Statistical shape analysis

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Statistical shape analysis is an analysis of the geometrical properties of some given set of shapes by statistical methods. For instance, it could be used to quantify differences between male and female Gorilla skull shapes, normal and pathological bone shapes, etc. Important aspects of shape analysis are to obtain a measure of distance between shapes, to estimate mean shapes from (possibly random) samples, to estimate shape variability within samples, to perform clustering and to test for differences between shapes.[1][2] One of the main methods used is principal component analysis. Statistical shape analysis has applications in various fields, including medical imaging, computer vision, sensor measurement, and geographical profiling.[3]


The first step after collecting a set of shapes is to create a proper shape model for further statistical analysis. In the point distribution model, a shape is determined by a finite set of coordinate points, known as landmark points; the Cartesian coordinate system is the most commonly used one. Alternatively, shapes can be represented by curves or surfaces representing their contours,[4] by the spacial region they occupy,[5] etc.

Shape deformations[edit]

Differences between shapes can be quantified by investigating deformations transforming one shape into another.[6] Deformations can be interpreted as resulting from a force applied to the shape. Mathematically, a deformation is defined as a mapping from a shape x to a shape y by a transformation function \Phi, i.e., y = \Phi(x) .[7] Given a notion of size of deformations, the distance between two shapes can be defined as the size of the smallest deformation between these shapes. For example, deformations could be diffeomorphisms of the ambient space, resulting in the LDDMM (Large Deformation Metric Mappings) framework for shape comparison.[8]

See also[edit]


  1. ^ I.L. Dryden and K.V. Mardia (1998). Statistical Shape Analysis. John Wiley & Sons. ISBN 0-471-95816-6. 
  2. ^ H. Ziezold (1994). Mean Figures and Mean Shapes Applied to Biological Figure and Shape Distributions in the Plane. Biometrical Journal, 36, p. 491-510. 
  3. ^ S. Giebel (2011). Zur Anwendung der Formanalyse. AVM, M\"unchen. 
  4. ^ M. Bauer, M. Bruveris, P. Michor (2014). "Overview of the Geometries of Shape Spaces and Diffeomorphism Groups". Journal of Mathematical Imaging and Vision 50 (490). 
  5. ^ D. Zhang, G. Lu (2004). "Review of shape representation and description techniques". Pattern Recognition 37 (1): 1–19. 
  6. ^ D'Arcy Thompson (1942). On Growth and Form. Cambridge University Press. 
  7. ^ Definition 10.2 in I.L. Dryden and K.V. Mardia (1998). Statistical Shape Analysis. John Wiley & Sons. ISBN 0-471-95816-6. 
  8. ^ F. Beg, M. Miller, A. Trouvé, L. Younes (February 2005). "Computing Large Deformation Metric Mappings via Geodesic Flows of Diffeomorphisms". International Journal of Computer Vision 61 (2).