Current Issues in Statistical Shape Analysis

The Leeds Conference "Current Issues in Statistical Shape Analysis" from April 5-7, 1995 was a great success.

Held in Fairbairn house - a complete facility for housing, conference, breakfast and lunch. One didn't have to stir far to be comfortably involved in the conference. The Leeds staff and especially Kanti Mardia and C. A. Gill made it a wonderful exciting experience for all fortunate enough to attend. In his welcoming talk Kanti set the tone of the entire 3 days by invoking the Jain idea of ANEKANTVAD - freely translated as "Striving for Balance". That was accomplished as much as at any conference I have attended where there was a mix of professions, viewpoints, and attitudes. Among the 80+ attendees were 14 each from the USA, Canada, and non UK Europe, a sizable contingent from the UK and a few others from as far away as Australia.

The Opening Address "Looking at Geodesics" by Professor David Kendall was on the 5th of April. Then followed sessions on Procrustes, Shape Geometry, Image Analysis and Computer Vision. The complete program was given earlier on MORPHMET and is in the archive there. A panel discussion was the last event of the conference in which several issues raised at the conference were discussed. Two poster sessions gave an opportunity for additional presentations, and especially applications of morphometrics to a variety of data sets in biology, anthropology, and paleontology. The Proceedings were available to all registered, and can be obtained as detailed at the end of this review. I borrowed heavily from the presentations and printed papers of Kanti Mardia, John Kent and especially that of Ian Dryden as they presented fine previews, summaries, and a review of accomplishments. D. G. Kendall's opening address gave some insight to looking at projections of shape space for 4 landmarks in 3 space - all tetrahedra. This shape space has 5 dimensions.

Session I. Procrustes and Mathematical Statistics Issues

Gower offered some new thoughts on distances presented in Euclidean Distance Matrices for landmarks. Goodall updated his 1991 paper with additional points on consistency, likelihood and estimation of covariance parameters. He described Euclidean Shape Tensor Analysis (ESTA)- a method based on subsets of landmarks - Euclidean Distance Matrix Analysis is a special case. A poster presentation of ESTA provided additional material and an example using data on Apert's Syndrome in humans. Le considered a special form of mean shape. Dryden and co-workers considered a case of a series of triangles sharing points over a set of objects in 2D. Some new results were obtained and applied to regularity in human muscle fiber cross-sections. Lele and Cole summarized EDMA and proposed a new version based on differences (rather than ratios) of forms. They studied the power of a new test using simulations.

Session II. Overview and Shape Geometry

Kanti Mardia gave one of his sterling rapid fire overviews of statistical shape analysis - including developments, distributions, tangent plane approximations, principal component analysis, Kriging with derivatives and also touched on image analysis. Bhavnagi defined a Markov process to use the results to classify objects as simple shapes in vision applications. Molchanov considered shape analysis of more abstract sets than the usual ones containing equal numbers of landmarks - a most intriguing development. Small and Lewis have developed a landmark free method which superimposes pixel lattices of objects. The method was applied to Iron Age broaches.

Session III. Image Analysis and Computer Vision

Markov Chain Monte Carlo methodology was discussed by Green - which can be applied to situations where the dimension of the parameter space is unknown - for example where an image contains an unknown number of objects. Practical problems such as object recognition in Bayesian image analysis can be addressed. A fun presentation was given by Marchant using "snakes" for locating objects in images. A snake is a physical analog for finding boundaries and compartments. "Sidewinders" also come into action. Applications were for locating and determining location and shape of pigs from dorsal photographs - an obvious economic application. Cootes described object recognition using Active Shape Models. This is a principal components approach to the decomposition of shape variability in a Procrustes tangent space. New refinements were given. The last three papers dealt with computer vision tasks - edge recognition, tracking objects and depth recognition.

Session IV. Morphometrics and Shape Geometry

Bookstein gave the opening talk on the synthesis of multivariate analysis and geometrical deformations. The use of relative warps was reviewed with two examples - Foraminifera landmarks, and brain scan images which can be analyzed to identify schizophrenics and normals. Rohlf summarized a simulation study that showed that two group multivariate analysis of variance using partial warp scores and multiple regression of uniformly distributed random numbers on the scores in tangent space gave correct significance levels under a variety of point configurations and covariance structure for Gaussian data. Sampson looked at shape changes in the heart left ventricle using Procrustes superimposition of points along the outline. He was able to decompose the shape variation. Finally W. D. K. Green gave an alternative construction of Kendall's shape space for triangles, with a new proof and some incites into distribution results for triangle shapes. Kent gave the concluding paper - summarizing current statistical approaches to shape analysis, with outstanding issues, and pointers toward further development - focusing on landmark-based methods. Consistency, tangent spaces and different modeling strategies were considered. This paper best summarized many of the current issues and will be discussed in greater length than the others. Three situations of landmark distributions are considered by Kent 1) the covariance matrix is such that the coordinates of all the landmarks have identical variances (isotropy - no correlations); 2) their are structured correlations between and within landmarks; 3) the most general case where the covariance matrix is unrestricted. Kent has worked out some elegant distributions for these situations, but they are not practical for applications. For "concentrated data" (occupying a small part of shape space - it has been suggested that at least in biological comparisons though shapes may look very different to the biologist, they really only occupy a small part of all possible shapes. A mammal skull is by and large the same over most mammals.), an alternative model is to consider case 3) in the tangent space to shape space. "It should be emphasized that for concentrated data any possible inconsistencies will usually be swamped by the variability in the data. Thus possible inconsistency of these methods is not usually an important statistical problem". He then went on to state the consistency properties of several of the popular methods, but with this caveat in mind. The issue of consistency therefore no longer seems so important.


The posters were divided into two sessions, and there was a 5 minute presentation for each one by the authors. I will briefly summarize the posters paralleling Session IV based on their abstracts. These were applications. Corti et al. applied relative warp analysis to a super species of mole rats and found shape differences associated with chromosome number, soil type and locality. Dean et al. continue work on deformable templates - finding space curves (ridge curves) of maximum curvature in order to tile the human skull surface patches. These lead to a deficient coordinate notion. The templates allow superimposition and precise topological reference to significant features. Kucera looked at random walks of shape coordinates in lineages of Foraminifera. Penin and Baylac have recorded 29 3D landmarks on 140 skulls of great apes. Principal components of a Procrustes fit were used for further analysis. Similar patterns of growth were observed, and a functional descriptive framework elucidated. Wood and Wood are beginning an analysis of articulating surfaces of the ankle using laser scanning in apes and man. The goal is to compare fossil hominids and quantify differing locomotor regimes. In an earlier session Renaud found it useful to study outlines of a lineage of fossil rodent teeth using Zahn and Roskies "inverse of the curvature radius" to avoid problems of standardization, and claimed interpretable phylogeny.

The high point of the meeting was supposed to be the Panel Discussion at the end - however the biological audience was somewhat frustrated. A discussion of efficiency of the methods by Bookstein and Lele considered different ideas of efficiency, and therefore the discussion wasn't! Lele presented some simulation results for one simple data set for a fixed number of landmarks; while Bookstein discussed the affect of increasing numbers of landmarks. However, much of the differences and a great deal of clarification has appeared on MORPHMET since the meeting (see the archives for April if you want to review the discussions) The meeting reflected much of the state of shape analysis, and it is clear that much more work remains to be done. I for one, would like to see a continuation of the simulations begun by Rohlf, and extended to calculations of power. The data presentations were representative of what is being done with landmark shape analysis, eg. in biology, and it is useful for statisticians to see more of the kinds of biological data applications and the problems they entail for the biologist.

Leslie F. Marcus, Professor of Biology, Queens College of CUNY. Department of Invertebrates, American Museum of Natural History, CPW at 79th, New York, NY 10024

Proceedings in CURRENT ISSUES IN STATISTICAL SHAPE ANALYSIS International Conference, held in Leeds, UK, 5-7 April 1995
Edited by K.V.Mardia, C.A.Gill Department of Statistics, University of Leeds, UK
Leeds University Press ISBN 0 85316 161 5

To obtain a copy of the Proceedings, apply to Dr C. A. Gill, Department of Statistics, University of Leeds, Leeds LS2 9JT, UK. email: Tel. 44 113 2335157 (international), 0113 2335157 (UK). Fax: 44 113 2335102 (international), 0113 2335102 (UK).

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