## Semidefinite programming heuristics for surface reconstruction ambiguities (2008)

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Venue: | In ECCV |

Citations: | 15 - 1 self |

### BibTeX

@INPROCEEDINGS{Ecker08semidefiniteprogramming,

author = {Ady Ecker and Allan D. Jepson and Kiriakos N. Kutulakos},

title = {Semidefinite programming heuristics for surface reconstruction ambiguities},

booktitle = {In ECCV},

year = {2008}

}

### OpenURL

### Abstract

Abstract. We consider the problem of reconstructing a smooth surface under constraints that have discrete ambiguities. These problems arise in areas such as shape from texture, shape from shading, photometric stereo and shape from defocus. While the problem is computationally hard, heuristics based on semidefinite programming may reveal the shape of the surface. 1

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Citation Context ...ints into two groups, and picks the partition with the lowest cost. We noticed that these angular sweeps can be made more efficient by careful bookkeeping similar to the Kernighan-Lin (K-L) algorithm =-=[23]-=-. Note that the cost of splitting n points in IR n based on a single normal is O(n 2 ). However, scanning a series of n normals, where at each transition a single point moves to the other side of the ... |

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Citation Context ...ep sets the discrete variables and proposes a 3D shape. The problems we deal with are larger than those considered by Zhu and Shi, and standard Goemans-Williamson random hyperplane rounding technique =-=[12]-=- will usually produce sub-optimal results. We describe several heuristics that can improve the solution considerably. D. Forsyth, P. Torr, and A. Zisserman (Eds.): ECCV 2008, Part I, LNCS 5302, pp. 12... |

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Citation Context ...at each transition a single point moves to the other side of the sweep line, can also be carried out in O(n 2 ). 3. The k-best results from the circular sweep phase are refined with the K-L algorithm =-=[23,24,25,26]-=-. This is a local search procedure that will clean up a small number of misplaced vertices. We terminate this algorithm early if no progress is made in 50 consecutive iterations [25]. The lowest cost ... |

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Citation Context ...at each transition a single point moves to the other side of the sweep line, can also be carried out in O(n 2 ). 3. The k-best results from the circular sweep phase are refined with the K-L algorithm =-=[23,24,25,26]-=-. This is a local search procedure that will clean up a small number of misplaced vertices. We terminate this algorithm early if no progress is made in 50 consecutive iterations [25]. The lowest cost ... |

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Citation Context ...ld ambiguity for the depth, with one depth in front of the in-focus plane and the other behind. The formulas relating σ and the camera parameters to the two possible depths were developed by Pentland =-=[18]-=-. In Sect. 4 we experiment with a first-order simplified model that assumes the depth is proportional to ±σ. This model falls naturally into the form (4), where now the matrix A ′ contains the depths ... |

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Citation Context ...hing between r1,r2 gives another solution. In the general case, the four solutions are of the form ±(p1,q1) and ±(p2,q2). In the SDP literature, the max-k-cut problem was studied by Frieze and Jerrum =-=[28]-=- and de Klerk, Pasechnik, and Warners [29]. While an ideal encoding requires two bits to encode four possibilities, their encoding uses four bits: a single indicator bit set to 1 and the rest 0. Since... |

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Citation Context ...ce under ambiguous constraints can be addressed with semidefinite programming (SDP). SDP has been applied to a wide range of combinatorial optimization problems. For a general introduction to SDP see =-=[1,2,3]-=-. Recently, SDP-based approximation algorithms have been developed for several computer vision problems, such as image restoration [4,5], segmentation [4,6], graph matching [7,8,9] and finding corresp... |

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Citation Context ...ent and additional unambiguous constraints are required. A natural extension to the approach presented here would be to replace the spline with a shape basis, e.g. for particular shapes such as faces =-=[31]-=-. The general form of (1) allows for many other variations, such as adding linear constraints (e.g. specifying depths or normals at specific points [32]), or using shading information to disambiguate ... |

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Citation Context ...ce under ambiguous constraints can be addressed with semidefinite programming (SDP). SDP has been applied to a wide range of combinatorial optimization problems. For a general introduction to SDP see =-=[1,2,3]-=-. Recently, SDP-based approximation algorithms have been developed for several computer vision problems, such as image restoration [4,5], segmentation [4,6], graph matching [7,8,9] and finding corresp... |

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Citation Context ...ce under ambiguous constraints can be addressed with semidefinite programming (SDP). SDP has been applied to a wide range of combinatorial optimization problems. For a general introduction to SDP see =-=[1,2,3]-=-. Recently, SDP-based approximation algorithms have been developed for several computer vision problems, such as image restoration [4,5], segmentation [4,6], graph matching [7,8,9] and finding corresp... |

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Citation Context ... are sufficient. This occurs when the two circles on the unit sphere touch at a point, or when one of the intersection points of the two circles is on the occluded half-hemisphere. Onn and Bruckstein =-=[15]-=- studied photometric stereo of Lambertian surfaces using two lights. Their method uses the points that are uniquely determined by two lights to divide the image into regions. Inside each region integr... |

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Citation Context ... project the points embedded in IR n on this plane (the first plane we check is the one spanned by the two principle singular vectors). Then we perform a circular sweep in this plane, as described in =-=[22]-=-. Basically, the sweep rotates a line through the origin that separates the points into two groups, and picks the partition with the lowest cost. We noticed that these angular sweeps can be made more ... |

33 | A.: Shape from texture without boundaries
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Citation Context ...biguity, and a continuous ambiguity in absolute depth. However, if we make the assumption that the surface is smooth, we can identify the more probable shapes of the surface. In related work, Forsyth =-=[13,14]-=- proposed alternating between optimizing surface smoothness and selecting the normals. We show in Sect. 3 that by using a quadratic smoothness term the problem can be converted into an entirely discre... |

33 |
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Citation Context ... For example, if we had to choose a single plane, a good choice for the normal would be the principle singular vector. Such “inertial” splitting methods have been previously used for other embeddings =-=[20,21]-=-, but to our best knowledge not for the SDP embedding. To widen the choices of planes, we randomly pick normals as a weighted linear combination of the singular vectors that correspond to the k-larges... |

30 | Binary partitioning, perceptual grouping, and restoration with semidefinite programming
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(Show Context)
Citation Context ... optimization problems. For a general introduction to SDP see [1,2,3]. Recently, SDP-based approximation algorithms have been developed for several computer vision problems, such as image restoration =-=[4,5]-=-, segmentation [4,6], graph matching [7,8,9] and finding correspondences in stereo [10]. Zhu and Shi [11] used SDP to solve in/out reversal ambiguities of surface patches in shape from shading. In thi... |

25 | Shape from texture and integrability
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- 2001
(Show Context)
Citation Context ...biguity, and a continuous ambiguity in absolute depth. However, if we make the assumption that the surface is smooth, we can identify the more probable shapes of the surface. In related work, Forsyth =-=[13,14]-=- proposed alternating between optimizing surface smoothness and selecting the normals. We show in Sect. 3 that by using a quadratic smoothness term the problem can be converted into an entirely discre... |

25 | On nonapproximability for quadratic programs
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Citation Context ...g the discrete vector d∈{−1, 1} n which minimizes C •(ddt ).Oncedisfound, v is given by (9) and the 3D shape is given by (2). Unfortunately the general problem is NP-hard and difficult to approximate =-=[19]-=-. Semidefinite programming is widely used to find approximate solutions for problems of this kind. The standard SDP relaxation requires the matrix X to be symmetric positive semidefinite (instead of r... |

22 | Solving markov random fields using semi definite programming
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Citation Context ...ed approximation algorithms have been developed for several computer vision problems, such as image restoration [4,5], segmentation [4,6], graph matching [7,8,9] and finding correspondences in stereo =-=[10]-=-. Zhu and Shi [11] used SDP to solve in/out reversal ambiguities of surface patches in shape from shading. In this paper we show that a similar mathematical formulation applies to other surface recons... |

22 | Solving Max-Cut to optimality by intersecting semidefinite and polyhedral relaxations
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(Show Context)
Citation Context ...ty of SDP is that there is no need for initialization. However, if a good starting vector d is available, it can be exploited by performing the rounding phase on a linear combination of X and d · d t =-=[30]-=-. Our focus has been on developing a general framework for solving problems of the form (1), involving ambiguous discrete constraints. In practice, after binary decisions are made, the surface can be ... |

22 | Combining cues: Shape from shading and texture - White, Forsyth - 2006 |

18 | Approximate graph colouring and max-k-cut algorithms based on the theta function
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Citation Context ... In the general case, the four solutions are of the form ±(p1,q1) and ±(p2,q2). In the SDP literature, the max-k-cut problem was studied by Frieze and Jerrum [28] and de Klerk, Pasechnik, and Warners =-=[29]-=-. While an ideal encoding requires two bits to encode four possibilities, their encoding uses four bits: a single indicator bit set to 1 and the rest 0. Since the matrix X has O(n 2 ) entries, redunda... |

17 | Probabilistic subgraph matching based on convex relaxation
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Citation Context ...oduction to SDP see [1,2,3]. Recently, SDP-based approximation algorithms have been developed for several computer vision problems, such as image restoration [4,5], segmentation [4,6], graph matching =-=[7,8,9]-=- and finding correspondences in stereo [10]. Zhu and Shi [11] used SDP to solve in/out reversal ambiguities of surface patches in shape from shading. In this paper we show that a similar mathematical ... |

13 | Geometric spectral partitioning
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Citation Context ... For example, if we had to choose a single plane, a good choice for the normal would be the principle singular vector. Such “inertial” splitting methods have been previously used for other embeddings =-=[20,21]-=-, but to our best knowledge not for the SDP embedding. To widen the choices of planes, we randomly pick normals as a weighted linear combination of the singular vectors that correspond to the k-larges... |

11 | A semidefinite programming based polyhedral cut and price approach for the Max-Cut problem
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(Show Context)
Citation Context ...at each transition a single point moves to the other side of the sweep line, can also be carried out in O(n 2 ). 3. The k-best results from the circular sweep phase are refined with the K-L algorithm =-=[23,24,25,26]-=-. This is a local search procedure that will clean up a small number of misplaced vertices. We terminate this algorithm early if no progress is made in 50 consecutive iterations [25]. The lowest cost ... |

8 |
Algorithm 875: DSDP5 – Software for semidefinite programming
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(Show Context)
Citation Context ...rates reconstruction from ambiguous normals. To simplify texture extraction we used square texture elements (see Sect. 5 for derivation of normals from parallelograms). The SDP solver we used is DSDP =-=[27]-=-. In Fig. 3 we computed a surface from a pair of synthetic images of a Lambertian surface using two-light photometric stereo. The two possibilities for the surface normal are computed on a 29 × 29 gri... |

5 |
Multiclass image labeling with semidefinite programming
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Citation Context ... optimization problems. For a general introduction to SDP see [1,2,3]. Recently, SDP-based approximation algorithms have been developed for several computer vision problems, such as image restoration =-=[4,5]-=-, segmentation [4,6], graph matching [7,8,9] and finding correspondences in stereo [10]. Zhu and Shi [11] used SDP to solve in/out reversal ambiguities of surface patches in shape from shading. In thi... |

5 | F.: Solving large scale binary quadratic problems: Spectral methods vs. semidefinite programming
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Citation Context ...ms. For a general introduction to SDP see [1,2,3]. Recently, SDP-based approximation algorithms have been developed for several computer vision problems, such as image restoration [4,5], segmentation =-=[4,6]-=-, graph matching [7,8,9] and finding correspondences in stereo [10]. Zhu and Shi [11] used SDP to solve in/out reversal ambiguities of surface patches in shape from shading. In this paper we show that... |

4 |
Graph seriation using semi-definite programming. Graph based Pattern Recognition
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Citation Context ...oduction to SDP see [1,2,3]. Recently, SDP-based approximation algorithms have been developed for several computer vision problems, such as image restoration [4,5], segmentation [4,6], graph matching =-=[7,8,9]-=- and finding correspondences in stereo [10]. Zhu and Shi [11] used SDP to solve in/out reversal ambiguities of surface patches in shape from shading. In this paper we show that a similar mathematical ... |

3 | E.: Graph matching using spectral embedding and semidefinite programming
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(Show Context)
Citation Context ...oduction to SDP see [1,2,3]. Recently, SDP-based approximation algorithms have been developed for several computer vision problems, such as image restoration [4,5], segmentation [4,6], graph matching =-=[7,8,9]-=- and finding correspondences in stereo [10]. Zhu and Shi [11] used SDP to solve in/out reversal ambiguities of surface patches in shape from shading. In this paper we show that a similar mathematical ... |

3 | Shape from shading: recognizing the mountains through a global view
- Zhu, Shi
(Show Context)
Citation Context ...lgorithms have been developed for several computer vision problems, such as image restoration [4,5], segmentation [4,6], graph matching [7,8,9] and finding correspondences in stereo [10]. Zhu and Shi =-=[11]-=- used SDP to solve in/out reversal ambiguities of surface patches in shape from shading. In this paper we show that a similar mathematical formulation applies to other surface reconstruction problems.... |

3 |
Shape from random planar features
- Naito, Rosenfeld
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(Show Context)
Citation Context ... orthographicSemidefinite Programming Heuristics for Surface Reconstruction Ambiguities 131 view, each segment can have a front/back reversal. Similar problems were considered by Naito and Rosenfeld =-=[16]-=- and Koenderink and van Doorn [17]. The depth difference of the segment’s endpoints is constrained by √ l2 − r2 ij = Δijdij , (7) zi − zj = dij where dij = ±1, l is the 3D length of the segment and ri... |

2 |
A.: Shape from chebyshev nets
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(Show Context)
Citation Context ...ming Heuristics for Surface Reconstruction Ambiguities 131 view, each segment can have a front/back reversal. Similar problems were considered by Naito and Rosenfeld [16] and Koenderink and van Doorn =-=[17]-=-. The depth difference of the segment’s endpoints is constrained by √ l2 − r2 ij = Δijdij , (7) zi − zj = dij where dij = ±1, l is the 3D length of the segment and rij is the observed length in the im... |