Hierarchical Variance Reduction Techniques for Monte Carlo Rendering

Ever since the first three-dimensional computer graphics appeared half a century ago, the goal has been to model and simulate how light interacts with materials and objects to form an image. The ultimate goal is photorealistic rendering, where the created images reach a level of accuracy that makes them indistinguishable from photographs of the real world. There are many applications ñ visualization of products and architectural designs yet to be built, special effects, computer-generated films, virtual reality, and video games, to name a few. However, the problem has proven tremendously complex; the illumination at any point is described by a recursive integral to which a closed-form solution seldom exists. Instead, computer simulation and Monte Carlo methods are commonly used to statistically estimate the result. This introduces undesirable noise, or variance, and a large body of research has been devoted to finding ways to reduce the variance. I continue along this line of research, and present several novel techniques for variance reduction in Monte Carlo rendering, as well as a few related tools. The research in this dissertation focuses on using importance sampling to pick a small set of well-distributed point samples. As the primary contribution, I have developed the first methods to explicitly draw samples from the product of distant high-frequency lighting and complex reflectance functions. By sampling the product, low noise results can be achieved using a very small number of samples, which is important to minimize the rendering times. Several different hierarchical representations are explored to allow efficient product sampling. In the first publication, the key idea is to work in a compressed wavelet basis, which allows fast evaluation of the product. Many of the initial restrictions of this technique were removed in follow-up work, allowing higher-resolution uncompressed lighting and avoiding precomputation of reflectance functions. My second main contribution is to present one of the first techniques to take the triple product of lighting, visibility and reflectance into account to further reduce the variance in Monte Carlo rendering. For this purpose, control variates are combined with importance sampling to solve the problem in a novel way. A large part of the technique also focuses on analysis and approximation of the visibility function. To further refine the above techniques, several useful tools are introduced. These include a fast, low-distortion map to represent (hemi)spherical functions, a method to create high-quality quasi-random points, and an optimizing compiler for analyzing shaders using interval arithmetic. The latter automatically extracts bounds for importance sampling of arbitrary shaders, as opposed to using a priori known reflectance functions. In summary, the work presented here takes the field of computer graphics one step further towards making photorealistic rendering practical for a wide range of uses. By introducing several novel Monte Carlo methods, more sophisticated lighting and materials can be used without increasing the computation times. The research is aimed at domain-specific solutions to the rendering problem, but I believe that much of the new theory is applicable in other parts of computer graphics, as well as in other fields

Efficient product sampling using hierarchical thresholding

We present an efficient method for importance sampling the product of multiple functions. Our algorithm computes a quick approximation of the product on the fly, based on hierarchical representations of the local maxima and averages of the individual terms. Samples are generated by exploiting the hierarchical properties of many low-discrepancy sequences, and thresholded against the estimated product. We evaluate direct illumination by sampling the triple product of environment map lighting, surface reflectance, and a visibility function estimated per pixel. Our results show considerable noise reduction compared to existing state-of-the-art methods using only the product of lighting and BRD

Efficient multi-view ray tracing using edge detection and shader reuse

Stereoscopic rendering and 3D stereo displays are quickly becoming mainstream. The natural extension is autostereoscopic multi-view displays, which by the use of parallax barriers or lenticular lenses, can accommodate many simultaneous viewers without the need for active or passive glasses. As these displays, for the foreseeable future, will support only a rather limited number of views, there is a need for high-quality interperspective antialiasing. We present a specialized algorithm for efficient multi-view image generation from a camera line using ray tracing, which builds on previous methods for multi-dimensional adaptive sampling and reconstruction of light elds. We introduce multi-view silhouette edges to detect sharp geometrical discontinuities in the radiance function. These are used to significantly improve the quality of the reconstruction. In addition, we exploit shader coherence by computing analytical visibility between shading points and the camera line, and by sharing shading computations over the camera line

Efficient Product Importance Sampling using Hierarchical Thresholding

We present an efficient method for importance sampling the product of multiple functions. Our algorithm computes a quick approximation of the product on-the-fly, based on hierarchical representations of the Local maxima and averages of the individual terms. Samples are generated by exploiting the hierarchical properties of many low-discrepancy sequences, and thresholded against the estimated product. We evaluate direct illumination by sampling the triple product of environment map lighting, surface reflectance, and a visibility function estimated per pixel. Our results show considerable noise reduction compared to existing state-of-the-art methods using only the product of lighting and BRDF

Fast equal-area mapping of the (hemi)sphere using SIMD

We present a fast vectorized implementation of a transform that maps points in the unit square to the surface of the sphere, while preserving fractional area. The mapping uses the octahedral map combined with an equal-area param- eterization and has many desirable features such as low distortion, straightforward interpolation, and fast inverse and forward transforms. Our SIMD implementation completely avoids branching and uses polynomial approximations for the trigono- metric operations, along with other tricks. This results in up to 9 times speed-up over a traditional scalar implementation. Source code is available onlin

Importance Sampling of Many Lights on the GPU

The introduction of standardized APIs for ray tracing, together with hardware acceleration, opens up possibilities for physically based lighting in real-time rendering. Light importance sampling is one of the fundamental operations in light transport simulations, applicable to both direct and indirect illumination. This chapter describes a bounding volume hierarchy data structure and associated sampling methods to accelerate importance sampling of local light sources. The work is based on recently published methods for light sampling in production rendering, but it is evaluated in a real-time implementation using Microsoft DirectX Raytracing

Exploiting visibility correlation in direct illumination

The visibility function in direct illumination describes the binary visibility over a light source, e.g., an environment map. Intuitively, the visibility is often strongly correlated between nearby locations in time and space, but exploiting this correlation without introducing noticeable errors is a hard problem. In this paper, we first study the statistical characteristics of the visibility function. Then, we propose a robust and unbiased method for using estimated visibility information to improve the quality of Monte Carlo evaluation of direct illumination. Our method is based on the theory of control variates, and it can be used on top of existing state-of-the-art schemes for importance sampling. The visibility estimation is obtained by sparsely sampling and caching the 4D visibility field in a compact bitwise representation. In addition to Monte Carlo rendering, the stored visibility information can be used in a number of other applications, for example, ambient occlusion and lighting design

Practical product importance sampling for direct illumination

We present a practical algorithm for sampling the product of environment map lighting and surface reflectance. Our method builds on wavelet-based importance sampling, but has a number of important advantages over previous methods. Most importantly, we avoid using precomputed reflectance functions by sampling the BRDF on-the-fly. Hence, all types of materials can be handled, including anisotropic and spatially varying BRDFs, as well as procedural shaders. This also opens up for using very high resolution, uncompressed, environment maps. Our results show that this gives a significant reduction of variance compared to using lower resolution approximations. In addition, we study the wavelet product, and present a faster algorithm geared for sampling purposes. For our application, the computations are reduced to a simple quadtree-based multiplication. We build the BRDF approximation and evaluate the product in a single tree traversal, which makes the algorithm both faster and more flexible than previous methods

Dynamic Many-Light Sampling for Real-Time Ray Tracing

Monte Carlo ray tracing offers the capability of rendering scenes with large numbers of area light sources---lights can be sampled stochastically and shadowing can be accounted for by tracing rays, rather than using shadow maps or other rasterization-based techniques that do not scale to many lights or work well with area lights. Current GPUs only afford the capability of tracing a few rays per pixel at real-time frame rates, making it necessary to focus sampling on important light sources. While state-of-the-art algorithms for offline rendering build hierarchical data structures over the light sources that enable sampling them according to their importance, they lack efficient support for dynamic scenes. We present a new algorithm for maintaining hierarchical light sampling data structures targeting real-time rendering. Our approach is based on a two-level BVH hierarchy that reduces the cost of partial hierarchy updates. Performance is further improved by updating lower-level BVHs via refitting, maintaining their original topology. We show that this approach can give error within 6% of recreating the entire hierarchy from scratch at each frame, while being two orders of magnitude faster, requiring less than 1 ms per frame for hierarchy updates for a scene with thousands of moving light sources on a modern GPU. Further, we show that with spatiotemporal filtering, our approach allows complex scenes with thousands of lights to be rendered with ray-traced shadows in 16.1 ms per frame