Analytical methods for polynomial weighted convolution surfaces with various kernels

Convolution surface has the advantage of being crease-free and bulge-free over other kinds of implicit surfaces. Among the various types of skeletal elements, line segments can be considered one of the most fundamental as they can approximate curve skeletons. This paper presents analytical solutions for convolving line segments with varying kernels modulated by polynomial weighted functions. We derive the closed-form formulae for most classical kernel functions, namely Gaussian, inverse linear, inverse squared, Cauchy, and quartic functions, and compare their computational complexity. These analytical solutions can be incorporated into existing implicit surface modeling systems for more convenient modeling of generalized cylindrical shapes. We demonstrate their high potentials for modeling and animating branching and tubular organic objects with some examples. We also propose a new competitive kernel function that has a smoothness control parameter.