## Computing Local Surface Orientation and Shape from Texture for Curved Surfaces (1997)

### Cached

### Download Links

- [www.cs.berkeley.edu]
- [www.cs.berkeley.edu]
- [www.eecs.berkeley.edu]
- [HTTP.CS.Berkeley.EDU]
- DBLP

### Other Repositories/Bibliography

Citations: | 90 - 4 self |

### BibTeX

@MISC{Malik97computinglocal,

author = {Jitendra Malik and Ruth Rosenholtz},

title = {Computing Local Surface Orientation and Shape from Texture for Curved Surfaces},

year = {1997}

}

### Years of Citing Articles

### OpenURL

### Abstract

Shape from texture is best analyzed in two stages, analogous to stereopsis and structure from motion: (a) Computing the `texture distortion' from the image, and (b) Interpreting the `texture distortion' to infer the orientation and shape of the surface in the scene. We model the texture distortion for a given point and direction on the image plane as an affine transformation and derive the relationship between the parameters of this transformation and the shape parameters. We have developed a technique for estimating affine transforms between nearby image patches which is based on solving a system of linear constraints derived from a differential analysis. One need not explicitly identify texels or make restrictive assumptions about the nature of the texture such as isotropy. We use non-linear minimization of a least squares error criterion to recover the surface orientation (slant and tilt) and shape (principal curvatures and directions) based on the estimated affine transforms in a number of different directions. A simple linear algorithm based on singular value decomposition of the linear parts of the affine transforms provides the initial guess for the minimization procedure. Experimental results on both planar and curved surfaces under perspective projection demonstrate good estimates for both orientation and shape. A sensitivity analysis yields predictions for both computer vision algorithms and human perception of shape from texture.

### Citations

1283 |
Robust Regression and Outlier Detection
- Rousseeuw, Leroy
- 1987
(Show Context)
Citation Context ...ints typically correspond to noise. The choice of this threshold should be based on the signal-to-noise ratio. 13s2. Reject outliers, using a reweighted least squares technique from robust statistics =-=[23]-=-. In this method, we obtain an initial estimate, ~a 0, for ~a, and calculate the resulting residual, ~r = D ~a 0 , ~ b.Weexpect a higher residual at the peaks of the spectrogram than in the low areas,... |

446 |
The Perception of the Visual World
- Gibson
- 1950
(Show Context)
Citation Context ... shape estimates in Section 6 and present ideal observer predictions for shape from texture. 2 Relationship to previous work Modern developments in shape from texture originate with the work of Gibson=-=[12]-=-. He coined the term texture gradient to describe the phenomenon in which neighboring surface patches which have identical, or su ciently similar, texture in the scene project in the image plane to pa... |

191 | Elementary differential geometry - O’Neill - 1966 |

141 |
Solid shape
- Koenderink
- 1990
(Show Context)
Citation Context ...rame eld This section is based on Garding [10], to which the reader is referred for proofs of the various assertions. Relevant di erential geometry concepts may be found in O'Neill [21] or Koenderink =-=[16]-=-. The basic geometry is illustrated in Figure 1. A smooth surface S is mapped by central projection to a unit sphere centered at the focal point. The backprojection map F from to S is de ned as F (p) ... |

132 |
M.: Riemannian Geometry
- Carmo
- 1992
(Show Context)
Citation Context ...s of constant negative Gaussian curvature. Rigorous results were obtained by Cartan who showed that in some sense, the Riemannian metric is determined locally by the curvature. We refer the reader to =-=[9]-=-, pp. 156-159 for the proof and the following corollary; Corollary 2.3 Let M be aspace ofconstant gaussian curvature and let p and q be any two points of M. Let fejg and ffjg be arbitrary orthonormal ... |

129 |
Recovering surface shape and orientation from texture
- Witkin
- 1981
(Show Context)
Citation Context ...e model given sample measurements in the image{and can be approached in either a Bayesian or maximum likelihood framework. The model most often used is that of isotropy orweak isotropy of the texture =-=[27, 8, 6, 4, 11]-=-. Under projection, the texture will not generally appear isotropic, and thus they use the deviation from isotropy in the projection to infer 3D shape and orientation. There are two major weaknesses o... |

70 |
Introduction to Shannon Sampling and Interpolation Theory
- Marks
- 1991
(Show Context)
Citation Context ...present a method for nding the a ne transform between a pair of image patches. Rather than estimating the transform in the space domain, we estimate it in the frequency domain. We can do this because =-=[20]-=- if G is the Fourier transform of g, which we write as G = F(g), then F[g(Ax)] = 1 j A j G(A,T !) Thus if we can nd the a ne transformation in the frequency domain, we can nd the a ne tranformation in... |

68 | Texture gradient as a depth cue - Bajcsy, Lieberman - 1976 |

67 | Three gradients and the perception of flat and curved surfaces - Cutting, Millard - 1984 |

66 |
Shape from texture: Estimation, isotropy and moments
- Blake, Marinos
- 1990
(Show Context)
Citation Context ...e model given sample measurements in the image{and can be approached in either a Bayesian or maximum likelihood framework. The model most often used is that of isotropy orweak isotropy of the texture =-=[27, 8, 6, 4, 11]-=-. Under projection, the texture will not generally appear isotropic, and thus they use the deviation from isotropy in the projection to infer 3D shape and orientation. There are two major weaknesses o... |

61 | Shape from texture using local spectral moments - Super, Bovik - 1995 |

52 | Shape from texture for smooth curved surfaces in perspective projection
- Gårding
- 1992
(Show Context)
Citation Context ... We will explain the remaining shape parameters in Section 3. Our derivation of the relationship between the texture distortion and the surface parameters makes use of previous results due to Garding =-=[10]-=-. A major motivation for using the texture distortion map formalism becomes evident in Section 4. It has proven di cult to develop algorithms for estimating the individual texture gradients which do n... |

50 |
Shape from Texture
- Aloimonos
- 1988
(Show Context)
Citation Context ...oit the change in size of the projected texture. The other assumption which has been used in the literature is that of rst order homogeneity, i.e. that the texture pattern has constant area or density=-=[13, 1, 15, 25, 19]-=-. This is a reasonable assumption for natural textures, and our rst criticism does not apply. However, this assumption is too weak{it fails to exploit the systematic change in shape of the texture ele... |

47 |
Shape from texture: General principle
- Kanatani, Chou
- 1989
(Show Context)
Citation Context ...oit the change in size of the projected texture. The other assumption which has been used in the literature is that of rst order homogeneity, i.e. that the texture pattern has constant area or density=-=[13, 1, 15, 25, 19]-=-. This is a reasonable assumption for natural textures, and our rst criticism does not apply. However, this assumption is too weak{it fails to exploit the systematic change in shape of the texture ele... |

42 | Shape from texture from a multi-scale perspective
- Lindeberg, Garding
- 1993
(Show Context)
Citation Context .... 3s2. It is not clear how these gradients could be measured in the image. Explicit texel identi cation as used by Blostein and Ahuja[5] is not feasible in many or most contexts. Lindeberg and Garding=-=[18]-=- present a technique applicable for measuring area gradient, but that is inadequate by itself. The other major family of approaches to shape from texture in the computer vision literature is based on ... |

39 |
ªSurface Orientation from Projective Foreshortening of Isotopic Texture Autocorrelation,º
- Brown, Shvaytser
- 1990
(Show Context)
Citation Context ...e model given sample measurements in the image{and can be approached in either a Bayesian or maximum likelihood framework. The model most often used is that of isotropy orweak isotropy of the texture =-=[27, 8, 6, 4, 11]-=-. Under projection, the texture will not generally appear isotropic, and thus they use the deviation from isotropy in the projection to infer 3D shape and orientation. There are two major weaknesses o... |

38 |
Shape from texture: Integrating texture-element extraction and surface estimation
- Blostein, Ahuja
- 1989
(Show Context)
Citation Context ...of complete local surface curvature e.g. sign of Gaussian curvature. 3s2. It is not clear how these gradients could be measured in the image. Explicit texel identi cation as used by Blostein and Ahuja=-=[5]-=- is not feasible in many or most contexts. Lindeberg and Garding[18] present a technique applicable for measuring area gradient, but that is inadequate by itself. The other major family of approaches ... |

36 |
The information content of texture gradients
- Stevens
- 1981
(Show Context)
Citation Context ...ar-valued functions such as foreshortening, area, density, compression or scaling. The mathematical relationship between these gradients and scene geometry has been developed both for planar surfaces =-=[24]-=- and curved surfaces [10]. There are two di culties in the use of these gradients: 1. Garding has shown that these simple distortion gradients do not contain enough information for measurement of comp... |

30 |
Shape from texture: Ideal observers and human psychophysics
- Blake, Bülthoff, et al.
- 1993
(Show Context)
Citation Context ... human visual system. In the context of shape from texture models based on discrete texels using isotropy and rst order homogeneity assumptions, such ideal observers have been developed by Blake et al=-=[3]-=-. The uncertainty in the shape estimates depends upon the measurement errors in the earlier stages of processing; here, in the estimation of the a ne transforms. The errors in the a ne transforms will... |

28 | Shape from texture and contour by weak isotropy
- Gårding
- 1993
(Show Context)
Citation Context |

26 | Determining three-dimensional shape from orientation and spatial frequency disparities
- Jones, Malik
- 1992
(Show Context)
Citation Context ...ve for the a ne transformation directly. It may be noted that our a ne transform estimation procedure can be applied to other problems in vision, e.g. in the context of stereopsis; see Jones and Malik=-=[14]-=-. 5 Shape Recovery Algorithms To estimate the texture distortion map at a point p, we nd the spectrograms for that point and for neighboring points in a number of di erent directions, ~vi =( ti bi) T ... |

26 | Shape from periodic texture using the spectrogram
- Krumm, Shafer
- 1992
(Show Context)
Citation Context ...hose moments. By contrast, our method uses the entire spectrogram, rather than only the peaks or the moments, to estimate the a ne transformation and ultimately the shape parameters. Krumm and Shafer =-=[17]-=- also nd the a ne transformation by using the entire spectrogram, but they nd it by testing every possible combination of slant and tilt in a discrete set, and choosing the one which gives them the sm... |

25 | ªA Differential Method for Computing Local Shape-from-Texture for Planar and Curved - Malik, Rosenholtz - 1993 |

24 | Differential techniques for optical flow - Verri, Girosi, et al. - 1990 |

20 |
Efficient recovery of shape from texture
- L, Dunn
- 1983
(Show Context)
Citation Context |

19 | Shape from regular patterns
- Ikeuchi
- 1984
(Show Context)
Citation Context ...oit the change in size of the projected texture. The other assumption which has been used in the literature is that of rst order homogeneity, i.e. that the texture pattern has constant area or density=-=[13, 1, 15, 25, 19]-=-. This is a reasonable assumption for natural textures, and our rst criticism does not apply. However, this assumption is too weak{it fails to exploit the systematic change in shape of the texture ele... |

18 | ªRecovering Surface Curvature and Orientation from Texture Distortion, a Least Squares Algorithm and Sensitive Analysis,º - Malik, Rosenholtz - 1994 |

17 |
Shape from texture: The homogeneity hypothesis
- C, Blake
- 1990
(Show Context)
Citation Context |

16 |
Elementary Di®erential Geometry
- O'Neill
- 1966
(Show Context)
Citation Context ....1 The slant-tilt frame eld This section is based on Garding [10], to which the reader is referred for proofs of the various assertions. Relevant di erential geometry concepts may be found in O'Neill =-=[21]-=- or Koenderink [16]. The basic geometry is illustrated in Figure 1. A smooth surface S is mapped by central projection to a unit sphere centered at the focal point. The backprojection map F from to S ... |

4 | Three gradients and the perception of at and curved surfaces - Cutting, Millard - 1984 |

4 |
Di erential techniques for optical ow
- Verri, Girosi, et al.
- 1990
(Show Context)
Citation Context ...on. We derive these constraints 2sfrom an assumption of stationarity under translations for the scene texture. The method bears strong resemblances to di erential techniques for estimating optical ow =-=[26]-=-. In Section 5 of the paper, we develop a new algorithm for recovering surface orientation and shape based on the estimated a ne transforms in a number of di erent directions. The method uses nonlinea... |

2 |
Shape-from-Texture byWavelet-Based Measurement of Local Spectral Moments
- Super, Bovik
- 1992
(Show Context)
Citation Context |

1 |
Texture gradient as a depth cue." CGIP
- Bajcsy, Lieberman
- 1976
(Show Context)
Citation Context ...ansformation in the frequency domain, we can nd the a ne tranformation in the space domain. The advantage of working in the frequency domain for texture analysis was rst noted by Bajcsy and Leiberman =-=[2]-=-: If we keep only the magnitude of the frequency response, our algorithm will be insensitive to small changes of position, since a small change in position causes only a change in phase. Otherwise, we... |