Statistical Shape Theory Research Articles

Recognition of Planar Objects Using the Density of Affine Shape

In this paper, we will study the recognition problem for finite point configurations, in a statistical manner. We study the statistical theory of shape for ordered finite point configurations, or otherwise stated, the uncertainty of geometric invariants. Here, a general approach for defining shape and finding its density, expressed in the densities for the individual points, is developed. No approximations are made, resulting in an exact expression of the uncertainty region. In particular, we will concentrate on the affine shape, where often analytical computations is possible. In this case confidence intervals for invariants can be obtained from a priori assumptions on the densities of the detected points in the images. However, the theory is completely general and can be used to compute the density of any invariant (Euclidean, affine, similarity, projective, etc.) from arbitrary densities of the individual points. These confidence intervals can be used in such applications as geometrical hashing, recognition of ordered point configurations, and error analysis of reconstruction algorithms. Finally, an example will be given, illustrating the theory for the problem of recognizing planar point configurations from images taken by an affine camera. This case is of particular importance in applications, where details on a conveyor belt are captured by a camera, with image plane parallel to the conveyor belt and extracted feature points from the images are used to sort the objects.

Multidimensional scaling of simplex shapes

Bookstein (Statist. Sci. 1 (1986) 181–242; Morphometric Tools for Landmark Data: Geometry and Biology, Cambridge University Press, Cambridge, 1991) has proposed a method for the representation of triangle shape as points in the Poincaré half plane – a space of constant negative curvature. Small (The Statistical Theory of Shape, Springer, New York, 1996) provided an extension of the Bookstein representation by representing the shapes of n-simplexes on manifolds. These manifolds are quite distinct from those proposed by D.G. Kendall based upon Procrustes arguments. In this paper, we examine the geometrical properties of these simplex shape spaces in greater detail. In particular, explicit formulas are given for the geodesic distance between any two points in these spaces. Such formulas permit the implementation of multidimensional scaling methods for the statistical shape analysis of two- and three-dimensional objects. In addition, the curvatures of the simplex shape spaces are examined. It is shown that the spaces of n-simplex shapes are not of constant curvature unless n=2.

BOOK REVIEWS

Book reviewed in this article:Statistical Methods for Industrial Process Control. By David Drain. New YorkInto Statistics: A Guide to Understanding Statistical Concepts in Engineering and the Sciences. By P.J. Smith. SingaporeBeyond ANOVA: Basics of Applied Statistics. By Rupert G. Miller, Jr. LondonPlane Answers to Complex Questions: The Theory of Linear Models. By Ronald Christensen. New YorkThe Statistical Theory of Shape. By C.G. Small. New York: Springer

Small, C. G.: The Statistical Theory of Shape. Springer‐Verlag Berlin, Heidelberg, New York, London, Paris, Tokyo, Hong Kong, 1996. 46 Fig., × + 227 pages, hardcover, DM 78,00, ISBN 0–387–94729–9

Biometrical JournalVolume 39, Issue 7 p. 862-862 Book Review Small, C. G.: The Statistical Theory of Shape. Springer-Verlag Berlin, Heidelberg, New York, London, Paris, Tokyo, Hong Kong, 1996. 46 Fig., × + 227 pages, hardcover, DM 78,00, ISBN 0–387–94729–9 D. Stoyan, D. StoyanSearch for more papers by this author D. Stoyan, D. StoyanSearch for more papers by this author First published: 18 January 2007 https://doi.org/10.1002/bimj.4710390714AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat No abstract is available for this article. Volume39, Issue71997Pages 862-862 RelatedInformation

Discussion of the Paper by Grenander and Miller

Discussion of the Paper by Grenander and Miller

Discussion of the Paper by Goodall

Discussion of the Paper by Goodall

A Survey of the Statistical Theory of Shape

This is a review of the current state of the theory of introduced by the author in 1977. It starts with a definition of for a set of $k$ points in $m$ dimensions. The first task is to identify the shape spaces in which such objects naturally live, and then to examine the probability structures induced on such a shape space by corresponding structures in $\mathbf{R}^m$. Against this theoretical background one formulates and solves statistical problems concerned with shape characteristics of empirical sets of points. Some applications (briefly sketched here) are to archeology, astronomy, geography and physical chemistry. We also outline more recent work on size-and-shape, on shapes of sets of points in riemannian spaces, and on shape-theoretic aspects of random Delaunay tessellations.

A Survey of the Statistical Theory of Shape]: Comment: Some Contributions to Shape Analysis

A Survey of the Statistical Theory of Shape]: Comment: Some Contributions to Shape Analysis

Further Developments and Applications of the Statistical Theory of Shape

Previous article Next article Further Developments and Applications of the Statistical Theory of ShapeDavid G. KendallDavid G. Kendallhttps://doi.org/10.1137/1131055PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] R. V. Ambartzumian, B. Ranneby, Random shapes by factorization, Statistics in Theory and Practice, New York: Umeå, 1982 0575.60009 Google Scholar[2] F. L. Bookstein, Size and shape spaces for landmark data in two dimensions, Statist. Science, 1 (1986), 181–242 0614.62144 CrossrefGoogle Scholar[3] F. O. Bower, Size and Form in Plants, Macmillian, London, 1930 Google Scholar[4] S. V. M. Clube and , A. S. Trew, Observational results on quasar alignments in five fields, Quasars and Gravitational Lenses, 24th Liège Astrophys. Coll., 1983, 374–377 Google Scholar[5] M. G. Edmunds and , G. H. George, On the statistics of quasar alignments, Mon. Not. R. Astr. Soc., 213 (1985), 905–915 CrossrefGoogle Scholar[6] D. G. Kendall, The diffusion of shape, Adv. Appl. Probab., 9 (1977), 428–430 CrossrefGoogle Scholar[7] David G. Kendall, Shape manifolds, Procrustean metrics, and complex projective spaces, Bull. London Math. Soc., 16 (1984), 81–121 86g:52010 0579.62100 CrossrefGoogle Scholar[8] D. G. Kendall, Exact distributions for shapes of random triangles in convex sets, Adv. in Appl. Probab., 17 (1985), 308–329 86m:60033 0566.60007 CrossrefGoogle Scholar[9] David G. Kendall and , Wilfrid S. Kendall, Alignments in two-dimensional random sets of points, Adv. in Appl. Probab., 12 (1980), 380–424 81d:60014 0425.60009 CrossrefGoogle Scholar[10] David G. Kendall and , Hui-Lin Le, Exact shape-densities for random triangles in convex polygons, Adv. in Appl. Probab., (1986), 59–72 88h:60022 0607.60009 Google Scholar[11] W. G. McGinley and , R. Sibson, Dissociated random variables, Math. Proc. Cambridge Philos. Soc., 77 (1975), 185–188 51:6940 0353.60018 CrossrefGoogle Scholar[12] B. W. Silvermann and , T. C. Brown, Short distances, flat triangles and Poisson limits, J. Appl. Probab., 15 (1978), 815–825 80c:60042 0396.60029 CrossrefGoogle Scholar[13] C. G. Small, Ph.D. Thesis, Distributions of shape, and maximal invariant statistics, University of Cambridge, 1981 Google Scholar[14] C. G. Small, Random uniform triangles and the alignment problem, Math. Proc. Cambridge Philos. Soc., 91 (1982), 315–322 83b:62033 0482.60014 CrossrefGoogle Scholar[15] D'Arcy W. Thompson, On Growth and Form, Cambridge University Press, 1917 CrossrefGoogle Scholar[16] Peter Wakeman, Plane polygons and a conjecture of Blaschke's, Adv. in Appl. Probab., 17 (1985), 774–793, (See also C. L. Mallows and J. M. C. Clarke, ibid., 7 (1970), pp. 240–244.) 87d:52008 CrossrefGoogle Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Exploring Errors in Reading a Visualization via Eye Tracking Models Using Stochastic Geometry14 June 2019 Cross Ref Face verification through tracking facial features1 December 2001 | Journal of the Optical Society of America A, Vol. 18, No. 12 Cross Ref Volume 31, Issue 3| 1987Theory of Probability & Its Applications History Submitted:08 April 1986Published online:28 July 2006 InformationCopyright © 1987 © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1131055Article page range:pp. 407-412ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics

The Statistical Theory of Nuclear Magnetic Resonance Line Shape in Crystals with Paramagnetic Impurities

physica status solidi (b)Volume 35, Issue 1 p. K71-K73 Short Note The Statistical Theory of Nuclear Magnetic Resonance Line Shape in Crystals with Paramagnetic Impurities M. N. Aliev, M. N. Aliev Faculty of Physics, Azerbaidzhan University, Baku, and Faculty of Physics, Kazan University, KazanSearch for more papers by this authorB. I. Kochelaev, B. I. Kochelaev Faculty of Physics, Azerbaidzhan University, Baku, and Faculty of Physics, Kazan University, KazanSearch for more papers by this author M. N. Aliev, M. N. Aliev Faculty of Physics, Azerbaidzhan University, Baku, and Faculty of Physics, Kazan University, KazanSearch for more papers by this authorB. I. Kochelaev, B. I. Kochelaev Faculty of Physics, Azerbaidzhan University, Baku, and Faculty of Physics, Kazan University, KazanSearch for more papers by this author First published: 1969 https://doi.org/10.1002/pssb.19690350177AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat No abstract is available for this article. Volume35, Issue11969Pages K71-K73 RelatedInformation