连续框架下二维标量场Morse-Smale复形分割

A Continuous Framework of Morse-Smale Complex Segmentation for Two-Dimensional Scalar Fields

挖掘拓扑结构是获取数据场关键信息的有效途径,Morse-Smale(MS)复形是二维标量场重要拓扑结构,传统方法均采用分片线性模型,利用离散Morse理论进行MS复形分割,精度较低,积分曲线呈锯齿状,特征冗余信息多,需经过烦琐的删减过程才能得到可用结果.为此提出连续框架下的二维标量场数据MS复形分割的方法.首先为二维标量场数据建立double ZP样条拟插值连续模型;然后利用连续Morse理论,在连续模型上提取临界点、精确计算Hessian矩阵对临界点分类;最后通过精确计算梯度信息构建积分曲线完成MS复形分割.本文采用的double ZP样条具有高阶连续性,满足连续Morse理论至少C2连续的条件,具有数学完备性,其上的各阶微分运算都是精确的,因此能保证提取高精度的特征点、有严格的分类标准和光滑的积分曲线,不需要大量的后处理过程.实验结果表明,与离散框架下的方法相比,该方法流程简单,能得到高精度、平滑性好的MS复形分割结果,能更清晰地呈现数据场蕴含的关键特征结构.

Mining topology structure is an effective way of obtaining key information of data fields, and Morse-Smale （MS） complex is an important topology of two dimensional scalar fields. The traditional methods wereall based on PL models and discrete Morse theory. The results generated from a discrete frame have lower accuracy,zigzag integral lines and much redundant feature information, which have to be cut repeatedly. This articleintroduced a new method of MS complex segmentation for two dimensional scalar data fields. Firstly, wereconstruct a quasi-interpolation model based on double ZP splines for the data field; Then we use the continuousMorse theory to extract feature points, to compute Hessian matrix for classifying features accurately;Finally, we calculate gradient information to build integral lines and MS complex. Double ZP splines adoptedin this paper have higher order of continuity and satisfy the condition of at least C2-continuity of Morse theory.The continuous framework is mathematically completeness, on which all differential computations are exact. Itcan extract high accurate feature points, and obey a strict standard of classification and generate smooth integrallines without massive post-processing. The experiments show that the flow of the method is simpler thanany of traditional methods. Its MS complex results are more accuracy and smoother, which can demonstrate key feature structures contained in the data field more clearly.