Point-Set Alignment Using Multidimensional Scaling - Semantic Scholar

Point-set Alignment using Multidimensional Scaling Marco Carcassoni and Edwin R. Hancock Department of Computer Science University of York York, Y01 5DD, UK email: marco,[email protected]

Abstract In this paper, we show how to perform point-set alignment by applying multidimensional scaling to the interpoint distance matrix. The idea is that alignment can be effected by transforming different point-sets into a common embedding space, and correspondences located on a nearest-neighbour basis. The method offers the advantage over conventional Procrustes analysis that it extends the range of rotational angles over which it is effective. Moreover, it does not require separate, and explicit, centering, scaling and rotation steps. It also proves robust under severe levels of point-set noise and corruption.

1 Introduction The task of aligning of point-patterns is one which arises in many high-level problems in computer vision. Examples include stereopsis, motion analysis and perspective reconstruction. Generally speaking, there are two approaches. In the case of rigid alignment, the points are constrained to deform under a similarity, affine or perspective transformation [10]. Non-rigid alignment, on the other hand, permits the points to undergo more complex transformations. Examples here include spline warps [2] and point distribution models [5]. In this paper we are interested rigid point set alignment. Here one of the most elegant techniques is Procrustes analysis [6]. This is a three-step process. It commences by co-centering the point-sets so that their centroids coincide. Next, the point-sets are scaled so that their variances are identical. Finally, singular value decomposition is performed on the point correlation matrix for the translated and scaled point-sets. The singular vectors are used to define the final rotation that brings the point-sets into alignment. There have been several extensions of this basic idea. For instance, Kendal [8] has shown how to generalise the method for affine and more complex transforms.

Luo and Hancock [12], have developed an iterative method which uses the apparatus of the EM algorithm to overcome problems associated with different size of the point-sets. Although the method does not accommodate perspective transformations or non-rigid deformations, it is a powerful tool since it can provide an initial alignment which can then be refined using more complex analysis. Unfortunately, one of the drawbacks of the method is its poor performance for large rotation angles. In this paper we aim to provide a new alignment method which offers a number of advantages over the Procrustes method. Our approach is as follows. Rather than working with the point correlation matrix, we use a proximity representation for the individual point-sets. This is obtained by computing the distance between points. We then apply multidimensional scaling to the proximity data. This transforms the points into a new space. By comparing the transformed position vectors we locate correspondences. The advantages over Procrustes analysis are that we do not need separate centering, scaling and rotation steps. These are handled implicitly by multidimensional scaling. Moreover, the method is not sensitive to the rotations of the point-sets.

2 Procrustes Analysis Our goal is to align two sets of feature-points and . For convenience, we refer to these respectively as the data and the model, however this does not imply a precedence. We represent each point in the image data set by a position where is the point index. We will vector assume that all these points lie on a single plane in the image. In the interests of brevity we will denote the entire set where is the point of image points by index-set. The points constituting the model are similarly where denotes the represented by index-set for the model feature-points . To perform Procrustes analysis, we construct two coordinate matrices from the point position vectors. The data-points are represented by the following matrix whose














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The matrices and define a rotation matrix which aligns the principal component directions of the point-sets and . The rotation matrix is equal to 





Multidimensional scaling(MDS)[4] is a procedure which allows data specified in terms of a matrix of pairwise distances to be embedded in a Euclidean space. The classical multidimensional scaling method was proposed by Torgenson [13] and Gower[7]. Shepard and Kruskal developed a different scaling technique called ordinal scaling[9]. Our aim in this paper is to use the method to align pointsets. The idea is simple one. For each point-set we compute a between-point distance matrix. From the distances we compute a similarity matrix. By performing multidimensional scaling, we transform the point-positions into a standardised co-ordinate frame. In this transformed co-ordinate system, the point-sets are aligned and correspondences can be computed using a nearest neighbour rule. To commence we require pairwise distances between pairs of points. We do this by computing the Euclidean distance. For the points and belonging the set of data-points the distance is (10) &



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With the rotation matrix to hand we can find the Procrustes alignment which maximises the correlation of the two point sets. The procedure is to first bring the centroids of the two point-sets into correspondence. Next the data points are scaled so that they have the same variance as those of the model. Finally, the scaled and translated point-sets are rotated so that their correlation is maximised. To be more formal the centroids of the two point-sets are





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is the average similarity value over all rows and columns of . For the model-points the analothe similarity matrix . gous matrix is We subject the matrices and to an eigenvector analysis to obtain two matrices of normalised co-ordinates and . If the rank of is , then we non-zero eigenvalues. We arrange these will have non-zero eigenvalues in descending order, i.e. . The corresponding ordered eigenvectors are denoted by where is the th eigenvalue. The embedding co-ordinate system for the the different point-sets where are the is scaled eigenvectors. For the point indexed , the embedded 

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vector of co-ordinates is This procedure is repeated to find embeeded co-ordinates for the model points. For the point indexed the co-ordinate vector is denoted by . To align the point-sets, we find closest point correspondences between the transformed positions of the two sets of points. Suppose that is a mtrix of correspondence indiof this matrix take on the value cators. The element unity if the points and are closest neighbour correspondences, and are zero otherwise. In other words Q

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4 Experiments In this section we describe our experimental evaluation of multidimensional scaling for re-aligning and normalizing distorted point-sets. To do this we conduct experiments with synthetic data. We average our results over 50 tests. In Figure 1 we illustrate the embeddings returned by the method. The left-hand panel in the figure shows two pointsets which have been translated, scaled by a factor of [1,3] and rotated by an angle of 90 degrees with respect to one another. In the right-hand panel we show the embeddings produced by multidimensional scaling. The two transformed point-sets are in perfect alignment. Figures 2, 3 and 4 show the results obtained when we add controlled jitter, contamination and scaling to the point-sets under study. In Figure 2 we study the effect of adding Gaussian position errors. Here we have randomly sampled the position errors from a Gaussian distribution of zero mean and specified standard deviation. The figure shows the average distance between corresponding transformed points as a function of the standard deviation of the position errors. In Figure 3 we turn our attention to the effect of adding clutter points. Here we plot the average distance between corresponding transformed points as a function of the fraction of added clutter. In Figure 4, we extend the study to scaling. Here we plot the average transformed corresponding point-distance as a function of scale-factor. The different curves are for different position error standard deviation. The curves are constant with scale-factor. This shows that the average difference scales with the jitter and not the scale-factor. In figure 5 we plot the average distance as function of rotation angle between the point-sets. The upper curve (i.e. the one with larger error) is obtained by applying Procrustes alignement. The lower curve (i.e. the one with smaller error) is obtained with multidimensional scaling. On the case of Procrustes alignment, the error curve is relatively flat. The other model, on the other hand, returns good reand . sults.The performance degrades between ;



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Figure 2. Alignment error (adding Gaussian noise)

5 Conclusions In this paper we have described a new point-set alignment method. To perform alignment we transform the point-sets into a new co-ordinate space by applying multidimensional scaling to the matrix of inter-point distances. This avoids the need to perform separate and explicit centering, scaling and rotation steps, which is the case in Procrustes alignment. We locate correspondences using closest neighbours in the transformed point co-ordinate space.

References [1] F. Attneave, ”Dimension of Similarity”, American Journal of Psychology, 63, 516-556, 1950.

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Figure 3. Alignment error (adding clutter noise)

Figure 4. Alignment error (scaling and adding noise)

[2] F.L. Bookstein, Morphometric Tools for Landmark Data, Cambridge, Cambridge University Press, 1991. [3] M.Carcassoni and E.R.Hancock, ”Point Pattern Matching with Robust Spectral Correspondence”, IEEE Computer Society Conference on Computer Vision and Pattern Recognition, IEEE Computer Society Press, pp. 649-655, 2000. [4] C. Chatfield and A.J.Collins, Introduction to Multivariate Analysis Chapman & Hall, 1980. [5] T.F Cootes and C.J. Taylor and D.H. Cooper and J. Graham, ”Active Shape Models - Their Training and Application”, CVIU, 61(1995), 38–59. [6] I.L. Dryden and K. V. Mardia, Statistical Shape Analysis, John Wiley & Son Ltd, 1998. [7] Some distance Properties of Latent Root and Vector Methods used in Multivariate Analysis”, Biometrika 53, pp.325-328, 1966. [8] D.G. Kendall, ”Shape manifolds,Procrustean metrics, and complex projective spaces”, Bulletin of the London Mathematical Society, 16, 81-121, 1984. [9] J.B. Kruskal, ”Non-metric Multidimensional Scaling: a Numerical Method”, Psychometrika, 29, pp.115129, 1964. [10] C.P. Lu, G.D. Hager and E. Mjoslness, ”Fast and Globally Convergent Pose Estimation from Video Images”, PAMI, 22:(6), pp. 610-622, Jun 2000.

Figure 5. Comparison between Procrustes and MDS methods

[11] B. Luo, R. Wilson and E.R.Hancock, ”Eigenspaces for graphs”, Proceedings of the 5th Asian Conference on Computer Vision, Australia, 23-25 Jan 2002. [12] B. Luo and E.R. Hancock, ”Matching Point-sets using Procrustes Alignmentand the EMAlgorithm”, Proceedings of the 10th British Machine Vision Conference, T. Pridmore and D. Elliman eds. ,British Machine Vision Association, Nottingham, pp. 43-52 , 1999. [13] W.S. Torgerson, Theory and methods of scaling, Wiley, New York, 1958. [14] G. Young and A.S. Householder, Discussion of a set of points in terms of their mutual distances”, Psychometrika, 3, ”19-22, 1938.

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