## Graph Matching With a Dual-Step EM Algorithm (1998)

### Cached

### Download Links

- [www.cs.york.ac.uk]
- [www-users.cs.york.ac.uk]
- [www.cs.york.ac.uk]
- [eprints.whiterose.ac.uk]
- DBLP

### Other Repositories/Bibliography

Venue: | IEEE Transactions on Pattern Analysis and Machine Intelligence |

Citations: | 86 - 5 self |

### BibTeX

@ARTICLE{Cross98graphmatching,

author = {Andrew D. J. Cross and Edwin R. Hancock},

title = {Graph Matching With a Dual-Step EM Algorithm},

journal = {IEEE Transactions on Pattern Analysis and Machine Intelligence},

year = {1998},

volume = {20},

pages = {1236--1253}

}

### Years of Citing Articles

### OpenURL

### Abstract

Abstract—This paper describes a new approach to matching geometric structure in 2D point-sets. The novel feature is to unify the tasks of estimating transformation geometry and identifying point-correspondence matches. Unification is realized by constructing a mixture model over the bipartite graph representing the correspondence match and by affecting optimization using the EM algorithm. According to our EM framework, the probabilities of structural correspondence gate contributions to the expected likelihood function used to estimate maximum likelihood transformation parameters. These gating probabilities measure the consistency of the matched neighborhoods in the graphs. The recovery of transformational geometry and hard correspondence matches are interleaved and are realized by applying coupled update operations to the expected log-likelihood function. In this way, the two processes bootstrap one another. This provides a means of rejecting structural outliers. We evaluate the technique on two real-world problems. The first involves the matching of different perspective views of 3.5-inch floppy discs. The second example is furnished by the matching of a digital map against aerial images that are subject to severe barrel distortion due to a line-scan sampling process. We complement these experiments with a sensitivity study based on synthetic data.

### Citations

8094 | Maximum likelihood from incomplete data via the em algorithm
- Dempster, Laird, et al.
- 1977
(Show Context)
Citation Context ...parameter estimates. In the maximization phase, the model parameters are recomputed to maximize the expected value of the incomplete data likelihood. According to the original work of Dempster et al. =-=[12]-=- the expected likelihood function is computed by weighting the current log-probability density by the a posteriori measurement probabilities estimated from the preceding maximum likelihood parameters.... |

467 |
The Interpretation of Visual Motion
- Ullman
- 1979
(Show Context)
Citation Context ...nstraints in recovering structure from motion sequences. As a concrete example, McReynolds and Lowe show how rigidity constraints can be used in perspective matching [27]. Historically, it was Ullman =-=[39]-=- who was one of the first to recognize the importance of exploiting rigidity constraints in the correspondence matching of point-sets. Recently, several authors have drawn inspiration from Ullman’s id... |

380 |
Active shape models-their training and application. Computer vision and image understanding
- Cootes, Taylor, et al.
- 1995
(Show Context)
Citation Context ... EM ALGORITHM 1237 comparing the modal eigenstructure of the intraimage proximity matrix. In fact, these techniques provide the basic ground-work on which the deformable shape models of Cootes et al. =-=[8]-=- and Sclaroff and Pentland [34] build. This work on the coordinate proximity matrix is closely akin to that of Umeyama [40] who shows how point-sets abstracted in a structural manner using weighted ad... |

293 |
Affine structure from motion
- Koenderink, Doom
(Show Context)
Citation Context ...predefined geometry. The process is a critical ingredient in intermediate level vision [36], [35], [33]. Concrete applications include camera calibration [6], object recognition [17], motion analysis =-=[25]-=-, and image mosaicing [21]. Depending on the imaging geometry, the transformation may be Euclidean, affine, or perspective. The simplest case is the Euclidean similarity transformation, which involves... |

248 | A Paraperspective Factorization Method for Shape and Motion Recovery
- Poelman, Kanade
- 1997
(Show Context)
Citation Context ..., relational constraints, Delaunay graph, discrete relaxation. 1 INTRODUCTION T HE estimation of transformational geometry from pointsets is key to many problems of computer vision and robotics [29], =-=[31]-=-. Broadly speaking, the aim is to recover a matrix representation of the transformation between image and model coordinate systems. Estimating the matrix requires a set of correspondence matches betwe... |

219 | The Development and Comparison of Robust Methods for Estimating the Fundamental Matrix
- Torr, Murray
- 1997
(Show Context)
Citation Context ...a matrix representation of the transformation between image and model coordinate systems. Estimating the matrix requires a set of correspondence matches between features in the two coordinate systems =-=[37]-=-. In other words, the feature points must be labeled. Posed in this way, there is a basic chicken-and-egg problem. Before good correspondences can be estimated, there needs to be reasonable bounds on ... |

216 |
Least-squares estimation of transformation parameters between two point patterns
- Umeyama
(Show Context)
Citation Context ...sets abstracted in a structural manner using weighted adjacency graphs can be matched using an eigendecomposition method. These ideas have been extended to accommodate parametererized transformations =-=[41]-=-, which can be applied to the matching of articulated objects [42]. More recently, there have been several attempts at modeling the structural deformation of point-sets. For instance, Amit and Kong [5... |

191 |
Analysis and solutions of the three point perspective pose estimation problem
- Haralick, Lee, et al.
- 1991
(Show Context)
Citation Context ... f321 , , , , is a column-vector formed from the elements in the bottom row of the transformation matrix. Because the transformation equations are nonlinear, the recovery of perspective geometry [4], =-=[16]-=-, [11], [29], [20] is more difficult than the affine case. The main problems stem from the numerical instabilities associated with the denominator of the transformation equations. Haralick et al. [16]... |

181 | Modal matching for correspondence and recognition
- Sclaroff, Pentland
(Show Context)
Citation Context ...he modal eigenstructure of the intraimage proximity matrix. In fact, these techniques provide the basic ground-work on which the deformable shape models of Cootes et al. [8] and Sclaroff and Pentland =-=[34]-=- build. This work on the coordinate proximity matrix is closely akin to that of Umeyama [40] who shows how point-sets abstracted in a structural manner using weighted adjacency graphs can be matched u... |

166 | Geometric structures for threedimensional shape representation - Boissonnat - 1984 |

166 |
Shape manifolds, procrustean metrics, and complex projective spaces
- Kendall
- 1984
(Show Context)
Citation Context ...r consideration. 1.1 Related Literature The problem of point pattern matching has attracted sustained interest in both the vision and statistics communities for several decades. For instance, Kendall =-=[24]-=- has generalized the process to projective manifolds using the concept of Procrustes distance. In the vision literature, the problem has attracted increased recent interest because of the pivotal role... |

148 |
An eigendecomposition approach to weighted graph matching problems
- Umeyama
- 1988
(Show Context)
Citation Context ...e the basic ground-work on which the deformable shape models of Cootes et al. [8] and Sclaroff and Pentland [34] build. This work on the coordinate proximity matrix is closely akin to that of Umeyama =-=[40]-=- who shows how point-sets abstracted in a structural manner using weighted adjacency graphs can be matched using an eigendecomposition method. These ideas have been extended to accommodate parameterer... |

135 | In defense of the eight-point algorithm - Hartley - 1997 |

105 | Structural matching by discrete relaxation
- Wilson, Hancock
- 1997
(Show Context)
Citation Context ...ased on their consistency with the assigned matches on neighboring nodes of the graph. This model draws on a refinement of the relational consistency measure originally reported by Wilson and Hancock =-=[45]-=-. Our goal is to compute the probability of assigning the correspondence match f (n+1) (i) = j to the center node i of the data-graph D clique Ci at iteration n + 1. In order to draw on contextual inf... |

103 |
An algorithm for associating the features of two images
- Scott, Longuet-Higgins
- 1991
(Show Context)
Citation Context ...urpose correspondence matching algorithms using the Gaussian weighted proximity matrix. There are two contrasting uses of the proximity-matrix which deserve special mention. Scott and Longuet-Higgins =-=[33]-=- locate correspondences by finding a singular value decomposition of the interimage proximity matrix. Shapiro and Brady [36], [35], on the other hand, match bysCROSS AND HANCOCK: GRAPH MATCHING WITH A... |

93 |
Hierarchical Mixtures of Experts and
- Jordan, Jacobs
- 1994
(Show Context)
Citation Context ...meters and refining correspondence matches. The framework for this study is furnished by a variant of the EM algorithm. Specifically, we use a structural gating process inspired by Jordan and Jacob’s =-=[23]-=- hierarchical mixture of experts architecture to control contributions to the log-likelihood function for the transformation parameters. The structural gating is based on the consistency of the corres... |

79 | Projective reconstruction and invariants from multiple images
- Hartley
- 1994
(Show Context)
Citation Context ...l geometry. Yet, this geometry is, after all, the ultimate goal of computation. This problem is usually overcome by invoking constraints to bootstrap the estimation of feasible correspondence matches =-=[17]-=-, [27]. One of the most popular ideas is to use the epipolar constraint to prune the space of potential correspondences [17]. If reliable correspondences are not available, then a robust fitting metho... |

62 | Combining point distribution models with shape models based on finite element analysis - Cootes, Taylor - 1995 |

58 | Object Pose: The Link between Weak Perspective
- Horaud, Dornaika, et al.
- 1997
(Show Context)
Citation Context ... column-vector formed from the elements in the bottom row of the transformation matrix. Because the transformation equations are nonlinear, the recovery of perspective geometry [4], [16], [11], [29], =-=[20]-=- is more difficult than the affine case. The main problems stem from the numerical instabilities associated with the denominator of the transformation equations. Haralick et al. [16] review the origin... |

52 | Graphical Templates for Model Registration
- Amit, Kong
- 1996
(Show Context)
Citation Context ...1], which can be applied to the matching of articulated objects [42]. More recently, there have been several attempts at modeling the structural deformation of point-sets. For instance, Amit and Kong =-=[5]-=- have used a graph-based representation (graphical templates) to model deforming two-dimensional shapes in medical images. Lades et al. [26] have used a dynamic mesh to model intensity-based appearanc... |

48 |
Iterative pose estimation using coplanar feature points. Computer Vision and Image Understanding
- Oberkampf, DeMenthon, et al.
- 1996
(Show Context)
Citation Context ...ometry, relational constraints, Delaunay graph, discrete relaxation. 1 INTRODUCTION T HE estimation of transformational geometry from pointsets is key to many problems of computer vision and robotics =-=[29]-=-, [31]. Broadly speaking, the aim is to recover a matrix representation of the transformation between image and model coordinate systems. Estimating the matrix requires a set of correspondence matches... |

43 | A stratified approach to metric self-calibration
- Pollefeys, Gool
- 1997
(Show Context)
Citation Context ... associated covariance matrix, it offers significant advantages in terms of its adaptive capabilities. When viewed in this way, our method has some conceptual similarity with Pollefeys and Van Gool’s =-=[32]-=- stratified self-calibration. Here, the calibration process is bootstrapped by interleaving the estimation of affine geometry and refining correspondences by imposing rigidity constraints. The outline... |

42 | Dot Pattern Processing Using Voronoi Neighbourhoods
- Ahuja
(Show Context)
Citation Context ...launay triangulation as our basic representation of image structure [38], [13]. We establish Delaunay triangulations on the data and the model by seeding Voronoi tessellations from the feature-points =-=[1]-=-, [2], [3]. The process of Delaunay triangulation generates relational graphs from the two sets of point-features. An example is shown in Fig. 1. More formally, the point-sets are the nodes of a data ... |

38 |
Parameterized Point Pattern Matching and Its Application to Recognition of Object Families
- Umeyama
- 1993
(Show Context)
Citation Context ...aphs can be matched using an eigendecomposition method. These ideas have been extended to accommodate parametererized transformations [41], which can be applied to the matching of articulated objects =-=[42]-=-. More recently, there have been several attempts at modeling the structural deformation of point-sets. For instance, Amit and Kong [5] have used a graph-based representation (graphical templates) to ... |

30 |
Representing stereo data with the delaunay triangulation
- Faugeras, Bras-Mehlman, et al.
- 1990
(Show Context)
Citation Context ...ATCHING WITH A DUAL-STEP EM ALGORITHM 1239 of its well documented robustness to noise and change of viewpoint, we adopt the Delaunay triangulation as our basic representation of image structure [38], =-=[13]-=-. We establish Delaunay triangulations on the data and the model by seeding Voronoi tessellations from the feature-points [1], [2], [3]. The process of Delaunay triangulation generates relational grap... |

29 | Rigidity checking of 3D point correspondences under perspective projection
- McReynolds, Lowe
- 1996
(Show Context)
Citation Context ...etry. Yet, this geometry is, after all, the ultimate goal of computation. This problem is usually overcome by invoking constraints to bootstrap the estimation of feasible correspondence matches [17], =-=[27]-=-. One of the most popular ideas is to use the epipolar constraint to prune the space of potential correspondences [17]. If reliable correspondences are not available, then a robust fitting method must... |

26 | Learning with pre-knowledge: Clustering with point and graph-matching distance measures
- Gold, Rangarajan, et al.
- 1996
(Show Context)
Citation Context ...del to provide structural constraints on parameter estimation is a task of generic importance. Although the EM algorithm has been used to extract affine and Euclidean parameters from point-sets [43], =-=[15]-=- or line-sets [28], there has been no attempt to impose structural constraints on the correspondence matches. Viewed from the perspective of graphical template matching [5], [26], our EM algorithm all... |

26 |
Rejecting outliers and estimating errors in an orthogonal regression framework,” Philos
- Shapiro, Brady
- 1995
(Show Context)
Citation Context ...most popular ideas is to use the epipolar constraint to prune the space of potential correspondences [17]. If reliable correspondences are not available, then a robust fitting method must be employed =-=[36]-=-, [35]. This involves removing rogue correspondences through outlier rejection. An example is furnished by the recent work of Torr and Murray [37]. In this paper, we adopt a somewhat different approac... |

20 | Labeled point pattern matching by Delaunay triangulation and maximal cliques - Ogawa - 1986 |

16 |
Navigation using affine structure and motion
- Beardsley, Zisserman, et al.
- 1994
(Show Context)
Citation Context ...ing the coordinates of the point-sets under a predefined geometry. The process is a critical ingredient in intermediate level vision [36], [35], [33]. Concrete applications include camera calibration =-=[6]-=-, object recognition [17], motion analysis [25], and image mosaicing [21]. Depending on the imaging geometry, the transformation may be Euclidean, affine, or perspective. The simplest case is the Eucl... |

15 |
An energy function and continuous edit process for graph matching
- Finch, Wilson, et al.
- 1998
(Show Context)
Citation Context ...ous pointdeformation models. In fact, we have recently taken some steps in this direction by demonstrating how the optimization of our relational consistency measure can be realized using soft-assign =-=[14]-=-. Finally, although we have demonstrated our matching process for the problem of image registration, the problem of calibration may well provide a more topical vehicle where our new technique could be... |

11 |
Image representation using Voronoi Tessellation
- Ahuja, An, et al.
- 1985
(Show Context)
Citation Context ...y triangulation as our basic representation of image structure [38], [13]. We establish Delaunay triangulations on the data and the model by seeding Voronoi tessellations from the feature-points [1], =-=[2]-=-, [3]. The process of Delaunay triangulation generates relational graphs from the two sets of point-features. An example is shown in Fig. 1. More formally, the point-sets are the nodes of a data graph... |

11 |
Mosaic-based representations of video sequences and their applications
- IRANI, ADANAN, et al.
- 1995
(Show Context)
Citation Context ...rocess is a critical ingredient in intermediate level vision [36], [35], [33]. Concrete applications include camera calibration [6], object recognition [17], motion analysis [25], and image mosaicing =-=[21]-=-. Depending on the imaging geometry, the transformation may be Euclidean, affine, or perspective. The simplest case is the Euclidean similarity transformation, which involves only translation, rotatio... |

10 |
Extraction of Early Perceptual Structure
- Ahuja, Tuceryan
- 1989
(Show Context)
Citation Context ...angulation as our basic representation of image structure [38], [13]. We establish Delaunay triangulations on the data and the model by seeding Voronoi tessellations from the feature-points [1], [2], =-=[3]-=-. The process of Delaunay triangulation generates relational graphs from the two sets of point-features. An example is shown in Fig. 1. More formally, the point-sets are the nodes of a data graph G D ... |

10 |
Feature-based correspondence—An eigenvector approach
- Shapiro, Brady
- 1992
(Show Context)
Citation Context ...opular ideas is to use the epipolar constraint to prune the space of potential correspondences [17]. If reliable correspondences are not available, then a robust fitting method must be employed [36], =-=[35]-=-. This involves removing rogue correspondences through outlier rejection. An example is furnished by the recent work of Torr and Murray [37]. In this paper, we adopt a somewhat different approach to t... |

8 |
Distortion-invariant object recognition in the dynamic link architecture
- Lades, Vorbruggen, et al.
- 1993
(Show Context)
Citation Context ...tural deformation of point-sets. For instance, Amit and Kong [5] have used a graph-based representation (graphical templates) to model deforming two-dimensional shapes in medical images. Lades et al. =-=[26]-=- have used a dynamic mesh to model intensity-based appearance in images. However, these contributions fall well short of formally integrating structural constraints into the recovery of transformation... |

7 |
Structural matching with active triangulations
- Wilson, Cross, et al.
- 1998
(Show Context)
Citation Context ... the points under the current estimate of the transformation parameters. We have recently shown how this process of active graph reconfiguration or editing can be realized as a MAP estimation process =-=[44]-=-. However, rather than confining itself purely to node relabeling, the algorithm now encompasses the possibility of node insertions or deletions together with the implied modification of the Delaunay ... |

3 |
3D Pose from Three Points Using Weak-perspective
- Alter
- 1994
(Show Context)
Citation Context ...= f31 f321 , , , , is a column-vector formed from the elements in the bottom row of the transformation matrix. Because the transformation equations are nonlinear, the recovery of perspective geometry =-=[4]-=-, [16], [11], [29], [20] is more difficult than the affine case. The main problems stem from the numerical instabilities associated with the denominator of the transformation equations. Haralick et al... |

3 | ªMixture Models and EM Algorithms for Object Recognition within Compositional Hierarchies,º ICSI Berkeley
- Utans
- 1993
(Show Context)
Citation Context ...cal model to provide structural constraints on parameter estimation is a task of generic importance. Although the EM algorithm has been used to extract affine and Euclidean parameters from point-sets =-=[43]-=-, [15] or line-sets [28], there has been no attempt to impose structural constraints on the correspondence matches. Viewed from the perspective of graphical template matching [5], [26], our EM algorit... |

2 | Robust Statistics: The Approach Based on Influence Functions - Hempel - 1986 |

2 |
Optimal matching of planar models
- Jacobs
- 1991
(Show Context)
Citation Context ... correspondences between 3D features and 2D image points are known. Horaud et al. [20] have extended these ideas to develop an iterative algorithm for recovering paraperspective pose. Finally, Jacobs =-=[22]-=- has an efficient voting algorithm which uses a hashing technique based on image triangles to recover the perspective pose of planar objects in 3D scenes. Notwithstanding these important contributions... |

2 |
Registering incomplete radar images with the em algorithm, image and vision computing
- Moss, Hancock
- 1997
(Show Context)
Citation Context ...uctural constraints on parameter estimation is a task of generic importance. Although the EM algorithm has been used to extract affine and Euclidean parameters from point-sets [43], [15] or line-sets =-=[28]-=-, there has been no attempt to impose structural constraints on the correspondence matches. Viewed from the perspective of graphical template matching [5], [26], our EM algorithm allows an explicit de... |

2 |
Relative Sensitivity of a Family of Closest
- Tuceryan, Chorzempa
- 1991
(Show Context)
Citation Context ...RAPH MATCHING WITH A DUAL-STEP EM ALGORITHM 1239 of its well documented robustness to noise and change of viewpoint, we adopt the Delaunay triangulation as our basic representation of image structure =-=[38]-=-, [13]. We establish Delaunay triangulations on the data and the model by seeding Voronoi tessellations from the feature-points [1], [2], [3]. The process of Delaunay triangulation generates relationa... |

1 | Perspective Pose Recovery With a Dual Step EM Algorithm - Cross, Hancock - 1998 |

1 |
Exact and Approximate Solutions of the Perspective Three-Point Problem
- DeMenthon, Davis
- 1992
(Show Context)
Citation Context ..., , , , is a column-vector formed from the elements in the bottom row of the transformation matrix. Because the transformation equations are nonlinear, the recovery of perspective geometry [4], [16], =-=[11]-=-, [29], [20] is more difficult than the affine case. The main problems stem from the numerical instabilities associated with the denominator of the transformation equations. Haralick et al. [16] revie... |