Computing the Distance between Piecewise-Linear Bivariate Functions

TL;DR

This paper presents an efficient algorithm to compute the L2 distance between two piecewise-linear bivariate functions, reducing the complexity from quadratic to near-linear time, with applications in terrain matching.

Contribution

It introduces a novel algorithm that computes the L2 distance between such functions in O(n log^4 n) time, improving over the naive quadratic approach.

Findings

The algorithm reduces computational complexity from Θ(n^2) to O(n log^4 n).

It employs multi-point polynomial evaluation techniques.

Application demonstrated in terrain matching scenarios.

Abstract

We consider the problem of computing the distance between two piecewise-linear bivariate functions $f$ and $g$ defined over a common domain $M$. We focus on the distance induced by the $L_2$-norm, that is $\|f-g\|_2=\sqrt{\iint_M (f-g)^2}$. If $f$ is defined by linear interpolation over a triangulation of $M$ with $n$ triangles, while $g$ is defined over another such triangulation, the obvious na\"ive algorithm requires $\Theta(n^2)$ arithmetic operations to compute this distance. We show that it is possible to compute it in $\O(n\log^4 n)$ arithmetic operations, by reducing the problem to multi-point evaluation of a certain type of polynomials. We also present an application to terrain matching.