Statistical Shape Analysis
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Statistical shape analysis is an analysis of the
geometrical Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
properties of some given set of
shape A shape or figure is a graphical representation of an object or its external boundary, outline, or external surface, as opposed to other properties such as color, texture, or material type. A plane shape or plane figure is constrained to lie on ...
s by
statistical Statistics (from German: '' Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industr ...
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. One of the main methods used is
principal component analysis Principal component analysis (PCA) is a popular technique for analyzing large datasets containing a high number of dimensions/features per observation, increasing the interpretability of data while preserving the maximum amount of information, and ...
(PCA). Statistical shape analysis has applications in various fields, including medical imaging,
computer vision Computer vision is an Interdisciplinarity, interdisciplinary scientific field that deals with how computers can gain high-level understanding from digital images or videos. From the perspective of engineering, it seeks to understand and automate t ...
, computational anatomy, sensor measurement, and geographical profiling.


Landmark-based techniques

In the point distribution model, a shape is determined by a finite set of coordinate points, known as
landmark point In morphometrics, landmark point or shortly landmark is a point in a shape object in which correspondences between and within the populations of the object are preserved. In other disciplines, landmarks may be known as vertices, anchor points, co ...
s. 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 Fred may refer to: People * Fred (name), including a list of people and characters with the name Mononym * Fred (cartoonist) (1931–2013), pen name of Fred Othon Aristidès, French * Fred (footballer, born 1949) (1949–2022), Frederico Rodr ...
for
geometric morphometrics in anthropology The study of geometric morphometrics in anthropology has made a major impact on the field of morphometrics by aiding in some of the technological and methodological advancements. Geometric morphometrics is an approach that studies shape using Cart ...
. Or an approach like
Procrustes analysis In statistics, Procrustes analysis is a form of statistical shape analysis used to analyse the distribution of a set of shapes. The name ''Procrustes'' ( el, Προκρούστης) refers to a bandit from Greek mythology who made his victims fit ...
which finds an average shape.
David George Kendall David George Kendall FRS (15 January 1918 – 23 October 2007) was an English statistician and mathematician, known for his work on probability, statistical shape analysis, ley lines and queueing theory. He spent most of his academic lif ...
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. Statistical distribution like the
Kent distribution In directional statistics, the Kent distribution, also known as the 5-parameter Fisher–Bingham distribution (named after John T. Kent, Ronald Fisher, and Christopher Bingham), is a probability distribution on the unit sphere (2-sphere ''S''2 in ...
can be used to analyse the distribution of such spaces. Alternatively, shapes can be represented by curves or surfaces representing their contours, by the spatial region they occupy.


Shape deformations

Differences between shapes can be quantified by investigating deformations transforming one shape into another. In particular a
diffeomorphism In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable. Definition Given tw ...
preserves smoothness in the deformation. This was pioneered in D'Arcy Thompson's
On Growth and Form ''On Growth and Form'' is a book by the Scottish mathematical biology, mathematical biologist D'Arcy Wentworth Thompson (1860–1948). The book is long – 793 pages in the first edition of 1917, 1116 pages in the second edition of 1942. The ...
before the advent of computers. Deformations can be interpreted as resulting from a
force In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a ...
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) . 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 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 \varphi \in \operato ...
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. Diffeomorphic registration, introduced in the 90's, is now an important player with existing codes bases organized around ANTS, DARTEL, DEMONS,
LDDMM 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, t ...
, StationaryLDDMM, and FastLDDMM are examples of actively used computational codes for constructing correspondences between coordinate systems based on sparse features and dense images.
Voxel-based morphometry Voxel-based morphometry is a computational approach to neuroanatomy that measures differences in local concentrations of brain tissue, through a voxel-wise comparison of multiple brain images. In traditional morphometry, volume of the whole bra ...
(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.


See also

*
Active shape model Active shape models (ASMs) are statistical models of the shape of objects which iteratively deform to fit to an example of the object in a new image, developed by Tim Cootes and Chris Taylor in 1995. The shapes are constrained by the PDM ( point dis ...
* Geometric data analysis *
Shape analysis (disambiguation) Shape analysis may refer to: * Shape analysis (digital geometry) * Shape analysis (program analysis), a type of method to analyze computer programs without actually executing the programs * Statistical shape analysis * Computational anatomy#Stat ...
*
Procrustes analysis In statistics, Procrustes analysis is a form of statistical shape analysis used to analyse the distribution of a set of shapes. The name ''Procrustes'' ( el, Προκρούστης) refers to a bandit from Greek mythology who made his victims fit ...
* Computational anatomy * Large Deformation Diffeomorphic Metric Mapping *
Bayesian Estimation of Templates in Computational Anatomy 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 o ...
*
Bayesian model of computational anatomy 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, ...


References

{{reflist Statistical data types Spatial analysis Computer vision Geometric shapes