## Global optimization for shape fitting (2007)

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Venue: | In CVPR, 2007. 6 |

Citations: | 30 - 3 self |

### BibTeX

@INPROCEEDINGS{Lempitsky07globaloptimization,

author = {Victor Lempitsky and Yuri Boykov},

title = {Global optimization for shape fitting},

booktitle = {In CVPR, 2007. 6},

year = {2007}

}

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### Abstract

We propose a global optimization framework for 3D shape reconstruction from sparse noisy 3D measurements frequently encountered in range scanning, sparse featurebased stereo, and shape-from-X. In contrast to earlier local or banded optimization methods for shape fitting, we compute global optimum in the whole volume removing dependence on initial guess and sensitivity to numerous local minima. Our global method is based on two main ideas. First, we suggest a new regularization functional with a data alignment term that maximizes the number of (weaklyoriented) data points contained by a surface while allowing for some measurement errors. Second, we propose a touchexpand algorithm for finding a minimum cut on a huge 3D grid using an automatically adjusted band. This overcomes prohibitively high memory cost of graph cuts when computing globally optimal surfaces at high-resolution. Our results for sparse or incomplete 3D data from laser scanning and passive multi-view stereo are robust to noise, outliers, missing parts, and varying sampling density. 1.

### Citations

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Citation Context ...h to surface representation. While explicit mesh-based representations are very common, e.g. [5], to avoid mesh-related numerical problems, many methods represent surfaces implicitly using level-sets =-=[6, 21, 22, 18]-=-. Both meshes and level-sets can be locally optimized via gradient descent. We use a different approach where surfaces are represented as binary partitionings (s/t cuts) on discrete grids [2, 13]. Thi... |

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Citation Context ... version of a more general optical flow (2D motion) problem. Interestingly, the basic problem illustrated in Fig. 4 connects to some known properties of standard optical flow methods. As widely known =-=[1, 6]-=-, sparse (Lucas-Kanade) techniques that focus on points with sufficiently discriminant features can be more accurate and robust to noise than dense (Horn-Schunck) techniques that enforce colorconsiste... |

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Citation Context ...h to surface representation. While explicit mesh-based representations are very common, e.g. [5], to avoid mesh-related numerical problems, many methods represent surfaces implicitly using level-sets =-=[6, 21, 22, 18]-=-. Both meshes and level-sets can be locally optimized via gradient descent. We use a different approach where surfaces are represented as binary partitionings (s/t cuts) on discrete grids [2, 13]. Thi... |

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Citation Context ...ay according to a binary MRF formulation with pairwise smoothness between neighboring grid points. Such MRFs posterior energies (possibly super-modular) can be optimized using graph-cut, QPBO methods =-=[20]-=-, and/or other powerful combinatorial optimization techniques. Yet, in this work we study a different approach to surface fitting. In contrast to [25], we do not want to brake the surface fitting prob... |

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Citation Context ...itfalls of local optimization such as dependence on initial guess (Fig. 2,3). Graph cuts typically require a lot of memory for high resolution volumes. Previous hierarchical or narrow band techniques =-=[16, 10]-=- do not guarantee global minima and generally behave as local optimization methods, see Fig. 2,3. In order to do globally optimal surface fitting at high resolutions, we propose a new memory-efficient... |

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Citation Context ...dding points from other views gives good overall accuracy (bottom). terms (e.g. photoconsistency). A wide class of such combined geometric surface functionals can be globally optimized via graph cuts =-=[2, 13]-=-, avoiding the pitfalls of local optimization such as dependence on initial guess (Fig. 2,3). Graph cuts typically require a lot of memory for high resolution volumes. Previous hierarchical or narrow ... |

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Citation Context ... techniques and the surface functionals that they use. 2.1. Explicit and implicit surface representation First, there is a group of methods using explicit representation of surfaces via triangle mesh =-=[9, 11]-=-. This approach allows optimization of geometrically-motivated energies using gradient descent but it can be highly dependent on good initialization. Also, if the shape undertakes significant changes ... |

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Citation Context ...put, the results on Dino reveal metrication artifacts, which are probably the major limitation of our method. For such problems where filling of large holes is required, the use of continuous maxflow =-=[1]-=- rather than discrete version may be considered (note that touch-expand method can be used for continuous maxflow as well). 6. Future Work This work focused on global optimization of geometric surface... |

32 |
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Citation Context ...itfalls of local optimization such as dependence on initial guess (Fig. 2,3). Graph cuts typically require a lot of memory for high resolution volumes. Previous hierarchical or narrow band techniques =-=[16, 10]-=- do not guarantee global minima and generally behave as local optimization methods, see Fig. 2,3. In order to do globally optimal surface fitting at high resolutions, we propose a new memory-efficient... |

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Citation Context ...o data. Recently, a number of multiview reconstruction methods have been proposed that produce high-accuracy “quasi-dense” output, leaving holes where the stereo correspondence can not be established =-=[15, 8, 9]-=-. Our method can post-process such outputs turning them into complete watertight meshes, while preserving the high level of detailization. To test this, we considered incomplete meshes produced by mut... |

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Citation Context ...biguities while flux enforces alignment with data. Combining flux with regularization is commonly used in vision for reconstruction of elongated structures, narrow protrusions, and other fine details =-=[19, 12, 13, 4]-=-. Section 3.1 further motivates our data fit functional statistically and geometrically. Section 3.2 combines it with surface regularization (generic shape priors). 3.1. Quality of Fit to Data Ideally... |

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Citation Context ...pand method was based on the max-flow algorithm [3] for computing minimum cuts. To extract an isosurface from binary segmented grids without aliasing artifacts, we used the modification of the method =-=[20]-=- (details are given in [14]). In general, our algorithm produced watertight meshes of correct genus but when the resolution was too high, the algorithm might give a few floating voxels. The algorithm ... |

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Citation Context ...ut framework to local optimization essentially equivalent to variational approach in [29]. Connection between narrow-band graph-cut methods and variational optimization (e.g. level-sets) was shown in =-=[4]-=-. Savadjiev, Ferrie, and Siddiqi [25] formulate surface fitting as a problem of estimating a dense vector field {vp} of surface normals from sparse data points. Then, a continuous surface can be recov... |

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Citation Context ...o data. Recently, a number of multiview reconstruction methods have been proposed that produce high-accuracy “quasi-dense” output, leaving holes where the stereo correspondence can not be established =-=[15, 8, 9]-=-. Our method can post-process such outputs turning them into complete watertight meshes, while preserving the high level of detailization. To test this, we considered incomplete meshes produced by mut... |

4 | Surface recovery from 3d point data using a combined parametric and geometric flow approach
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