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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, leaf outlines with and without herbivory by insects, 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(PCA). Statistical shape analysis has applications in various fields, including medical imaging, computer vision, sensor measurement, and geographical profiling.[3]

Methods based on diffeomorphic flows and metric space constructions of shape are now emerging in the field of Computational anatomy[4] Diffeomorphometry[5] as a tool for constructing metric spaces of shapes and forms is becoming a mainstay in Medical Imaging. Diffeomorphometry and its tools diverge from methods such as PCA in that in general the space of shapes and forms is not additive, rather it is a Riemannian manifold in which the fundamental tools of transformation are based on compositions of functions corresponding to flows of diffeomorphisms. The statistics of shapes is transformed to the study of local charts and flows associated to local coordinate system representations.

Modeling

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,[6] by the spatial region they occupy,[7] etc.

Shape deformations

Differences between shapes can be quantified by investigating deformations transforming one shape into another.[8] 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 , i.e., .[9] 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 Diffeomorphic Metric Mapping) framework for shape comparison.[10] On such deformations is the right invariant metric of Computational Anatomy which generalizes the metric of non-compressible Eulerian flows but to include the Sobolev norm ensuring smoothness of the flows.[11] Other metrics have now been defined associated to Hamiltonian controls of diffeomorphic flows.[12]

See also

References

  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. ^ Grenander, Ulf; Miller, Michael I. (1998-12-01). "Computational Anatomy: An Emerging Discipline". Q. Appl. Math. LVI (4): 617–694. ISSN 0033-569X.
  5. ^ Miller, Michael I.; Younes, Laurent; Trouvé, Alain (2013-11-18). "Diffeomorphometry and geodesic positioning systems for human anatomy". TECHNOLOGY. 02 (01): 36–43. doi:10.1142/S2339547814500010. ISSN 2339-5478. PMC 4041578. PMID 24904924.
  6. ^ 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): 60–97. doi:10.1007/s10851-013-0490-z.{{cite journal}}: CS1 maint: multiple names: authors list (link)
  7. ^ D. Zhang, G. Lu (2004). "Review of shape representation and description techniques". Pattern Recognition. 37 (1): 1–19. doi:10.1016/j.patcog.2003.07.008.
  8. ^ D'Arcy Thompson (1942). On Growth and Form. Cambridge University Press.
  9. ^ Definition 10.2 in I.L. Dryden and K.V. Mardia (1998). Statistical Shape Analysis. John Wiley & Sons. ISBN 0-471-95816-6.
  10. ^ 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): 139–157. doi:10.1023/b:visi.0000043755.93987.aa.{{cite journal}}: CS1 maint: multiple names: authors list (link)
  11. ^ Miller, M. I.; Younes, L. (2001-01-01). "Group Actions, Homeomorphisms, And Matching: A General Framework". International Journal of Computer Vision. 41: 61–84.
  12. ^ Miller, Michael I.; Trouvé, Alain; Younes, Laurent (2015-01-01). "Hamiltonian Systems and Optimal Control in Computational Anatomy: 100 Years Since D'Arcy Thompson". Annual Review of Biomedical Engineering. 17: 447–509. doi:10.1146/annurev-bioeng-071114-040601. ISSN 1545-4274. PMID 26643025.


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