We present an integral equation which generalizes a variety of known rendering algorithms. In the course of discussing a monte carlo solution we also present a new form of variance reduction, called Hierarchical sampling and give a number of elaborations shows that it may be an efficient new technique for a wide variety of monte carlo procedures. The resulting rendering algorithm extends the range of optical phenomena which can be effectively simulated.

/pdf/the-rendering-equation-123av1m206.pdf

The rendering equation

We present a method that achieves state-of-the-art results for synthesizing novel views of complex scenes by optimizing an underlying continuous volumetric scene function using a sparse set of input views. Our algorithm represents a scene using a fully-connected (non-convolutional) deep network, whose input is a single continuous 5D coordinate (spatial location $(x,y,z)$ and viewing direction $(\theta, \phi)$) and whose output is the volume density and view-dependent emitted radiance at that spatial location. We synthesize views by querying 5D coordinates along camera rays and use classic volume rendering techniques to project the output colors and densities into an image. Because volume rendering is naturally differentiable, the only input required to optimize our representation is a set of images with known camera poses. We describe how to effectively optimize neural radiance fields to render photorealistic novel views of scenes with complicated geometry and appearance, and demonstrate results that outperform prior work on neural rendering and view synthesis. View synthesis results are best viewed as videos, so we urge readers to view our supplementary video for convincing comparisons.

NeRF: Representing Scenes as Neural Radiance Fields for View Synthesis

We present a data structure and an algorithm for efficient and exact interference detection amongst complex models undergoing rigid motion. The algorithm is applicable to all general polygonal models. It pre-computes a hierarchical representation of models using tight-fitting oriented bounding box trees (OBBTrees). At runtime, the algorithm traverses two such trees and tests for overlaps between oriented bounding boxes based on a separating axis theorem, which takes less than 200 operations in practice. It has been implemented and we compare its performance with other hierarchical data structures. In particular, it can robustly and accurately detect all the contacts between large complex geometries composed of hundreds of thousands of polygons at interactive rates. CR

/pdf/obbtree-a-hierarchical-structure-for-rapid-interference-377ihhq344.pdf

OBBTree: a hierarchical structure for rapid interference detection

The theory of elasticity describes deformable materials such as rubber, cloth, paper, and flexible metals. We employ elasticity theory to construct differential equations that model the behavior of non-rigid curves, surfaces, and solids as a function of time. Elastically deformable models are active: they respond in a natural way to applied forces, constraints, ambient media, and impenetrable obstacles. The models are fundamentally dynamic and realistic animation is created by numerically solving their underlying differential equations. Thus, the description of shape and the description of motion are unified.

/pdf/elastically-deformable-models-7253i89xh6.pdf

Elastically deformable models

A technique for rendering images of volumes containing mixtures of materials is presented. The shading model allows both the interior of a material and the boundary between materials to be colored. Image projection is performed by simulating the absorption of light along the ray path to the eye. The algorithms used are designed to avoid artifacts caused by aliasing and quantization and can be efficiently implemented on an image computer. Images from a variety of applications are shown.

Volume rendering

This paper presents new algorithms to trace objects represented by densities within a volume grid, e.g. clouds, fog, flames, dust, particle systems. We develop the light scattering equations, discuss previous methods of solution, and present a new approximate solution to the full three-dimensional radiative scattering problem suitable for use in computer graphics. Additionally we review dynamical models for clouds used to make an animated movie.

http://www.cs.duke.edu/courses/spring03/cps296.8/papers/RayTracingVolumeDensities.pdf

Ray tracing volume densities

A quadtree algorithm is developed to triangulate deformed, intersecting parametric surfaces. The biggest problem with adaptive sampling is to guarantee that the triangulation is accurate within a given tolerance. A new method guarantees the accuracy of the triangulation, given a "Lipschitz" condition on the surface definition. The method constructs a hierarchical set of bounding volumes for the surface, useful for ray tracing and solid modeling operations. The task of adaptively sampling a surface is broken into two parts: a subdivision mechanism for recursively subdividing a surface, and a set of subdivision criteria for controlling the subdivision process.An adaptive sampling technique is said to be robust if it accurately represents the surface being sampled. A new type of quadtree, called a restricted quadtree, is more robust than the traditional unrestricted quadtree at adaptive sampling of parametric surfaces. Each sub-region in the quadtree is half the width of the previous region. The restricted quadtree requires that adjacent regions be the same width within a factor of two, while the traditional quadtree makes no restriction on neighbor width. Restricted surface quadtrees are effective at recursively sampling a parametric surface. Quadtree samples are concentrated in regions of high curvature, and along intersection boundaries, using several subdivision criteria. Silhouette subdivision improves the accuracy of the silhouette boundary when a viewing transformation is available at sampling time. The adaptive sampling method is more robust than uniform sampling, and can be more efficient at rendering deformed, intersecting parametric surfaces.

Accurate triangulations of deformed, intersecting surfaces

We develop an algorithm to detect geometric collisions between pairs of time-dependent parametric surfaces. The algorithm works on surfaces that are continuous and have bounded derivatives, and includes objects that move or deform as a function of time. The algorithm numerically solves for the parametric values corresponding to coincident points and near-misses between the surfaces of two parametric functions.Upper bounds on the parametric derivatives make it possible to guarantee the successful detection of collisions and near-misses; we describe a method to find the derivative bounds for many surface types. To compute collisions between new types of surfaces, the mathematical collision analysis is needed only once per surface type, rather than analyzing for each pair of surface types.The algorithm is hierarchical, first finding potential collisions over large volumes, and then refining the solution to smaller volumes. The user may specify the desired accuracy of the solution. A C-code implementation is described, with results for several non-bicubic and bicubic time-dependent parametric functions. An animation of the collision computation demonstrates collisions between complex parametric functions.

/pdf/geometric-collisions-for-time-dependent-parametric-surfaces-eibt19ddl8.pdf

Brian Von Herzen

Papers

Ray tracing volume densities

Accurate triangulations of deformed, intersecting surfaces

Geometric collisions for time-dependent parametric surfaces