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C00002 00002 {F1F2F3⊂C<NαWINGED EDGE.λ30P13JUFA
C00005 00003 ⊂6. Use of Polyhedra in Computer Vision.⊃
C00010 00004 {W0,675JUFA} An example of a purely descriptive vision technique
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{F1;F2;F3;⊂C;<N;αWINGED EDGE.;λ30;P13;JUFA
}⊂5. Coordinate Free Polyhedron Representation.⊃
As in general relativity, all geometric entities can be
represented in a coordinate free form. In particular, the vertex
coordinates of a polyhedron can be recovered from edge lengths and
dihedral angles (the angle formed by the two faces at each edge).
Having the geometry carried by only two numbers per edge rather than
by three numbers per vertex does not necessarily yield a more concise
representation because edges always outnumber vertices two for one,
and in the case of a triangulated polyhedron edges outnumber vertices
by three to one.
One application of a coordinate free representation arises
when it is necessary to measure a shape with simple tools such as a
caliper and straight edge. For example, one way to go about recording
the topology and geometry of an arbitrary object is to draw a
triangulated polyhedron on its surface with serial numbered vertices
and to record for each edge its length, its two vertices and its
<signed dihedral length>. The dihedral length is the distance
between the vertices opposite the edge in each of the edge's two
triangles; the length can be given a sign convention to indicate
whether the edge is concave or convex. The required dihedral angles
can then be computed from the signed dihedral lengths.
⊂6. Use of Polyhedra in Computer Vision.⊃
My approach to computer vision is best characterized as
inverse computer graphics. In computer graphics, the world is
represented in sufficient detail so that the image forming process
can be numerically simulated to generate synthetic television images;
in the inverse, perceived television pictures (from a real TV camera)
are analysed to compute detailed geometric models. For example, the
polyhedra in Figure 6 was computed from views of a plastic horse on a
turntable. It is hoped, that visually acquired 3-D geometric models
can be of use to other robotic processes such as manipulation,
navigation or recognition.
Once acquired, a 3-D model can be used to anticipate the
appearance of an object in a scene, making feasible a quantitative
form of visual feedback. For example, the appearance of the machine
{W0,675;JUFA
}\parts depicted in Figure 5 can be computed and analyzed
and compared with an anaylsis of an actual video image of the parts.
By comparing the predicted image with a perceived image, the
correspondence between features of the internal model and features of
the external reality can be established and a corrected location of
the parts and the camera can be measured.{
W0,1250;↓;I200,1050;FA} FIGURE 5 {H2;L400,580;*PUMP02;↑;JU;FA;}
{W0,675;JUFA} An example of a purely descriptive vision technique
is the silhouette cone intersection method, which is a conceptually
simple form of wide angle stereo reconstruction. The idea arose out
of an original intention {W0,1260;I∂-56,220;
}\to do "blob" oriented visual model
acquisition, however a 2-D blob came to be represented by a
silhouette polygon and a 3-D blob consequently came to be represented
by a polyhedron. The present implementation requires a very favorably
arranged viewing environment (white objects on dark backgrounds or
vice versa); application to more natural situations might be possible
if the necessary hardware (and software) were available for
extracting depth discontinuities by bulk correlation. Furthermore,
the restriction to turntable rotation is for the sake of easy camera
solving; this restriction could be lifted by providing stronger
feature tracking for camera calibration.
Like in the joke about carving a statue by cutting away
everything that does not look like the subject, the approximate shape
of the horse is hewed out of 3-D space by cutting away everything
that falls outside of the silhouettes. An example of silhouette
cone intersection is depicted in Figure 6; the model was made from
three silhouettes of the horse facing to the left which may be
compared with a video image and a final view of the result
of the horse facing to the right - a plausible (maximal) backside has
been constructed that is consistent with the front views.
The silhouette cone intersection method can construct
concave objects and even objects with holes in them - what are missed
are concavities with a full rim, that is points on the surface of the
object whose tangent plane cuts the surface in a loop that encloses
the point.