The construction of networks of maps among shapes in a collection enables a variety of applications in data-driven geometry processing. A key task in network construction is to make the maps consistent with each other. This consistency constraint, when properly defined, leads not only to a concise representation of such networks, but more importantly, it serves as a strong regularizer for correcting and improving noisy initial maps computed between pairs of shapes in isolation. Up-to-now, however, the consistency constraint has only been fully formulated for point-based maps or for shape collections that are fully similar. In this paper, we introduce a framework for computing consistent functional maps within heterogeneous shape collections. In such collections not all shapes share the same structure --- different types of shared structure may be present within different (but possibly overlapping) sub-collections. Unlike point-based maps, functional maps can encode similarities at multiple levels of detail (points or parts), and thus are particularly suitable for coping with such diversity within a shape collection. We show how to rigorously formulate the consistency constraint in the functional map setting. The formulation leads to a powerful tool for computing consistent functional maps, and also for discovering shared structures, such as meaningful shape parts. We also show how to adapt the procedure for handling very large-scale shape collections. Experimental results on benchmark datasets show that the proposed framework significantly improves upon state-of-the-art data-driven techniques. We demonstrate the usefulness of the framework in shape co-segmentation and various shape exploration tasks.

/pdf/functional-map-networks-for-analyzing-and-exploring-large-2g3on5vbac.pdf

Functional map networks for analyzing and exploring large shape collections

We propose to use deep neural networks for generating samples in Monte Carlo integration. Our work is based on non-linear independent components estimation (NICE), which we extend in numerous ways to improve performance and enable its application to integration problems. First, we introduce piecewise-polynomial coupling transforms that greatly increase the modeling power of individual coupling layers. Second, we propose to preprocess the inputs of neural networks using one-blob encoding, which stimulates localization of computation and improves inference. Third, we derive a gradient-descent-based optimization for the Kullback-Leibler and the χ2 divergence for the specific application of Monte Carlo integration with unnormalized stochastic estimates of the target distribution. Our approach enables fast and accurate inference and efficient sample generation independently of the dimensionality of the integration domain. We show its benefits on generating natural images and in two applications to light-transport simulation: first, we demonstrate learning of joint path-sampling densities in the primary sample space and importance sampling of multi-dimensional path prefixes thereof. Second, we use our technique to extract conditional directional densities driven by the product of incident illumination and the BSDF in the rendering equation, and we leverage the densities for path guiding. In all applications, our approach yields on-par or higher performance than competing techniques at equal sample count.

Neural Importance Sampling

Monte Carlo integration is firmly established as the basis for most practical realistic image synthesis algorithms because of its flexibility and generality. However, the visual quality of rendered images often suffers from estimator variance, which appears as visually distracting noise. Adaptive sampling and reconstruction algorithms reduce variance by controlling the sampling density and aggregating samples in a reconstruction step, possibly over large image regions. In this paper we survey recent advances in this area. We distinguish between "a priori" methods that analyze the light transport equations and derive sampling rates and reconstruction filters from this analysis, and "a posteriori" methods that apply statistical techniques to sets of samples to drive the adaptive sampling and reconstruction process. They typically estimate the errors of several reconstruction filters, and select the best filter locally to minimize error. We discuss advantages and disadvantages of recent state-of-the-art techniques, and provide visual and quantitative comparisons. Some of these techniques are proving useful in real-world applications, and we aim to provide an overview for practitioners and researchers to assess these approaches. In addition, we discuss directions for potential further improvements.

/pdf/recent-advances-in-adaptive-sampling-and-reconstruction-for-zfbba7tefu.pdf

Recent Advances in Adaptive Sampling and Reconstruction for Monte Carlo Rendering

Recent advances in ultra-fast imaging have triggered many promising applications in graphics and vision, such as capturing transparent objects, estimating hidden geometry and materials, or visualizing light in motion. There is, however, very little work regarding the effective simulation and analysis of transient light transport, where the speed of light can no longer be considered infinite. We first introduce the transient path integral framework, formally describing light transport in transient state. We then analyze the difficulties arising when considering the light's time-of-flight in the simulation (rendering) of images and videos. We propose a novel density estimation technique that allows reusing sampled paths to reconstruct time-resolved radiance, and devise new sampling strategies that take into account the distribution of radiance along time in participating media. We then efficiently simulate time-resolved phenomena (such as caustic propagation, fluorescence or temporal chromatic dispersion), which can help design future ultra-fast imaging devices using an analysis-by-synthesis approach, as well as to achieve a better understanding of the nature of light transport.

/pdf/a-framework-for-transient-rendering-3wn5h1kby2.pdf

A framework for transient rendering

(1971). Monte Carlo Principles and Neutron Transport Problems. Nuclear Science and Engineering: Vol. 46, No. 3, pp. 439-440.

Monte Carlo Principles and Neutron Transport Problems

Monte Carlo techniques for light transport simulation rely on importance sampling when constructing light transport paths. Previous work has shown that suitable sampling distributions can be recovered from particles distributed in the scene prior to rendering. We propose to represent the distributions by a parametric mixture model trained in an on-line (i.e. progressive) manner from a potentially infinite stream of particles. This enables recovering good sampling distributions in scenes with complex lighting, where the necessary number of particles may exceed available memory. Using these distributions for sampling scattering directions and light emission significantly improves the performance of state-of-the-art light transport simulation algorithms when dealing with complex lighting.

On-line learning of parametric mixture models for light transport simulation

Efficiently computing light transport in participating media in a manner that is robust to variations in media density, scattering albedo, and anisotropy is a difficult and important problem in realistic image synthesis. While many specialized rendering techniques can efficiently resolve subsets of transport in specific media, no single approach can robustly handle all types of effects. To address this problem we unify volumetric density estimation, using point and beam estimators, and Monte Carlo solutions to the path integral formulation of the rendering and radiative transport equations. We extend multiple importance sampling to correctly handle combinations of these fundamentally different classes of estimators. This, in turn, allows us to develop a single rendering algorithm that correctly combines the benefits and mediates the limitations of these powerful volume rendering techniques.

Unifying points, beams, and paths in volumetric light transport simulation

Efficiently simulating light transport in various scenes with a single algorithm is a difficult and important problem in computer graphics. Two major issues have been shown to hinder the efficiency of the existing solutions: light transport due to multiple highly glossy or specular interactions, and scenes with complex visibility between the camera and light sources. While recent bidirectional path sampling methods such as vertex connection and merging/unified path sampling (VCM/UPS) efficiently deal with highly glossy or specular transport, they tend to perform poorly in scenes with complex visibility. On the other hand, Markov chain Monte Carlo (MCMC) methods have been able to show some excellent results in scenes with complex visibility, but they behave unpredictably in scenes with glossy or specular surfaces due to their fundamental issue of sample correlation. In this paper, we show how to fuse the underlying key ideas behind VCM/UPS and MCMC into a single, efficient light transport solution. Our algorithm is specifically designed to retain the advantages of both approaches, while alleviating their limitations. Our experiments show that the algorithm can efficiently render scenes with both highly glossy or specular materials and complex visibility, without compromising the performance in simpler cases.

Robust light transport simulation via metropolised bidirectional estimators

Multiple importance sampling (MIS) has become an indispensable tool in Monte Carlo rendering, widely accepted as a near-optimal solution for combining different sampling techniques. But an MIS combination, using the common balance or power heuristics, often results in an overly defensive estimator, leading to high variance. We show that by generalizing the MIS framework, variance can be substantially reduced. Specifically, we optimize one of the combined sampling techniques so as to decrease the overall variance of the resulting MIS estimator. We apply the approach to the computation of direct illumination due to an HDR environment map and to the computation of global illumination using a path guiding algorithm. The implementation can be as simple as subtracting a constant value from the tabulated sampling density done entirely in a preprocessing step. This produces a consistent noise reduction in all our tests with no negative influence on run time, no artifacts or bias, and no failure cases.

MIS compensation: optimizing sampling techniques in multiple importance sampling

Two decades have passed since the introduction of Markov chain Monte Carlo (MCMC) into light transport simulation by Veach and Guibas, and numerous follow-up works have been published since then. However, up until now no survey has attempted to cover the majority of these methods. The aim of this paper is therefore to offer a first comprehensive survey of MCMC algorithms for light transport simulation. The methods presented in this paper are categorized by their objectives and properties, while we point out their strengths and weaknesses. We discuss how the methods handle the main issues of MCMC and how they could be combined or improved in the near future. To make the paper suitable for readers unacquainted with MCMC methods, we include an introduction to general MCMC and its demonstration on a simple example.

Martin Šik

Papers

On-line learning of parametric mixture models for light transport simulation

Unifying points, beams, and paths in volumetric light transport simulation

Robust light transport simulation via metropolised bidirectional estimators

MIS compensation: optimizing sampling techniques in multiple importance sampling

Survey of Markov Chain Monte Carlo Methods in Light Transport Simulation