Statistical shape analysis

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. 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.

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, by the spacial region they occupy, etc.

Shape deformations

Differences between shapes can be quantified by investigating deformations transforming one shape into another. 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., $y = \Phi(x)$. 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.

See also

• Geometric data analysis
• Shape analysis (disambiguation)
• Procrustes analysis