Realism in computer-generated images requires accurate input models for lighting, textures and BRDFs. One of the best ways of obtaining high-quality data is through measurements of scene attributes from real photographs by inverse rendering. However, inverse rendering methods have been largely limited to settings with highly controlled lighting. One of the reasons for this is the lack of a coherent mathematical framework for inverse rendering under general illumination conditions. Our main contribution is the introduction of a signal-processing framework which describes the reflected light field as a convolution of the lighting and BRDF, and expresses it mathematically as a product of spherical harmonic coefficients of the BRDF and the lighting. Inverse rendering can then be viewed as deconvolution. We apply this theory to a variety of problems in inverse rendering, explaining a number of previous empirical results. We will show why certain problems are ill-posed or numerically ill-conditioned, and why other problems are more amenable to solution. The theory developed here also leads to new practical representations and algorithms. For instance, we present a method to factor the lighting and BRDF from a small number of views, i.e. to estimate both simultaneously when neither is known.

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.

Figure 1: These images, showing many different lighting conditions and BRDFs, were each rendered at approximately 30 frames per second using our Spherical Harmonic Reflection Map (SHRM) representation. From left to right, a simplified microfacet BRDF, krylon blue (using McCool et al.’s reconstruction from measurements at Cornell), orange and velvet (CURET database), and an anisotropic BRDF (based on the Kajiya-Kay model). The environment maps are the Grace Cathedral, St. Peter’s Basilica, the Uffizi gallery, and a Eucalyptus grove, courtesy Paul Debevec. The armadillo model is from Venkat Krishnamurthy. We present a new method for real-time rendering of objects with complex isotropic BRDFs under distant natural illumination, as specified by an environment map. Our approach is based on spherical frequency space analysis and includes three main contributions. Firstly, we are able to theoretically analyze required sampling rates and resolutions, which have traditionally been determined in an ad-hoc manner. We also introduce a new compact representation, which we call a spherical harmonic reflection map (SHRM), for efficient representation and rendering. Finally, we show how to rapidly prefilter the environment map to compute the SHRM—our frequency domain prefiltering algorithm is generally orders of magnitude faster than previous angular (spatial) domain approaches.

We introduce structured importance sampling, a new technique for efficiently rendering scenes illuminated by distant natural illumination given in an environment map. Our method handles occlusion, high-frequency lighting, and is significantly faster than alternative methods based on Monte Carlo sampling. We achieve this speedup as a result of several ideas. First, we present a new metric for stratifying and sampling an environment map taking into account both the illumination intensity as well as the expected variance due to occlusion within the scene. We then present a novel hierarchical stratification algorithm that uses our metric to automatically stratify the environment map into regular strata. This approach enables a number of rendering optimizations, such as pre-integrating the illumination within each stratum to eliminate noise at the cost of adding bias, and sorting the strata to reduce the number of sample rays. We have rendered several scenes illuminated by natural lighting, and our results indicate that structured importance sampling is better than the best previous Monte Carlo techniques, requiring one to two orders of magnitude fewer samples for the same image quality.

Lambertian object

We present a signal-processing framework for light transport. We study the frequency content of radiance and how it is altered by phenomena such as shading, occlusion, and transport. This extends previous work that considered either spatial or angular dimensions, and it offers a comprehensive treatment of both space and angle. We show that occlusion, a multiplication in the primal, amounts in the Fourier domain to a convolution by the spectrum of the blocker. Propagation corresponds to a shear in the space-angle frequency domain, while reflection on curved objects performs a different shear along the angular frequency axis. As shown by previous work, reflection is a convolution in the primal and therefore a multiplication in the Fourier domain. Our work shows how the spatial components of lighting are affected by this angular convolution. Our framework predicts the characteristics of interactions such as caustics and the disappearance of the shadows of small features. Predictions on the frequency content can then be used to control sampling rates for rendering. Other potential applications include precomputed radiance transfer and inverse rendering.

We present an algorithm for determining quadrature rules for computing the direct illumination of predominantly diffuse objects by high dynamic range images. The new method precisely reproduces fine shadow detail, is much more efficient as compared to Monte Carlo integration, and does not require any manual intervention.

Precomputed Radiance Transfer allows interactive rendering of objects illuminated by low-frequency environment maps, including self-shadowing and interreflections. The expensive integration of incident lighting is partially precomputed and stored as matrices.Incorporating anisotropic, glossy BRDFs into precomputed radiance transfer has been previously shown to be possible, but none of the previous methods offer real-time performance. We propose a new method, matrix radiance transfer, which significantly speeds up exit radiance computation and allows anisotropic BRDFs. We generalize the previous radiance transfer methods to work with a matrix representation of the BRDF and optimize exit radiance computation by expressing the exit radiance in a new, directionally locally supported basis set instead of the spherical harmonics. To determine exit radiance, our method performs four dot products per vertex in contrast to previous methods, where a full matrix-vector multiply is required. Image quality can be controlled by adapting the number of basis functions. We compress our radiance transfer matrices through principal component analysis (PCA). We show that it is possible to render directly from the PCA representation, which also enables the user to trade interactively between quality and speed.

Previous methods for soft shadows numerically integrate over many light directions at each receiver point, testing blocker visibility in each direction. We introduce a method for real-time soft shadows in dynamic scenes illuminated by large, low-frequency light sources where such integration is impractical. Our method operates on vectors representing low-frequency visibility of blockers in the spherical harmonic basis. Blocking geometry is modeled as a set of spheres; relatively few spheres capture the low-frequency blocking effect of complicated geometry. At each receiver point, we compute the product of visibility vectors for these blocker spheres as seen from the point. Instead of computing an expensive SH product per blocker as in previous work, we perform inexpensive vector sums to accumulate the log of blocker visibility. SH exponentiation then yields the product visibility vector over all blockers. We show how the SH exponentiation required can be approximated accurately and efficiently for low-order SH, accelerating previous CPUbased methods by a factor of 10 or more, depending on blocker complexity, and allowing real-time GPU implementation.

... In this paper, we formalize these notions, showing that the reflected light field can be thought of in a precise quantitative way as obtained by convolving the lighting and BRDF, i.e. by filtering the incident illumination using the BRDF. Mathematically, we are able to express the frequency-space coe#cients of the reflected light field as a product of the spherical harmonic coe#- cients of the illumination and the BRDF. These results are of practical importance in determining the well-posedness and conditioning of problems in inverse rendering---estimation of BRDF and lighting parameters from real photographs. Furthermore, we are able to derive analytic formulae for the spherical harmonic coe#cients of many common BRDF and lighting models. From this formal analysis, we are able to determine precise conditions under which estimation of BRDFs and lighting distributions are well posed and well-conditioned. Our mathematical analysis also has implications for forward rendering---especially the e#cient rendering of objects under complex lighting conditions specified by environment maps. The results, especially the analytic formulae derived for Lambertian surfaces, are also relevant in computer vision in the areas of recognition, photometric stereo and structure from motion.