Primary-Space Adaptive Control Variates using Piecewise-Polynomial Approximations

TL;DR

This paper introduces an adaptive control variate method using piecewise-polynomial approximations for efficient multidimensional numerical integration, combining quadrature and Monte Carlo techniques with importance sampling.

Contribution

It proposes a novel primary-space adaptive control variate approach that effectively handles low and high-frequency integrand features in complex multidimensional integrals.

Findings

Achieves faster convergence than previous methods.

Demonstrates effectiveness on four low-dimensional applications.

Extensible to higher-dimensional integrals with maintained accuracy.

Abstract

We present an unbiased numerical integration algorithm that handles both low-frequency regions and high frequency details of multidimensional integrals. It combines quadrature and Monte Carlo integration, by using a quadrature-base approximation as a control variate of the signal. We adaptively build the control variate constructed as a piecewise polynomial, which can be analytically integrated, and accurately reconstructs the low frequency regions of the integrand. We then recover the high-frequency details missed by the control variate by using Monte Carlo integration of the residual. Our work leverages importance sampling techniques by working in primary space, allowing the combination of multiple mappings; this enables multiple importance sampling in quadrature-based integration. Our algorithm is generic, and can be applied to any complex multidimensional integral. We demonstrate its effectiveness with four applications with low dimensionality: transmittance estimation in heterogeneous participating media, low-order scattering in homogeneous media, direct illumination computation, and rendering of distributed effects. Finally, we show how our technique is extensible to integrands of higher dimensionality, by computing the control variate on Monte Carlo estimates of the high-dimensional signal, and accounting for such additional dimensionality on the residual as well. In all cases, we show accurate results and faster convergence compared to previous approaches.