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.

中文翻译：

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Q-NET：神经代理的低维积分网络

多维函数的积分通常通过对多个位置的函数值求平均值来估计。如果需要重复评估，则对真实函数使用近似代理或代理是很有用的。如果代理本身的积分在分析上是已知的并且可以实际计算，则它甚至更有用。我们设计了一系列固定网络，我们称之为 Q-NET，可以计算由 sigmoidal 通用逼近器表示的函数的积分。Q-NET 对经过训练的代理的参数进行操作，并且可以计算输入域的任何维度子集上的精确积分。在对输入空间的某些转换下，Q-NET 还有助于方便地重新计算积分，而无需重新采样被积函数或重新训练代理。我们强调了该方案对各种渲染应用程序的好处，包括反向渲染、采样过程噪声和可视化。Q-NET 在以下情况下很有吸引力：维数低（< 10D）；采样函数的积分需要在不同的子域上重新计算；积分的估计需要与采样策略分离，例如在使用稀疏、自适应采样时；边际函数需要以函数形式已知；或者当强大的单指令多数据/线程 (SIMD/SIMT) 流水线可用时。积分的估计需要与采样策略分离，例如在使用稀疏、自适应采样时；边际函数需要以函数形式已知；或者当强大的单指令多数据/线程 (SIMD/SIMT) 流水线可用时。积分的估计需要与采样策略分离，例如在使用稀疏、自适应采样时；边际函数需要以函数形式已知；或者当强大的单指令多数据/线程 (SIMD/SIMT) 流水线可用时。