James Arvo

Barycentric parameterizations for isotropic BRDFs

A bidirectional reflectance distribution function (BRDF) is often expressed as a function of four real variables: two spherical coordinates in each of the incoming and outgoing directions. However, many BRDFs reduce to functions of fewer variables. For example, isotropic reflection can be represented by a function of three variables. Some BRDF models can be reduced further. In This work, we introduce new sets of coordinates which we use to reduce the dimensionality of several well-known analytic BRDFs as well as empirically measured BRDF data. The proposed coordinate systems are barycentric with respect to a triangular support with a direct physical interpretation. One coordinate set is based on the BRDF mode) proposed by Lafortune. Another set, based on a model of Ward, is associated with the halfway vector common in analytical BRDF formulas. Through these coordinate sets we establish lower bounds on the approximation error inherent in the models on which they are based. We present a third set of coordinates, not based on any analytical model, that performs well in approximating measured data. Finally, our proposed variables suggest novel ways of constructing and visualizing BRDFs.

Appearance-preserving manipulation of hand-drawn graphs

We describe a sketching system that allows users to create and manipulate directed graphs, such as those depicting state diagrams, using pen-input alone. The system exactly preserves the users strokes, which may be entered in any order, and depicts them with a chalk texture to evoke a blackboard metaphor. The system automatically interprets the geometry of the sketch, distinguishing vertices, edges, and arrow heads, then tacitly imparts the intended graph semantics based on the two-dimensional placement of these elements. Once drawn, the user can manipulate the directed graph gesturally using the pen. The system responds to vertices or edges being picked and dragged by adjusting all adjacent edges appropriately. The original appearance of the hand-drawn vertices and edges is maintained even while their shapes are continually morphed in response to rearrangement of these elements. All edges exhibit shape memory, which is the proclivity to return to their original hand-drawn shape despite repeated stretching and compression.

State of the art in Monte Carlo global illumination

Realistic image synthesis is increasingly important in areas such as entertainment (movies, special effects and games), design, architecture and more. A common trend in all these areas is the quest for more realistic images of increasingly complex models. Monte Carlo global illumination algorithms are the only methods that can handle this complexity. Recent advances in algorithms and compute power has made Monte Carlo algorithms very practical and a natural choice for most problems.The purpose of this course is to impart upon the attendees a thorough understanding of the principles of Monte Carlo path tracing methods, as well as a detailed overview of the most recently developed methods.Part 1 will cover the fundamentals of realistic image synthesis (radiometry, rendering equation). Part 2 focuses on the mathematical tools needed to compute integrals using Monte Carlo sampling. Parts 3 and 4 cover specific algorithms that have proven very useful in global illumination rendering.

Steerable Importance Sampling

We present an algorithm for efficient stratified importance sampling of environment maps that generates samples in the positive hemisphere defined by local orientation of arbitrary surfaces while accounting for cosine weighting. The importance function is dynamically adjusted according to the surface normal using steerable basis functions. The algorithm is easy to implement and requires no user-defined parameters. As a preprocessing step, we approximate the incident illumination from an environment map as a continuous piecewise linear function on P2 and represent this as a triangulated height field. The product of this approximation and a dynamically orientable steering function, viz. the cosine lobe, serves as an importance sampling function. Our method allows the importance function to be sampled with an asymptotic cost of O(logn) per sample where n is the number of triangles. The most novel aspect of the algorithm is its ability to dynamically compute normalization factors which are integrals of the illumination over the positive hemispheres defined by the local surface normals during shading. The key to this feature is that the weight variation of each triangle due to the clamped cosine steering function can be well approximated by a small number of spherical harmonic coefficients which can be accumulated over any collection of triangles, in any orientation, without introducing higher-order terms. Consequently, the weighted integral of the entire steerable piecewise-linear approximation is no more costly to compute than that of a single triangle, which makes re-weighting and re-normalizing with respect to any surface orientation a trivial constant-time operation. The choice of spherical harmonics as the set of basis functions for our steerable importance function allows for easy rotation between coordinate systems. Another novel element of our algorithm is an analytic parametrization for generating stratified samples with linearly-varying density over a triangular support.

Statistical Hypothesis Testing for Assessing Monte Carlo Estimators: Applications to Image Synthesis

The faceting signal, defined in this paper as the difference signal between a rendering of the original geometric model and a simplified version of the geometric model, is responsible for the faceting artifacts commonly observed in the renderings of coarse geometric models. In this paper, we analyze the source of the faceting signal and develop a perceptual metric for the visibility of the faceting signal.Image synthesis algorithms are commonly compared on the basis of running times and/or perceived quality of the generated images. In the case of Monte Carlo techniques, assessment often entails a qualitative impression of convergence toward a reference standard and severity of visible noise; these amount to subjective assessments of the mean and variance of the estimators, respectively. In this paper we argue that such assessments should be augmented by well-known statistical hypothesis testing methods. In particular, we show how to perform a number of such tests to assess random variables that commonly arise in image synthesis such as those estimating irradiance, radiance, pixel color, etc. We explore five broad categories of tests: 1) determining whether the mean is equal to a reference standard, such as an analytical value, 2) determining that the variance is bounded by a given constant, 3) comparing the means of two different random variables, 4) comparing the variances of two different random variables, and 5) verifying that two random variables stem from the same parent distribution. The level of significance of these tests can be controlled by a parameter. We demonstrate that these tests can be used for objective evaluation of Monte Carlo estimators to support claims of zero or small bias and to provide quantitative assessments of variance reduction techniques. We also show how these tests can be used to detect errors in sampling or in computing the density of an importance function in MC integrations.

Stratified Sampling of Convex Quadrilaterals

We derive a closed-form expression for a smooth uniform bijection from a unit square onto an arbitrary convex planar quadrilateral; that is, a smooth parameterization of the quadrilateral under which the image of equal areas remain of equal area. The properties of the mapping make it well-suited to stratified Monte Carlo sampling and therefore useful for illumination computations. We begin with a simple bilinear mapping from the unit square onto the quadrilateral, then derive a warping function, from the unit square to itself, which results in a uniform map onto the quadrilateral when composed with the original bilinear map. The resulting sampling algorithm requires only a few lines of code with no iteration or branching.

Geometric anticipation: assisting users in 2D layout tasks

We describe an experimental interface that anticipates a users intentions and accommodates predicted changes in advance. Our canonical example is an interactive version of ``magnetic poetry in which rectangular blocks containing single words can be juxtaposed to form arbitrary sentences or ``poetry. The user can rearrange the blocks at will, forming and dissociating word sequences. A crucial attribute of the blocks in our system is that they anticipate insertions and gracefully rearrange themselves in time to make space for a new word or phrase. The challenges in creating such an interface are three fold: 1) the users intentions must be inferred from noisy input, 2) arrangements must be altered smoothly and intuitively in response to anticipated changes, and 3) new and changing goals must be handled gracefully at any time, even in mid animation. We describe a general approach for handling the dynamic creation and deletion of organizational goals. Fluid motion is achieved by continually applying and correcting goal-directed forces to the objects. Future applications of this idea include the manipulation of text and graphical elements within documents and the manipulation of symbolic information such as equations.

Analytical solution for irradiance due to inhomogeneous Lambertian polygonal emitters

We present an analytic solution for the irradiance at a point due to a polygonal Lambertian emitter with radiant exitance that varies with position according to a polynomial of arbitrary degree. This is a basic problem that arises naturally in radiative transfer and more specifically in global illumination, a subfield of computer graphics. Our solution is closed form except for a single nonalgebraic special function known as the Clausen integral. We begin by deriving several useful formulas for high-order tensor analogs of irradiance, which are natural generalizations of the radiation pressure tensor. We apply the resulting tensor formulas to linearly varying emitters, obtaining a solution that exhibits the general structure of higher-degree cases, including the dependence on the Clausen integral. We then generalize to higher-degree polynomials with a recurrence formula that combines solutions for lower-degree polynomials; the result is a generalization of Lamberts formula for homogeneous diffuse emitters, a well-known formula with many applications in radiative transfer and computer graphics. Similar techniques have been used previously to derive closed-form solutions for the irradiance due to homogeneous polygonal emitters with directionally varying radiance. The present work extends this previous result to include inhomogeneous emitters, which proves to be significantly more challenging to solve in closed form. We verify our theoretical results with numerical approximations and briefly discuss their potential applications.