Intrinsic Integration

TL;DR

This paper introduces a novel, efficient numerical method using the coarea formula to compute integrals over entire families of level surfaces, significantly improving efficiency over traditional explicit extraction methods.

Contribution

The paper presents a new discretization technique and algorithms for integrating over multiple level surfaces simultaneously using the coarea formula, enhancing computational efficiency.

Findings

Accuracy comparable to explicit methods for single level surface integration

Significant speedup in computing integrals over multiple level surfaces

Coupled algorithms outperform sequential explicit methods

Abstract

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.