface of the original object to the surface of a unit sphere. The mapping is constrained by two requirements, minimization of distortions and preservation of area. The former is formulated as the goal function of a nonlinear optimization problem and the latter as its constraints. Practicable starting values are obtained by an initial mapping based on a heat conduction model. In contract to earlier approaches, the novel parameterization method provides a mapping of arbitrarily shaped simply connected objects, i.e., it performs an unfolding of convoluted surface structures. This global parameterization allows the systematical scanning of the object surface by the variation of two parameters. As one possible approach to shape analysis, it enables us to expand the object surface into a series of spherical harmonic functions, extending the concept of elliptical Fourier descriptors for 2-D closed curves. The novel parameterization overcomes the traditional limitations of expressing an object surface in polar coordinates, which restricts such descriptions to star-shaped objects. The numerical coefficients in the Fourier series form an object-centered, surface-oriented descriptor of the object's form. Rotating the coefficients in parameter space and object space puts the object into a standard position and yields a spherical harmonic descriptor which is invariant to translations, rotations, and scaling of the object. The series can be truncated after a number of harmonics chosen according to the amount of detail to be expressed. The new methods are illustrated with simple 3-D test objects. Potential applications are recognition, classification, and comparison of convoluted surfaces or parts of surfaces of 3-D shapes, e.g., of anatomical objects segmented from multidimensional medical image data.