106,722 research outputs found
Singular solutions, momentum maps and computational anatomy
This paper describes the variational formulation of template matching
problems of computational anatomy (CA); introduces the EPDiff evolution
equation in the context of an analogy between CA and fluid dynamics; discusses
the singular solutions for the EPDiff equation and explains why these singular
solutions exist (singular momentum map). Then it draws the consequences of
EPDiff for outline matching problem in CA and gives numerical examples
Higher-Order Momentum Distributions and Locally Affine LDDMM Registration
To achieve sparse parametrizations that allows intuitive analysis, we aim to
represent deformation with a basis containing interpretable elements, and we
wish to use elements that have the description capacity to represent the
deformation compactly. To accomplish this, we introduce in this paper
higher-order momentum distributions in the LDDMM registration framework. While
the zeroth order moments previously used in LDDMM only describe local
displacement, the first-order momenta that are proposed here represent a basis
that allows local description of affine transformations and subsequent compact
description of non-translational movement in a globally non-rigid deformation.
The resulting representation contains directly interpretable information from
both mathematical and modeling perspectives. We develop the mathematical
construction of the registration framework with higher-order momenta, we show
the implications for sparse image registration and deformation description, and
we provide examples of how the parametrization enables registration with a very
low number of parameters. The capacity and interpretability of the
parametrization using higher-order momenta lead to natural modeling of
articulated movement, and the method promises to be useful for quantifying
ventricle expansion and progressing atrophy during Alzheimer's disease
Qualitative Analysis of the Classical and Quantum Manakov Top
Qualitative features of the Manakov top are discussed for the classical and
quantum versions of the problem. Energy-momentum diagram for this integrable
classical problem and quantum joint spectrum of two commuting observables for
associated quantum problem are analyzed. It is demonstrated that the evolution
of the specially chosen quantum cell through the joint quantum spectrum can be
defined for paths which cross singular strata. The corresponding quantum
monodromy transformation is introduced.Comment: This is a contribution to the Vadim Kuznetsov Memorial Issue on
Integrable Systems and Related Topics, published in SIGMA (Symmetry,
Integrability and Geometry: Methods and Applications) at
http://www.emis.de/journals/SIGMA
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