Discrete shapes can be described and analyzed using Lie groups, which
are mathematical structures having both algebraic and geometrical
properties. These structures, borrowed from mathematical physics, are
both algebraic groups and smooth manifolds. A key property of a Lie
group is that a curved space can be studied, using linear algebra, by
local linearization with an exponential map.
Here, a discrete shape was a Euclidean-invariant computer
representation of an object. Highly variable shapes are known to
exist in non-linear spaces where linear analysis tools, such as
Pearson's decomposition of principal components, are inadequate. The
novel method proposed herein represented a shape as an ensemble of
homogenous matrix transforms. The Lie group of homogenous transforms
has elements that both represented a local shape and
acted as matrix operators on other local shapes. For the
manifold, a matrix transform was found to be equivalent to
a vector transform in a linear space. This combination of
representation and linearization gave a simple implementation for
solving a computationally expensive problem.
Two medical datasets were analyzed: 2D contours of femoral
head-neck cross-sections and 3D surfaces of proximal femurs. The
Lie-group method outperformed the established principal-component
analysis by capturing higher variability with fewer components. Lie
groups are promising tools for medical imaging and data analysis. / Thesis (Ph.D, Computing) -- Queen's University, 2014-01-30 09:49:03.293
| Identifer | oai:union.ndltd.org:LACETR/oai:collectionscanada.gc.ca:OKQ.1974/8597 |
| Date | 30 January 2014 |
| Creators | Hefny, Mohamed Salahaldin |
| Contributors | Queen's University (Kingston, Ont.). Theses (Queen's University (Kingston, Ont.)) |
| Source Sets | Library and Archives Canada ETDs Repository / Centre d'archives des thèses électroniques de Bibliothèque et Archives Canada |
| Language | English, English |
| Detected Language | English |
| Type | Thesis |
| Rights | This publication is made available by the authority of the copyright owner solely for the purpose of private study and research and may not be copied or reproduced except as permitted by the copyright laws without written authority from the copyright owner. |
| Relation | Canadian theses |
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