**I.L. Dryden, Department of Statistics, University of Leeds, Leeds, LS2 9JT, UK.**

** **

- Introduction and Opening Address
- Session I: Procrustes and Mathematical Statistics Issues
- Session II: Overview and Shape Geometry
- Session III: Image Analysis and Computer Vision
- Session II: Overview and Shape Geometry
- Session III: Image Analysis and Computer Vision
- Session IV: Morphometrics and Shape Geometry
- References

Statistical shape analysis is concerned with all aspects of analyzing objects where location, orientation and possibly scale information can be ignored. The subject has flourished since the pioneering work of D.G. Kendall (1984) and F.L. Bookstein (1986), both of whom we are privileged to have speaking at the conference. Interest in the area is widespread, as can be seen from this collection of papers - from theoretical issues in mathematical statistics, probability and differential geometry to practical issues in computer vision, medical imaging and biology. The papers have been collected in the order of presentation at the conference, with the poster abstracts at the end. Taken as a whole the papers contain a comprehensive survey of the theory and application of the current state of the art in shape analysis. As well as the presentation of some new results, many important discussion points and directions for further work are included.

The opening address of the conference is given by D.G. Kendall, whose seminal paper in 1984 laid the mathematical foundations for the subject. His shape space and its differential geometric properties (including the Procrustes distance) provide the environment for the work in most of the proceeding papers.

Procrustes analysis has proved to be a popular method of shape analysis. The generalized Procrustes procedure was developed by Gower (1975) and has been adapted for shape analysis by Goodall (1991). These two speakers open our first session and offer new thoughts on shape analysis. J.C. Gower considers shape analysis based on Euclidean Distance Matrices and offers geometrical insights into this multidimensional scaling type of approach. C.R. Goodall considers a review of Procrustes methods in shape, including some new developments and updates since his 1991 read paper, particularly with reference to consistency, likelihood and estimation of covariance parameters. Further, he proceeds to describe Euclidean Shape Tensor Analysis - an approach to shape analysis based on subsets of landmarks. A special case of this procedure is Euclidean Distance Matrix Analysis (EDMA).

To conclude the session further mathematical statistical issues are considered. H. Le considers a general notion of mean shape - the Karcher mean shape - and obtains such a mean under Bookstein's (1986) independent isotropic Gaussian model. I.L. Dryden, M.R. Faghihi and C.C. Taylor consider the analysis of subsets of points in a spatial tessellation. The approximate distribution of a Procrustes statistic is found under the isotropic Gaussian model, which is then used for investigating regularity in human muscle fibre cross sections, following the work of Mardia et al. (1977). Finally, S. Lele and T. Cole summarize the EDMA approach to shape analysis. They propose a new bootstrap based test for shape differences and investigate the significance level and power in a simulation study.

The second session opens with an overview of statistical shape analysis by K.V. Mardia. Summaries of the key developments for statistical inference are given, including discussion of distributional results, tangent plane approximations, principal components analysis, kriging with derivatives and image analysis issues. In particular, marginal isotropic normal shape distributions are discussed including the work of Mardia and Dryden (1989) and Dryden and Mardia (1991). A demonstration is given that isotropic landmark distributions lead to isotropic shape distributions in a suitable Procrustes tangent space. Bookstein's very recent work linking Procrustes analysis and bending energy analysis has been summarized.

Further mathematical issues are tackled in the rest of the session. B. Bhavnagri provides some calculations for distances in shape space to some special degenerate shapes. A Markov process is defined and the idea is to use the results to classify objects as simple shapes in a vision application. I.S. Molchanov considers shape analysis of more abstract sets than the usual finite equal number of landmarks on each object. The results could be applied to the shape analysis of random sets or to analysis with different numbers of landmarks on each object. C.G. Small and M.E. Lewis also consider the comparison of objects where identifiable landmarks are not available. They compute automatic homologies between objects with application to the comparison of Roman brooches. They also investigate new metrics based on Frobenius norms, generalizing to higher dimensions the planar shape space of negative curvature of Bookstein (1986).

The third session has an emphasis strongly on image analysis and computer vision. P.J. Green's address concerns the topic of reversible jump Markov Chain Monte Carlo (MCMC) methodology. Statistical inference may often be carried out when the dimension of the parameter space is unknown, for example in an image containing an unknown number of objects. MCMC has proved to be enormously popular and useful for a wide range of difficult inference problems and its adaptation to the unknown parameter dimension case is made possible through this work. Hence, practical problems such as object recognition in Bayesian image analysis can be addressed, where shape parameters in deformable templates often play an important role.

J.A. Marchant, C.M. Onyango and R.D. Tillet discuss their application and refinement of Kass et al.'s (1988) snakes for locating objects in images. The snake uses a physical analogy for finding boundaries and the use of compartments allows a larger range of possible shapes to be generated from the model. The methodology is applied to the automatic location of pigs in images.

T.F. Cootes and C.J. Taylor describe their highly successful approach to object recognition using Active Shape Models (ASM). Their principal components approach to decomposing shape variability in a Procrustes tangent space has led to a very effective way of summarizing the important aspects of shape variability. Recent advances are presented to improve the practical implementation of the procedure for object recognition in images, including the use of a multi-resolution search and directional weighting.

The last three papers in the session are concerned with important computer vision tasks - edge recognition, tracking objects and depth recognition. A. Baumberg and D.C. Hogg highlight an alternative to ASM, the physically based Finite Element Method of Pentland and co-workers. The model is then developed for low-dimensional descriptions of moving and deforming objects. J.D. Andersen and K. Hansen use angular information to locate the principal directions of edges in an image and M. Nielsen considers the estimation of depth information from stereo pairs of images.

The fourth session begins with the keynote address by F.L. Bookstein. A summary is given of his vision of shape analysis as the synthesis of multivariate analysis and deformations for geometrical visualization. The use of thin-plate splines for deformations (Bookstein, 1989) has proved popular and Bookstein's orthogonal decompositions of the deformation - relative warps - provide summaries of the key aspects of shape variation in a sample at large and small scales. The connection with principal components analysis in the Procrustes tangent plane is made.

F.J. Rohlf demonstrates the use of relative warps in the Procrustes tangent space. Standard multivariate analyses can be used on the important warp scores. A simulation study is carried out to demonstrate that significance levels of an independent two sample test are correct under a variety of covariance structures for Gaussian data.

P.D. Sampson considers an application in medical imaging studying the shape changes in the left ventricle over the cardiac cycle. Outlines are available and are registered in a Procrustes type of superimposition. An orthogonal decomposition of a matrix of normal deviations from the mean ventricle is used to summarize the principal modes of shape variation.

W.D.K. Green concludes the session with an alternative construction of Kendall's shape space for triangles. In particular an alternative proof of Kendall's result that is isometric to is given and insight given into certain distributional results for triangle shapes.

The concluding address of the conference is given by J.T. Kent. A review is given of the current statistical approaches to shape analysis and discussion of outstanding issues and pointers for future development are considered. The elegant results from the complex Bingham distribution (Kent, 1994) have provided great insight into many approaches to shape analysis for planar data including Procrustes methods. The topics of consistency, tangent spaces, different modelling strategies and the use of derivative information are considered in detail. It should be noted that Kent emphasises the similarity of the estimation approaches discussed throughout this volume for small variations.

Finally, abstracts from the poster session are included in this collection. Topics include mathematical characterizations of certain shape spaces, classification of shape using neural nets, Fourier series representations of outlines, superimposition of star shaped sets, Euclidean shape tensor analysis, ridge curves as well as further discussion of the subjects in the papers of this volume. The large variety of applications of shape analysis is emphasized in the posters - a diverse collection of examples are given from biology, medical imaging, palaeontology, agriculture and engineering.

**Bookstein, 1986**-
Bookstein, F. L. (1986).
Size and shape spaces for landmark data in two dimensions (with
discussion).
*Statist. Sci.*, 1:181-242. **Bookstein, 1989**-
Bookstein, F. L. (1989).
Principal warps: Thin-plate splines and the decomposition of
deformations.
*IEEE Trans. Pattern Anal. Machine Intell.*, 11:567-585. **Dryden and Mardia, 1991**-
Dryden, I. L. and Mardia, K. V. (1991).
General shape distributions in a plane.
*Adv. Appl. Prob.*, 23:259-276. **Goodall, 1991**-
Goodall, C. R. (1991).
Procrustes methods in the statistical analysis of shape (with
discussion).
*Journal of the Royal Statistical Society B*, 53:285-339. **Gower, 1975**-
Gower, J. C. (1975).
Generalized Procrustes analysis.
*Psychometrika*, 40:33-50. **Kass et al., 1988**-
Kass, M., Witkin, A., and Terzopoulos, D. (1988).
Snakes: Active contour models.
*Int. Jour. Comp. Vision*, 1:321-331. **Kendall, 1984**-
Kendall, D. G. (1984).
Shape manifolds, Procrustean metrics and complex projective spaces.
*Bull. Lond. Math. Soc.*, 16:81-121. **Kent, 1994**-
Kent, J. T. (1994).
The complex Bingham distribution and shape analysis.
*Journal of the Royal Statistical Society B*, 56:285-299. **Mardia and Dryden, 1989**-
Mardia, K. V. and Dryden, I. L. (1989).
Shape distributions for landmark data.
*Advances in Applied Probability*, 21:742-755. **Mardia et al., 1977**-
Mardia, K. V., Edwards, R., and Puri, M. L. (1977).
Analysis of Central Place Theory.
*Bulletin of the International Statistical Institute*, 47:93-110.