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C00002 00002	{≥G⊂CαLOCUS SOLVINGλ30P109JUFA}
C00006 00003	{P114JUFA}⊂9.4	Sun Locus Solving: A Simple Solar Ephemeris.⊃
C00010 00004	⊂9.5	Related and Future Locus Solving Work.⊃
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{≥G;⊂C;αLOCUS SOLVING;λ30;P109;JUFA}
⊂9.3	Object Locus Solving: Silhouette Cone Intersection.⊃

	After the  camera  location, orientation  and projection  are
known;  3-D object models  can be  constructed.  The  silhouette cone
intersection method  is a  conceptually simple  form  of wide  angle,
stereo reconstruction.  The idea arose out of an original intention 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.

	Figure  9.3 shows  four video  images  and the  corresponding
silhouette contours of a  baby doll on a turn table. Figure 9.4 is an
overhead view of the four  silhouette cones that were swept from  the
contours, the  circle in the middle  of Figure 9.4 is  the turntable.
Figure  9.5  gives  three  views (cross  eyed  stereo  pairs)  of the
polyhedron that  resulted  by taking  the  intersection of  the  four
silhouette cones. Like in the  joke about carving a statue by cutting
away everything that does not look like the subject, the  approximate
shape  of  the doll  is  hewed  out  of  3-D space  by  cutting  away
everything that falls outside of the silhouettes. A second example of
silhouette cone intersection is depicted in Figure 9.6; the model was
made from three silhouettes of the horse facing to the left which can
be  compared with  an initial  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 does indeed 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.{Q}
{P114;JUFA}⊂9.4	Sun Locus Solving: A Simple Solar Ephemeris.⊃

	The location of the  sun is useful to a  robot vehicle vision
system  both for sophisticated scene  interpretation and for avoiding
the  blunder of  burning  holes  in  the  television  vidicon.    The
approximate position of  the sun in the sky is  readily computed from
the  time,   date  and  latitude using  circular  approximations. The
longitude is implicitly used to compute Local Solar Time,   since the
Stanford A.I.   Lab is 122  degrees 10 minutes west  of the Greenwich
meridian, Local  Solar time  is 8  minutes, 44  seconds earlier  than
Pacific Standard  Time  (120 degrees  west). The  orientation of  the
earth with  respect to the sun follows  from remembering that the sun
is highest at noon. The tilt  of the earth with respect to its  orbit
is 23.45 degrees, so in earth centered coordinates the sun appears to
circle  the earth counterclockwise crossing the  plane of the equator
from south to  north on the  spring equinox, March 21.
The  SUNLOCUS procedure given below computes  the local azimuth
and altitude of the sun  in the sky, given  the number of days  since
March 21,  the time  in seconds  since midnight  and the latitude  in
radians.

{λ10;JAF3;}
PROCEDURE SUNLOCUS (REAL DAY,TIME,LAT; REFERENCE REAL SUNAZM,SUNALT);
BEGIN
	REAL RHO,PHI,TMP,ECLIPTIC,NORTH,EAST,ZENITH;
COMMENT POSITION OF THE SUN ON THE ECLIPTIC IN THE CELESTIAL SPHERE;
	ECLIPTIC←	((23+27/60)*PI);
	RHO	←	2*PI*DAY/365.25;
	EAST	←	SIN(RHO)*COS(ECLIPTIC);
	NORTH	←	SIN(RHO)*SIN(ECLIPTIC);
	ZENITH	←	COS(RHO);
COMMENT LOCAL SOLAR TIME, OVER THE MAST AT NOON;
	TIME	←	TIME - (8*60 + 44);
	PHI	←	PI*(1-TIME/(12*3600)) - ATAN2(EAST,ZENITH);
	TMP	←	ZENTITH*COS(PHI) - SIN(PHI)*EAST;
	EAST	←	EAST*COS(PHI) + SIN(PHI)*ZENITH;
	ZENITH	←	TMP;
COMMENT ROTATE CLOCKWISE IN THE NORTH/ZENITH PLANE TO LOCAL LATITUDE;
	TMP	←	COS(LAT)*ZENITH + SIN(LAT)*NORTH;
	NORTH	←	COS(LAT)*NORTH  - SIN(LAT)*ZENITH;
	ZENITH	←	TMP;
CONVERT TO ANGULAR MEASURES;
	SUNAZM	←	ATAN2(NORTH,EAST);	COMMENT AZIMUTH FROM DUE EAST;
	SUNALT	←	PI/2 - ACOS(ZENTIH);	COMMENT ALTITUDE ABOVE HORIZON;
END "SUNLOCUS";
{λ30;JUFAQ;}
⊂9.5	Related and Future Locus Solving Work.⊃

	The  camera solving  problem is  discussed  in Roberts  (63),
Sobel (70) and Quam (71). I have always disliked the many dimensional
hill  climbing approach  to  camera  solving  and  have  sought  more
geometric and intuitive  solutions to the problem.  Although the bulk
of  this chapter  concerned camera  solving using  one view  of three
points the multi view  camera calibration is probably  more important
to continuous image processing.