## Statistical shape analysis.(English)Zbl 0901.62072

Chichester: Wiley. xvii, 347 p. (1998).
Shape is all the geometrical information that remains when location, scale and rotational effects are filtered out from object. Statistical shape analysis is concerned with methodology for analysing shapes in the presence of randomness. The objects under study could be sampled at random from a population and the main aims of statistical shape analysis are to estimate population average shapes, to estimate the structure of population shape variability and to carry out inference on population quantities. This text focuses on the situation where the objects are summarized by key points called landmarks. The essential ingredients of the statistical shape analysis of landmark data are presented in this book.
Ch.1 of the book provides an introduction to the topic of shape analysis. There is a vast range of applications and the authors introduce the field to statisticians and applied researchers through important examples and data in Biology (mouse vertebrae, gorilla and macaque skulls, sooty mangabeys), Medicine (magnetic resonance brain scans of schizophrenic patients), Archaeology (alignments of standing stones), Geography (Central Place Theory), Geology (microfossils), Agriculture (fish recognition, robotic harvesting of mushrooms), Genetics (electrophoretic gels), Image Analysis (postcode recognition). Ch.2 provides some preliminary material on simple measures of size and shape, in order to familiarize the reader with the topic.
In Ch.3 the key concepts of shape distance, mean shape and shape variability for two dimensional data using Procrustes analysis are outlined. Complex arithmetics leads to neat solutions. In Ch.4 the shape space is introduced. Various distances in the shape space are described, together with some further choices of shape coordinates. Ch.5 provides further details on the Procrustes analysis of shape suitable for two and higher dimensions. Further discussion of principal components analysis for shape is also included. Ch.6 introduces some suitable distributions for shape analysis in two dimensions, notably the complex Bingham distribution, the complex Watson distribution and the various offset normal shape distributions. The offset normal distributions are referred to as “Mardia-Dryden” distributions in the literature.
Ch.7 develops some inference procedures for shape analysis, where variations are considered to be small. Three approaches are considered: tangent space methods, approximate distributions of Procrustes statistics and edge superimposition procedures. The two-sample tests for mean shape difference are particularly useful. Ch.8 discusses size-and-shape analysis – the situation where invariance is with respect to location and rotation, but not scale. The allometry which involves studying the relationship of shape and size is discussed. The geometry of the size-and-shape space is described and some size-and-shape distributions are discussed. Ch.9 involves the extension of the distributional results to higher than two dimensions, which is a more difficult situation to deal with than the planar case.
Ch.10 considers methods for describing the shape change between objects. A particularly useful tool is the thin-plate spline deformation used by F. L. Bookstein [see ‘Morphometric tools for landmark data: geometry and biology.’ (1991; Zbl 0770.92001)] in shape analysis. Pictures can be easily drawn for describing shape differences in the spirit of D’Arcy Thompson. Some of the historical developments and some recent work using derivative information and kriging are given. The method of relative warps is also described, which provides an alternative to principal components analysis emphasizing large or small scale shape variability.
Ch.11 is fundamentally different from the rest of the book. Shape plays an important part in high-level image analysis. Various prior modelling procedures in Bayesian image analysis where it is often convenient to model the similarity transformations and the shape parameters separately are discussed. Some recent work on image warping using deformations is also described. Finally, Ch.12 involves a brief description of alternative methods and issues in shape analysis, including consistency, distance-based methods, more general shape spaces, affine shape, robust methods, smoothing, unlabelled shape, probabilistic issues and landmark-free methods.
Of interest to statisticians and researchers in Biology, Medicine, Computer Science and Image Analysis.

### MSC:

 62H11 Directional data; spatial statistics 62-02 Research exposition (monographs, survey articles) pertaining to statistics 62H25 Factor analysis and principal components; correspondence analysis 92-02 Research exposition (monographs, survey articles) pertaining to biology 68T10 Pattern recognition, speech recognition 62P99 Applications of statistics

Zbl 0770.92001