双调和B样条的基础理论及其应用研究

Spline functions are ubiquitous in a large variety of scientific and engineering disciplines. The most fundamental spline functions are piecewise polynomials, that are founded upon univariate B-splines and/or multivariate B-splines. This project aims to significantly advance the research frontier of B-splines by articulating a brand new methodology deviating from prior B-spline paradigms, and devising a novel computational framework. At the theoretic level, our technical foci are on the novel theory that bridges multivariate biharmonic B-splines and Green’s functions towards dual representations, which have heretofore remained unexplored in data representation, modeling, processing, and analysis. At the algorithmic level, in sharp contrast to conventional B-splines, the knots of biharmonic B-splines are totally free without any constraint. They do not rely on any user-specified planar/volumetric parameterization and are free of singularity. Yet, their bases are constructed based on finite difference approximation, which is unstable when the knots are distributed unevenly. We devise novel optimization-centric algorithms for bases construction in order to improve the algorithm robustness. Moreover, current bases construction is computationally expensive, we shall take advantage of the intrinsic connection between biharmonic B-splines and Green’s functions towards better design and development of more effective algorithms. We envision that the computation of biharmonic B-splines could be significantly simplified by Green’s functions, hence enhancing their performance and flexibility with greater potentials never explored before. At the application level, we plan to integrate biharmonic B-splines with numerical solutions of popular partial differential equations (PDEs), and our goal is to expand biharmonic B-spline application scope for image vectorization, distance metric and clustering on graph far beyond the traditional boundary of geometry computing.

样条函数已广泛应用于多种科学和工程领域。最基本的单变量B样条和/或多元B样条函数可用分段多项式表示，该项目旨在阐明一个全新的B样条范式，并设计一种新的计算框架，来进一步推进B样条的前沿研究。在理论层面，拟采用全新的隐式表达来描述双调和B样条函数，架起多元双调和B样条和格林函数的桥梁。在算法层面，我们将设计以优化为中心的基函数构建算法来提高算法的鲁棒性，设计没有任何约束，完全自由的双调和B样条的节点，使样条不依赖于任何用户指定的平面/立体参数化，并且无奇异性。同时，我们将充分利用双调和样条和格林函数的内在联系，使双调和B样条的计算可以借助格林函数显著简化，从而提高其性能和灵活性。在应用层面，我们计划把双调和B样条与流形偏微分方程的数值解相结合，使其能够超越几何计算的传统界限，应用于图像矢量化，离散图上距离度量及聚类，来进一步扩大双调和B样条的实际应用范围。

本项目提出了基于格林函数的双调和B样条函数计算方法，从而简化双调和B样条函数计算。基于格林函数相关理论，我们提出格林聚类方法，模拟狭管效应，用格林函数完成数据嵌入，我们发展了泊松曲面算法，提出可微泊松重建算法，无需输入定向法向量，完成三维表面重建。在更多应用方面，提出泊松矢量图模型，能够更好地控制光照明暗变化，并将其用于人脸图像视频编辑，以及基于草图的三维建模。提出基于热方程的时变扩散曲线，模拟基于笔画的绘画过程。本项目共发表学术论文13篇，其中在ACM SIGGRAPH、IEEE TVCG发表CCF A类论文4篇，发表CCF B类论文2篇，申请国家发明专利1项。