Convolution Surfaces Modeling

Convolution surfaces modeling is one kind of implicit surfaces modeling techniques that is widely used in Computer Graphics. Convolution surface is actually an isosurface of a scalar field defined by convolving skeleton primitives. While convolution surfaces are attractive for modeling natural phenomena and objects of complex evolving topology, the analytical evaluation of integrals of convolution models still poses some open problems.In this thesis, we give analytical convolution solutions for points, line segments, arcs, quadratic Bezier curves and triangle segments, and present a fast ray tracing algorithm of convolution surfaces.The thesis is organized as follows:In chapter one, we introduce background knowledge of convolution surfaces modeling, analyze the existing work and the application field of the convolution surfaces modeling, and present the innovative contribution of this thesis.In chapter two, using piecewise quartic polynomial as kernel function, we give analytical convolution solutions for points, line segments, arcs, quadratic Bezier curves and triangle segments. Especially, we give a closed form solution of convolving a triangle skeleton with kernel of a finite support for the first time. This lead a solid foundation for using a large number of triangles as skeletons of convolution surfaces. For cubic B-Spline skeleton, we first reduce the cubic B-spline curve to quadratic B-splines using constrained optimization methods, then sum the potentials of each quadratic B-spline skeleton to obtain the potential of the whole cubic B-spline skeleton.In chapter three, we present a fast ray tracing algorithm of convolution surfaces. First, octree partitioning is used to partition the whole space of the scene. Second, for each ray, we note down the intersection intervals between the ray and the bounding box of the skeletons. Third, we use a six degree polynomial to approximate the field function along the ray. Finally, substitute the ray equation into the polynomial equation and solve it. We consider the solution as theintersection between the ray and the convolution surfaces.Finally, we conclude this dissertation and discuss the future work.