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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.
Testing whether a differentiable map is a diffeomorphism can be made locally under some mild restrictions. This is the Hadamard-Caccioppoli theorem: [1] If , are connected open subsets of such that is simply connected, a differentiable map : is a diffeomorphism if it is proper and if the differential: is bijective (and hence a linear isomorphism) at each point in .
The first algorithm for dense image mapping via diffeomorphic metric mapping was Beg's LDDMM [1] [2] for volumes and Joshi's landmark matching for point sets with correspondence, [3] [4] with LDDMM algorithms now available for computing diffeomorphic metric maps between non-corresponding landmarks [5] and landmark matching intrinsic to ...
[2] It follows that a map f : X → Y {\displaystyle f:X\to Y} between two manifolds of equal dimension ( dim X = dim Y {\displaystyle \operatorname {dim} X=\operatorname {dim} Y} ) is a local diffeomorphism if and only if it is a smooth immersion (smooth local embedding), or equivalently, if and only if it is a smooth submersion .
Recall that a topological manifold is a topological space which is locally homeomorphic to . Differentiable manifolds (also called smooth manifolds) generalize the notion of smoothness on in the following sense: a differentiable manifold is a topological manifold with a differentiable atlas, i.e. a collection of maps from open subsets of to the manifold which are used to "pull back" the ...
In mathematics, an exotic is a differentiable manifold that is homeomorphic (i.e. shape preserving) but not diffeomorphic (i.e. non smooth) to the Euclidean space. The first examples were found in 1982 by Michael Freedman and others, by using the contrast between Freedman's theorems about topological 4-manifolds, and Simon Donaldson's theorems about smooth 4-manifolds.
The relationship between general covariance and general relativity may be summarized by quoting a standard textbook: [3] Mathematics was not sufficiently refined in 1917 to cleave apart the demands for "no prior geometry" and for a geometric, coordinate-independent formulation of physics.
The cases n = 1 and 2 have long been known by the classification of manifolds in those dimensions. For a PL or smooth homotopy n-sphere, in 1960 Stephen Smale proved for n ≥ 7 {\displaystyle n\geq 7} that it was homeomorphic to the n -sphere and subsequently extended his proof to n ≥ 5 {\displaystyle n\geq 5} ; [ 3 ] he received a Fields ...