This paper proposes a new methodology for measuring the error of unbiased physically based rendering algorithms. The current state of the art includes mean squared error (MSE) based metrics and visual comparisons of equal‐time renderings of competing algorithms. Neither is satisfying as MSE does not describe behavior and can exhibit significant variance, and visual comparisons are inherently subjective. Our contribution is two‐fold: First, we propose to compute many short renderings instead of a single long run and use the short renderings to estimate MSE expectation and variance as well as per‐pixel standard deviation. An algorithm that achieves good results in most runs, but with occasional outliers is essentially unreliable, which we wish to quantify numerically. We use per‐pixel standard deviation to identify problematic lighting effects of rendering algorithms. The second contribution is the error spectrum ensemble (ESE), a tool for measuring the distribution of error over frequencies. The ESE serves two purposes: It reveals correlation between pixels and can be used to detect outliers, which offset the amount of error substantially.

This paper proposes a novel method for deep learning based on the analytical convolution of multidimensional Gaussian mixtures. In contrast to tensors, these do not suffer from the curse of dimensionality and allow for a compact representation, as data is only stored where details exist. Convolution kernels and data are Gaussian mixtures with unconstrained weights, positions, and covariance matrices. Similar to discrete convolutional networks, each convolution step produces several feature channels, represented by independent Gaussian mixtures. Since traditional transfer functions like ReLUs do not produce Gaussian mixtures, we propose using a fitting of these functions instead. This fitting step also acts as a pooling layer if the number of Gaussian components is reduced appropriately. We demonstrate that networks based on this architecture reach competitive accuracy on Gaussian mixtures fitted to the MNIST and ModelNet data sets. INTRODUCTIONConvolutional neural networks (CNNs) have led to the widespread adoption of deep learning in many disciplines. They operate on grids of discrete data samples, i.e., vectors, matrices, and tensors, which are structured representations of data. While this is approach is sufficiently efficient in one and two dimensions, it suffers the curse of dimensionality: The amount of data is O(d k ), where d is the resolution of data, and k is the dimensionality. The exponential growth permeates all layers of a CNN, which must be scaled appropriately to learn a sufficient number of k-dimensional features. In three or more dimensions, the implied memory requirements thus quickly become intractable (Maturana & Scherer, 2015;Zhirong Wu et al., 2015), leading researchers to propose specialized CNN architectures as a trade-off between mathematical exactness and performance (Riegler et al., 2017;Wang et al., 2017). In this work, we propose a novel deep learning architecture based on the analytical convolution of Gaussian mixtures (GMs). While maintaining the elegance of conventional CNNs, our architecture does not suffer from the curse of dimensionality and is therefore well-suited for application to higher-dimensional problems.GMs are inherently unstructured and sparse, in the sense that no storage is required to represent empty data regions. In contrast to discretized k-dimensional volumes, this allows for a more compact representation of data across dimensions, preventing exponential memory requirements. In this regard, they are similar to point clouds (Qi et al., 2017a). However, GMs also encode the notion of a spatial extent. The fidelity of the representation directly depends on the number of Gaussians in the mixture, which means that, given an adequate fitting algorithm and sufficient resources, it can be arbitrarily adjusted. Hence, GMs allow trading representation accuracy for memory requirements on a more fine-grained level than voxel grids.In this work, we present our novel architecture, the Gaussian mixture convolution network (GMCN), as an alternative to conventional CNNs. An impo...