Publication

Statistics on Manifolds applied to Shape Theory

2013
Student Projects

Abstract

In this report, we use a variety of tools from differential geometry to propose a nonlinear extension of the principal components analysis (PCA) into manifolds setting. This extension, that we shall call principal geodesics analysis (PGA), attempts to find analogs of the principal components by introducing the principal geodesic components. We then construct the shape space of triangles Σ^3_2 and find a convenient parametrization of it. Finally, we apply the PGA procedure previously designed to analyze the variability of a sample of shapes, randomly chosen onto the shape space of triangles.

Official source

https://infoscience.epfl.ch/entities/publication/5cfb0927-38e2-4a0f-b90f-e256c8996070

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2013
Student Projects

Abstract

Official source

https://infoscience.epfl.ch/entities/publication/5cfb0927-38e2-4a0f-b90f-e256c8996070

About this result

Related concepts (32)

Differential geometry of surfaces

In mathematics, the differential geometry of surfaces deals with the differential geometry of smooth surfaces with various additional structures, most often, a Riemannian metric. Surfaces have been extensively studied from various perspectives: extrinsically, relating to their embedding in Euclidean space and intrinsically, reflecting their properties determined solely by the distance within the surface as measured along curves on the surface.

Differential geometry

Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra. The field has its origins in the study of spherical geometry as far back as antiquity. It also relates to astronomy, the geodesy of the Earth, and later the study of hyperbolic geometry by Lobachevsky.

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 enabling the visualization of multidimensional data. Formally, PCA is a statistical technique for reducing the dimensionality of a dataset. This is accomplished by linearly transforming the data into a new coordinate system where (most of) the variation in the data can be described with fewer dimensions than the initial data.