00100	The fourier transform model of vision is a version  of  the  template
00200	matching  paradigm,  that  the  primary  task  of  a visual system is
00300	recognition, ie matching  an  image  (of  an  isolated  object)  with
00400	templates  from previous images.   The templates could be portions of
00500	images or quantities derived from images, quantities such as  moments
00600	or fourier coefficients. The set of problems motivating this paradigm
00700	is classification of isolated, two-dimensional forms  among  a  small
00800	set  of  possibilities;  character recognition is typical. We contend
00900	that template matching is inadequate for the visual requirements of a
01000	human;  even so, the fourier model is not very effective for template
01100	matching.  A model for template matching  is:  portions  of  previous
01200	images  are  matched  against  similar portions of the current image.
01300	What part of the original image should be taken as a  template?    It
01400	appears  that whatever system is capable of segmentation is a primary
01500	part of the visual system, and not the template  matching  mechanism.
01600	The  templates  could  be  supplied  by  revelation, or inferred from
01700	another (non template matching) facility  such  as  motion.   In  the
01800	paradigm,  the  chief  difficulty  is  the  computational  effort  in
01900	matching  templates  to  scenes  involving  rotation,  dilation   and
02000	translation  of objects.   We contend that real world visual problems
02100	involve articulation, obscuration, and  judgments  of  similarity  of
02200	objects  which  are  arbitrarily  different  according to non-trivial
02300	metrics of template matching.  The class of cups is not identified by
02400	any  unique  global  shape  or  by  enumeration,  but  by  being open
02500	containers (capable of holding liquid) of  a  certain  volume.  Thus,
02600	similarity  depends  on  a  description  of  three-dimensional  form,
02700	similarity  judgments  often  also  depend  on   the   facility   for
02800	segmentation,   and  local  description.  Even  within  the  template
02900	matching  paradigm,  the  fourier   transform   model   has   serious
03000	difficulties.   The  fourier  transform is equivalent to the original
03100	image; the model is useful only if there are  great  simplifications.
03200	The  supposed  advantage  of  the  fourier transform is translational
03300	invariance. However, the transform is translationally invariant  only
03400	for  periodic  functions.   Objects are finite and images are finite.
03500	In  this  case,  the  transforms  are  position  dependent.   Another
03600	difficulty  is  thatif  there  are  several  objects  in a scene, the
03700	fourier transform depends on  all  of  them.   The  spatial  template
03800	matching at least allows some localization to the template.
03900	
04000	Experiments   show   that  linear  systems  analysis  is  useful  for
04100	describing  response  of  the  visual  system  to  various   stimuli.
04200	Further,  there  appear to be separate channels with about one octave
04300	width.  The alternative, that the system  acts  as  a  single  linear
04400	filter,  seems  like  a  straw man set up only to be knocked down, in
04500	view  of  the  wide  range  of  tasks  facing  the   visual   system.
04600	Experiments show that the detection threshold for square waves can be
04700	predicted adequately from the component of the fundamental frequency.
04800	Further,  square  waves  are  distinguishable from sine waves only at
04900	contrasts such that the third  harmonic  is  above  threshold.  These
05000	results  do  not  discriminate  against  an edge detection mechanism,
05100	however, since the ratios of sensitivities for square  wave  to  sine
05200	wave   would   be  very  similar.  and  whatever  means  is  used  to
05300	discriminate betwee the two stimuli must  give  similar  results,  ie
05400	must  be  dominated by the third harmonic. Any edge detector of fixed
05500	size must have a frequency response with approx one octave width.  If
05600	for  example, the spatial weighting is unity across the detector, the
05700	response is zero for sine  waves  with  period  half  the  width  and
05800	response is half for sine waves with period twice the width.
     

00100	Much  work  has  been  addressed  to  what  might  be  called  global
00105	representations.  Typical are prototypes of blocks, cylinders, cones,
00110	etc.  The basic hope is that by description of these simple forms, we
00115	are starting out to enumerate all shapes.   We  would  contrast  this
00120	approach   with   that  of  locally  generated  representations.    A
00125	representation useful for our work with complex objects was chosen to
00130	satisfy design criteria which are relevant to human perception.   The
00135	representation must be generative, that is,  a  rich  set  of  shapes
00140	should be conveniently generated from local primitives.   For that to
00145	be effective, the representation  should  have  a  segmentation  into
00150	parts,  which  themselves  may  be  formed  of  other parts.  For the
00155	part/whole segmentation to be effective, the primitive parts must  be
00160	naturally  defined  and  computationally adequate.  We have chosen to
00165	define  primitive  parts  by  continuity.   This  implies  that   the
00170	primitives  are  volume  primitives and not surface primitives, since
00175	surfaces are discontinuous for  objects  with  what  we  conveniently
00180	think of as a single part (eg a block).
00185	
00190	The  representations  depend  upon  segmentation  to describe complex
00195	objects in terms of parts.  The representations are graph-structured;
00200	nodes correspond to parts, while arcs correspond to relations between
00205	parts.   Parts may be compound parts, ie graphs  of  the  same  form.
00210	Relations  include  relative  position  and  orientation,  degrees of
00215	freedom,  symmetry  and  any  special  knowledge  available.      The
00220	topological  operations  of  cutting  and pasting are used in joining
00225	parts.  Normally, we think of holes as made by cuts and protuberances
00230	as  made  by pasting; we choose to generalize that to a choice of the
00235	simpler of structures, depending upon  fabrication  techniques.   For
00240	example, metal cutting is often used to form protuberances.
00245	
00250	Primitve  parts are arm-like, described as "generalized local cones".
00255	These parts are described formally in  terms  of  "generalized  local
00260	translational  invariance", appropriate to parts whose cross sections
00265	change slowly along some space curve.  A general cylinder  is  formed
00270	by  an  arbitrary cross section translated along a straight line.   A
00275	cone has a linear variation of cross section along this axis. If  the
00280	scale of the cross section is varied smoothly along the axis, we have
00285	what might be called a local cone. If the cross section is allowed to
00290	vary by distortion or rotation, and the axis is allowed to be a space
00295	curve, then we have "generalized local cones". These are the  volumes
00300	swept  out  by  taking  an arbitrary cross section and translating it
00305	along a space  curve,  meanwhile  varying  the  cross  section  while
00310	holding the cross section normal to the path.
00315	
00320	Cross  sections are represented in the same fashion in two dimensions
00325	as objects are in three dimension.  Again, part/whole segmentation is
00330	crucial.   We again require that parts be defined by continuity. Thus
00335	the parts must be area parts,  not  curve  parts,  since  curves  are
00340	discontinuous  for plane figures which we normally think of as having
00345	a single part.  The primitives are described by a cross section (one-
00350	dimensional) varying smoothly along a plane curve.
00355	
00360	Each  element has a local coordinate system.  Each joint contains the
00365	transformation necessary to go from  the  coordinate  system  of  one
00370	element to that of the other.
00375	
00380	The process of segmentation allows non-unique representations.   This
00385	permits a  choice  of  simple  representations;  we  can  regard  the
00390	non-uniqueness as an advantage.