【作者】 张海林；

【导师】 金小刚；

【作者基本信息】 浙江大学 ， 应用数学， 2006， 硕士

【摘要】 卷积曲面造型是计算机图形学中广泛使用的一种隐式曲面造型方法。卷积曲面实质上是骨架通过卷积方式所定义生成的标量场的一个等值面。虽然卷积曲面在模拟自然现象和拓扑结构复杂、易变物体时具有很大的优势,但能否解析表达骨架的势函数,始终是我们面对的一个难题。 本文解析给出点,直线段,圆弧,二次曲线和三角面片等骨架的势函数,并且提出一种快速的光线跟踪绘制卷积曲面算法。 本文内容和整体架构如下: 在第一章中,主要介绍卷积曲面的背景知识及其应用领域,分析前人工作以及叙述本文主要研究工作和内容。 在第二章中,我们采用截断多项式函数为核函数,解析的给出点、直线段、圆弧、二次曲线和三角面片等骨架的势函数。特别是首次解析表达出具有局部支撑核函数下的三角面片骨架势函数,为采用大量的三角面片作为卷积曲面骨架奠定了基础。对于三次B-样条曲线骨架,我们提出采用降阶的方法,在给定误差下,将三次B-样条转化为二次B-样条,然后将二次B-样条分段转化为二次Bézier曲线,最后将各个二次Bézier曲线势函数相加。相加的结果近似表达原曲线的势函数。 在第三章中,我们提出了一种较快速的光线跟踪绘制卷积曲面的算法。首先,运用空间八叉树剖分技术对整个场景进行剖分。然后,对每条光线计算其和当前格子中每个骨架包围盒的交点,记录下相交区间。再次,用多项式函数来逼近当前区间上的势函数。最后,将光线方程带入到多项式方程,求解上面方程,将求得交点作为光线和曲面的交点。 在第四章中,我们对全文进行总结,并提出了未来工作的展望。更多还原

【Abstract】 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.更多还原