Browsing by Author "Christensen, Per"

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    Orthogonal Array Sampling for Monte Carlo Rendering
    (The Eurographics Association and John Wiley & Sons Ltd., 2019) Jarosz, Wojciech; Enayet, Afnan; Kensler, Andrew; Kilpatrick, Charlie; Christensen, Per; Boubekeur, Tamy and Sen, Pradeep
    We generalize N-rooks, jittered, and (correlated) multi-jittered sampling to higher dimensions by importing and improving upon a class of techniques called orthogonal arrays from the statistics literature. Renderers typically combine or ''pad'' a collection of lower-dimensional (e.g. 2D and 1D) stratified patterns to form higher-dimensional samples for integration. This maintains stratification in the original dimension pairs, but looses it for all other dimension pairs. For truly multi-dimensional integrands like those in rendering, this increases variance and deteriorates its rate of convergence to that of pure random sampling. Care must therefore be taken to assign the primary dimension pairs to the dimensions with most integrand variation, but this complicates implementations. We tackle this problem by developing a collection of practical, in-place multi-dimensional sample generation routines that stratify points on all t-dimensional and 1-dimensional projections simultaneously. For instance, when t=2, any 2D projection of our samples is a (correlated) multi-jittered point set. This property not only reduces variance, but also simplifies implementations since sample dimensions can now be assigned to integrand dimensions arbitrarily while maintaining the same level of stratification. Our techniques reduce variance compared to traditional 2D padding approaches like PBRT's (0,2) and Stratified samplers, and provide quality nearly equal to state-of-the-art QMC samplers like Sobol and Halton while avoiding their structured artifacts as commonly seen when using a single sample set to cover an entire image. While in this work we focus on constructing finite sampling point sets, we also discuss potential avenues for extending our work to progressive sequences (more suitable for incremental rendering) in the future.
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    Stochastic Generation of (t, s) Sample Sequences
    (The Eurographics Association, 2021) Helmer, Andrew; Christensen, Per; Kensler, Andrew; Bousseau, Adrien and McGuire, Morgan
    We introduce a novel method to generate sample sequences that are progressively stratified both in high dimensions and in lower-dimensional projections. Our method comes from a new observation that Owen-scrambled quasi-Monte Carlo (QMC) sequences can be generated as stratified samples, merging the QMC construction and random scrambling into a stochastic algorithm. This yields simpler implementations of Owen-scrambled Sobol', Halton, and Faure sequences that exceed the previous state-of-the-art sample-generation speed; we provide an implementation of Owen-scrambled Sobol' (0,2)-sequences in fewer than 30 lines of C++ code that generates 200 million samples per second on a single CPU thread. Inspired by pmj02bn sequences, this stochastic formulation allows multidimensional sequences to be augmented with best-candidate sampling to improve point spacing in arbitrary projections. We discuss the applications of these high-dimensional sequences to rendering, describe a new method to decorrelate sequences while maintaining their progressive properties, and show that an arbitrary sample coordinate can be queried efficiently. Finally we show how the simplicity and local differentiability of our method allows for further optimization of these sequences. As an example, we improve progressive distances of scrambled Sobol' (0,2)-sequences using a (sub)gradient descent optimizer, which generates sequences with near-optimal distances.