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C00002 00002	{⊂C<NαRESULTS AND CONCLUSIONS.λ30P100I425,0JCFA}   SECTION 10.
C00006 00003		As a  system design, the  present work  can be compared  with
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C00014 00005	⊂10.2	Critique: Errors and Ommissions.⊃
C00019 00006	⊂10.3	Suggestions for Future Work.⊃
C00022 00007		Future  development  of <Combination  Geometric  Models>  may
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{⊂C;<N;αRESULTS AND CONCLUSIONS.;λ30;P100;I425,0;JCFA}   SECTION 10.
{JCFD}  RESULTS AND CONCLUSIONS.
{λ10;W250;JAFA}
	10.1	Results: Accomplishments and Original Contributions.
	10.2	Critique: Errors and Ommissions.
	10.3	Suggestions for Future Work.
	10.4	Conclusion.
{λ30;W0;I800,0;JUFA}
⊂10.1	Results: Accomplishments and Original Contributions.⊃

	As a regular feature in a Ph.D.  dessertation, it is required
to explicitly state  what has been accomplished and what is original.
Some of what has been accomplished is itemized in box 10.1;  with the
so called <original contributions> marked  by asterisks.  Each of the
accomplishments has been elaborated in the indicated chapter.
{|;λ10;T150,165,900;JA;FA}
BOX 10.1{JC} ACCOMPLISHMENTS AND ORIGINAL CONTRIBUTIONS.
		0. The Geometric Feedback Vision Theory 	chapter 6.
	*	1. The Winged Edge Polyhedron Representation	chapter 2.
	*	2. The Euler Primitives for Polyhedron Construction	chapter 3.
		3. The Iron  Triangle Camera Locus Algorithm	chapter 9.
	*	4. The OCCULT hidden line elimination algorithm	chapter 4.
	*	5. The Polygon Nesting Algorithm	chapter 7.
	*	6. The Polygon Dekinking Method 	chapter 7.
		7. The Polygon Segmenting Method 	chapter 7.
		8. The Polygon Comparing Method 	chapter 8.
	*	9. Silhouette Cone Intersection 	chapters 5 and 9.
{|;T-1;λ30;JUFA}
	As a whole, the system described in  this thesis is the third
of  its kinds,   succeeding  the systems of  Roberts (1963)  and Falk
(1970). Although,   the modeling routines of  the present system  are
considerably more sophisticated than  were those of its predecessors;
improvement in the visual analysis routines is less dramatic and more
open to question.  The present image analysis differs  from the early
systems  in that emphasis is  placed on  the use  of multiple
images for the sake of parallax depth perception and in that several
spatially coherent image representations are combined (contour image,
mosaic image and raster image) to preserve the structure of the scene
through feature extraction rather  than following the earlier  design
paradigm  of  extracting  features   from  the  image  piecemeal  and
attempting to splice them together afterwards.

	As a  system design, the  present work  can be compared  with
earlier works  by comparing the block diagrams, the charcteristically
circular mandalas  of feedback  vision. Mandala like  diagrams
appear in (Newell), (Falk) Figure  4-7, page 78; (Grape) Figure 12.1,
page 242;  (Tenenbaum) Figure 1.13,  page 43; as well as in this work
Figure 6.1, page 70. The feedback mandala is  conspicuously absent in
the best  of the stimulus-response  visual parsing work,   (Waltz), as
well as in statistical recognition work, (Duda and Hart).
The important ideas depicted  in the feedback vision mandala  are
the duality of the simulated  and  physical worlds,  the  duality  of
the description  and verification,
the  duality of camera  and body locus
solving,  the dual opposing  flows of predicted and perceived  images
along  a  hieracry  of commensurate  abstractions.  Newell's  general
schema  of a  problem solver (embellished  with disembodied eyeballs)
has the two worlds dichotomy, but lacks the  compare steps.
Tenenbaum's  figure,   as well as  his thesis  as a  whole,
illustrates the basic feedback loop in the  immediate vicinity of the
visual  sensor. The  diagrams of Falk and  Grape are similar
mirrors of the overall system  design of the Stanford Hand/Eye  group
(1969  to  1973)  under  the leadership  of  Professor  Feldman.  The
two  diagrams depict an array of  relevant boxes (camera solver,
edge finder, world modeler and  so on) all sending messages  to each
other  under   the  benign  direction  of  a   box  labeled  "general
strategist".

	Among the elements composing the GEOMED/CRE system, the most
original accomplishment is the winged edge polyhedron representation.
In computer graphics  models are based  on face perimeter  lists
(or arrays), with an awareness  that more topological relations exist
but  with no  realization that  a substantial improvement  in surface
topology modeling is feasible with aprroximately the  same memory and
computational resources.

	The idea for the Euler primitives was based on a constructive
proof of the  Euler relation  found in (Coxeter  61), which  combined
with  a  fondness   for  sweep  operators  resulted  in   the  Euler
primitives.   Comparison  with other work  is difficult  since, other
graphics models lack a level of abstraction falling between the level
of  node/link  operations and  operations  with  solids.   The  Euler
primitives  were a blessing  in implementing OCCULT  and GEOMED sweep
and glue operations; the  Euler primitives were a deceptive  curse in
implementing the body intersector, BIN.

	A  pre-computer  form of  the  Iron  Triangle camera  solving
method appears in a paper by Berkay (59). Berkay described the method
as an analog procedure to be perform with paper, ruler and afew other
photogrammetric hand tools.  (The existence of this paper was pointed
out to me by Irwin Sobel).

	The original accomplishment  of the  hidden line  eliminator,
OCCULT lies in its unification of  methods and in its exploitation of
object and  image coherence made feasible by the Euler primitives and
the Winged Edge Representation.

	The last five accomplishments listed in box  10.1 are related
to vision.  The nesting and  dekinking problems have  been stated and
solved by  others,    the  present solutions  are  original  only  in
technical  detail the  nesting  for  its use  of  memory to  avoid  a
combinatorial number of compares and the dekinking in its achievement
of good  results  with  almost  no effort.    The  recursive  polygon
segmentation  idea and  the polygon  compare  idea have  been in  the
vision  and graphics  oral  tradition for  as long  as I  have (since
1967); although I have found no references for the methods.

⊂10.2	Critique: Errors and Ommissions.⊃

	The major weakness in the existing modeling system is that it
lacks  overall unity -  the modeling and  image anaylsis  are not yet
sufficiently integrated.    The second  major  weakness is  that  the
essential  subsystems   involving  comparing,    locus   solving  and
recognition  are  still in  a  primitive condition.  Consequently, an
unambiguous objective demonstation  of the relevance of  3-D modeling
to computer  vision is missing; the particular  demonstration which I
had in mind  was to  have a robot  vehicle drive  outside around  the
laboratory visually servoing along a trajectory given in advance.
{λ9;|;JA}
Box 10.2 {λ9;JAJC} ITEMS WHICH SHOULD HAVE BEEN DONE YESTERDAY.
	1.	System unity: Image Anaylsis with 3-D Geometric Models.
	2.	Image Compare Problems.
	3.	Locus Solving Problems.
	4.	3-D Geometric Recognition.
{|λ30;JUFA}
	In the course of this work,  technical failures have included
the  attempt to use  Euler primitive to  implement body intersection,
the attempt to bundle contour  images into mosiac images, as well  as
attempts  to  make the  Euler  kill  primitives logically  air  tight
without time consuming model checking. The worst system design errors
are of the  form of  misallocated effort. More  time might have  been
spent on image analysis  programming and less time on image synthesis
work and  so forth.  The research  suffers from  having no  objective
criterion for  deciding which of  several possibilities  deserves the
most immediate effort.

	A final great barrier  to progress in computer vision  is the
inadequacy of the hardware. It may be true that "It is a poor workman
who blames  his tools";  but for  me the  greatest  single source  of
personal  frustration has  been the  television cameras  and robotics
hardware: cart and turntable. At Stanford these devices have not been
implemented  or  maintained   with  sufficient  care  to   make  them
convenient to use.
{Q}
⊂10.3	Suggestions for Future Work.⊃

	The application of geometric modeling to vision and robotics
raises numerous interesting ideas and problems, box 10.3.
{|λ9;JA}
Box 10.3 {λ9;JAJC} SUGGESTIONS FOR FUTURE WORK.

SPATIAL MODELING WORK.
	0.	Combination Geometric Models - Converters.
	1.	Cellular Space Modeling - Tetrahedral Simplices.
	2.	Spatial Simulation: Collision Avoidance Problem.
	3.	Higher Dimensionality, 4-D GEOMED.
SIMULATIONS.
	4.	Mechanical Simulation.
	5.	Creature Simulations.
	6.	Geometric Task Planning.
	7.	Geometric/Semantics Modeling.
MATHEMATICALLY ORIENTED PROBLEMS.
	8.	The Manifold Resurfacing Problem.
	9.	The Curved Patchs Problem.
	10.	Prove the Correctness of a Hidden Line Eliminator.
GET RICH QUICK APPLICATIONS.
	11.	Automatic Machine Shop.
	12.	Animation for Entertainment Industry.
SYSTEMS SOFTWARE AND VISION HARDWARE WORK.
	13.	Better Loader and/or Incremental Assembler.
	14.	Better Cameras.
	15.	Image Oriented Number Crunching Computer Hardware.
	16.	Better Robot Vehicles.
{|λ30;JUFA}
	Future  development  of <Combination  Geometric  Models>  may
begin  by writing  converters between geometric  representations. For
example, there  is  a need  to  convert  polyhedra into  spine  cross
sections, space  points into  polyhedra,   contour maps  into faceted
surfaces   and   so   on.  Extramural   combination   models  include
<Geometric/Semantic Modeling> which will be needed to  cover the gulf
between  Minsky's  (1974)  notion  of  a  visual  frame-system  (e.g.
expectation of a room) and a geometric prediction of the features  to
be found in the image.  Although  the Minsky Frame-System theory does
not  explicitly   reveal  the  crucial  interface  between  numerical
geometric modeling and symbolic abstractions, that nexus is a central
part of the frame-system idea.

	The <Cellular  Space Modeling>  idea is  that both  space and
objects should be modeled using a space filling tesselation of cells;
perhaps  using  the tetrahedral  3-simplex.  The  difficultly  is  in
getting the Euclidean primitives to correctly update the geometry and
topology of empty space as an  object moves and rotates. The  rewards
might an include an elegant approach  to collision avoidance problems
in  vehicle navigation and arm  trajectory planning. Other approaches
to <spatail simulation> and <collision avoidance problems> that might
be pursued  is the use  of simulated  viewpoint to see  obstacle free
trajectories  by means  of  hidden line  elimination, this  method is
suggest in (Sutherland 69).

	In several recent Stanford  dissertations, (Falk, Yakimofsky,
Grape,  and so  on.) the  authors conclude  with the  prediction that
their essentially 2-D  techniques can readily be  extended to 3-D  in
future  work.   In my  turn,   I seriously  wish to  propose that  my
essentially  3-D techniques  can  be extended  to 4-D.  The resulting
models could be applied  to Regge Calculus for computing  the general
relativistic  geometric  models  of  such  systems as  two  or  three
colliding blackholes or on a less cosmic level a 4-D Geomed could  be
of service for  planning sequences of arm manipulations  viewing time
as a spatial  dimension. Collision of 3-D polyhdera moving in
time can be discribed as a static intersection of 4-D polytopes.

	Geometric  modeling is  also  applicable  to future  work  in
simulation. <Mechanical  Simulation> involves computing the Newtonian
mechanics  of everyday  objects,    problems  which  are  immediately
approachible  from  a  GEOMED  foundation  include  simulated  object
collision,  statics, and pseudo friction.  For example, consider what
is needed  to predict the  out come of  setting one  more block at  a
given  place on an  existing tower  or of throwing  one block  into a
tower of other blocks. <Geometric Task Planning> problems include the
old  A.I.   favorite  block stacking  as  well as  the  newer
research problems  related to industrial assembly. Existing solutions
to geometric tasks are notoriously restricted, for example I  know of
no blocks  stacking program  that handles  arbitrary rotations  - all
blocks  to   date  are  piled  on  the  square.

	Although,   it  has  been  recognized  early and  often  that
numerical  control of machine  tools should be  automated, the actual
implementation of  a system  that builds  artifacts  directly from  a
geometric  modeling program  still lies  in the  future. As  a start,
someone  at  any  of  the  research  labs  with  an  general  purpose
manipulator could begin by carving  models out of soap or  other soft
material with a rotating cutting tool.

	Advanced mechanical  simulations  as well  as <Animation  for
Entertainment> quickly  run into the problem of <Creature Simulation>
- given a multilegged bug,  what control program is required  to make
the bug  walk through a building.  Barring the darkness of  war,  the
greatest potential  future use of robotic simulation will be required
not by  governments, universities,  or  manufacturing industries  but
rather  by the entertainment  industry. When it  becomes economically
feasible to  generate  motion  pictures and  television  programs  by
computer graphics, great  progress will be made in  simulating visual
reality and in representing mundane situations in a computer.

	Theoretical  work  in geometric  modeling  will  continue  to
pursue  curved representations. Two problems  that I would especially
like to see solved involve  fitting curved surfaces to form a  smooth
object, (a  manifold), as  well as  resurfacing an  existing manifold
representation.   Both  problems   involve  more   segmentation  than
smoothing. It is easy to fit functions to facial patchs of an object,
it is hard to subdivide  an object into the proper set of patchs. The
one geometric algorithm  that seems most  ripe for future  quantative
study and logical analysis is the hidden line elimination process.

	Finally progress  in computer  vision and geometric  modeling
requires  progress in systems software  and computer systems.   In my
opinion, recent university based research in programming languages is
over concentrated  in very high  level language theory  and automatic
programming.    Future  language  and  systems  work  should  include
developing an  incremental loader/assembler/debugger/editor that  can
handle  algebraic  expressions,  block structure,  node/link  storage
notation  as  well  as  unvarnished  machine  instructions.  Although
special purpose image processing hardware has earned a bad reputation
(starting  with the Illiac-III); in  my opinion a  real vision system
will be composed of a large array of computer like elements  (4096 by
4096)  that  pipeline  a  stream  of  images  into  structured  image
representations.    The  perceived  images  are  then  compared  with
predicted images and a  detailed 3-D model is altered  or constructed
in real time (24 images per second) using a small number of computers
(32 or less) which by the standards  of our day (1974) would be  very
large and very fast  (ten megawords of main memory  and ten megahertz
instruction execution).

	Assuming  the continuation  of  civilization with  a  growing
technology   over  the  next   one  hundred  to   a  thousand  years.
developments in  Computer Vision  and Artificial  Intellegence  could
lead to  robots, androids and cyborgs  which will be able  to see, to
think  and to feel  conscious. The utility of  building (or becoming)
such entities  is that  as an  android one  would be  smarter,   more
sensitive  and would  live longer  - one  could live  long  enough to
explore the galaxy.

⊂10.4 Conclusions.⊃
	
	The particular technical conclusions of this work include the
methods,  system  designs and data structures for  geometric modeling
which have already  been elaborated.  Based on the details, one could
make such generalized observations as that: recursive windowing  is a
good technique for spatial  sorting, simple geometric representations
fall into space oriented and object oriented classes,  the essence of
an object representation is its coherence under various operators and
that the power of a vision system might be enhanced by application of
3-D modeling techniques.  However in  closing, I would  like to  draw
three rather more general conclusions,  conclusions which by contrast
to the technical ones might be constued as scientific conclusions.

	1.  ~<The Nature of Perception>~.  Perception is essential to
intelligence as it is the process which converts  external sensations
into internal  thoughts.   There are two  kinds of  simple perception
systems:   Stimulus-Response   and   Prediction-Correction  Feedback;
together S-R. and  P-C.F.  can be  formed into a compound  perception
system.

	2.  ~<The  Necessity  to  Experiment>~. Robotic  hardware  is
essential to Artificial Intelligence  as an experimental science.  It
is  misleading  to  study  only  theoretical  robotics  of  plausible
abstractions,   mathematics, puzzles, games and simulations. The real
physical world is the best test of adaptive general intelligence. The
complexity and subtlety of real world situations, even of a situation
as  seemingly finite  as a  digital television  picture,  can  not be
anticipated from  a  philosopher's armchair  or  from a  programmer's
console.

	3. ~<The Necessity  to Simulate Visual Reality>~. Modeling is
essential to  prediction-correction  feedback perception.    Although
simulated robot  environments  should not  be used  in  place of  the
external  physical reality,   such  environmental simulations  are an
essential  part of  a  robot's  internal  mental  reality.    In  the
particular case of vision,  geometric  models should be easy to adapt
to  the basic mental  abilities of present day  computer hardware. To
conclude,  perception  requires two worlds  one that is the  external
physical reality and the other which is the internal mental reality.