## Learning Graph Matching

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Citations: | 44 - 9 self |

### BibTeX

@MISC{Caetano_learninggraph,

author = {Tibério S. Caetano and Li Cheng and Quoc V. Le and Alex J. Smola},

title = {Learning Graph Matching},

year = {}

}

### Years of Citing Articles

### OpenURL

### Abstract

As a fundamental problem in pattern recognition, graph matching has found a variety of applications in the field of computer vision. In graph matching, patterns are modeled as graphs and pattern recognition amounts to finding a correspondence between the nodes of different graphs. There are many ways in which the problem has been formulated, but most can be cast in general as a quadratic assignment problem, where a linear term in the objective function encodes node compatibility functions and a quadratic term encodes edge compatibility functions. The main research focus in this theme is about designing efficient algorithms for solving approximately the quadratic assignment problem, since it is NP-hard. In this paper, we turn our attention to the complementary problem: how to estimate compatibility functions such that the solution of the resulting graph matching problem best matches the expected solution that a human would manually provide. We present a method for learning graph matching: the training examples are pairs of graphs and the “labels” are matchings between pairs of graphs. We present experimental results with real image data which give evidence that learning can improve the performance of standard graph matching algorithms. In particular, it turns out that linear assignment with such a learning scheme may improve over state-of-the-art quadratic assignment relaxations. This finding suggests that for a range of problems where quadratic assignment was thought to be essential for securing good results, linear assignment, which is far more efficient, could be just sufficient if learning is performed. This enables speed-ups of graph matching by up to 4 orders of magnitude while retaining state-of-the-art accuracy. 1.

### Citations

5885 | Distinctive image features from scale-invariant keypoints
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(Show Context)
Citation Context ...earning process finds how relevant is the rth feature of φ1. In our experiments to be described in the next section, we use two types of node features (i.e. Gi, G ′ i ′): the well-known SIFT features =-=[14]-=- and Shape Context features [4]. SIFT is a 128-dimensional rotation and scale-invariant descriptor which has also some invariance with respect to viewpoint and illumination changes and has been widely... |

2308 | Online learning with kernels
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- 2004
(Show Context)
Citation Context ...d/or clustering graphs, but not learning a matching criterion per se. Since graphs are eminently nonvectorial data structures, a substantial part of this literature has been focused on Kernel Methods =-=[2]-=-, [3], which comprise a principled framework for dealing with structured data using standard tools from linear analysis. We refer the reader to the recent unified treatment on these methods as applied... |

1394 |
Combinatorial Optimization: Algorithms and Complexity
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Citation Context ... map should be a function (many-to-one, that is � ′ = 0 i ′ yii ′ = 1 for all i). If dii ′ jj for all ii ′ jj ′ then (1) becomes a linear assignment problem, exactly solvable in worst case cubic time =-=[15]-=-. Although the compatibility functions c and d obviously depend on the attributes {Gi, G ′ i ′} and {Gij, G ′ i ′ j ′}, the functional form of this dependency is typically assumed to be fixed in graph... |

1378 | Shape matching and object recognition using shape contexts
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- 2002
(Show Context)
Citation Context ...nt is the rth feature of φ1. In our experiments to be described in the next section, we use two types of node features (i.e. Gi, G ′ i ′): the well-known SIFT features [14] and Shape Context features =-=[4]-=-. SIFT is a 128-dimensional rotation and scale-invariant descriptor which has also some invariance with respect to viewpoint and illumination changes and has been widely used in computer vision [14]. ... |

469 | D.: Max-margin Markov networks
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- 2003
(Show Context)
Citation Context ...render the problem of minimizing (2) more tractable is to replace the empirical risk by a convex upper bound on the empirical risk, an idea that has been exploited in Machine Learning in recent years =-=[25,27,28]-=-. By minimizing this convex upper bound, we hope to decrease the empirical risk as well. It is easy to show that the convex (in particular, linear) function 1 ∑ N for 1 ∑ N appropriately chosen constr... |

411 | Large margin methods for structured and interdependent output variables
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- 2005
(Show Context)
Citation Context ...trying a “better answer” to the same question.s3. Learning Graph Matching 3.1. General Problem Setting We approach the problem of learning the compatibility functions as a supervised learning problem =-=[19]-=-. The training set comprises N observations x from an input set X, N corresponding labels y from an output set Y, and can be represented by {(x 1 ; y 1 ), . . . , (x N ; y N )}. Critical in our settin... |

366 |
Training linear svms in linear time
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Citation Context ...NTELLIGENCE, VOL. 31, NO. 6, JUNE 2009 4.5 The Algorithm Instead of using the formulation in (9), which has n slack variables (used in [1] and [33]), we here use the (equivalent) formulation given in =-=[37]-=-, in which there is only a single slack variable: subject to 1 X hw; N n for all y 2Y: minimize w; n ðyÞi þ 2 kwk 2 1 X N n ðy; y n Þ ð10aÞ ð10bÞ Note that the number of constraints in (10) is given b... |

309 | A Graduated Assignment Algorithm for Graph Matching
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- 1996
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Citation Context ...[12], [13], relaxation labeling and probabilistic approaches [14], [15], [16], [17], [18], [19], [20], semidefinite relaxations [21], replicator equations [22], tree search [23], graduated assignment =-=[24]-=-, and RKHS methods [25]. Spectral methods consist of studying the similarities between the spectra of the adjacency or Laplacian matrices of the graphs and using them for matching. Relaxation and prob... |

281 |
Caltech-256 object category dataset
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- 2007
(Show Context)
Citation Context ...utperforms nonlearning for all baselines. The weight vector (Fig. 2b) is heavily peaked about particular angular bins. 6.4 Bikes For our final experiment, we used images from the Caltech 256 data set =-=[50]-=-. We chose to match images in the “touring bike” class, which contains 110 images of bicycles. Since the Shape Context features we are using are robust to only a small amount of rotation (and not to r... |

183 | Structural matching in computer vision using probabilistic relaxation
- Christmas, Kittler, et al.
- 1995
(Show Context)
Citation Context ...e A variety of approaches has been proposed to solve the attributed graph matching problem. An incomplete list includes spectral methods [13, 18], semidefinite programming [17], probabilistic methods =-=[10, 5, 7]-=- and the well-known graduated assignment method [9]. The above literature strictly focuses on trying better algorithms for solving the graph matching problem, but does not address the issue of how to ... |

164 |
A shortest augmenting path algorithm for dense and sparse linear assignment problems
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- 1987
(Show Context)
Citation Context ...ng and at test time are linear assignment problems, which can be solved efficiently in worst case cubic time. In our experiments, we solve the linear assignment problem with the efficient solver from =-=[11]-=-. For quadratic assignment, we developed a C implementation of the wellknown Graduated Assignment algorithm [9]. However it should be stressed that the learning scheme discussed here is completely ind... |

157 | A spectral technique for correspondence problems using pairwise constraints
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- 2005
(Show Context)
Citation Context ...m under the proposed learning framework. 1.1. Related Literature A variety of approaches has been proposed to solve the attributed graph matching problem. An incomplete list includes spectral methods =-=[13, 18]-=-, semidefinite programming [17], probabilistic methods [10, 5, 7] and the well-known graduated assignment method [9]. The above literature strictly focuses on trying better algorithms for solving the ... |

149 |
A relationship between arbitrary positive matrices and stochastic matrices
- Sinkhorn
- 1966
(Show Context)
Citation Context ...al have worstcase exponential complexity and work via sequential tests of compatibility of local parts of the graphs. Graduated Assignment combines the “softassign” method [26] with Sinkhorn’s method =-=[27]-=- and essentially consists of a series of first-order approximations to the quadratic assignment objective function. This method is particularly popular in computer vision since it produces accurate re... |

124 | Relational matching by discrete relaxation
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- 1994
(Show Context)
Citation Context ...roximations and heuristics for the quadratic assignment problem. An incomplete list includes spectral methods [9], [10], [11], [12], [13], relaxation labeling and probabilistic approaches [14], [15], =-=[16]-=-, [17], [18], [19], [20], semidefinite relaxations [21], replicator equations [22], tree search [23], graduated assignment [24], and RKHS methods [25]. Spectral methods consist of studying the similar... |

122 |
Feature-based correspondence: an eigenvector approach
- Shapiro, Brady
- 1992
(Show Context)
Citation Context ...m under the proposed learning framework. 1.1. Related Literature A variety of approaches has been proposed to solve the attributed graph matching problem. An incomplete list includes spectral methods =-=[13, 18]-=-, semidefinite programming [17], probabilistic methods [10, 5, 7] and the well-known graduated assignment method [9]. The above literature strictly focuses on trying better algorithms for solving the ... |

111 |
Kernel Methods for Pattern Analysis. Cambridge Univ
- Shawe-Taylor, Cristianini
- 2004
(Show Context)
Citation Context ...clustering graphs, but not learning a matching criterion per se. Since graphs are eminently nonvectorial data structures, a substantial part of this literature has been focused on Kernel Methods [2], =-=[3]-=-, which comprise a principled framework for dealing with structured data using standard tools from linear analysis. We refer the reader to the recent unified treatment on these methods as applied to g... |

108 |
A new algorithm for error-tolerant subgraph isomorphism detection
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- 1998
(Show Context)
Citation Context ...l methods [9], [10], [11], [12], [13], relaxation labeling and probabilistic approaches [14], [15], [16], [17], [18], [19], [20], semidefinite relaxations [21], replicator equations [22], tree search =-=[23]-=-, graduated assignment [24], and RKHS methods [25]. Spectral methods consist of studying the similarities between the spectra of the adjacency or Laplacian matrices of the graphs and using them for ma... |

91 |
R.: Numerical geometry of non-rigid shapes
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- 2008
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Citation Context ...l learning is able to choose those features which remain useful under these transformations. Our setup is similar to the previous experiments: We begin with the point set in Fig. 6a (image taken from =-=[44]-=-, [45], [46], with 35 landmarks identified using code provided 5. This should be interpreted with some caution: The features have different scales, meaning that their importances cannot be compared di... |

60 | A scalable modular convex solver for regularized risk minimization
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- 2007
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Citation Context ...with respect to w) is w 1 X n ð^ynÞ: ð14Þ N n Equations (13) and (14) define the new constraint to be added to the optimization problem. Pseudocode for this algorithm is described in Algorithm 1. See =-=[38]-=- for more details. Let us investigate the complexity of solving (11). Using the joint feature map as in (6) and the loss as in (7), the argument in (11) becomes h ðG; G 0 ;yÞ;wiþ ðy; y n Þ X X yii0c... |

54 | Replicator equations, maximal cliques and graph isomorphism, Neural Comput
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- 1999
(Show Context)
Citation Context ...t includes spectral methods [9], [10], [11], [12], [13], relaxation labeling and probabilistic approaches [14], [15], [16], [17], [18], [19], [20], semidefinite relaxations [21], replicator equations =-=[22]-=-, tree search [23], graduated assignment [24], and RKHS methods [25]. Spectral methods consist of studying the similarities between the spectra of the adjacency or Laplacian matrices of the graphs and... |

50 | Balanced graph matching
- Cour, Srinivasan, et al.
- 2007
(Show Context)
Citation Context ...word alignment. A recent paper of interest shows that very significant improvements on the performance of graph matching can be obtained by an appropriate normalization of the compatibility functions =-=[8]-=-; however, no learning is involved. 2. The Graph Matching Problem The notation used in this paper is summarized in table 1. In the following we denote a graph by G. We will often refer to a pair of gr... |

43 |
An Eigenspace Projection Clustering Method for Inexact Graph Matching
- Caelli, Kosinov
- 2004
(Show Context)
Citation Context ...t types of approaches have been proposed, which mainly focus on approximations and heuristics for the quadratic assignment problem. An incomplete list includes spectral methods [9], [10], [11], [12], =-=[13]-=-, relaxation labeling and probabilistic approaches [14], [15], [16], [17], [18], [19], [20], semidefinite relaxations [21], replicator equations [22], tree search [23], graduated assignment [24], and ... |

42 | Bundle methods for machine learning
- Smola, Vishwanathan, et al.
(Show Context)
Citation Context ...render the problem of minimizing (2) more tractable is to replace the empirical risk by a convex upper bound on the empirical risk, an idea that has been exploited in Machine Learning in recent years =-=[25,27,28]-=-. By minimizing this convex upper bound, we hope to decrease the empirical risk as well. It is easy to show that the convex (in particular, linear) function 1 ∑ N for 1 ∑ N appropriately chosen constr... |

41 | Learning compatibility coefficients for relaxation labeling processes
- Pelillo, Refice
- 1994
(Show Context)
Citation Context ... literature strictly focuses on trying better algorithms for solving the graph matching problem, but does not address the issue of how to determine the compatibility functions in a principled way. In =-=[16]-=- the authors learn compatibility functions for the relaxation labeling process; this is however a different problem than graph matching, and the “compatibility functions” have a different meaning. In ... |

39 |
Recent advances in the solution of quadratic assignment problems
- Anstreicher
(Show Context)
Citation Context ...↦→ i ′ j ′ . Then, a generic formulation of the graph matching problem consists of finding the optimal matching matrix y∗ given by the solution of the following (NP-hard) quadratic assignment problem =-=[3]-=- Table 1. Definitions and Notation G - generic graph (similarly, G ′ ); Gi - attribute of node i in G (similarly, G ′ i ′ for G′ ); Gij - attribute of edge ij in G (similarly, G ′ i ′ j ′ for G′ ); G ... |

38 | Word alignment via quadratic assignment
- Lacoste-Julien, Taskar, et al.
- 2006
(Show Context)
Citation Context ...ocess; this is however a different problem than graph matching, and the “compatibility functions” have a different meaning. In terms of methodology, possibly the paper most closely related to ours is =-=[12]-=-, which uses structured estimation tools in a quadratic assignment setting for word alignment. A recent paper of interest shows that very significant improvements on the performance of graph matching ... |

35 | Spectral correspondence for point pattern matching
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(Show Context)
Citation Context ...fferent types of approaches have been proposed, which mainly focus on approximations and heuristics for the quadratic assignment problem. An incomplete list includes spectral methods [9], [10], [11], =-=[12]-=-, [13], relaxation labeling and probabilistic approaches [14], [15], [16], [17], [18], [19], [20], semidefinite relaxations [21], replicator equations [22], tree search [23], graduated assignment [24]... |

34 | Graphical models and point pattern matching
- Caetano, Caelli, et al.
- 2006
(Show Context)
Citation Context ...eriments with the CMU ‘house’ dataset [1]. This dataset contains 111 frames of a video sequence of a toy house for which labeling of the same 30 landmark points is available across the whole sequence =-=[6]-=-. We can easily deal with outliers by augmenting the smaller graph with dummy nodes, in the same way as described in [4]. The sequence is such that the first and last frames are separated by a very wi... |

31 | Inverse optimization
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(Show Context)
Citation Context ...G, G ′ , y; w) = 〈w, Φ(G, G ′ , y)〉, so that our predictor has the form g(G, G ′ , w) = argmax 〈w, Φ(G, G y∈Y ′ , y)〉 . (3) Effectively we are solving an inverse optimization problem, as described by =-=[25, 26]-=-, that is, we are trying to find f such that g has desirable properties. Further specification of g(G, G ′ ; w) requires determining the joint feature map Φ(G, G ′ , y), which has to encode the proper... |

30 |
Combining evidence in probabilistic relaxation
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(Show Context)
Citation Context ...uristics for the quadratic assignment problem. An incomplete list includes spectral methods [9], [10], [11], [12], [13], relaxation labeling and probabilistic approaches [14], [15], [16], [17], [18], =-=[19]-=-, [20], semidefinite relaxations [21], replicator equations [22], tree search [23], graduated assignment [24], and RKHS methods [25]. Spectral methods consist of studying the similarities between the ... |

21 | Learning shape-classes using a mixture of tree-unions
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Citation Context ...ese methods as applied to graphs [4], as well as the references therein. Another line of work has been the use of generative models for graphs in the structural pattern recognition community, such as =-=[5]-=-, [6] and [7]. Also, learning the graph edit distance for purposes of graph classification has been introduced in [8]. 2.2 Graph Matching The graph matching literature is extensive, and many different... |

18 | A markov random field model for object matching under contextual constraints
- Li
- 1994
(Show Context)
Citation Context ...cs for the quadratic assignment problem. An incomplete list includes spectral methods [9], [10], [11], [12], [13], relaxation labeling and probabilistic approaches [14], [15], [16], [17], [18], [19], =-=[20]-=-, semidefinite relaxations [21], replicator equations [22], tree search [23], graduated assignment [24], and RKHS methods [25]. Spectral methods consist of studying the similarities between the spectr... |

18 |
A new class of corner finder
- Smith
- 1992
(Show Context)
Citation Context ...n a video sequence. We used a video sequence from the SAMPL data set [47]—a 108 frame sequence of a human face (see Fig. 2c). To identify landmarks for these scenes, we used the SUSAN corner detector =-=[48]-=-, [49]. This detector essentially identifies points as corners if their neighbors within a small radius are dissimilar. This detector was tuned such that no more than 200 landmarks were identified in ... |

17 |
A RKHS interpolator-based graph matching algorithm
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Citation Context ...labeling and probabilistic approaches [14], [15], [16], [17], [18], [19], [20], semidefinite relaxations [21], replicator equations [22], tree search [23], graduated assignment [24], and RKHS methods =-=[25]-=-. Spectral methods consist of studying the similarities between the spectra of the adjacency or Laplacian matrices of the graphs and using them for matching. Relaxation and probabilistic methods defin... |

11 | Automatic learning of cost functions for graph edit distance
- Neuhaus, Bunke
(Show Context)
Citation Context ...erative models for graphs in the structural pattern recognition community, such as [5], [6] and [7]. Also, learning the graph edit distance for purposes of graph classification has been introduced in =-=[8]-=-. 2.2 Graph Matching The graph matching literature is extensive, and many different types of approaches have been proposed, which mainly focus on approximations and heuristics for the quadratic assign... |

10 |
Feature-based correspondence—An eigenvector approach
- Shapiro, Brady
- 1992
(Show Context)
Citation Context ...any different types of approaches have been proposed, which mainly focus on approximations and heuristics for the quadratic assignment problem. An incomplete list includes spectral methods [9], [10], =-=[11]-=-, [12], [13], relaxation labeling and probabilistic approaches [14], [15], [16], [17], [18], [19], [20], semidefinite relaxations [21], replicator equations [22], tree search [23], graduated assignmen... |

7 |
Graph-based methods for vision: A yorkist manifesto
- Hancock, Wilson
(Show Context)
Citation Context ...e A variety of approaches has been proposed to solve the attributed graph matching problem. An incomplete list includes spectral methods [13, 18], semidefinite programming [17], probabilistic methods =-=[10, 5, 7]-=- and the well-known graduated assignment method [9]. The above literature strictly focuses on trying better algorithms for solving the graph matching problem, but does not address the issue of how to ... |

6 | Convergence properties of the softassign quadratic assignment algorithm
- Rangarajan, Yuille, et al.
- 1999
(Show Context)
Citation Context ...e-search techniques in general have worstcase exponential complexity and work via sequential tests of compatibility of local parts of the graphs. Graduated Assignment combines the “softassign” method =-=[26]-=- with Sinkhorn’s method [27] and essentially consists of a series of first-order approximations to the quadratic assignment objective function. This method is particularly popular in computer vision s... |

5 |
Convex mathematical programs for relational matching of object views
- Schellewald
- 2005
(Show Context)
Citation Context ...work. 1.1. Related Literature A variety of approaches has been proposed to solve the attributed graph matching problem. An incomplete list includes spectral methods [13, 18], semidefinite programming =-=[17]-=-, probabilistic methods [10, 5, 7] and the well-known graduated assignment method [9]. The above literature strictly focuses on trying better algorithms for solving the graph matching problem, but doe... |

5 |
Flexible Filter Neighbourhood Designation
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(Show Context)
Citation Context ...deo sequence. We used a video sequence from the SAMPL data set [47]—a 108 frame sequence of a human face (see Fig. 2c). To identify landmarks for these scenes, we used the SUSAN corner detector [48], =-=[49]-=-. This detector essentially identifies points as corners if their neighbors within a small radius are dissimilar. This detector was tuned such that no more than 200 landmarks were identified in each s... |

5 | A kernel view of spectral point pattern matching - Wang, Hancock |

4 |
Spectral Generative Models for Graphs
- White, Wilson
- 2007
(Show Context)
Citation Context ...ethods as applied to graphs [4], as well as the references therein. Another line of work has been the use of generative models for graphs in the structural pattern recognition community, such as [5], =-=[6]-=- and [7]. Also, learning the graph edit distance for purposes of graph classification has been introduced in [8]. 2.2 Graph Matching The graph matching literature is extensive, and many different type... |

3 |
Graphical models for graph matching
- Caetano, Caelli, et al.
- 2004
(Show Context)
Citation Context ...e A variety of approaches has been proposed to solve the attributed graph matching problem. An incomplete list includes spectral methods [13, 18], semidefinite programming [17], probabilistic methods =-=[10, 5, 7]-=- and the well-known graduated assignment method [9]. The above literature strictly focuses on trying better algorithms for solving the graph matching problem, but does not address the issue of how to ... |

2 |
A Kernel View of Spectral Point
- Wang, Hancock
- 2004
(Show Context)
Citation Context ... and many different types of approaches have been proposed, which mainly focus on approximations and heuristics for the quadratic assignment problem. An incomplete list includes spectral methods [9], =-=[10]-=-, [11], [12], [13], relaxation labeling and probabilistic approaches [14], [15], [16], [17], [18], [19], [20], semidefinite relaxations [21], replicator equations [22], tree search [23], graduated ass... |

2 |
A New Logical Framework for Deductive Planning
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(Show Context)
Citation Context ... the problem of minimizing (2) more tractable is to replace the empirical risk by a convex upper bound on the empirical risk, an idea that has been exploited in machine learning in recent years [33], =-=[35]-=-, [36]. By minimizing this convex upper bound, we hope to decrease the empirical risk as well. It is easy to show that the convex (in particular, linear) function 1 P N n n is an upper bound for 1 P N... |

2 |
data set, http://vasc.ri.cmu.edu/idb/html/motion/house/index.html
- “house”
- 2006
(Show Context)
Citation Context ...G0 i0j0ÞGijG 0 i0j0 (so that w2 is a scalar). 6 EXPERIMENTS 6.1 House/Hotel Sequence For our first experiment, we consider the CMU “house” sequence—a data set consisting of 111 frames of a toy house =-=[42]-=-. Each frame in this sequence has been hand-labeled, with the same 30 landmarks identified in each frame [43]. We explore the performance of our method as the baseline (separation between frames) vari... |

2 | Combining evidence in probabilistic relaxation,” Int - Kittler, Hancock - 1989 |

1 |
Training Structural SVMs when Exact
- Finley, Joachims
- 2008
(Show Context)
Citation Context ...te that, in standard graph matching without learning, we typically have cii0 expð kGi G0 i0k2Þ, which can be 3. Recent work has been done on structured learning when exact inference is not feasible =-=[40]-=-. In that paper, the authors analyze both theoretically and empirically the class of models represented by a fully connected Markov random field. This sheds some light on structured prediction problem... |

1 |
motion dataset: http://sampl.ece.ohio-state.edu/database.htm
- SAMPLE
(Show Context)
Citation Context ...th learning and it is significantly slower. 5.2 Video Sequence For our second experiment, we consider matching features of a human in a video sequence. We used a video sequence from the SAMPL dataset =-=[34]-=- – a 108 frame sequence of a human face (see figure 2, bottom). To identify landmarks for these scenes, we used the SUSAN corner detector [35, 36]. This detector essentially identifies points as corne... |