The theory of light transport on Gaussian process implicit surface (GPIS) provides a unified framework for rendering surfaces, participating media, and the intermediate spectrum. However, previous approaches rely on brute-force ray marching for surface intersections, requiring full noise evaluations at each marching point, whether using multivariate Gaussian sampling or sparse convolution noise approximation. This imposes a severe limitation on the rendering efficiency. In this paper, we derive bounds to significantly reduce the total number of full noise evaluations, leading to efficient ray marching for ray-surface intersections. We introduce stratified Bernoulli impulses, enabling a fast point-level bound for individual realizations to replace unnecessary full noise evaluations. To further reduce the number of point-level bound evaluations, we propose a region-level bound, leveraging a spatial acceleration structure to prune probabilistically empty regions, thereby avoiding unnecessary marching points in advance. By combining these two bounds, our bounded ray marching accelerates ray-surface intersections in GPIS, and consequently significantly improves overall GPIS rendering efficiency. Code for this paper are at https://github.com/Cchen-77/bounded-gpis.

Systematically simulating specular light transport requires an exhaustive search for triangle tuples containing admissible paths. Given the extreme inefficiency of enumerating all combinations, we significantly reduce the search domain by stochastically sampling such tuples. The challenge is to design proper sampling probabilities that keep the noise level controllable. Our key insight is that by bounding the irradiance contributed by each triangle tuple at a given position, we can sample a subset of triangle tuples with potentially high contributions. Although low-contribution tuples are assigned a negligible probability, the overall variance remains low. Therefore, we derive position and irradiance bounds for caustics casted by each triangle tuple, introducing a bounding property of rational functions on a Bernstein basis. When formulating position and irradiance expressions into rational functions, we handle non-rational parts through remainder variables to maintain bounding validity. Finally, we carefully design the sampling probabilities by optimizing the upper bound of the variance, expressed only using the position and irradiance bounds. The bound-driven sampling of triangle tuples is intrinsically unbiased even without defensive sampling. It can be combined with various unbiased and biased root-finding techniques within a local triangle domain. Extensive evaluations show that our method enables the fast and reliable rendering of complex caustics effects. Yet, our method is efficient for no more than two specular vertices, where complexity grows sublinearly to the number of triangles and linearly to that of emitters, and does not consider the Fresnel and visibility terms. We also rely on parameters to control subdivisions.

Contemporary path guiding employs an iterative training scheme to fit radiance distributions. However, existing methods combine the estimates generated in each iteration merely within image space, overlooking differences in the convergence of distribution fitting over individual light paths. This paper formulates the estimation combination task as a path reweighting process. To compute spatio-directional varying combination weights, we propose multiple importance reweighting , leveraging the importance distributions from multiple guiding iterations. We demonstrate that our proposed path-level reweighting makes guiding algorithms less sensitive to noise and overfitting in distributions. This facilitates a finer subdivision of samples both spatially and temporally (i.e., over iterations), which leads to additional improvements in the accuracy of distributions and samples. Inspired by adaptive multiple importance sampling (AMIS), we introduce a simple yet effective mixture-based weighting scheme with theoretically guaranteed consistency, demonstrating good practical performance compared to alternative weighting schemes. To further foster usage with high sample rates, we introduce a hyperparameter that controls the size of sample storage. When this size limit is exceeded, low-valued samples are splatted during rendering and reweighted using a partial mixture of distributions. We found limiting the storage size reduces memory overhead and keeps variance reduction and bias comparable to the unlimited ones. Our method is largely agnostic to the underlying guiding method and compatible with conventional pixel reweighting techniques. Extensive evaluations underscore the feasibility of our approach in various scenes, achieving variance reduction with negligible bias over state-of-the-art solutions within equal sample rates and rendering time.

Monte Carlo rendering often faces a dilemma, namely, whether to choose an unbiased estimator or a biased one. Although different integrators have been developed to address various scenarios, no single method can effectively manage all situations. Thus, finding a good approach to combine different integrators has always been a topic that warrants exploration. This work proposes DSCombiner, a new shrinkage estimator that flexibly combines unbiased and biased estimators (typically generated by different integrators) in image space into a single estimating procedure, strategically utilizing the strengths of different integrators while minimizing their weaknesses. DSCombiner overcomes the limitation of single shrinkage combiners by introducing a two-step shrinkage towards a noise-free radiance prior. We derive optimal shrinkage factors for the two steps within a hierarchical Bayesian framework, and provide a deep learning-based method to improve the results. Comprehensive qualitative and quantitative validations across diverse scenes demonstrate visible improvements in image quality, as compared with previous image-space and path-space combiners.

Finding valid light paths that involve specular vertices in Monte Carlo rendering requires solving many non-linear, transcendental equations in high-dimensional space. Existing approaches heavily rely on Newton iterations in path space, which are limited to obtaining at most a single solution each time and easily diverge when initialized with improper seeds. We propose specular polynomials , a Newton iteration-free methodology for finding a complete set of admissible specular paths connecting two arbitrary endpoints in a scene. The core is a reformulation of specular constraints into polynomial systems, which makes it possible to reduce the task to a univariate root-finding problem. We first derive bivariate systems utilizing rational coordinate mapping between the coordinates of consecutive vertices. Subsequently, we adopt the hidden variable resultant method for variable elimination, converting the problem into finding zeros of the determinant of univariate matrix polynomials. This can be effectively solved through Laplacian expansion for one bounce and a bisection solver for more bounces. Our solution is generic, completely deterministic, accurate for the case of one bounce, and GPU-friendly. We develop efficient CPU and GPU implementations and apply them to challenging glints and caustic rendering. Experiments on various scenarios demonstrate the superiority of specular polynomial-based solutions compared to Newton iteration-based counterparts. Our implementation is available at https://github.com/mollnn/spoly.

The efficiency of inverse optimization in physically based differentiable rendering heavily depends on the variance of Monte Carlo estimation. Despite recent advancements emphasizing the necessity of tailored differential sampling strategies, the general approaches remain unexplored. In this paper, we investigate the interplay between local sampling decisions and the estimation of light path derivatives. Considering that modern differentiable rendering algorithms share the same path for estimating differential radiance and ordinary radiance, we demonstrate that conventional guiding approaches, conditioned solely on the last vertex, cannot attain this density. Instead, a mixture of different sampling distributions is required, where the weights are conditioned on all the previously sampled vertices in the path. To embody our theory, we implement a conditional mixture path guiding that explicitly computes optimal weights on the fly. Furthermore, we show how to perform positivization to eliminate sign variance and extend to scenes with millions of parameters. To the best of our knowledge, this is the first generic framework for applying path guiding to differentiable rendering. Extensive experiments demonstrate that our method achieves nearly one order of magnitude improvements over state-of-the-art methods in terms of variance reduction in gradient estimation and errors of inverse optimization. The implementation of our proposed method is available at https://github.com/mollnn/conditional-mixture.

Complex visual effects such as caustics are often produced by light paths containing multiple consecutive specular vertices (dubbed specular chains) , which pose a challenge to unbiased estimation in Monte Carlo rendering. In this work, we study the light transport behavior within a sub-path that is comprised of a specular chain and two non-specular separators. We show that the specular manifolds formed by all the sub-paths could be exploited to provide coherence among sub-paths. By reconstructing continuous energy distributions from historical and coherent sub-paths, seed chains can be generated in the context of importance sampling and converge to admissible chains through manifold walks. We verify that importance sampling the seed chain in the continuous space reaches the goal of importance sampling the discrete admissible specular chain. Based on these observations and theoretical analyses, a progressive pipeline, manifold path guiding , is designed and implemented to importance sample challenging paths featuring long specular chains. To our best knowledge, this is the first general framework for importance sampling discrete specular chains in regular Monte Carlo rendering. Extensive experiments demonstrate that our method outperforms state-of-the-art unbiased solutions with up to 40 × variance reduction, especially in typical scenes containing long specular chains and complex visibility.