Pattern Recognition 39 (2006) 1188 – 1198 www.elsevier.com/locate/patcog
A spectral approach to learning structural variations in graphs Bin Luo, Richard C. Wilson, Edwin R. Hancock∗ Department of Computer Science, University of York, Heslington, York, YO10 5DD, UK Received 23 June 2005; received in revised form 4 October 2005; accepted 3 January 2006
Abstract This paper shows how to construct a linear deformable model for graph structure by performing principal components analysis (PCA) on the vectorised adjacency matrix. We commence by using correspondence information to place the nodes of each of a set of graphs in a standard reference order. Using the correspondences order, we convert the adjacency matrices to long-vectors and compute the long-vector covariance matrix. By projecting the vectorised adjacency matrices onto the leading eigenvectors of the covariance matrix, we embed the graphs in a pattern-space. We illustrate the utility of the resulting method for shape-analysis. 䉷 2006 Pattern Recognition Society. Published by Elsevier Ltd. All rights reserved. Keywords: Generative model; Graph; Covariance matrix; Clustering
1. Introduction Many shape and scene analysis problems in computer vision can be abstracted using relational graphs. Examples include the use of shock graphs [1] to represent the differential structure of boundary contours and view graphs to represent 3D object structure. The main advantage of the graph-representation is that it captures the structural variation of shape in a parsimonious way. However, the task of learning structural descriptions from sets of examples has proved to be an elusive one. The obstacle to this endeavour is that graphs are not vectorial in nature. As a result the apparatus of statistical pattern recognition may not be applied to them directly to construct shape-spaces. To convert graphs into vectors, correspondences between nodes must be established. This task is frustrated by the fact that graphs are notoriously susceptible to the effects of noise and clutter. Hence, the addition or loss of a few nodes and edges can result in graphs of significantly different structure, and this in turn frustrates the task of correspondence analysis and hence vectorisation. As a result it is difficult to characterise
∗ Corresponding author. Tel.: +44 1904 43 3374; fax: +44 1904 43 2767.
E-mail address:
[email protected] (E.R. Hancock).
and hence learn the distribution of structural variations in sets of graphs. The ability to learn the modes of variation of the adjacency matrix is an important one. The reason for this is that it allows the statistical significance of changes in the edgestructure of graphs to be assessed. This is crucial capability in measuring the similarity of graphs, matching them to oneanother or clustering them. 1.1. Related literature The literature describes a number of attempts at developing probabilistic models for variations in graph-structure. Some of the earliest work was that of Wong et al. [2], who capture the variation in graph-structure using a discretely defined probability distribution. Bagdanov and Worring [3] have overcome some of the computational difficulties associated with this method by using continuous Gaussian distributions. For problems of graph matching Christmas et al. [4], and Wilson and Hancock [5] have used simple probability distributions to measure the similarity of graphs. There is a considerable body of related literature in the graphical models community concerned with learning the structure of Bayesian networks from data [6]. Recently there has been some research aimed at applying central clustering techniques to cluster graphs.
0031-3203/$30.00 䉷 2006 Pattern Recognition Society. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.patcog.2006.01.001
B. Luo et al. / Pattern Recognition 39 (2006) 1188 – 1198
However, rather than characterising them in a statistical manner, a structural characterisation is adopted. For instance, both Lozano and Escolano [7], and Bunke et al. [8] summarise the data using a supergraph. Each sample can be obtained from the super-graph using edit operations. However, the way in which the super-graph is learned or estimated is not statistical in nature. Jain and Wysotzki, adopt a geometric approach which aims to embed graphs in a high dimensional space by means of the Schur–Hadamard inner product [10]. Central clustering methods are then deployed to learn the class structure of the graphs. The embedding offers the advantage that it is guaranteed to preserve structural information present. Unfortunately, the algorithm does not provide a means of statistically characterising the modes of structural variation encountered. Hence, the methods described in the literature fall well short of constructing genuine generative models from which explicit graph structures can be sampled. In this respect the study of graph-structures is less advanced than the study of pattern spaces for images [11] or shapes [12–14]. Here a well established route to constructing a pattern space for the data is to use principal components analysis. This commences by encoding the image data or shape landmarks as a long-vector. The data is then projected into a low-dimensional space by projecting the long-vectors onto the leading eigenvectors of the sample covariance matrix. This approach has proved to be particularly effective, especially for face data, and has lead the development of more sophisticated analysis methods capable of dealing with quite complex pattern spaces. This field of study is sometimes referred to a manifold learning theory. Recently developed algorithms include locally linear embedding [15], isomap [16], the Laplacian eigenmap [17] and locality preserving projections [18]. However, all these methods are applicable only to data that exists in a vectorial form. To overcome the problems that result from the nonvectorial nature of graphs, in a recent paper [19] we have explored how ideas from spectral graph theory can be used to construct pattern-spaces for sets of graphs. The idea here has been to extract features that are permutation invariants from the adjacency matrices of the graphs under study. Pattern spaces may then be constructed from the feature-vectors using techniques such as principal components analysis, or the more recently developed ones from manifold learning theory described above. However, this work does not lead to a generative model. The reason for this is that it is not possible to reconstruct the graph adjacency matrices from the feature-vectors used to construct the pattern-spaces. Hence, although the embedding procedure can account for the statistical variations in the graph feature vectors, it does not account for the statistical variation in graph-structure that gave rise to the feature-vectors. In other words, the statistical variation in graph-structure remains hidden or implicit.
1189
1.2. Contribution The aim in this paper is to combine ideas from the spectral analysis of graphs and linear deformable models to construct a simple and explicit generative model for graph-structure. One of the problems that limits the use of the structural clustering methods described above [7,8] is that they suffer from exponential complexity and are therefore not easily sampled from. To overcome this problem of exponential complexity we turn to the shape-analysis literature where principal components analysis has proved to be a powerful way of capturing the variations in sets of landmark points for 2D and 3D objects [14]. The idea underpinning the model is to convert graphs to pattern vectors by stacking the columns of their adjacency matrices to form long-vectors. From the covariance matrices for the standardised vectors, there are a number of ways in which to construct pattern-spaces. The simplest of these is to perform PCA and project the standardised adjacency matrix long-vectors onto the leading eigenvectors of the covariance matrix. The distribution of graphs so produced can be further simplified by fitting a manifold or a mixture model. However, here we use the eigenvectors of the covariance matrix to construct a linear model for variations in the adjacency matrices. To do this we borrow ideas from point distribution models. Here Cootes and Taylor [14] have shown how to construct a linear shape-space for sets of landmark points for 2D shapes. We use a variant of this idea to model variations in the long-vectors for the standardised covariance matrices. We commence by computing the leading eigenvectors for the cluster covariance matrices. The graphs are deformed by displacing the mean adjacency matrix long-vectors in the directions of the leading eigenvectors of the covariance matrix. Our method allows the pattern of edge-deformations to be learned and applied at the global level. In principle, edge edit costs can be obtained from our model via a process of averaging the deformations. In this way we construct a generative model of graph-structure. This model may be both fitted to data and sampled. The main practical issue that must be addressed in developing this method is that of how to vectorise the graph adjacency matrices. Here we turn to our own prior work and make use of the algorithm of Luo and Hancock [20]. This method is a variant of Umeyama’s singular value decomposition method [21]. The Umeyama method locates correspondences between graphs by performing singular value decomposition on the adjacency matrices under match, and then locates the permutation matrix that brings the nodes of the graph into correspondence by taking the outer-product of the left singular vectors of the adjacency matrices of the two graphs being matched. However, the method only works for graphs with the same numbers of nodes and a slight difference in edge structure. Luo and Hancock render the method robust to structural difference using the apparatus of the EM algorithm.
1190
B. Luo et al. / Pattern Recognition 39 (2006) 1188 – 1198
The outline of the remainder of the paper is as follows. In Section 2 we describe the formal ingredients of the method, and briefly outline our method for establishing correspondences and converting graphs to long-vectors. Section 3 describes our linear deformable model that can be used to account for the statistical distribution of the long-vectors using the eigenvectors of their covariance matrix. Section 4 describes experiments on synthetic and real-world data. Finally Section 5 offers some conclusions.
1 0
if (i, j ) ∈ Ek , otherwise.
(1)
To construct our generative model of variations in graph structure, we will convert the adjacency matrices into longvectors where the entries have a standardised order. To do this we need to permute the order of the rows and columns of the adjacency matrices. We represent the set of correspondences between the nodes in pairs of graphs using a correspondence matrix. For the graphs indexed k and l, the correspondence matrix is denoted by Sk,l . The elements of the matrix convey the following meaning:
1
Sk,l (i, j ) = 0
if node i ∈ Vk is in correspondence with node j ∈ Vl , otherwise.
(2)
To recover the correspondence matrices, we use the EM algorithm recently reported by Luo and Hancock [20]. This algorithm commences from a Bernoulli model for the correspondences indicators which are treated as missing data. From this distribution an expected log-likelihood function for the missing correspondence indicators is developed. In the maximisation step a singular value decomposition method is used to recover the correspondence matrix which satisfies the condition Sk,l = arg max Tr[ATk SAl S T ]. S
(3)
In other words, the maximum likelihood correspondence matrices are those that maximise the correlation of the two adjacency matrices. With the correspondences to land, we can compute the distance between graphs. This is the negative of the loglikelihood for the optimal set of correspondences and is
(4)
These distances can be used to select a reference graph. The reference graph is has minimum total distance to the remaining graphs in the sample. The index of the reference graph satisfies the condition
r
In this paper we are concerned with the set of graphs G1 , G2 , . . . , Gk , . . . , GN . The kth graph is denoted by Gk = (Vk , Ek ) where Vk is the set of nodes and Ek ⊆ Vk × Vk is the edge-set. Our approach in this paper is a graph-spectral one. For each graph Gk we compute the adjacency matrix Ak . This is a |Vk | × |Vk | matrix whose element with row index i and column index j is
i∈Vk j ∈Vk i ∈Vl j ∈Vl × (1 − Sk,l (j, j )).
r∗ = arg min
2. Background
Ak (i, j ) =
given by 2 = Ak (i, j )Al (i , j )(1 − Sk.l (i, i )) dk,l
N
2 dk,r .
k=1
We use the correspondence matrices to permute the nodeorder of the graphs into the order of the reference graph. For the graph indexed k, the permuted adjacency matrix is T A S . Mk = Sk,r ∗ k k,r∗
(5)
Once the adjacency matrices have been permuted, then we can convert them into pattern-vectors. We do this by stacking the columns of the adjacency matrix to form a long-vector. For the graph-indexed k and the cluster the long-vector is vk = (Mk (1, 1), Mk (1, 2), . . . , Mk (|Vk |, |Vk |))T .
(6)
Once the graphs have been vectorised in this way, we can compute the mean long-vector and the covariance matrix. The mean-vector is N 1 u= vk N
(7)
k=1
and the covariance matrix is =
N 1 (vk − u)(vk − u)T . N
(8)
k=1
3. Linear deformable model for graphs Our aim is to construct a linear deformable model which can be used as a generative model for the variations is graph edge-structure. To do this we use a variant of the point distribution model which has been used to represent the modes of variations for sets of shape landmarks by Cootes and Taylor [14]. To do this, we represent the variations present in the set of graphs using the mean long-vector and the covariance matrix for the long-vectors. Deformations in graph structure are modelled by perturbing the mean long-vector in the directions of the principal eigenvectors of the covariance matrix. We commence by computing the eigenvalues and eigenvectors for the covariance matrix . The eigenvalues 1 , 2 , . . . , |Vr∗ |2 are found by solving the polynomial equation | − I | = 0, where I is the identity matrix. The associated eigenvectors 1 , 2 , . . . are found by solving the
B. Luo et al. / Pattern Recognition 39 (2006) 1188 – 1198
linear eigenvector equation l = l l . From the eigenvectors we construct a modal matrix. The eigenvectors are ordered in decreasing eigenvalue order to form the columns of the modal matrix = (1 |2 | . . . |N 2 ). The linear deformation model allows the components of the adjacency matrix long-vectors to undergo displacement in the directions of the eigenvectors of the covariance matrix. For the longvector of the graph Gk , the displaced vector is given by v˜k = u + bk∗ .
(9)
The degree of displacement for the different vector components is controlled by the vector of parameters bk . The linear deformation model may be fitted to data. This is done by searching for the least squares parameter vector. Suppose that the model is to be fitted to the graph with standardised adjacency matrix vk . The least-squares parameter vector which satisfies the condition bk∗
= arg min (vk − u − b) (vk − u − b) T
(10)
[T ]−1 T {vk − u}.
(11)
b
and the solution is bk∗ =
1 2
The graphs can be projected onto the principal component axes of the covariance matrix. If the matrix formed with ˆ = (1 | . . . |m ), the leading m eigenvectors as columns is then the centred projection of the long-vector for the
1191
ˆ T vk . For each pair of graphs graph indexed k is Zk = 2 = Gk and Gl , we compute the Euclidean distance Dk,l T (Zk − Zl ) (Zk − Zl ).
4. Experiments Our experimental investigation is divided into two parts. We commence by illustrating the ability of the statistical model to capture variations in graph-structure for synthetic data where the correspondences are known. We then show how the method can used to deal with real-world data-sets where the correspondences are unknown. 4.1. Synthetic data In this subsection we investigate the effectiveness of the model for model variations in graph-structure for synthetic data. In this example the correspondences between nodes are known and the graphs have the same number of nodes, but a different edge structure. The set of synthetic images used in this study are shown in Fig. 1. This is a set of perspective views of a house as it rotates. The associated graphs are shown in Fig. 2. It is important to note that although the number of feature-points in this sequence remains the same, there are significant structural differences in the graphs in the different views.
Fig. 1. Model sequence with feature points.
1192
B. Luo et al. / Pattern Recognition 39 (2006) 1188 – 1198
Third eigenvector
1 4 17 3 2 2 3 1 4 16 0 6 5 15 -1 87 14 -2 9 4 13 11 12 10 -7 Se 2 con 0 -8 -7.5 d e -2 -8.5 ige -9 r nve -4 -6 -10 -9.5 vecto eigen cto First r
Distance
Fig. 2. Graph representation of the model sequence.
8 6 4 2 0 16 14 Gr 12 ap h I 10 8 nd 6 ex 4 2 2
2
4
6
8
10
ph Gra
Fig. 3. Left eigen-space analysis of unweighted adjacency matrix, right distance matrix.
Fig. 4. Face sequence.
12
14 x1
Inde
16
B. Luo et al. / Pattern Recognition 39 (2006) 1188 – 1198
Fig. 5. Graph representation of the face sequence.
Fig. 6. Three modes of graph set BioID.
1193
1194
B. Luo et al. / Pattern Recognition 39 (2006) 1188 – 1198 Fitted graph
Test graph
Mean graph 120
120
120
140
140
140
160
160
160
180
180
180
200
200
200
220
220
220
240
240 140
160
180
200
240 140
220
160
180
200
220
140
160
180
200
220
Fig. 7. Mean, test and fitted graphs.
14
0.6
6 26 5 4 9 1 21
Third eigenvector
0.4 10
0.2
17 2
16
8
0
13 7 18 24
23 12 15 25
-0.2
3
-0.4
11
-0.6
0.5 Se
20
22
19
27 0.5
0
con
de
0
ige
nve -0.5 cto r
-0.5 -1
-1
First
eigen
vecto
r
Fig. 8. Graph embedding.
In Fig. 3 we show the eigen-space projection of the graphs. The trajectory is well behaved and does not exhibit kinks, or fold back on itself. In addition, the points corresponding to neighbouring views are always closer to one-another than views that are not adjacent. This feature is underlined by the inter-point distance function which is shown in the right panel of the figure. 4.2. Real world experiments We now turn our attention to real-world data with unknown correspondences. Our first example is furnished by images in the BioID face databases [22]. For a subset of 27 images from the data-bases we have extracted 20 feature points. The graphs used in out studies are again the Delaunay triangulations of the feature points. In Fig. 4 we show a sample of the face images used, and the corresponding graphs generated are shown in Fig. 5. Due to changes in expression, the relationships between the feature points changes and hence they give rise to different graph structures.
Once, the matrix is estimated, then we can generate new graph-instances by selecting a parameter vector b. The corresponding long-vector adjacency matrix is v = u + b. The long-vector v can be folded to give the adjacency matrix A. The parameter vector b can be sampled from a prior distribution. However, here we simply vary its components by hand. By setting the components corresponding to all but one of the covariance matrix vectors to zero, we can systematically explore the deformation modes of the learned structural model. The rows in Fig. 6 shows the variations modes of along the three eigenvector directions corresponding the largest three eigenvalues. The different panels in the rows are obtained by varying the relevant component of b. There are clear differences in the structures captured by the different eigen-modes. The first mode captures separately the complex edge-structure around the eyes and the lips. The second eigen-mode, represents the edge-connections between the eyes and the lips. The third eigen-mode introduces left-right asymmetries. Next we investigate the fitting of the model to data. The left panel of Fig. 7 shows the mean graph of the BioID graph data-set. The middle panel is the test graph which is the 10th of the subset of BioID face images (and was not used in training). The right panel shows the least squares fit of the model to the test graph. Bold lines are used to show the differences of the mean graph with the mode graphs. The model fits the test graph well. Finally, we show the projection of the face-graphs onto the space spanned by the three leading eigenvectors of the adjacency matrix in Fig. 8. Two clear clusters emerge. Examination of the raw data shows that these correspond to the differences between neutral and exaggerated expressions. Our second real-world example is provided by 2D views of 3D objects. The objects studied are three different toy houses. For each object we have collected sequences of views under slowly varying changes in viewer angle. From each image in each view sequence, we extract corner features. We use the extracted corner points to construct Delaunay graphs. In our experiments we use three different
B. Luo et al. / Pattern Recognition 39 (2006) 1188 – 1198
1195
Fig. 9. Image sequences and corresponding graphs.
sequences. Each sequence contains images with equally spaced viewing directions. For each sequence we show a sample of five images. Fig. 9 shows example images from the three sequences and the associated graphs. The number of feature points varies significantly from image to image, for the first (CMU) sequence there are about 30 points, for the second (MOVI) sequence there are about 140 and for the final (Chalet) sequence there are about 100. There are a number of “events” in the sequences. For instance
in the MOVI sequence, the right-hand gable wall disappears after the 12th frame, and the left-hand gable wall appears after the 17th frame. Several of the background objects also disappear and reappear. In the Swiss chalet sequence, the front face of the house disappears after the 15th frame. We commence by providing some examples to illustrate the behaviour of the learning process on the CMU house. The top-left panel of Fig. 10 shows the reference graph from the
1196
B. Luo et al. / Pattern Recognition 39 (2006) 1188 – 1198
Fig. 10. Modal graph, test graph, mean graph and fitted graph for the CMU sequence.
sequence. The top-right panel shows the test graph which is generated from the eighth image in the sequence. The mean graph is displayed in bottom-left panel. This graph is generated using the method described in Section 4. The model fitting result is shown in the bottom-right panel. In the lower two panels of the figure, the darkness of the edges is proportional to the magnitude of the corresponding element of the adjacency matrix. In the case of the mean-graph, this quantity is proportional to the number of times the corresponding edge appears in the training data. It is interesting to note the similarities and differences in the structure of the mean and modal graphs. In the first instance, all the strong edges in the mean graph are common with the modal graph. Second, in the mean graph edges are pruned away from the high degree nodes. Hence, the learning process would appear to locate common salient structure, but remove ephemeral detail. We have applied our method to each house data-set in turn. Here the aim is to determine whether the our method can identify the finer view structure of the different objects.
At this point it is important to note that we have used Delaunay graphs to abstract the corner features. Our view-based analysis explores how the edge-structure of the graphs changes with varying object pose. Since the Delaunay graph is the neighbourhood graph of the Voronoi tessellation, i.e. the locus of the median line between adjacent points, it may be expected to reflect changes in the shape of the arrangement of corner features. We aim to determine whether the different views are organised into a well structured view trajectory, in which subsequent views are adjacent to one-another. In Fig. 11 the top row shows the results obtained for the CMU sequence, the second row those for the MOVI house and the bottom row shows those for the chalet sequence. In the left-hand column of each figure, we show the matrix of pairwise L2 distances between the projected vectors i.e. 2 . Here the entries are ordered according to view number Dk,l in the sequence. The second column shows the eigen-space found by projecting the fitted long-vectors onto the basis
B. Luo et al. / Pattern Recognition 39 (2006) 1188 – 1198
1197
10 9 0.05
8
6 5 4
8
10
Third eigenvector
Graph Index2
7
1 9
0
7 5 4 3
-0.05 2
3 0.1
2
Se
2
3
4
6 5 Graph Index1
7
8
9
ec
10
0.05 0 tor -0.05 nvec e ig e First
0
nv
1
0.1
6
co 0.05 nd eig e
1
-0.05
tor
-0.1
-0.15
40 27 24 x 10
26
2.5
20 15
1.5
1417 13
1
1 0.5 0
12
-0.5
3
2 2
10
15
25 20 Graph Index1
35
30
38 36 39 31 34 35
18 20 19
4
-2
5
30 29
0.3 0.2 0.1
37
-1.5
10
11
8
6
-1
40
21
15 16
7
4
5 6
40
33 32
9 10 8
10 12 14 First eigenve ctor
16
18
20
0 -0.1 -0.2 -0.3
Se co nd
25
5
23 22 28 25
2 Third eigenvector
Graph Index2
30
ei ge nv ec to r
35
x 10 56 59 57 4960 52 54 53 58
1.5
70 1
Third eigenvector
50 Graph Index2
55 41 47 37 40 48 50 51 46 61 45 44 42 62 34 39 43
0.5
60
40
0
29 30 33 38 32 36 35 31 28
2
-0.5
1
-1
4
26
-1.5 3
-2
5
30
7
-0.5
6
co
Se
nd
20
0
27 24 11 22 23 1621 12 119 7 15 14 25 13 20 10 18
68 66 64 65 72 69 7 1 70 63 67
en
eig
0.5
20
40 30 Graph Index1
50
60
70
r
10
cto
ve
10
-2
-4
-6
8 9 -10 -12 -14 -16 -8 First eigenvector
-18
-20
Fig. 11. Eigen-spaces from graphs belonging to separate sequences.
spanned by the leading three eigenvectors of the covariance matrix. As the difference in viewing position changes then so the distance increases. Also, adjacent views appear close to each other in the eigenspace.
5. Conclusions In this paper, we have presented a framework for leaning a linear model of the modes of structural variation in sets of graphs. We commence by locating correspondences between the nodes in different graphs and using the correspondence order to vectorise the adjacency matrices. From the eigen-modes of the cluster covariance matrices we
construct a linear model of the modes of structural variation in the graphs. There are a number of ways in which we intend to develop this work. First, the experiments have been presented on relatively simple data-sets, and it still needs to be demonstrate how the method performs when there is large variance in the training data and significant structural error on the test data. Second, we aim to integrate correspondence and clustering steps into a single process to develop a mixture model over different object categories. Here there will be both cluster indicators and correspondence indicators, and the maximisation step will involve optimisation over these indicators. Third, we aim to use the cluster covariance matrices to construct piecewise subspace models for the graphs.
1198
B. Luo et al. / Pattern Recognition 39 (2006) 1188 – 1198
References [1] K. Siddiqi, A. Shokoufandeh, S.J. Dickinson, S.W. Zucker, Shock graphs and shape matching, IJCV 35 (1) (1999) 13–32 17(8) (1995) 749–764. [2] A.K. C Wong, J. Constant, M.L. You, Random Graphs Syntactic and Structural Pattern Recognition, World Scientific, Singapore, 1990. [3] A.D. Bagdanov, M. Worring, First order Gaussian graphs for efficient structure classification, Pattern Recognition 36 (2003) 1311–1324. [4] W.J. Christmas, J. Kittler, M. Petrou, Structural matching in computer vision using probabilistic relaxation, IEEE Trans. Pattern Anal. Mach. Intell. 17 (8) (1995) 749–764. [5] R.C. Wilson, E.R. Hancock, Structural matching by discrete relaxation, IEEE Trans. Pattern Anal. Mach. Intell. 19 (6) (1997) 634–648. [6] D. Heckerman, D. Geiger, D.M. Chickering, Learning Bayesian networks: the combination of knowledge and statistical data, Mach. Learn. 20 (1995) 197–243. [7] M.A. Lozano, F. Escolano, ACM attributed graph clustering for learning classes of images, in: Graph Based Representations in Pattern Recognition, Lecture Notes in Computer Science, vol. 2726, 2003, pp. 247–258. [8] H. Bunke, et al., Graph clustering using the weighted minimum common supergraph, in: Graph Based Representations in Pattern Recognition, Lecture Notes in Computer Science, vol. 2726, 2003, pp. 235–246. [10] B.J. Jain, F. Wysotzki, Central clustering of attributed graphs, Machine Learn. 56 (2004) 169–207. [11] H. Murase, S.K. Nayar, Illumination planning for object recognition using parametric eigenspaces, IEEE Trans. Pattern Anal. Mach. Intell. 16 (12) (1994) 1219–1227. [12] E. Klassen, A. Srivastava, W. Mio, S.H. Joshi, Analysis of planar shapes using geodesic paths on shape spaces, IEEE Trans. Pattern Anal. Mach. Intell. 26 (2004) 372–383.
[13] H. Le, C.G. Small, Multidimensional scaling of simplex shapes, Pattern Recognition 32 (1999) 1601–1613. [14] T.F. Cootes, C.J. Taylor, D.H. Cooper, J. Graham, Active shape models—their training and application, Comput. Vision Image Understanding 61 (1) (1995) 38–59. [15] S. Roweis, L. Saul, Nonlinear dimensionality reduction by locally linear embedding, Science 290 (5500) (2000) 2323–2326. [16] J.B. Tenenbaum, V.D. Silva, J.C. Langford, A global geometric framework for non-linear dimensionality reduction, Science 290 (2000) 586–591. [17] M. Belkin, P. Niyogi, Laplacian eigenmaps for dimensionality reduction and data representation, Neural Comput. 15 (2003) 1373– 1396. [18] X. He, P. Niyogi, “Locality preserving projections”, NIPS03. [19] B. Luo, R.C. Wilson, E.R. Hancock, Spectral embedding of graphs, Pattern Recognition 36 (2003) 2213–2230. [20] B. Luo, E.R. Hancock, Structural graph matching using the em algorithm and singular value decomposition, IEEE Pattern Anal. Mach. Intell. 23 (10) (2001) 1120–1136. [21] S. Umeyama, An eigen decomposition approach to weighted graph matching problems, IEEE Trans. Pattern Anal. Mach. Intell. 10 (1988) 695–703. [22] R. Frischholz, U. Dieckmann, BioID: a multimodal biometric identification system, IEEE Comput. 33 (2) (2000) 64–68.
Further reading [9] H. Bunke, K. Shearer, A graph distance metric based on the maximal common subgraph, Pattern Recognition Lett. 19 (1998) 255–259.
About the Author—BIN LUO received his B.Eng. degree in electronics and M.Eng. degree in Computer Science from Anhui university of China in 1984 and 1991, respectively. From 1996 to 1997, he was working as a British Council visiting scholar at the University of York under the Sino-British Friendship Scholarship Scheme (SBFSS). In 2002, he was awarded the Ph.D. degree in Computer Science from the University of York, the United Kingdom. He is at present a professor at Anhui University of China. He has published some 60 papers in journals, edited books and refereed conferences. His current research interests include graph spectral analysis, large image database retrieval, image and graph matching, statistical pattern recognition and image feature extraction. About the Author—RICHARD WILSON received the B.A. degree in Physics from the University of Oxford in 1992. In 1996, he completed a Ph.D. degree at the University of York in the area of pattern recognition. From 1996 to 1998, he worked as a Research Associate at the University of York. After a period of postdoctoral research, he was awarded an Advanced Research Fellowship in 1998. In 2003, he took up a lecturing post and he is now a Reader in the Department of Computer Science at the University of York. He has published some 100 papers in journals, edited books and refereed conferences. He received an outstanding paper award in the 1997 Pattern Recognition Society awards and has won the best paper prize in ACCV 2002. He is currently an Associate Editor of the journal Pattern Recognition. His research interests are in statistical and structural pattern recognition, graph methods for computer vision, high-level vision and scene understanding. He is a member of the IEEE computer society. About the Author—EDWIN HANCOCK studied physics as an undergraduate at the University of Durham and graduated with honours in 1977. He remained at Durham to complete a Ph.D. in the area of high-energy physics in 1981. Following this, he worked for 10 years as a researcher in the fields of high-energy nuclear physics and pattern recognition at the Rutherford-Appleton Laboratory (now the Central Research Laboratory of the Research Councils). In 1991, he moved to the University of York as a lecturer in the Department of Computer Science. He was promoted to Senior Lecturer in 1997 and to Reader in 1998. In 1998, he was appointed to a Chair in Computer Vision. Professor Hancock now leads a group of some 15 faculty, research staff and Ph.D. students working in the areas of computer vision and pattern recognition. His main research interests are in the use of optimisation and probabilistic methods for high and intermediate level vision. He is also interested in the methodology of structural and statistical pattern recognition. He is currently working on graph-matching, shape-from-X, image data bases and statistical learning theory. Professor Hancock has published some 100 journal papers and 400 refereed conference publications. He was awarded the Pattern Recognition Society medal in 1991 and an outstanding paper award in 1997 by the journal Pattern Recognition. In 1998, he became a fellow of the International Association for Pattern Recognition. Professor Hancock has been a member of the Editorial Boards of the journals IEEE Transactions on Pattern Analysis and Machine Intelligence, and Pattern Recognition. He has also been a guest editor for special editions of the journals Image and Vision Computing and Pattern Recognition.
