Abstract
This thesis deals with two unrelated issues, restricting curve evolution to subspaces and computing image
patches in the equivalence class of Histogram of Gradient orientation based features using nonlinear
projection methods.
Curve evolution is a well known method used in various applications like tracking interfaces, active
contour based segmentation methods and others. It can also be used to study shape spaces, as deforming
a shape can be thought of as evolving its boundary curve. During curve evolution a curve traces out
a path in the infinite dimensional space of curves. Due to application specific requirements like shape
priors or a given data model, and due to limitations of the computer, the computed curve evolution forms
a path in some finite dimensional subspace of the space of curves. We give methods to restrict the curve
evolution to a finite dimensional linear or implicitly defined nonlinear subspace of curves. We also deal
with cases where a non-Euclidean metric is induced on such a subspace. We build differential geometric
tools like the Exponential map and Log map which are essential for the study of such nonlinear spaces.
We demonstrate these tools on a particular implicitly defined subspace, the N-links bicycle chain space,
i.e. the space of curves with equidistant neighboring landmark points. This in itself is a useful shape
space for medical image analysis applications.
The Histogram of Gradient orientation based features are many in number and are widely used in
applications like object recognition which is a vital component of any computer vision system. In order
to get some intuition behind their success, we attempt to explore the metameric class of a basic version
of such features. We characterize such an equivalence class using implicitly defined constraints over
the statistical moments of the gradient orientations. This is another case for use of nonlinear projection
methods since such an equivalence class is nonlinear. We use an approximation of the Exponential map
developed in the first part of the thesis to evolve a given patch in the equivalence class. Specifically, given
two initial visually different patches, we evolve one patch into another patch that visually looks like the
other given patch, while still preserving its Histogram of Gradient orientation features.
patches in the equivalence class of Histogram of Gradient orientation based features using nonlinear
projection methods.
Curve evolution is a well known method used in various applications like tracking interfaces, active
contour based segmentation methods and others. It can also be used to study shape spaces, as deforming
a shape can be thought of as evolving its boundary curve. During curve evolution a curve traces out
a path in the infinite dimensional space of curves. Due to application specific requirements like shape
priors or a given data model, and due to limitations of the computer, the computed curve evolution forms
a path in some finite dimensional subspace of the space of curves. We give methods to restrict the curve
evolution to a finite dimensional linear or implicitly defined nonlinear subspace of curves. We also deal
with cases where a non-Euclidean metric is induced on such a subspace. We build differential geometric
tools like the Exponential map and Log map which are essential for the study of such nonlinear spaces.
We demonstrate these tools on a particular implicitly defined subspace, the N-links bicycle chain space,
i.e. the space of curves with equidistant neighboring landmark points. This in itself is a useful shape
space for medical image analysis applications.
The Histogram of Gradient orientation based features are many in number and are widely used in
applications like object recognition which is a vital component of any computer vision system. In order
to get some intuition behind their success, we attempt to explore the metameric class of a basic version
of such features. We characterize such an equivalence class using implicitly defined constraints over
the statistical moments of the gradient orientations. This is another case for use of nonlinear projection
methods since such an equivalence class is nonlinear. We use an approximation of the Exponential map
developed in the first part of the thesis to evolve a given patch in the equivalence class. Specifically, given
two initial visually different patches, we evolve one patch into another patch that visually looks like the
other given patch, while still preserving its Histogram of Gradient orientation features.
| Original language | English |
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| Place of Publication | København |
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| Publisher | Faculty of Science, University of Copenhagen |
| Number of pages | 121 |
| Publication status | Published - Jan 2010 |