## Shape from Shading: Provably Convergent Algorithms and Uniqueness Results (1994)

Venue: | In Computer Vision-ECCV 94 |

Citations: | 9 - 0 self |

### BibTeX

@INPROCEEDINGS{Dupuis94shapefrom,

author = {Paul Dupuis and John Oliensis},

title = {Shape from Shading: Provably Convergent Algorithms and Uniqueness Results},

booktitle = {In Computer Vision-ECCV 94},

year = {1994},

pages = {259--268},

publisher = {Springer-Verlag}

}

### Years of Citing Articles

### OpenURL

### Abstract

. An explicit representation for the surface corresponding to a shaded image is presented and proven to be correct (under standard conditions) . Uniqueness of the surface is an immediate consequence. Using this representation, various iterative algorithms for shape reconstruction are derived. It has been proven that all these algorithms converge monotonically to the correct surface reconstruction, and they have been shown experimentally to be fast and robust. Some of the results of this paper extend previous ones to the case of illumination from a general direction. 1 Introduction Shape from shading has proven to be a difficult problem, even under the standard idealizing assumptions of a Lambertian surface, known light source, and no shadowing or occlusion. Recently, a new approach has been developed, based on relating shape from shading to an "equivalent" optimal control problem [7,8,4,9,13,14,1]. The advantages of this approach are both theoretical and practical. First, it makes pos...

### Citations

3293 | Convex Analysis - Rockafellar - 1972 |

321 |
eds.): Shape from Shading
- Horn, Brooks
- 1989
(Show Context)
Citation Context ...nder standard conditions) provably convergent to the correct surface. In contrast, traditional algorithms typically require thousands of iterations for reconstruction with no guarantee of convergence =-=[5]-=-. In this paper we prove the correctness of an optimal control representation for shape from shading, extending previous work to the case of illumination from a general direction (Section 2). Uniquene... |

199 |
A viscosity solutions approach to shape-from-shading
- Rouy, Tourin
- 1992
(Show Context)
Citation Context ...f a Lambertian surface, known light source, and no shadowing or occlusion. Recently, a new approach has been developed, based on relating shape from shading to an "equivalent" optimal contro=-=l problem [7,8,4,9,13,14,1]-=-. The advantages of this approach are both theoretical and practical. First, it makes possible an easy uniqueness proof for the surface corresponding to a shaded image (under standard conditions); thu... |

61 |
Direct method for reconstructing shape from shading
- Dupuis, Oliensis
- 1992
(Show Context)
Citation Context ...f a Lambertian surface, known light source, and no shadowing or occlusion. Recently, a new approach has been developed, based on relating shape from shading to an "equivalent" optimal contro=-=l problem [7,8,4,9,13,14,1]-=-. The advantages of this approach are both theoretical and practical. First, it makes possible an easy uniqueness proof for the surface corresponding to a shaded image (under standard conditions); thu... |

58 | A simple algorithm for shape from shading
- Bichsel, Pentland
- 1992
(Show Context)
Citation Context ...f a Lambertian surface, known light source, and no shadowing or occlusion. Recently, a new approach has been developed, based on relating shape from shading to an "equivalent" optimal contro=-=l problem [7,8,4,9,13,14,1]-=-. The advantages of this approach are both theoretical and practical. First, it makes possible an easy uniqueness proof for the surface corresponding to a shaded image (under standard conditions); thu... |

48 |
The eikonal equation: Some results applicable to computer vision
- Bruss
- 1982
(Show Context)
Citation Context ...e so written. Singular points---those image points where the intensity achieves its maximal brightness I(\Delta) = 1---play a critical role in constraining the surface corresponding to a shaded image =-=[8,11,16,2]-=-. Only at these points is the local surface orientation determined from the intensity alone. Let S denote the set of singular points in the image. It is easy to see that rf = 0 on S, so that S include... |

46 |
An optimal control formulation and related numerical methods for a problem in shape reconstruction
- Dupuis, Oliensis
- 1994
(Show Context)
Citation Context ... Foundation grants DMS9115762, IRI-9113690, and CDA-8922572. of the presented algorithms has a simple theoretical interpretation and can be used to prove the correctness of the surface reconstruction =-=[3]-=-, while the other is more efficient computationally. Note that although the algorithms described here require some information about the imaged surface in order to reconstruct it, they have recently b... |

46 |
Uniqueness in shape from shading
- Oliensis
- 1991
(Show Context)
Citation Context ...e so written. Singular points---those image points where the intensity achieves its maximal brightness I(\Delta) = 1---play a critical role in constraining the surface corresponding to a shaded image =-=[8,11,16,2]-=-. Only at these points is the local surface orientation determined from the intensity alone. Let S denote the set of singular points in the image. It is easy to see that rf = 0 on S, so that S include... |

35 |
Shape from shading as a partially well-constrained problem
- Oliensis
- 1991
(Show Context)
Citation Context ...r; \Delta) is C 1 implies �� u(r) is continuous on D. An explicit calculation shows that �� u(r) is proportional to the projection in the (x; y) plane of the steepest descent direction on the =-=surface [10], where &q-=-uot;steepest descent" is defined with respect to the light directionsL, rather than the vertical direction (0; 0; 1). We consider subsets G of D satisfying the following assumption. A2.1 Assume t... |

7 | Numerical Methods for Stochastic - Kushner, Dupuis - 1992 |

6 |
A Modern Differential Geometric Approach to Shape from Shading
- Saxberg
- 1989
(Show Context)
Citation Context ...e so written. Singular points---those image points where the intensity achieves its maximal brightness I(\Delta) = 1---play a critical role in constraining the surface corresponding to a shaded image =-=[8,11,16,2]-=-. Only at these points is the local surface orientation determined from the intensity alone. Let S denote the set of singular points in the image. It is easy to see that rf = 0 on S, so that S include... |

4 |
global algorithm for shape from shading,”pp
- Oliensis, Dupuis, et al.
- 1993
(Show Context)
Citation Context |

4 | An Application of Dynamical Systems Theory to Shape From Shading - Saxberg - 1989 |