Affine shape analysis and image analysis 1 Summary

1、Affine shape analysis and image analysis 1 SummaryAf?ne shape analysis and image analysisVic Patrangenaru,Kanti V.MardiaDepartment of Mathematics and Statistics,Georgia State University,Atlanta,USADepartment of Statistics,University of Leeds1SummaryA study of af?ne shape is needed in certain problem

2、s that arise in bioinformatics and pattern recognition.In particular af?ne shape analysis is useful in the analysis of2D electrophoresis images and in the reconstruction of a larger area from a number aerial images.Two aerial images taken from different distances of a ground scene are given in Figur

3、e1.;the problem asks to reconstruct a larger contiguous image that contains the information in both these images.Figure1.Aerial photographs of two sections of a ground scene.Another classical example consists in matching two labelled electrophoresis gels(see Figure 2,based on data from http:/www.wen

4、/doc/7ea0b8c4aa00b52acfc7ca77.html/?icker/)./doc/7ea0b8c4aa00b52acfc7ca77.htmlbelled electrophoresis gels.Such simple problems involve image warping based on mean af?ne shapes of conFigura-tions of convenient landmarks.When it comes to af?ne shape,researchers from di

5、fferent areas are apparently using two different de?nitions for af?ne shape.In this paper we give two such de?nitions,and show that they are are equivalent.As a consequence we show that the af?ne shape space of con?gurations of points in general position is a real Grassmann manifold,therefore statis

6、tical analysis of af?ne shape is statistical analysis on a real Grasmann manifold.Grassmann manifolds are not apriori submanifolds of a numerical space.For an extrinsic statistical analysis of probability distributions,such equivariant embeddings are sought,and such an equivariant embedding is de?ne

7、d here.This preliminary report ends with a discussion.2The af?ne shape spaceIn this section we will assume that An af?ne transformation of is a linear automorphism of,followed by a translation.denotes the group of af?ne trans-formations.In af?ne shape analysis,for we consider the set of all systems

8、of points The group of af?ne transformations acts diagonally on the left on the action is given by(1)DEFINITION2.1.a.The af?ne shape in statistical shape analysis ofis the orbit of via the action(1).Another de?nition of shape is given by A.Heyden(1995)and G.Sparr(1996):DEFINITION2.1.b.The af?ne shap

9、e in computer vision of is the linear subspace of given by(2)We can prove the following:PROPOSITION2.1.There is a natural one to one correspondence between af?ne shapes in statistical shape analysis and af?ne shapes in computer vision.DEFINITION2.2.The af?ne shape space,or space of af?ne-ads in is t

10、he quotient .The Grassmann manifold of-dimensional vector subspaces of will be denoted by Namely,rank where is a matrix.As a consequence of Proposition2.1one may show thatTHEOREM2.1.The af?ne shape space has a strati?cation,where the -stratum is diffeomorphic to the Grassmann manifold of-dimensional

11、 vector subspaces of In particular,the stratum of af?ne shapes of ads in general position is diffeomorphic to the Grassmann manifold3Extrinsic means of af?ne shapes and reconstruction of larger planar scenesAf?ne shape distributions have been considered by Goodall and Mardia(1993),Leung,Burl and Per

12、ona(1998),Berthilsson and Heyden(1999)et al.In view of Theorem2.1,extrinsic means of distributions of af?ne shapes can be determined using the general approach in Bhattacharya and Patrangenaru(2021),for certain convenient equivariant embeddings of in an Euclidean space.Such an embedding can be de?ne

13、d as follows:let be the set of symmetric matrices endowed with the canonical Euclidean square norma natural embedding of into is obtained by identifying each-dimensional vector subspace with the matrix of orthogonal projection into .Dimitric(1996)proved that this embedding is equivariant,has paralle

14、l second fundamental form and embeds the Grassmannian minimally into a hypersphere.This is an extension of the Veronese-Whitney embedding of projective spaces(see Bhattacharya and Patrangenaru(2021), that is commonly used for axial data(see Mardia and Jupp,1999),or for multivariate axial data (Mardi

15、a and Patrangenaru,2021).Assume a probability distribution of af?ne shapes of conFigurations in general position is nonfocal w.r.t.this embedding In this case,the mean of the corresponding distribution of symmetric matrices of rank has the eigenvaluessuch that The extrinsic mean of is the vector sub

16、space spanned by unit eigenvectors corresponding to the?rst eigenvalues of Assume is a sample of size of-vector subspaces in and the subspace is spanned by the orthonormal unit vectors and set The extrinsic sample mean is of this sample,when it exists,is the-vector subspaceception.The extrinsic samp

17、le mean is useful in averaging images of remote planar scenes,by adapting the standard method of image averaging of Dryden and Mardia(1998)as shown in Mardia et al.(2021).This method can be used in reconstruction of larger planar scenes as in Faugeras and Luong(2021),as shown in Figure3.Figure3.Reco

18、nstruction of a larger view of the scene in images in Figure1,based on anextrinsic mean af?ne shape.4DiscussionIn summary,in af?ne shape analysis,the real Grasmann manifolds play the key r?o le in the same way as that of the complex projective spaces in similarity shape analysis(Kendall,1984). While

19、 the?rst large sample results on Grassmann manifolds are about distributions without an extrinsic mean,data driven analysis is needed for concentrated distributions on such manifolds. Therefore large sample and nonparametric bootstrap methods should be developed for extrinsic sample means on Grassma

20、nn manifolds and applied in practice.The method of image warping was used to align microscope images by Glasbey and Mardia(2021).It would be interesting touse such results or the methods brie?y described in Section3when applied to electrophoresis gels,in matching of a new gel image with an image ave

21、rage based on mean af?ne shapes of conFigurations of marked spots in existent samples of such images,such as those in Figure2 (see http:/www.cmis.csiro.au/iap/RecentProjects/gelregistration.htm).In this paper we showed that Grassmann manifolds are useful in image analysis;other applications of stati

22、stical analysis on such manifolds are in signal subspace estimation(Srivastava and Klassen,2021).It should be noted that af?ne shape is applicable only for analysis of remote scenes,otherwise projective shape analysis should be used(see Mardia and Patrangenaru,2021)5AcknowledgementsWe would like to

23、thank Gordana Derado for the implementation of our algorithm of image reconstruction.ReferencesBerthilsson,R;Heyden,A.(1999).Recognition of Planar Objects using the Density of Af?ne Shape Computer Vision and Image Understanding76,135-145.Bhattacharya,R.N.and Patrangenaru,V(2021)Large sample theory o

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26、N.S.)59,131-137.Dryden,I.L.and Mardia,K.V.(1998).Statistical Shape Analysis Wiley:Chichester. Faugeras,Olivier and Luong,Quang-Tuan(2021).The geometry of multiple images.With contributions from Theo Papadopoulo.MIT Press,Cambridge,MAGlasbey C.A.and Mardia,K.V.(2021).A penalized likelihood approach t

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