Importance sampling techniques significantly reduce variance in physically based rendering. In this article, we propose a novel online framework to learn the spatial-varying distribution of the full product of the rendering equation, with a single small neural network using stochastic ray samples. The learned distributions can be used to efficiently sample the full product of incident light. To accomplish this, we introduce a novel closed-form density model, called the Normalized Anisotropic Spherical Gaussian mixture, that can model a complex light field with a small number of parameters and that can be directly sampled. Our framework progressively renders and learns the distribution, without requiring any warm-up phases. With the compact and expressive representation of our density model, our framework can be implemented entirely on the GPU, allowing it to produce high-quality images with limited computational resources. The results show that our framework outperforms existing neural path guiding approaches and achieves comparable or even better performance than state-of-the-art online statistical path guiding techniques.

The quantum-walk-based spatial search problem aims to find a marked vertex using a quantum walk on a graph with marked vertices. We describe a framework for determining the computational complexity of spatial search by continuous-time quantum walk on arbitrary graphs by providing a recipe for finding the optimal running time and the success probability of the algorithm. The quantum walk is driven by a Hamiltonian derived from the adjacency matrix of the graph modified by the presence of the marked vertices. The success of our framework depends on the knowledge of the eigenvalues and eigenvectors of the adjacency matrix. The spectrum of the Hamiltonian is subsequently obtained from the roots of the determinant of a real symmetric matrix M , the dimensions of which depend on the number of marked vertices. The eigenvectors are determined from a basis of the kernel of M . We show each step of the framework by solving the spatial searching problem on the Johnson graphs with a fixed diameter and with two marked vertices. Our calculations show that the optimal running time is \(O(\sqrt {N})\) with an asymptotic probability of 1+ o (1), where N is the number of vertices.