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Figure:
The model test002 rendered solid and with wireframe.

Figure:
The model test002 fully decimated, rendered solid and with wireframe.

However, when we try to fully decimate another model, specifically
test003 (see Figure ), our method
for edge detection gets greedy and marks edges that can be removed as
nonremovable (see Figure
. In this figure,
note that there are vertices along the edges. These are considered boundary
vertices[#!shroeder!#], and can be removed safely. The ideal solution for this
mesh on any simplification algorithm should be to remove all vertices (except
for the four corners), therefore creating a four vertex and two triangle
rectangle.
Figure:
The model test003 rendered solid and with wireframe.

Figure:
The model test003 fully decimated, rendered solid and with wireframe.

When we simplify a mesh we expect a visually accurate depiction of
the mesh, within a suitable threshold. Our algorithm can create ideal
representations from certain geometrically generic meshes. In Figure
, we use a geometrically basic
model: a cube. This cube has many similar vertices; more specifically, most
vertices lie flat on one of the six sides of the model. With help from edge
detection, decimating the model using our program produces the ideal result. In
Figure , we see that
the model has become minimal, since a cube that is represented by a triangle
mesh can contain at the very least eight vertices and 12 triangles.
Figure:
The model test004 rendered solid and with wireframe.

Figure:
The model test004 fully decimated, rendered solid and with wireframe.

Our algorithm sometimes decimates with less than ideal results. In Figure
, we see the original mesh of a
hollow cylinder, or a tube. Unlike the cube from the previous figures, when we
decimate the model, the mesh starts to collapse upon itself (see Figure
). After only completing eighty
percent of the vertex removal process, the model starts to produce jagged edges.
This is the result of our corner detection algorithm, which only marks vertices
as corners when all angles between faces incident upon the vertex are smaller
than a threshold. Since the top and bottom of the cylinder are round, there are
some angles between adjacent faces here that are not corners. If we did not
remove these vertices, we would also not remove the edges of a cube, thus
limiting the ability to simplify in many geometrically primitive cases.
Figure:
The model test005 rendered solid and with wireframe.

Figure:
The model test005 decimated to eighty percent, rendered solid and with
wireframe.

Here we explore how rendering a simplified mesh with smooth shading
enabled gives a better visual result. Figure
shows the original mesh of the test009 model. This image has many hard edges
that are a result of the representation of the model as a mesh of flat
triangles. In the next image (Figure ),
the original mesh appears to have less sharp edges: look at the tabletop in
particular. In the second image, the OpenGL option GL_SMOOTH has been enabled.
This means that the colors of all adjacent triangles are blended across each
triangle to make a smoother representation; hence, the mesh appears smoother.
Figure:
The model test009 rendered solid and with flat shading.

Figure:
The model test009 rendered solid and with smooth shading enabled.

The GL_SMOOTH option becomes even more valuable as we decimate the
model. In the next image (Figure ), the
mesh has been decimated to sixty percent of its original size. Removing 34,542
vertices from the model leaves 23,028 vertices that are still rendered. With
this few vertices still being rendering, the model looks much more blocky. In
Figure , we enable smooth shading:
the mesh appears less rough.
Figure:
The decimated model rendered solid and with flat shading.

Figure:
The decimated model rendered solid and with smooth shading enabled.

Figure shows a comparison of these models
in greater detail. This figure is an image of the two levels of detail (original
and sixty percent), each with flat shading and smooth shading enabled. The top
left is the original model with flat shading. The top right is the sixtypercent
decimated model with flat shading. Here we clearly see the difference in polygon
count. The model in the top right has significantly fewer polygons and looks
jagged. The bottom two viewports show two visually similar models. The bottom
left viewport is the original 114,880 polygon mesh, and the bottom right
viewport contains the decimated model with only 45,792 polygons. Both were
rendered with smooth shading; the only difference between the top right and
bottom right viewports is that smooth shading has been enabled in the bottom
viewport. We conclude that the GL_SMOOTH feature in OpenGL helps to increase
the visual quality of a simplified mesh.
Figure:
The model test009 rendered with flat shading on the top and smooth
shading on the bottom. The two levels of detail are the original model on the
left, and the decimated model on the right.

Next: Geometric Error
Up: Analysis
Previous: Analysis
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Tim Garthwaite
20020203