Q-NET: A Network for Low-dimensional Integrals of Neural Proxies

Integrals of multidimensional functions are often estimated by averaging function values at multiple locations. The use of an approximate surrogate or proxy for the true function is useful if repeated evaluations are necessary. A proxy is even more useful if its own integral is known analytically and can be calculated practically. We design a family of fixed networks, which we call Q-NETs, that can calculate integrals of functions represented by sigmoidal universal approximators. Q-NETs operate on the parameters of the trained proxy and can calculate exact integrals over any subset of dimensions of the input domain. Q-NETs also facilitate convenient recalculation of integrals without resampling the integrand or retraining the proxy, under certain transformations to the input space. We highlight the benefits of this scheme for diverse rendering applications including inverse rendering, sampled procedural noise and visualization. Q-NETs are appealing in the following contexts: the dimensionality is low (< 10D); integrals of a sampled function need to be recalculated over different sub-domains; the estimation of integrals needs to be decoupled from the sampling strategy such as when sparse, adaptive sampling is used; marginal functions need to be known in functional form; or when powerful Single Instruction Multiple Data/Thread (SIMD/SIMT) pipelines are available.