GEOMETRIC ANALYSIS OF PLANAR SHAPES WITH APPLICATIONS TO CELL DEFORMATIONS
DOI:
https://doi.org/10.5566/ias.1151Keywords:
cell deformation, geodesics, planar closed curves, radius-vector function, shape spaceAbstract
Shape analysis is of great importance in many fields such as computer vision, medical imaging, and computational biology. In this paper we focus on a shape space in which shapes are represented by means of planar closed curves. In this shape space a new metric was recently introduced with the result that this shape space has the property of being isometric to an infinite-dimensional Grassmann manifold of 2-dimensional subspaces. Using this isometry it is possible, from Younes et al. (2008), to explicitly describe geodesics, a task that previously was not at all easy. Our aim is twofold, namely: to use this general theory in order to show some applications to the study of erythrocytes, using digital images of peripheral blood smears, in the treatment of sickle cell disease; and, since normal erythrocytes are almost circular and many Sickle cells have elliptical shape, to particularize the computation of geodesics and distances between shapes using this metric to planar objects considered as deformations of a template (circle or ellipse). The applications considered include: shape interpolation, shape classification, and shape clustering.References
Asakura T, Hirota T, Nelson AT, Reilly MP, Ohene-Frempong K (1996). Percentage of reversibly and irreversibly sickled cells are altered by the method of blood drawing and storage conditions. Blood Cell Mol Dis 22:297-306.
Bacus JW, Belanger MG, Aggarwal RK, Trobaugh FG (1976). Image processing for automated erythrocytes classification. Journal Hystochem Cytochem 24:195-201.
Bacus JW, Yasnoff WA, Belanger MG (1977). Computer techniques for cell analysis in hematology. Proceedings of the First Annual Symposium on Computer Applications in Medical Care (SCAMC), 1977 Oct, 24-35.
Cover TM, Hart PE (1967). Nearest neighbor pattern classification. IEEE Trans Inform Theory 13:21-7.
Frejlichowski D (2010). Pre-processing, extraction and recognition of binary erythrocyte shapes for computer-assisted diagnosis based on MGG images. International Conference on Computer Vision and Graphics, Poland 2010, Part I. LNCS 6374:368-75.
Grenander U, Manbeck KM (1993). A stochastic model for defect detection in potatoes. J Comp Graph Statist 2:131-51.
Hobolth A, Vedel-Jensen EB (2000). Modelling stochastic changes in curve shape, with an application to cancer diagnostics. Adv Appl Prob (SGSA) 32:344-62.
Hobolth A, Pedersen J, Vedel-Jensen EB (2002). A deformable template model, with special reference to elliptical templates. J Math Imaging Vis 17:131-7.
Hobolth A, Pedersen J, Vedel-Jensen EB (2003). A continuous parametric shape model. Ann Inst Statist Math 55:227-42.
Horiuchi K, Ohata J, Hirano Y, Asakura T (1990). Morphologic studies of sickle erythrocytes by image analysis. J Lab Clin Med 115:613-20.
Huckemann SF (2011). Intrinsic inference on the mean geodesic of planar shapes and tree discrimination by leaf growth. Ann Statist 39:1098-124.
Jayavanth S, Lee DH, Pak BC (2010). Multi-shape erythrocyte deformability analysis by imaging technique. J Mech Science Technology 24:931-5.
Kaufman L, Rousseeuw PJ (1990). Finding groups in data: an introduction to cluster analysis. John Wiley. New York.
Kavitha A, Ramakrishnan S (2005). Analysis on the erythrocyte shape changes using wavelet transforms. Clin Hemorheol Micro 33:327-35.
Kindratenko V (2003). On using functions to describe shapes. J Math Imaging Vis 18:225-45.
Klassen E, Srivastava A, Mio W, Joshi SH (2004). Analysis of planar shapes using geodesic paths on shape spaces. IEEE T Pattern Anal 26:372-83.
Lestrel PE (1997). Fourier descriptors and their applications in biology. Cambridge University Press. New York.
Loncaric S (1998). A survey of shape analysis techniques. Pattern recogn 31:983-1001.
Neretin YA (2001). On Jordan angles and the triangle inequality in Grassmann manifold. Geometriae Dedicata 86:81-92.
Osher S, Sethian JA (1998). Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulation. J Comput Phys 79:12-49.
Pennec X (2006). Intrinsic statistics on Riemannian manifolds: basic tools for geometric measurements. J Math Imaging Vis 25:127-54.
R Development Core Team (2009). R: A language and environment for statistical computing. R Foundation for Statistical Computing. Vienna, Austria. http://www.R-project.org.
Rao CR (2009). Linear statistical inference and its applications. John Wiley & Sons.
Sauer T (2011). Numerical analysis. 2nd Edition. Pearson.
Santaló LA (1976). Integral geometry and geometric probability. Addison-Wesley Publishing Company Inc. London.
Srivastava A, Klassen E, Joshi SH, Jermyn H (2011). Shape analysis of elastic curves in Euclidean spaces. IEEE T Pattern Anal 33:1415-28.
Wheeless L, Robinson R, Lapets OP, Cox C, Rubio A, Weintraub M, Benjamin LJ (1994). Classification of red-blood-cells as normal, sickle, or other abnormal, using a single image-analysis feature. Cytometry 17:59-166.
Woods RP (2003). Characterizing volume and surface deformations in an atlas framework: theory, applications, and implementation. NeuroImage 18:769-88.
Younes L, Michor PW, Shah J, Mumford D (2008). A metric on shape space with explicit geodesics. Rend Lincei Mat Appl 39:25-57.
