Methods for detection and resolution of self-intersection in simulation of lattice-based deformation

Abstract

This dissertation presents self-intersection detection and resolution techniques in lattice-based deformation. The lattice-based deformation indirectly deforms an object by warping the space in which the object is embedded. This approach allows efficient simulation of high resolution deformable objects by adjusting a few number of control points of the lattice. A mapping function defined by the control points determines the deformed shape of the object by mapping from parameter space to world space. Different points in parameter space may be mapped to the same point in world space according to configurations of the lattice. The self-intersection due to the non-injective mapping causes visual anomalies such as surface infiltration and distortion, which deteriorate the visual realism in interactive applications such as computer games and medical simulation. Thus, an accurate and efficient method is required to detect and resolve the self-intersection problem in simulation of the lattice-based deformation. This dissertation derives a geometrically intuitive sufficient condition for ensuring that the mapping function is a homeomorphism (injective, onto, and continuous invertible) and there is no self-intersection. The mapping functions are defined by linear and quadratic B-Spline functions with the control points of hexahedral and cylindrical lattice cells. In a hexahedral lattice cell, the sufficient condition is satisfied if each trilinear function of the 27 hexahedrons derived from the lattice structure has a positive Jacobian determinant. The positivity is satisfied if the 32 tetrahedrons derived from the hexahedron have positive volumes. In a cylindrical lattice cell, the sufficient condition is satisfied if each trilinear function of the nine prism-shaped pentahedrons derived from the cell has a positive Jacobian determinant. The positivity is satisfied if the 12 tetrahedrons derived from the pentahedron have positive volumes. The self-intersection dete...