COMMENT ⊗   VALID 00004 PAGES
C REC  PAGE   DESCRIPTION
C00001 00001
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
C00013 ENDMK
C⊗;
{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.