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29
Lambertian Reflectance and Linear Subspaces
, 2000
"... 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 wi ..."
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Cited by 345 (19 self)
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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 lowdimensional 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 nonnegative lighting functions. Finally, we show a simple way to enforce nonnegative lighting when the images of an object lie near a 4D linear space. Research conducted w...
Separating style and content with bilinear models
 NEURAL COMPUTATION
, 2000
"... PERCEPTUAL systems routinely separate content from style, classifying familiar words spoken in an unfamiliar accent, identifying a font or handwriting style across letters, or recognizing a familiar face or object seen under unfamiliar viewing conditions. Yet a general and tractable computational mo ..."
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Cited by 174 (3 self)
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PERCEPTUAL systems routinely separate content from style, classifying familiar words spoken in an unfamiliar accent, identifying a font or handwriting style across letters, or recognizing a familiar face or object seen under unfamiliar viewing conditions. Yet a general and tractable computational model of this ability to untangle the underlying factors of perceptual observations remains elusive. Existing factor models are either insufficiently rich to capture the complex interactions of perceptually meaningful factors such as phoneme and speaker accent or letter and font, or do not allow efficient learning algorithms. Here we show how perceptual systems may learn to solve these crucial tasks using surprisingly simple bilinear models. We report promising results in three realistic perceptual domains: spoken vowel classification with a benchmark multispeaker database, extrapolation of fonts to unseen letters, and translation of faces to novel illuminants.
Bayesian color constancy
 Journal of the Optical Society of America A
, 1997
"... The problem of color constancy may be solved if we can recover the physical properties of illuminants and surfaces from photosensor responses. We consider this problem within the framework of Bayesian decision theory. First, we model the relation among illuminants, surfaces, and photosensor response ..."
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Cited by 143 (21 self)
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The problem of color constancy may be solved if we can recover the physical properties of illuminants and surfaces from photosensor responses. We consider this problem within the framework of Bayesian decision theory. First, we model the relation among illuminants, surfaces, and photosensor responses. Second, we construct prior distributions that describe the probability that particular illuminants and surfaces exist in the world. Given a set of photosensor responses, we can then use Bayes’s rule to compute the posterior distribution for the illuminants and the surfaces in the scene. There are two widely used methods for obtaining a single best estimate from a posterior distribution. These are maximum a posteriori (MAP) and minimum meansquarederror (MMSE) estimation. We argue that neither is appropriate for perception problems. We describe a new estimator, which we call the maximum local mass (MLM) estimate, that integrates local probability density. The new method uses an optimality criterion that is appropriate for perception tasks: It finds the most probable approximately correct answer. For the case of low observation noise, we provide an efficient approximation. We develop the MLM estimator for the colorconstancy problem in which flat matte surfaces are uniformly illuminated. In simulations we show that the MLM method performs better than the MAP estimator and better than a number of standard colorconstancy algorithms. We note conditions under which even the optimal estimator produces poor estimates: when the spectral properties of the surfaces in the scene are biased. © 1997 Optical Society of America [S07403232(97)016074] 1.
Photometric Stereo with General, Unknown Lighting
 In IEEE Conference on Computer Vision and Pattern Recognition
, 2001
"... Work on photometric stereo has shown how to recover the shape and reflectance properties of an object using multiple images taken with a fixed viewpoint and variable lighting conditions. This work has primarily relied on the presence of a single point source of light in each image. In this paper we ..."
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Cited by 95 (8 self)
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Work on photometric stereo has shown how to recover the shape and reflectance properties of an object using multiple images taken with a fixed viewpoint and variable lighting conditions. This work has primarily relied on the presence of a single point source of light in each image. In this paper we show how to perform photometric stereo assuming that all lights in a scene are isotropic and distant from the object but otherwise unconstrained. Lighting in each image may be an unknown and arbitrary combination of diffuse, point and extended sources. Our work is based on recent results showing that for Lambertian objects, general lighting conditions can be represented using low order spherical harmonics. Using this representation we can recover shape by performing a simple optimization in a lowdimensional space. We also analyze the shape ambiguities that arise in such a representation. 1.
Learning bilinear models for twofactor problems in vision
 Proc. IEEE Computer Soc. Conf. on Computer Vision and Pattern Recognition (CVPR
, 1997
"... In many vision problems, we want to infer two (or more) hidden factors which interact to produce our observations. We may want to disentangle illuminant and object colors in color constancy; rendering conditions from surface shape in shapefromshading; face identity and head pose in face recognitio ..."
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Cited by 51 (2 self)
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In many vision problems, we want to infer two (or more) hidden factors which interact to produce our observations. We may want to disentangle illuminant and object colors in color constancy; rendering conditions from surface shape in shapefromshading; face identity and head pose in face recognition; or font and letter class in character recognition. We refer to these two factors
Separating style and content
 ADV. IN NEURAL INFO. PROC. SYSTEMS, VOLUME 9, MIT PRESS, 1997.
, 1997
"... We seek to analyze and manipulate two factors, which we generically call style and content, underlying a set of observations. We fit training data with bilinear models which explicitly represent the twofactor structure. These models can adapt easily during testing to new styles or content, allowing ..."
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Cited by 44 (2 self)
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We seek to analyze and manipulate two factors, which we generically call style and content, underlying a set of observations. We fit training data with bilinear models which explicitly represent the twofactor structure. These models can adapt easily during testing to new styles or content, allowing us to solve three general tasks: extrapolation of a new style to unobserved content; classification of content observed in a new style; and translation of new content observed in a new style. For classification, we embed bilinear models in a probabilistic framework, Separable Mixture Models (SMMs), which generalizes earlier work on factorial mixture models (Hinton ´94, Ghahramani ´95). Significant performance improvement on a benchmark speech dataset shows the benefits of our approach.
Estimation of nonlinear errorsinvariables models for computer vision applications
 IEEE Trans. Patt. Anal. Mach. Intell
, 2006
"... Abstract—In an errorsinvariables (EIV) model, all the measurements are corrupted by noise. The class of EIV models with constraints separable into the product of two nonlinear functions, one solely in the variables and one solely in the parameters, is general enough to represent most computer visi ..."
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Cited by 24 (4 self)
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Abstract—In an errorsinvariables (EIV) model, all the measurements are corrupted by noise. The class of EIV models with constraints separable into the product of two nonlinear functions, one solely in the variables and one solely in the parameters, is general enough to represent most computer vision problems. We show that the estimation of such nonlinear EIV models can be reduced to iteratively estimating a linear model having point dependent, i.e., heteroscedastic, noise process. Particular cases of the proposed heteroscedastic errorsinvariables (HEIV) estimator are related to other techniques described in the vision literature: the Sampson method, renormalization, and the fundamental numerical scheme. In a wide variety of tasks, the HEIV estimator exhibits the same, or superior, performance as these techniques and has a weaker dependence on the quality of the initial solution than the LevenbergMarquardt method, the standard approach toward estimating nonlinear models. Index Terms—Nonlinear least squares, heteroscedastic regression, camera calibration, 3D rigid motion, uncalibrated vision. 1 MODELING COMPUTER VISION PROBLEMS SOLVING most computer vision problems requires the estimation of a set of parameters from noisy measurements using a statistical model. A statistical model provides a mathematical description of a problem in terms of a constraint equation relating the measurements to the
Specularities Reduce Ambiguity of Uncalibrated Photometric Stereo
, 2002
"... Lambertian photometric stereo with uncalibrated light directions and intensities determines the surface normals only up to an invertible linear transformation. We show that if object reflectance is a sum of Lambertian and specular terms, the ambiguity reduces into a 2dof group of transformations ..."
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Cited by 18 (1 self)
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Lambertian photometric stereo with uncalibrated light directions and intensities determines the surface normals only up to an invertible linear transformation. We show that if object reflectance is a sum of Lambertian and specular terms, the ambiguity reduces into a 2dof group of transformations (compositions of isotropic scaling, rotation around the viewing vector, and change in coordinate frame handedness).
Bilinearity, rules, and prefrontal cortex
"... Humans can be instructed verbally to perform computationally complex cognitive tasks; their performance then improves relatively slowly over the course of practice. Many skills underlie these abilities; in this paper, we focus on the particular question of a uniform architecture for the instantiatio ..."
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Cited by 11 (3 self)
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Humans can be instructed verbally to perform computationally complex cognitive tasks; their performance then improves relatively slowly over the course of practice. Many skills underlie these abilities; in this paper, we focus on the particular question of a uniform architecture for the instantiation of habitual performance and the storage, recall, and execution of simple rules. Our account builds on models of gated working memory, and involves a bilinear architecture for representing conditional inputoutput maps and for matching rules to the state of the input and working memory. We demonstrate the performance of our model on two paradigmatic tasks used to investigate prefrontal and basal ganglia function.
Can two specular pixels calibrate photometric stereo
 In Proc. of Int’l Conf. on Computer Vision
, 2005
"... Lambertian photometric stereo with unknown light source parameters is ambiguous. Provided that the object imaged constitutes a surface, the ambiguity is represented by the group of Generalised BasRelief (GBR) transformations. We show that this ambiguity is resolved when specular reflection is prese ..."
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Cited by 9 (0 self)
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Lambertian photometric stereo with unknown light source parameters is ambiguous. Provided that the object imaged constitutes a surface, the ambiguity is represented by the group of Generalised BasRelief (GBR) transformations. We show that this ambiguity is resolved when specular reflection is present in two images taken under two different light source directions. We identify all configurations of the two directional lights which are singular and show that they can easily be tested for. While previous work used optimisation algorithms to apply the constraints implied by the specular reflectance component, we have developed a linear algorithm to achieve this goal. Our theory can be utilised to construct fast algorithms for automatic reconstruction of smooth glossy surfaces. 1.