关于等位面的分形维数。
On the fractal dimension of isosurfaces.
机构信息
Computer and Information Science Department, The Ohio State University, USA.
出版信息
IEEE Trans Vis Comput Graph. 2010 Nov-Dec;16(6):1198-205. doi: 10.1109/TVCG.2010.182.
A (3D) scalar grid is a regular n1 x n2 x n3 grid of vertices where each vertex v is associated with some scalar value sv. Applying trilinear interpolation, the scalar grid determines a scalar function g where g(v) = sv for each grid vertex v. An isosurface with isovalue σ is a triangular mesh which approximates the level set g(-1)(σ). The fractal dimension of an isosurface represents the growth ;in the isosurface as the number of grid cubes increases. We define and discuss the fractal isosurface dimension. Plotting the fractal ;dimension as a function of the isovalues in a data set provides information about the isosurfaces determined by the data set. We present statistics on the average fractal dimension of 60 publicly available benchmark data sets. We also show the fractal dimension is highly correlated with topological noise in the benchmark data sets, measuring the topological noise by the number of connected components in the isosurface. Lastly, we present a formula predicting the fractal dimension as a function of noise and validate the formula with experimental results.
一个（3D）标量网格是一个规则的 n1 x n2 x n3 网格顶点，其中每个顶点 v 都与某个标量值 sv 相关联。通过应用三次线性插值，标量网格确定了一个标量函数 g，对于每个网格顶点 v，g(v) = sv。等 值 σ 的等 值面是一个三角网格，它近似于水平集 g(-1)(σ)。等 值面的分形维数表示增长；随着网格立方体数量的增加，等 值面的增长。我们定义并讨论分形等 值面维度。在数据集中等 值面维度绘制为函数，提供了有关数据集确定的等 值面的信息。我们展示了 60 个公开可用的基准数据集的平均分形维数的统计信息。我们还表明，分形维数与基准数据集中的拓扑噪声高度相关，通过等 值面中的连通分量数来衡量拓扑噪声。最后，我们提出了一个预测分形维数作为噪声函数的公式，并通过实验结果验证了该公式。
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