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94
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 ..."
Abstract

Cited by 339 (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...
A SignalProcessing Framework for Inverse Rendering
 In SIGGRAPH 01
, 2001
"... Realism in computergenerated images requires accurate input models for lighting, textures and BRDFs. One of the best ways of obtaining highquality data is through measurements of scene attributes from real photographs by inverse rendering. However, inverse rendering methods have been largely limit ..."
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Cited by 189 (18 self)
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Realism in computergenerated images requires accurate input models for lighting, textures and BRDFs. One of the best ways of obtaining highquality 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 signalprocessing 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 illposed or numerically illconditioned, 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.
An Efficient Representation for Irradiance Environment Maps
, 2001
"... We consider the rendering of diffuse objects under distant illumination, as specified by an environment map. Using an analytic expression for the irradiance in terms of spherical harmonic coefficients of the lighting, we show that one needs to compute and use only 9 coefficients, corresponding to th ..."
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Cited by 160 (10 self)
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We consider the rendering of diffuse objects under distant illumination, as specified by an environment map. Using an analytic expression for the irradiance in terms of spherical harmonic coefficients of the lighting, we show that one needs to compute and use only 9 coefficients, corresponding to the lowestfrequency modes of the illumination, in order to achieve average errors of only 1%. In other words, the irradiance is insensitive to high frequencies in the lighting, and is well approximated using only 9 parameters. In fact, we show that the irradiance can be procedurally represented simply as a quadratic polynomial in the cartesian components of the surface normal, and give explicit formulae. These observations lead to a simple and efficient procedural rendering algorithm amenable to hardware implementation, a prefiltering method up to three orders of magnitude faster than previous techniques, and new representations for lighting design and imagebased rendering.
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 94 (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.
Frequency space environment map rendering
 ACM Transactions on Graphics (SIGGRAPH
, 2002
"... 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 reconstructio ..."
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Cited by 91 (8 self)
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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 KajiyaKay 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 realtime 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 adhoc 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.
Analytic PCA Construction for Theoretical Analysis of Lighting Variability in Images of a Lambertian Object
 IEEE Trans. Pattern Analysis and Machine Intelligence
, 2002
"... Lambertian object ..."
Total variation models for variable lighting face recognition and uneven background correction
 IEEE Transactions on Pattern Analysis and Machine Intelligence
, 2005
"... In this paper, we present the logarithmic total variation (LTV) model for face recognition under varying illumination, including natural lighting condition, where we can hardly know the strength, the directions, and the number of light sources. The proposed LTV model has the capability to factorize ..."
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Cited by 41 (6 self)
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In this paper, we present the logarithmic total variation (LTV) model for face recognition under varying illumination, including natural lighting condition, where we can hardly know the strength, the directions, and the number of light sources. The proposed LTV model has the capability to factorize a single face image and obtain the illumination invariant facial structure, which is then used for face recognition. The merit of this model is that neither does it require any lighting assumption nor does it need any training process. Besides, there is only one parameter which could be easily set. The LTV model is able to reach very high recognition rates on both Yale and CMU PIE face databases as well as on a face database containing 765 subjects under outdoor lighting conditions. Keywords: I.5.4.d Face and gesture recognition; I.5.4.m Signal processing; I.4 Image Processing and Computer Vision; I.5.2.c Pattern analysis;
Exposing digital forgeries in complex lighting environments
 IEEE Transactions on Information Forensics and Security
"... The availability of sophisticated digital imaging technology has given rise to digital forgeries that are increasing in sophistication and frequency. We describe a technique for exposing such fakes by detecting inconsistencies in lighting. We show how to approximate complex lighting environments wit ..."
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Cited by 36 (8 self)
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The availability of sophisticated digital imaging technology has given rise to digital forgeries that are increasing in sophistication and frequency. We describe a technique for exposing such fakes by detecting inconsistencies in lighting. We show how to approximate complex lighting environments with a lowdimensional model and, further, how to estimate the model’s parameters from a single image. Inconsistencies in the lighting model are then used as evidence of tampering. I.
A SignalProcessing Framework for Reflection
 ACM TRANSACTIONS ON GRAPHICS
, 2004
"... ... 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 frequencyspac ..."
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Cited by 35 (4 self)
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... 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 frequencyspace 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 wellposedness and conditioning of problems in inverse renderingestimation 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 wellconditioned. Our mathematical analysis also has implications for forward renderingespecially 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.
A fourier theory for cast shadows
 IEEE Transactions on Pattern Analysis and Machine Intelligence
, 2004
"... Abstract. Cast shadows can be significant in many computer vision applications such as lightinginsensitive recognition and surface reconstruction. However, most algorithms neglect them, primarily because they involve nonlocal interactions in nonconvex regions, making formal analysis difficult. Wh ..."
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Cited by 25 (7 self)
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Abstract. Cast shadows can be significant in many computer vision applications such as lightinginsensitive recognition and surface reconstruction. However, most algorithms neglect them, primarily because they involve nonlocal interactions in nonconvex regions, making formal analysis difficult. While general cast shadowing situations can be arbitrarily complex, many real instances map closely to canonical configurations like a wall, a Vgroove type structure, or a pitted surface. In particular, we experiment on 3D textures like moss, gravel and a kitchen sponge, whose surfaces include canonical cast shadowing situations like Vgrooves. This paper shows theoretically that many shadowing configurations can be mathematically analyzed using convolutions and Fourier basis functions. Our analysis exposes the mathematical convolution structure of cast shadows, and shows strong connections to recently developed signalprocessing frameworks for reflection and illumination. An analytic convolution formula is derived for a 2D Vgroove, which is shown to correspond closely to many common shadowing situations, especially in 3D textures. Numerical simulation is used to extend these results to general 3D textures. These results also provide evidence that a common set of illumination basis functions may be appropriate for representing lighting variability due to cast shadows in many 3D textures. We derive a new analytic basis suited for 3D textures to represent illumination on the hemisphere, with some advantages over commonly used Zernike polynomials and spherical harmonics. New experiments on analyzing the variability in appearance of real 3D textures with illumination motivate and validate our theoretical analysis. Empirical results show that illumination eigenfunctions often correspond closely to Fourier bases, while the eigenvalues drop off significantly slower than those for irradiance on a Lambertian curved surface. These new empirical results are explained in this paper, based on our theory. 1