We prove that the set of all reflectance functions (the mapping from surface normals to intensities) produced by Lambertian objects under distant, isotropic lighting lies close to a 9D linear subspace. This implies that, in general, the set of images of a convex Lambertian object obtained under a wide variety of lighting conditions can be approximated accurately by a low-dimensional linear subspace, explaining prior empirical results. We also provide a simple analytic characterization of this linear space. We obtain these results by representing lighting using spherical harmonics and describing the effects of Lambertian materials as the analog of a convolution. These results allow us to construct algorithms for object recognition based on linear methods as well as algorithms that use convex optimization to enforce non-negative lighting functions. Finally, we show a simple way to enforce non-negative lighting when the images of an object lie near a 4D linear space. Research conducted w...

Wavelets have proven to be powerful bases for use in numerical analysis and signal processing. Their power lies in the fact that they only require a small number of coefficients to represent general functions and large data sets accurately. This allows compression and efficient computations. Classical constructions have been limited to simple domains such as intervals and rectangles. In this paper we present a wavelet construction for scalar functions defined on the sphere. We show how biorthogonal wavelets with custom properties can be constructed with the lifting scheme. The bases are extremely easy to implement and allow fully adaptive subdivisions. We give examples of functions defined on the sphere, such as topographic data, bidirectional reflection distribution functions, and illumination, and show how they can be efficiently represented with spherical wavelets.

We introduce a new class of primitive functions with non-linear parameters for representing light reflectance functions. The functions are reciprocal, energy-conserving and expressive. They can capture important phenomena such as off-specular reflection, increasing reflectance and retro-reflection. We demonstrate this by fitting sums of primitive functions to a physically-based model and to actual measurements. The resulting representation is simple, compact and uniform. It can be applied efficiently in analytical and Monte Carlo computations. CR Categories: I.3.7 [Computer Graphics]: Three-Dimensional Graphics and Realism; I.3.3 [Computer Graphics]: Picture/Image Generation Keywords: Reflectance function, BRDF representation 1 INTRODUCTION The bidirectional reflectance distribution function (BRDF) of a material describes how light is scattered at its surface. It determines the appearance of objects in a scene, through direct illumination and global interreflection effects. Local r...

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The reflection of light from most materials consists of two major terms: the specular and the diffuse. Specular reflection may be modeled from first principles by considering a rough surface consisting of perfect reflectors, or micro-facets. Diffuse reflection is generally considered to result from multiple scattering either from a rough surface or from within a layer near the surface. Accounting for diffuse reflection by Lambert's Cosine Law, as is universally done in computer graphics, is not a physical theory based on first principles. This paper presents

We present a generative model for isotropic bidirectional reflectance distribution functions (BRDFs) based on acquired reflectance data. Instead of using analytical reflectance models, we represent each BRDF as a dense set of measurements. This allows us to interpolate and extrapolate in the space of acquired BRDFs to create new BRDFs. We treat each acquired BRDF as a single high-dimensional vector taken from a space of all possible BRDFs. We apply both linear (subspace) and non-linear (manifold) dimensionality reduction tools in an effort to discover a lowerdimensional representation that characterizes our measurements. We let users define perceptually meaningful parametrization directions to navigate in the reduced-dimension BRDF space. On the low-dimensional manifold, movement along these directions produces novel but valid BRDFs.

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A separable decomposition of bidirectional reflectance distributions (BRDFs) is used to implement arbitrary reflectances from point sources on existing graphics hardware. Two-dimensional texture mapping and compositing operations are used to reconstruct samples of the BRDF at every pixel at interactive rates. A change of variables, the Gram-Schmidt halfangle/difference vector parameterization, improves separability. Two decomposition algorithms are also presented. The singular value decomposition (SVD) minimizes RMS error. The normalized decomposition is fast and simple, using no more space than what is required for the final representation.

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.