## The Bas-Relief Ambiguity (1997)

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### BibTeX

@MISC{Belhumeur97thebas-relief,

author = {Peter N. Belhumeur and David J. Kriegman and Alan L. Yuille},

title = {The Bas-Relief Ambiguity },

year = {1997}

}

### Years of Citing Articles

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### Abstract

### Citations

1406 |
Robot vision
- Horn
- 1986
(Show Context)
Citation Context ...eralized bas-relief transformations. The fact that an object and every GBR transformation of the object produce the same shadow regions arises from an implicit duality. For each image of a Lambertian =-=[20, 11]-=- surface viewed under orthographic projection (parallel lines of sight) and illuminated by a distant light source (e.g., the sun), there exists an identical image of a GBR surface with transformed alb... |

632 |
A computer algorithm for reconstructing a scene from two projections
- Longuet-Higgins
- 1981
(Show Context)
Citation Context ...both the depth and shape of viewed objects. For an object undergoing rigid motion and viewed under perspective projection, the object’s Euclidean structure can be determined from as few as two images =-=[22, 25, 33]-=-. If the object is viewed orthographically in two images, its structure can only be recovered up to a one parameter family of affine distortions [13]. For infinitesimal motion under orthographic proje... |

410 |
Recognition by linear combinations of models
- Ullman, Basri
- 1991
(Show Context)
Citation Context ...s and is often unresolved in the human visual system. Likewise, the results suggest that the aim of structure recovery might be a weaker non-Euclidean representation, such as an affine representation =-=[17, 26, 27, 31]-=-, a projective representation [7], or an ordinal representation [8]; for many applications, machine vision systems need not resolve this ambiguity. In summary, this paper will make the following point... |

324 | What is the set of images of an object under all possible illumination conditions
- Belhumeur, Kriegman
- 1998
(Show Context)
Citation Context ...l, irrespective of the material type. If the material can modeled as having Lambertian reflectance, then the set of possible images including shadowing under any lighting condition (illumination cone =-=[3, 4]-=-) for an object and a GBR transformed object are identical. (Note the GBR transformation alters both the surface geometry and surface albedo.) Therefore, these objects cannot be distinguished by any r... |

293 |
Affine structure from motion
- Koenderink, Doorn
- 1992
(Show Context)
Citation Context ...s-relief ambiguity: the shape of the surface can only be recovered up to a scale factor in the direction of the camera’s optical axis, i.e., a classical bas-relief transformation for which is unknown =-=[17]-=-.The Bas-Relief Ambiguity 3 Fig. 2. Three-dimensional data for the human head (top row) was obtained using a laser scan (Cyberware) and rendered as a Lambertian surface with constant albedo (equal gr... |

115 |
Geometric Algebra. Interscience
- Artin
- 1957
(Show Context)
Citation Context ...ries passing through in the image plane to the surface normal . Under these two mappings, we have a projective isomorphism between and which in turn is a projective transformation (collineation), see =-=[1]-=-. Because and are the duals of and respectively, the surface normals of are also related to the surface normals of by a projective transformation, i.e., where is a matrix in the general projective gro... |

111 | The variational approach to shape from shading - Horn, Brooks - 1986 |

97 | Illumination cones for recognition under variable lighting: faces
- Georghiades, Kriegman, et al.
- 1998
(Show Context)
Citation Context ...dean structure. These nonEuclidean representations (affine [17, 26, 27, 31], projective [7] or ordinal [8]), can still be used to solve numerous vision-based tasks such as object and face recognition =-=[9]-=-, vehicle navigation, robotic manipulation and synthetic image generation without resolution of these ambiguities. Appendix: Uniqueness of the Generalized BasRelief Transformation Here we prove that t... |

89 |
Handbuch der physiologischen Optik. English. Helmholtz's treatise on physiological optics. Translated from the 3d German ed. Edited by James P.C
- Helmholtz
(Show Context)
Citation Context ...ng or shading of the surface can resolve this. Yet, if the viewer moves relative to the surface, or the surface moves relative to the viewer, can this ambiguity be resolved? As discussed by Helmholtz =-=[32]-=-, image changes produced by an observer’s motion reveal both the depth and shape of viewed objects. For an object undergoing rigid motion and viewed under perspective projection, the object’s Euclidea... |

73 | Geometry and photometry in 3D visual recognition
- Shashua
- 1992
(Show Context)
Citation Context ...p 1 n s p2 p 2 n p 1 n n s surfaces are Lambertian, the set of all possible images of both surfaces are identical. 2.1. Shadows We can identify two types of shadows: attached shadows and cast shadows =-=[2, 28]-=-. See Figure 3. A surface point lies in an attached shadow for light source direction * iff . This definition leads to the following lemma. Lemma 1. A point lies in an attached shadow for light source... |

64 |
Photometric stereo under a light-source with arbitrary motion,” JOSA
- Hayakawa
- 1994
(Show Context)
Citation Context ...ely disguise the distortions produced by the bottom image in the second row * where ; here we rotated the head * by around the vertical axis and then by about the line of sight. linear transformation =-=[10, 5, 26]-=- (5) Note that numerous vision problems have this bilinear form and a similar linear ambiguity [18]. If is the true vector field of surface normals, then the recovered vector field is any vector field... |

60 | 3D motion recovery via affine epipolar geometry - Shapiro, Zisserman, et al. - 1995 |

59 | fitting with missing data: Applications to structure-from-motion and to characterizing intensity images - Linear - 1997 |

55 |
Motion and structure from orthographic projections
- Huang, Lee
(Show Context)
Citation Context ...an be determined from as few as two images [22, 25, 33]. If the object is viewed orthographically in two images, its structure can only be recovered up to a one parameter family of affine distortions =-=[13]-=-. For infinitesimal motion under orthographic projection, there is a genuine bas-relief ambiguity: the shape of the surface can only be recovered up to a scale factor in the direction of the camera’s ... |

50 | Shape ambiguities in structure from motion
- Szeliski, Kang
- 1997
(Show Context)
Citation Context ...ject’s motion is small. For infinitesimal motion under perspective projection, the structure estimates are sensitive to noise, producing an implicit error in the estimate of the relief of the surface =-=[24, 30]-=-. For small (infinitesimal) unknown motion under orthographic projection, there is a genuine bas-relief ambiguity: the shape of the surface can only be recovered up to a scale factor in the direction ... |

48 | Stratification of 3-d vision: projective, affine, and metric representations
- Faugeras
- 1995
(Show Context)
Citation Context ...system. Likewise, the results suggest that the aim of structure recovery might be a weaker non-Euclidean representation, such as an affine representation [17, 26, 27, 31], a projective representation =-=[7]-=-, or an ordinal representation [8]; for many applications, machine vision systems need not resolve this ambiguity. In summary, this paper will make the following points: The set of cast and attached s... |

47 |
Analysing images of curved surfaces
- Woodham
- 1981
(Show Context)
Citation Context ...iewpoint but with different light source direction, we can solve Eq. 4 for when the light source strengths and directions are known. This, of course, is the standard photometric stereo technique, see =-=[11, 29, 34]-=-. However, if the light source strengths and directions are not known, then we can only determine the vector field of surface normals and albedos up to a linear transformation. For any invertible wher... |

46 |
Uniqueness in shape from shading
- Oliensis
- 1991
(Show Context)
Citation Context ...ilar argument holds when the light source intensities are known. Thus, we can determine the true surface up to a sign, i.e., . This is the classical in-out ambiguity that occurs in shape from shading =-=[11, 23]-=-. Note however, that the shadowing configurations change when changes sign, and if shadowing is present, this ambiguity can be resolved. In contrast to [10], it is worth noting that the surface cannot... |

44 |
Determining Shape and Reflectance Using Multiple Images
- Silver
- 1980
(Show Context)
Citation Context ...iewpoint but with different light source direction, we can solve Eq. 4 for when the light source strengths and directions are known. This, of course, is the standard photometric stereo technique, see =-=[11, 29, 34]-=-. However, if the light source strengths and directions are not known, then we can only determine the vector field of surface normals and albedos up to a linear transformation. For any invertible wher... |

35 |
Surface structure and three dimensional motion from image flow kinematics
- WAXMAN, ULLMAN
- 1985
(Show Context)
Citation Context ...both the depth and shape of viewed objects. For an object undergoing rigid motion and viewed under perspective projection, the object’s Euclidean structure can be determined from as few as two images =-=[22, 25, 33]-=-. If the object is viewed orthographically in two images, its structure can only be recovered up to a one parameter family of affine distortions [13]. For infinitesimal motion under orthographic proje... |

34 | A ne Structure from Motion - Koenderink, Doorn - 1992 |

34 |
D.: Shape and albedo from multiple images using integrability
- Yuille, Snow
- 1997
(Show Context)
Citation Context ...s – each acquired under different light source directions – of a Lambertian surface , then by imposing the integrability constraint in Eq. 6, we can recover the surface up to a GBR transformation See =-=[35]-=- for a method to estimate an integrable from image data. Note that no information given in the image shadows can resolve this ambiguity, as Section 2 showed that the set of all possible images of a su... |

32 |
Photometria sive de mensura et gradibus luminus, colorum et umbrae,” Eberhard Klett
- Lambert
(Show Context)
Citation Context ...eralized bas-relief transformations. The fact that an object and every GBR transformation of the object produce the same shadow regions arises from an implicit duality. For each image of a Lambertian =-=[20, 11]-=- surface viewed under orthographic projection (parallel lines of sight) and illuminated by a distant light source (e.g., the sun), there exists an identical image of a GBR surface with transformed alb... |

20 |
Surface Curvature and Shape Reconstruction from Unknown Multiple Illumination and Integrability
- Fan, Wolff
- 1997
(Show Context)
Citation Context ...ould be noted that Fan and Wolff showed that the Hessian of can be determined from three images up to scale factor using the integrability constraint to enforce equality of the off-diagonal elements, =-=[6]-=-. The unknown scale of the Hessian corresponds to the parameter and the unknown initial conditions of Fan and Wolff’s differential equation correspond to the parameters and . If, however, we have addi... |

18 | Structure from linear or planar motions
- Oliensis
- 1996
(Show Context)
Citation Context ...ject’s motion is small. For infinitesimal motion under perspective projection, the structure estimates are sensitive to noise, producing an implicit error in the estimate of the relief of the surface =-=[24, 30]-=-. For small (infinitesimal) unknown motion under orthographic projection, there is a genuine bas-relief ambiguity: the shape of the surface can only be recovered up to a scale factor in the direction ... |

17 |
fitting with missing data: Applications to structure from motion and characterizing intensity images
- Linear
- 1997
(Show Context)
Citation Context ... with no surface point in shadow, can be estimated from a collection of images using singular value decomposition; when some of the surface points are shadowed, Jacobs’ method can be used to estimate =-=[14]-=-. Note, however, that not all vector fields correspond to continuous (or even piecewise continuous) surfaces. We will use this observation to restrict the group of allowable transformations on [5]. If... |

11 |
The Science of Art: Optical Themes in Western Art from Brunelleschi to Seurat
- Kemp
- 1990
(Show Context)
Citation Context ...he figures to the proper relative depths, sculptors like Donatello and Ghiberti employed rules of perspective to determine the size and location of figures, sculpting each figure to the proper relief =-=[16]-=-. Barring the effects of constant albedo, the shading and shadowing for each figure is self consistent; however, the shadows cast from one figure onto another are incorrect. Furthermore, the shadows c... |

11 |
What is a light source
- Langer, Zucker
- 1997
(Show Context)
Citation Context ... at infinity, with the magnitude of proportional to the intensity of the light source. (For a more general model of illumination, e.g., one that does not restrict light sources to be at infinity, see =-=[21]-=-.) We first show that shadowing on a surface for some light source is identical to that on a GBR transformed surface with an appropriate light source ; we then show that if the p 1 n s p2 p 2 n p 1 n ... |

11 |
Affine structure and photometry
- Rosenholtz, Koenderink
- 1996
(Show Context)
Citation Context ...s and is often unresolved in the human visual system. Likewise, the results suggest that the aim of structure recovery might be a weaker non-Euclidean representation, such as an affine representation =-=[17, 26, 27, 31]-=-, a projective representation [7], or an ordinal representation [8]; for many applications, machine vision systems need not resolve this ambiguity. In summary, this paper will make the following point... |

10 |
Ordinal representations of visual space
- Fermuller, Aloimonos
- 1996
(Show Context)
Citation Context ...est that the aim of structure recovery might be a weaker non-Euclidean representation, such as an affine representation [17, 26, 27, 31], a projective representation [7], or an ordinal representation =-=[8]-=-; for many applications, machine vision systems need not resolve this ambiguity. In summary, this paper will make the following points: The set of cast and attached shadows produced by a surface and a... |

10 |
Shape constancy in pictorial relief
- Koenderink, Doorn, et al.
- 1996
(Show Context)
Citation Context ...ambiguity also has implications for our understanding of human surface perception and for the development of computational vision algorithms. Our results support the recent psychophysical findings of =-=[19]-=- that for a variety of surfaces this ambiguity exists and is often unresolved in the human visual system. Likewise, the results suggest that the aim of structure recovery might be a weaker non-Euclide... |

8 | 3D Motion Recovery via A ne Epipolar Geometry - Shapiro, Zisserman, et al. - 1995 |

7 |
Shadows and Enlightenment
- Baxandall
- 1995
(Show Context)
Citation Context ...p 1 n s p2 p 2 n p 1 n n s surfaces are Lambertian, the set of all possible images of both surfaces are identical. 2.1. Shadows We can identify two types of shadows: attached shadows and cast shadows =-=[2, 28]-=-. See Figure 3. A surface point lies in an attached shadow for light source direction * iff . This definition leads to the following lemma. Lemma 1. A point lies in an attached shadow for light source... |

5 |
Learning and recognizing objects using illumination subspaces
- Epstein, Yuille, et al.
- 1996
(Show Context)
Citation Context ...ives of with respect to and respectively. Consider transforming the surface to a new surface in the following manner. We first flatten (or scale) it along the axis and then add a plane, i.e., where * =-=[5]-=-. We call this transformation the generalized bas-relief (GBR) transformation. See Figures 2 and 3. When + * and * , we refer to this as the classical bas-relief transformation, since for! " The Bas-... |

4 | Strati cation of 3-d vision: projective, a ne, and metric representations - Faugeras - 1995 |

4 | What is a light soruce - Langer, Zucker - 1997 |

3 |
Computer tracking of objects from a sequence of images
- Roach, Aggarwal, et al.
- 1979
(Show Context)
Citation Context ...both the depth and shape of viewed objects. For an object undergoing rigid motion and viewed under perspective projection, the object’s Euclidean structure can be determined from as few as two images =-=[22, 25, 33]-=-. If the object is viewed orthographically in two images, its structure can only be recovered up to a one parameter family of affine distortions [13]. For infinitesimal motion under orthographic proje... |

2 | Leonardo on painting - Kemp - 1989 |

1 |
The generic bilinear calibration estimation problem
- Koenderink, Doorn
- 1997
(Show Context)
Citation Context ...head * by around the vertical axis and then by about the line of sight. linear transformation [10, 5, 26] (5) Note that numerous vision problems have this bilinear form and a similar linear ambiguity =-=[18]-=-. If is the true vector field of surface normals, then the recovered vector field is any vector field in the orbit of under the group . For a pixelated image with no surface point in shadow, can be es... |

1 | Determining Shape andRe ectance UsingMultiple Images - Silver - 1980 |