If we wish to integrate a function $h|\Omega\subset\Re^{n}\to\Re$ along a single $T$-level surface of a function $\psi |\Omega\subset\Re^{n}\to\Re$, then a number of different methods for extracting finite elements appropriate to the dimension of the level surface may be employed to obtain an explicit representation over which the integration may be performed using standard numerical quadrature techniques along each element. However, when the goal is to compute an entire continuous family $m(T)$ of integrals over all the $T$-level surfaces of $\psi$, then this method of explicit level set extraction is no longer practical. We introduce a novel method to perform this type of numerical integration efficiently by making use of the coarea formula. We present the technique for discretization of the coarea formula and present the algorithms to compute the integrals over families of T-level surfaces. While validation of our method in the special case of a single level surface demonstrates accuracies close to more explicit isosurface integration methods, we show a sizable boost in computational efficiency in the case of multiple T-level surfaces, where our coupled integration algorithms significantly outperform sequential one-at-a-time application of explicit methods.

翻译：如果我们希望将一个函数($+ ⁇ Omega\subset\ re ⁇ n ⁇ to\ re$) 整合成一个函数 $+%Omega\subset\ re ⁇ n ⁇ to\ re$,那么,如果我们想将一个函数($>Omega\subset\ subset\ re ⁇ @re$) 的单一T$水平表面上一个函数($\psi$) 的连续家属($(T)美元),那么,这种清晰水平的提取方法就不再实用了。我们采用一种新的方法,通过使用共域公式来高效地进行这种类型的数字整合。我们介绍区域公式的离散化技术,并介绍在T级表面各组群中进行整合的算法。在单层表面特殊情况下验证我们的方法表明接近于更清晰的表面整合方法,我们展示了在多层集成的地面集成法中,我们展示了一种显著的地面集成法。