The appearance of a particular object depends on both the viewpoint from which it is observed and the light sources by which it is illuminated. If the appearance of two objects is never identical for any pose or lighting conditions, then- in theory- the objects can always be distinguished or recognized. The question arises: What is the set of images of an object under all lighting conditions and pose? In this paper, ive consider only the set of images of an object under variable allumination (including multiple, extended light sources and attached shadows). We prove that the set of n-pixel images of a convex object with a Lambertian reflectance function, illuminated by an arbitrary number of point light sources at infinity, forms a convex polyhedral cone in IR " and that the dimension of this illumination cone equals the number of distinct surface normals. Furthermore, we show that the cone for a particular object can be constructed from three properly chosen images. Finally, we prove that the set of n-pixel images of an object of any shape and with an arbitrary reflectance function, seen under all possi-ble illumination conditions, still forms a convex cone in Rn. Th.ese results immediately suggest certain approaches to object recognition. Throughout this paper, we ofler results demonstrating the empirical validity of the illumination cone representation. 1

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...

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...

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.

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.

We compress storage and accelerate performance of precomputed radiance transfer (PRT), which captures the way an object shadows, scatters, and reflects light. PRT records over many surface points a transfer matrix. At run-time, this matrix transforms a vector of spherical harmonic coefficients representing distant, low-frequency source lighting into exiting radiance. Per-point transfer matrices form a high-dimensional surface signal that we compress using clustered principal component analysis (CPCA), which partitions many samples into fewer clusters each approximating the signal as an affine subspace. CPCA thus reduces the high-dimensional transfer signal to a low-dimensional set of perpoint weights on a per-cluster set of representative matrices. Rather than computing a weighted sum of representatives and applying this result to the lighting, we apply the representatives to the lighting per-cluster (on the CPU) and weight these results perpoint (on the GPU). Since the output of the matrix is lowerdimensional than the matrix itself, this reduces computation. We also increase the accuracy of encoded radiance functions with a new least-squares optimal projection of spherical harmonics onto the hemisphere. We describe an implementation on graphics hardware that performs real-time rendering of glossy objects with dynamic self-shadowing and interreflection without fixing the view or light as in previous work. Our approach also allows significantly increased lighting frequency when rendering diffuse objects and includes subsurface scattering.

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Abstract. We present a method – termed Helmholtz stereopsis – for reconstructing the geometry of objects from a collection of images. Unlike most existing methods for surface reconstruction (e.g., stereo vision, structure from motion, photometric stereo), Helmholtz stereopsis makes no assumptions about the nature of the bidirectional reflectance distribution functions (BRDFs) of objects. This new method of multinocular stereopsis exploits Helmholtz reciprocity by choosing pairs of light source and camera positions that guarantee that the ratio of the emitted radiance to the incident irradiance is the same for corresponding points in the two images. The method provides direct estimates of both depth and field of surface normals, and consequently weds the advantages of both conventional and photometric stereopsis. Results from our implementations lend empirical support to our technique. 1

Figure 1: A model rendered at real-time rates (approximately half the performance of the standard per-vertex lighting model on an NVIDIA GeForce 3) with several BRDFs approximated using the technique in this paper. From left to right: satin (anisotropic Poulin-Fournier model), krylon blue, garnet red, cayman, mystique (Cornell measured data), leather, and velvet (CURET measured data). A bidirectional reflectance distribution function (BRDF) describes how a material reflects light from its surface. To use arbitrary BRDFs in real-time rendering, a compression technique must be used to represent BRDFs using the available texture-mapping and computational capabilities of an accelerated graphics pipeline. We present a numerical technique, homomorphic factorization, that can decompose arbitrary BRDFs into products of two or more factors of lower dimensionality, each factor dependent on a different interpolated geometric parameter. Compared to an earlier factorization technique based on the singular value decomposition, this new technique generates a factorization with only positive factors (which makes it more suitable for current graphics hardware accelerators), provides control over the smoothness of the result, minimizes relative rather than absolute error, and can deal with scattered, sparse data without a separate resampling and interpolation algorithm.

. The measurement of accurate material properties is an important step towards photorealistic rendering. Many real-world objects are composed of a number of materials that often show subtle changes even within a single material. Thus, for photorealistic rendering both the general surface properties as well as the spatially varying effects of the object are needed. We present an image-based measuring method that robustly detects the different materials of real objects and fits an average bidirectional reflectance distribution function (BRDF) to each of them. In order to model the local changes as well, we project the measured data for each surface point into a basis formed by the recovered BRDFs leading to a truly spatially varying BRDF representation. A high quality model of a real object can be generated with relatively few input data. The generated model allows for rendering under arbitrary viewing and lighting conditions and realistically reproduces the appearance of the original object. 1