In this paper, we present a new mathematical foundation for image-based lighting. Using a simple manipulation of the local coordinate system, we derive a closed-form solution to the light integral equation under distant environment illumination. We derive our solution for different BRDF's such as lambertian and Phong-like. The method is free of noise, and provides the possibility of using the full spectrum of frequencies captured by images taken from the environment. This allows for the color of the rendered object to be toned according to the color of the light in the environment. Experimental results also show that one can gain an order of magnitude or higher in rendering time compared to Monte Carlo quadrature methods and spherical harmonics.
Closed-form solutions were also proposed for special cases of non-constant lighting such as the work done by @cite_57 , which provides a solution for linearly-varying luminaires. The most common application for non-constant lighting is in environment maps. @cite_144 used spherical harmonics to solve the light integral for diffuse materials lit by environment maps, and achieved real-time rendering.
@cite_57 @cite_144
2027509884 2111999986 2026173400 2076669957
We present a method, based on pre-computed light transport, for real-time rendering of objects under all-frequency, time-varying illumination represented as a high-resolution environment map. Current techniques are limited to small area lights, with sharp shadows, or large low-frequency lights, with very soft shadows. Our main contribution is to approximate the environment map in a wavelet basis, keeping only the largest terms (this is known as a non-linear approximation). We obtain further compression by encoding the light transport matrix sparsely but accurately in the same basis. Rendering is performed by multiplying a sparse light vector by a sparse transport matrix, which is very fast. For accurate rendering, using non-linear wavelets is an order of magnitude faster than using linear spherical harmonics, the current best technique. Real-time shading using general (e.g., anisotropic) BRDFs has so far been limited to a few point or directional light sources. We extend such shading to smooth, area lighting using a low-order spherical harmonic basis for the lighting environment. We represent the 4D product function of BRDF times the cosine factor (dot product of the incident lighting and surface normal vectors) as a 2D table of spherical harmonic coefficients. Each table entry represents, for a single view direction, the integral of this product function times lighting on the hemisphere expressed in spherical harmonics. This reduces the shading integral to a simple dot product of 25 component vectors, easily evaluatable on PC graphics hardware. Non-trivial BRDF models require rotating the lighting coefficients to a local frame at each point on an object, currently forming the computational bottleneck. Real-time results can be achieved by fixing the view to allow dynamic lighting or vice versa. We also generalize a previous method for precomputed radiance transfer to handle general BRDF shading. This provides shadows and interreflections that respond in real-time to lighting changes on a preprocessed object of arbitrary material (BRDF) type. Real-time shading with environment maps requires the ability to rotate the global lighting to each surface point's local coordinate frame. Although extensive previous work has studied rotation of functions represented by spherical harmonics, little work has investigated efficient rotation of wavelets. Wavelets are superior at approximating high frequency signals such as detailed high dynamic range lighting and very shiny BRDFs, but present difficulties for interactive rendering due to the lack of an analytic solution for rotation. In this paper we present an efficient computational solution for wavelet rotation using precomputed matrices. Each matrix represents the linear transformation of source wavelet bases defined in the global coordinate frame to target wavelet bases defined in sampled local frames. Since wavelets have compact support, these matrices are very sparse, enabling efficient storage and fast computation at run-time. In this paper, we focus on the application of our technique to interactive environment map rendering. We show that using these matrices allows us to evaluate the integral of dynamic lighting with dynamic BRDFs at interactive rates, incorporating efficient non-linear approximation of both illumination and reflection. Our technique improves on previous work by eliminating the need for prefiltering environment maps, and is thus significantly faster for accurate rendering of dynamic environment lighting with high frequency reflection effects. This paper focuses on efficient rendering based on pre-computed light transport, with realistic materials and shadows under all-frequency direct lighting such an environment maps. The basic difficulty is representation and computation in the 6D space of light direction, view direction, and surface position. While image-based and synthetic methods for real-time rendering have been proposed, they do not scale to high sampling rates with variation of both lighting and viewpoint. Current approaches are therefore limited to lower dimensionality (only lighting or viewpoint variation, not both) or lower sampling rates (low frequency lighting and materials). We propose a new mathematical and computational analysis of pre-computed light transport. We use factored forms, separately pre-computing and representing visibility and material properties. Rendering then requires computing triple product integrals at each vertex, involving the lighting, visibility and BRDF. Our main contribution is a general analysis of these triple product integrals, which are likely to have broad applicability in computer graphics and numerical analysis. We first determine the computational complexity in a number of bases like point samples, spherical harmonics and wavelets. We then give efficient linear and sublinear-time algorithms for Haar wavelets, incorporating non-linear wavelet approximation of lighting and BRDFs. Practically, we demonstrate rendering of images under new lighting and viewing conditions in a few seconds, significantly faster than previous techniques.