THESIS
2015
xxii, 168 pages : illustrations ; 30 cm
Abstract
Geometric model representation schemes have become more advanced in
structure and mathematical complexity over the years. Boundary representations
are widely accepted in geometric modeling due to their effectiveness in graphics
rendering. Complex geometries are formed by polygonal mesh or patching
analytical surfaces. Polygonal mesh representation is favoured due to its flexibility
in shape modeling and ease in rendering while it suffers from large data size and
failure to represent sharp features accurately. Patched analytical surface modelling
has a much small data size thus favoring file transfer in mobile network, but
complex shape modelling demanding higher order representation with stringent
smoothness requirement remains a challenge. Individual extensions of polygonal
m...[
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Geometric model representation schemes have become more advanced in
structure and mathematical complexity over the years. Boundary representations
are widely accepted in geometric modeling due to their effectiveness in graphics
rendering. Complex geometries are formed by polygonal mesh or patching
analytical surfaces. Polygonal mesh representation is favoured due to its flexibility
in shape modeling and ease in rendering while it suffers from large data size and
failure to represent sharp features accurately. Patched analytical surface modelling
has a much small data size thus favoring file transfer in mobile network, but
complex shape modelling demanding higher order representation with stringent
smoothness requirement remains a challenge. Individual extensions of polygonal
mesh and patched analytical surfaces for complex object modeling are continuously
being proposed while build complex shape model by integrating the two
representation have not been fully exploited.
In this thesis, shapes are classified into features with two separate levels of
shape complexity. For low complexity shape features, they are defined using
parametric patched analytical surface with up to C
2 continuity requirement. On the other hand, complex shape features such as sculptures, streamlined objects, organic shape, are represented by polygonal mesh. Two separate methods are proposed respectively for two-dimensional and three-dimensional complex shape models.
For two dimensional shape models, a Non-Uniform Degree B-Spline Scheme
has been proposed to represent objects with sharp features embedded in a
parametric curve. By allowing non-uniform degree distribution along the
polynomial basis, the proposed parametric spline can combine different continuity
shapes into one single compact form. The scheme also works in higher dimensional
space by taking the tensor product of lower dimensional domains. The proposed
scheme generalizes the current basic-spline representation and provides a simple and efficient mathematical formulation in comparison with prior work. The work
has been extended to three-dimensional shape modelling showing significant
limitations which remain to be resolved.
For complex three-dimensional shape models, a Hybrid PN Mesh/Parametric
Surface Scheme has been proposed to blend a low level shape feature represented
by a patched analytical surface and a higher level shape feature represented by a
triangular mesh using a transitional surface. The planar mesh elements in the
triangular mesh shape model are transformed into PN triangles of cubic degree. This
transformation facilitates the blending of the trimmed low level shape feature to the
complex high level meshed shape feature with prescribed constraints. Energy
minimization is also applied to the transition surface to ensure smooth blending
between the features. The resulting shape model will be smooth, continuous and a
2-manifold model.
This work is the first step in an effort to construct complex shape model
using a mixed representation scheme of minimum data size which is much needed
in the modern world of wireless communication.
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