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, [3] computer vision, computational anatomy, sensor measurement, and geographical profiling. [4]

Contents

Landmark-based techniques

In the point distribution model, a shape is determined by a finite set of coordinate points, known as landmark points. These landmark points often correspond to important identifiable features such as the corners of the eyes. Once the points are collected some form of registration is undertaken. This can be a baseline methods used by Fred Bookstein for geometric morphometrics in anthropology. Or an approach like Procrustes analysis which finds an average shape.

David George Kendall investigated the statistical distribution of the shape of triangles, and represented each triangle by a point on a sphere. He used this distribution on the sphere to investigate ley lines and whether three stones were more likely to be co-linear than might be expected. [5] Statistical distribution like the Kent distribution can be used to analyse the distribution of such spaces.

Alternatively, shapes can be represented by curves or surfaces representing their contours, [6] by the spatial region they occupy. [7]

Shape deformations

Differences between shapes can be quantified by investigating deformations transforming one shape into another. In particular a diffeomorphism preserves smoothness in the deformation. This was pioneered in D'Arcy Thompson's On Growth and Form before the advent of computers. [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.

Diffeomorphometry [10] is the focus on comparison of shapes and forms with a metric structure based on diffeomorphisms, and is central to the field of Computational anatomy. [11] Diffeomorphic registration, [12] introduced in the 90's, is now an important player with existing codes bases organized around ANTS, [13] DARTEL, [14] DEMONS, [15] LDDMM, [16] StationaryLDDMM, [17] and FastLDDMM [18] are examples of actively used computational codes for constructing correspondences between coordinate systems based on sparse features and dense images. Voxel-based morphometry (VBM) is an important technology built on many of these principles. Methods based on diffeomorphic flows are also used. For example, deformations could be diffeomorphisms of the ambient space, resulting in the LDDMM (Large Deformation Diffeomorphic Metric Mapping) framework for shape comparison. [19]

See also

Related Research Articles

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<span class="mw-page-title-main">Diffeomorphism</span> Isomorphism of smooth manifolds; a smooth bijection with a smooth inverse

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<span class="mw-page-title-main">Image registration</span> Mapping of data into a single system

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<span class="mw-page-title-main">Shape</span> Form of an object or its external boundary

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<span class="mw-page-title-main">Morphometrics</span> Quantitative study of size and shape

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In neuroimaging, spatial normalization is an image processing step, more specifically an image registration method. Human brains differ in size and shape, and one goal of spatial normalization is to deform human brain scans so one location in one subject's brain scan corresponds to the same location in another subject's brain scan.

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This article describes shape analysis to analyze and process geometric shapes.

Brain morphometry is a subfield of both morphometry and the brain sciences, concerned with the measurement of brain structures and changes thereof during development, aging, learning, disease and evolution. Since autopsy-like dissection is generally impossible on living brains, brain morphometry starts with noninvasive neuroimaging data, typically obtained from magnetic resonance imaging (MRI). These data are born digital, which allows researchers to analyze the brain images further by using advanced mathematical and statistical methods such as shape quantification or multivariate analysis. This allows researchers to quantify anatomical features of the brain in terms of shape, mass, volume, and to derive more specific information, such as the encephalization quotient, grey matter density and white matter connectivity, gyrification, cortical thickness, or the amount of cerebrospinal fluid. These variables can then be mapped within the brain volume or on the brain surface, providing a convenient way to assess their pattern and extent over time, across individuals or even between different biological species. The field is rapidly evolving along with neuroimaging techniques—which deliver the underlying data—but also develops in part independently from them, as part of the emerging field of neuroinformatics, which is concerned with developing and adapting algorithms to analyze those data.

Spectral shape analysis relies on the spectrum of the Laplace–Beltrami operator to compare and analyze geometric shapes. Since the spectrum of the Laplace–Beltrami operator is invariant under isometries, it is well suited for the analysis or retrieval of non-rigid shapes, i.e. bendable objects such as humans, animals, plants, etc.

Medical image computing (MIC) is an interdisciplinary field at the intersection of computer science, information engineering, electrical engineering, physics, mathematics and medicine. This field develops computational and mathematical methods for solving problems pertaining to medical images and their use for biomedical research and clinical care.

Computational anatomy is an interdisciplinary field of biology focused on quantitative investigation and modelling of anatomical shapes variability. It involves the development and application of mathematical, statistical and data-analytical methods for modelling and simulation of biological structures.

Group actions are central to Riemannian geometry and defining orbits. The orbits of computational anatomy consist of anatomical shapes and medical images; the anatomical shapes are submanifolds of differential geometry consisting of points, curves, surfaces and subvolumes,. This generalized the ideas of the more familiar orbits of linear algebra which are linear vector spaces. Medical images are scalar and tensor images from medical imaging. The group actions are used to define models of human shape which accommodate variation. These orbits are deformable templates as originally formulated more abstractly in pattern theory.

Statistical shape analysis and statistical shape theory in computational anatomy (CA) is performed relative to templates, therefore it is a local theory of statistics on shape. Template estimation in computational anatomy from populations of observations is a fundamental operation ubiquitous to the discipline. Several methods for template estimation based on Bayesian probability and statistics in the random orbit model of CA have emerged for submanifolds and dense image volumes.

Computational anatomy (CA) is the study of shape and form in medical imaging. The study of deformable shapes in computational anatomy rely on high-dimensional diffeomorphism groups which generate orbits of the form . In CA, this orbit is in general considered a smooth Riemannian manifold since at every point of the manifold there is an inner product inducing the norm on the tangent space that varies smoothly from point to point in the manifold of shapes . This is generated by viewing the group of diffeomorphisms as a Riemannian manifold with , associated to the tangent space at . This induces the norm and metric on the orbit under the action from the group of diffeomorphisms.

Large deformation diffeomorphic metric mapping (LDDMM) is a specific suite of algorithms used for diffeomorphic mapping and manipulating dense imagery based on diffeomorphic metric mapping within the academic discipline of computational anatomy, to be distinguished from its precursor based on diffeomorphic mapping. The distinction between the two is that diffeomorphic metric maps satisfy the property that the length associated to their flow away from the identity induces a metric on the group of diffeomorphisms, which in turn induces a metric on the orbit of shapes and forms within the field of Computational Anatomy. The study of shapes and forms with the metric of diffeomorphic metric mapping is called diffeomorphometry.

<span class="mw-page-title-main">Michael I. Miller</span> American biomedical engineer and neuroscientist

Michael Ira Miller is an American-born biomedical engineer and data scientist, and the Bessie Darling Massey Professor and Director of the Johns Hopkins University Department of Biomedical Engineering. He worked with Ulf Grenander in the field of Computational Anatomy as it pertains to neuroscience, specializing in mapping the brain under various states of health and disease by applying data derived from medical imaging. Miller is the director of the Johns Hopkins Center for Imaging Science, Whiting School of Engineering and codirector of Johns Hopkins Kavli Neuroscience Discovery Institute. Miller is also a Johns Hopkins University Gilman Scholar.

Computational anatomy (CA) is a discipline within medical imaging focusing on the study of anatomical shape and form at the visible or gross anatomical scale of morphology. The field is broadly defined and includes foundations in anatomy, applied mathematics and pure mathematics, including medical imaging, neuroscience, physics, probability, and statistics. It focuses on the anatomical structures being imaged, rather than the medical imaging devices. The central focus of the sub-field of computational anatomy within medical imaging is mapping information across anatomical coordinate systems most often dense information measured within a magnetic resonance image (MRI). The introduction of flows into CA, which are akin to the equations of motion used in fluid dynamics, exploit the notion that dense coordinates in image analysis follow the Lagrangian and Eulerian equations of motion. In models based on Lagrangian and Eulerian flows of diffeomorphisms, the constraint is associated to topological properties, such as open sets being preserved, coordinates not crossing implying uniqueness and existence of the inverse mapping, and connected sets remaining connected. The use of diffeomorphic methods grew quickly to dominate the field of mapping methods post Christensen's original paper, with fast and symmetric methods becoming available.

Diffeomorphometry is the metric study of imagery, shape and form in the discipline of computational anatomy (CA) in medical imaging. The study of images in computational anatomy rely on high-dimensional diffeomorphism groups which generate orbits of the form , in which images can be dense scalar magnetic resonance or computed axial tomography images. For deformable shapes these are the collection of manifolds , points, curves and surfaces. The diffeomorphisms move the images and shapes through the orbit according to which are defined as the group actions of computational anatomy.

References

  1. ↑ I.L. Dryden & K.V. Mardia (1998). Statistical Shape Analysis. John Wiley & Sons. ISBN   978-0-471-95816-1.
  2. ↑ H. Ziezold (1994). "Mean Figures and Mean Shapes Applied to Biological Figure and Shape Distributions in the Plane". Biometrical Journal. Biometrical Journal, 36, pp. 491–510. 36 (4): 491–510. doi:10.1002/bimj.4710360409.
  3. ↑ G. Zheng; S. Li; G. Szekely (2017). Statistical Shape and Deformation Analysis. Academic Press. ISBN   9780128104941.
  4. ↑ S. Giebel (2011). Zur Anwendung der Formanalyse. AVM, M\"unchen.
  5. ↑ Bingham, N. H. (1 November 2007). "Professor David Kendall" . The Independent. Archived from the original on 2022-05-24. Retrieved 5 April 2016.
  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. arXiv: 1305.1150 . doi:10.1007/s10851-013-0490-z. S2CID   2866580.
  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 & K.V. Mardia (1998). Statistical Shape Analysis. John Wiley & Sons. ISBN   978-0-471-95816-1.
  10. ↑ Miller, Michael I.; Younes, Laurent; Trouvé, Alain (2013-11-18). "Diffeomorphometry and geodesic positioning systems for human anatomy". Technology. 2 (1): 36–43. doi:10.1142/S2339547814500010. ISSN   2339-5478. PMC   4041578 . PMID   24904924.
  11. ↑ Grenander, Ulf; Miller, Michael I. (1998-12-01). "Computational Anatomy: An Emerging Discipline". Q. Appl. Math. LVI (4): 617–694. doi: 10.1090/qam/1668732 . ISSN   0033-569X.
  12. ↑ Christensen, G. E.; Rabbitt, R. D.; Miller, M. I. (1996-01-01). "Deformable templates using large deformation kinematics". IEEE Transactions on Image Processing. 5 (10): 1435–1447. Bibcode:1996ITIP....5.1435C. doi:10.1109/83.536892. ISSN   1057-7149. PMID   18290061.
  13. ↑ "stnava/ANTs". GitHub. Retrieved 2015-12-11.
  14. ↑ Ashburner, John (2007-10-15). "A fast diffeomorphic image registration algorithm". NeuroImage. 38 (1): 95–113. doi:10.1016/j.neuroimage.2007.07.007. ISSN   1053-8119. PMID   17761438. S2CID   545830.
  15. ↑ "Software – Tom Vercauteren". sites.google.com. Retrieved 2015-12-11.
  16. ↑ "NITRC: LDDMM: Tool/Resource Info". www.nitrc.org. Retrieved 2015-12-11.
  17. ↑ "Publication:Comparing algorithms for diffeomorphic registration: Stationary LDDMM and Diffeomorphic Demons". www.openaire.eu. Archived from the original on 2016-02-16. Retrieved 2015-12-11.
  18. ↑ Zhang, Miaomiao; Fletcher, P. Thomas (2015). "Finite-Dimensional Lie Algebras for Fast Diffeomorphic Image Registration". Information Processing in Medical Imaging. Lecture Notes in Computer Science. Vol. 24. pp. 249–259. doi:10.1007/978-3-319-19992-4_19. ISBN   978-3-319-19991-7. ISSN   1011-2499. PMID   26221678. S2CID   10334673.
  19. ↑ 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. S2CID   17772076.
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