## Tensor Voting: A Perceptual Organization Approach to Computer Vision and Machine LearningThis lecture presents research on a general framework for perceptual organization that was contacted mainly at the Institute for Robotics and Intelligent Systems of the University of Southern California. It is not written as a historical recount of the work, since the sequence of the presentation is not in chronological order. It aims at presenting an approach to a wide range of problems in computer vision and machine learning that is data-driven, local and requires a minimal number of assumptions. The tensor voting framework combines these properties and provides a unified perceptual organization methodology applicable in situations that may seem heterogeneous initially. Authors Philippos Mordohai and Gerard Medioni show how several problems can be posed as the organization of the inputs into salient perceptual structures, which are inferred via tensor voting. The book extends the original tensor voting framework with the addition of boundary inference capabilities, a novel re-formulation of the framework applicable to high-dimensional spaces and the development of algorithms for computer vision and machine learning problems. The authors provide complete analysis for some problems and briefly outline the approach for other applications and provide references to relevant sources. |

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### Contents

1 | |

4 | |

6 | |

TENSOR VOTING | 9 |

22 TENSOR VOTING IN 2D | 12 |

222 Second Order Voting in 2D | 13 |

223 Voting Fields | 16 |

224 Vote Analysis | 18 |

432 The Voting Process | 52 |

433 Vote Analysis | 55 |

45 COMPUTER VISION PROBLEMS IN HIGH DIMENSIONS | 59 |

453 Texture Synthesis | 61 |

46 DISCUSSION | 62 |

Dimensionality Estimation Manifold Learning and Function Approximation | 63 |

51 RELATED WORK | 65 |

52 DIMENSIONALITY ESTIMATION | 69 |

226 Quantitative Evaluation of Saliency Estimation | 19 |

23 TENSOR VOTING IN 3D | 20 |

231 Representation in 3D | 21 |

232 Voting in 3D | 23 |

233 Vote Analysis | 25 |

234 Results in 3D | 26 |

Stereo Vision from a Perceptual Organization Perspective | 27 |

32 RELATED WORK | 29 |

33 OVERVIEW OF OUR APPROACH | 31 |

34 INITIAL MATCHING | 32 |

35 SELECTION OF MATCHES AS SURFACE INLIERS | 35 |

36 SURFACE GROUPING AND REFINEMENT | 37 |

37 DISPARITY ESTIMATION FOR UNMATCHED PIXELS | 40 |

38 EXPERIMENTAL RESULTS | 41 |

39 DISCUSSION | 43 |

310 OTHER 3D COMPUTER VISION RESEARCH | 46 |

3102 Tracking | 47 |

Tensor Voting in N D | 49 |

42 LIMITATIONS OF ORIGINAL IMPLEMENTATION | 50 |

43 TENSOR VOTING IN HIGHDIMENSIONAL SPACES | 51 |

53 MANIFOLD LEARNING | 71 |

54 MANIFOLD DISTANCES AND NONLINEAR INTERPOLATION | 73 |

55 GENERATION OF UNOBSERVED SAMPLES AND NONPARAMETRIC FUNCTION APPROXIMATION | 78 |

56 DISCUSSION | 83 |

Boundary Inference | 87 |

62 FIRSTORDER REPRESENTATION AND VOTING | 89 |

621 FirstOrder Voting in High Dimensions | 92 |

63 VOTE ANALYSIS | 93 |

64 RESULTS USING FIRSTORDER INFORMATION | 97 |

65 DISCUSSION | 99 |

Figure Completion | 101 |

72 OVERVIEW OF THE APPROACH | 103 |

73 TENSOR VOTING ON LOW LEVEL INPUTS | 104 |

75 EXPERIMENTAL RESULTS | 106 |

76 DISCUSSION | 111 |

Conclusions | 113 |

115 | |

Author Biographies | 125 |

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### Common terms and phrases

algorithm align amodal completion approach ball component ball tensors ball vote boundaries candidate matches chapter color Computer Vision Conf correct matches curvature curve saliency dataset deﬁned detected dimensionality estimation distance edgels eigenvalues eigenvectors embedding endpoints epipolar epipolar geometry ﬁnal ﬁrst ﬁrst-order votes function approximation global high-dimensional spaces IEEE Trans illusory contour image pairs implementation infer initial matching inliers instance-based learning intersection Isomap Laplacian Eigenmaps layer linear machine learning manifold learning Medioni methods modal completion neighbors noise normal space occluded orientation orthogonal osculating circle outliers outputs parameters Pattern Anal perceptual organization perceptual structure pixel correspondences pixels polarity vector problem regions representation robustness saliency maps samples second order second-order vote Section seen in Fig segmentation signiﬁcant smooth stereo stick tensor stick vote surface normal surface saliency T-junctions tangent space tensor voting framework unmatched pixels vote cast voter and receiver voting ﬁeld window