00100	Neurophysiological studies suggest that cells in the mammalian visual
00200	cortex  extract  edges from retinal data [].   However, edges between
00300	regions  of  uniform  intensity  are  only  one   useful   perceptual
00400	construct.
00500	Surfaces of hair, leaves, feathers,and foam appear irregular;
00600	 others  are  textured.   Gibson
00700	argues   that   textures  and  gradients  of  texture  are  important
00800	perceptual constructs [].   AI  has just  begun to  deal  with
00900	textured  scenes  [].   Our  perceptual  goals are 
01000	interpretations  in  terms  of  objects  and  their  spatial
01100	relations.  Stereo  and  motion  parallax  depth  perception  provide
01200	information most closely related to these  perceptual  goals. 
01300	These abilities are apparently
01400	developed at birth [].  Perhaps other visual learning depends  on
01500	the segmentation structure provided by depth perception.
01600	
01700	Natural scenes usually  contain  objects
01800	with  well-defined  surface  boundaries.   The retinal projections of
01900	such scenes are patchworks of areas of obscuring objects. Some  edges
02000	correspond  to  boundaries between objects, some to interior edges of
02100	objects,  others  to  surface  markings,  others   to   shadows   and
02200	reflections. All signify features of potential interest, but there is
02300	no  direct  connection  between  intensity  boundaries  and   spatial
02400	interpretations (which are our perceptual goal). Thus, it is possible
02500	to find intensity discontinuities on a local basis, even though  some
02600	edges with low contrast will be missed.    It is not possible to find
02700	the "meaningful edges" on a local basis, ie to make a  "perfect  line
02800	drawing".  Present  techniques  of  finding  intensity discontinuities
02900	require extensive calculation and are not really adequate.
03000	 We  expect  that
03100	performance approaching human visual acuity inherently requires great
03200	computation  cost  (eg the retina has 1000 times as many cells as our
03300	tv  images),  and  that  substantial  economies  in  computation  are
03400	possible  only using sampling strategies which ignore much of the image.
03500	Thus, sampling coarsely allows detection of extended boundaries, from
03600	which  many  finer  details (but not all) can be determined. For most
03700	problems of interest, however, those reduced abilities are presumably
03800	adequate.
03900	We see much attention to strategies which take account of extension
04000	in space or continuity in time, or for which context greatly limits
04100	the resolution necessary, or the field of view.
04200	
04300	
04400	We prefer to discuss boundary finding (for regions  without  texture)
04500	in  terms  of computational components, local edge operators and edge
04600	organization procedures.  In the biological system,  the  local  edge
04700	operators are Hubel-Wiesel cells, defined over small disks (typically
04800	1 degree in  the  monkey).   These  cells  are  strongly  directional
04900	[angular  resolution]  and  there  are  cells for various directions.
05000	There are approximately [] of them.  A  computational  equivalent  is
05100	the  Hueckel operator [], defined over a small disk. This operator is
05200	directional in that it calculates moments of the  intensity  function
05300	in  a  few  directions  and determines the direction of best fit of a
05400	single edge on the disk. We can estimate the  computational  cost  of
05500	sampling  a  picture  coarsely with this operator.  We want to detect
05600	edges at all directions. We look  for  long  features,  thus  we  are
05700	content  to  sample  at  a  few  positions  along  edges, but at many
05800	positions perpendicular to an edge.   It  is  enough  to  sample  the
05900	Hueckel operator at every other point along widely separated vertical
06000	and horizontal lines. Let us say we use 8 horizontal and  8  vertical
06100	lines.   At  about 7 msec per application, this amounts to 15 sec for a
06200	256 by 256 image, or 90 sec in three colors and stereo.
06300	This is about a factor of 1000 slower than comparable human operations,
06400	such as stereopsis [].
06500	
06600	
06700	Consider design criteria for a local edge operator.  It describes the
06800	light intensity surface over a small disk by a step  function  normal
06900	to  a  line  through  the  disk.   We  evaluate  its  utility  by its
07000	sensitivity and computational cost for a  specified  error  rate  (eg
07100	sensitivity  defined  by  the  contrast  required  for  50%  positive
07200	response at an edge for a threshold set for 5% false positives  where
07300	there  is  no  edge).  We can model light intensity at an edge by the
07400	edge signal with detector noise added.  Detector noise can be  easily
07500	measured and modelled.  Surface specks and markings could be regarded
07600	as noise. Their contrast can be arbitrarily large,  however,  and  to
07700	attempt  to  eliminate  response to them by simply raising thresholds
07800	would lower sensitivity.  They are systematic phenomena  at  a  local
07900	scale,  but  are  not  extended;  the  edge organization process (and
08000	texture organization) can deal with such specks.
08100	 There  is  a  tradeoff  between
08200	sensitivity  on  the one hand, and computation cost and resolution on
08300	the other.  This suggests a range of sizes of  operators  from  those
08400	for  specks  to  those for extended edges.  Unfortunately,
08500	the Hueckel operator is sensitive primarily to  the  intensity gradient.
08600	 This  means  that  in non-uniform
08700	illumination, predominantly from reflections,  the  operator  returns
08800	false positive results over an entire surface.  The biological system is known
08900	to be insensitive to smooth gradients; lateral inhibition  serves  to
09000	eliminate these effects.  This technique has been used by Binford [],
09100	and elucidated by Horn and Marr [].
09200	
09300	To what extent  is  improvement  in  sensitivity  possible  for  edge
09400	operators?
09500	The Hueckel operator requires a threshold only 1.5 times optimum
09600	(we assume that the difference of means across an edge is nearly
09700	optimal).  If we assume that two multiply-add operations per point
09800	is minimal, the Hueckel operator is about a factor of 4 slower than
09900	optimal.  There seems little room for dramatic improvements in such
10000	edge operators.
     

00100	Finding local edge fragments is local in the sense that the support
00200	of the calculation is a small disk.
00300	One operation of edge organization is also
00400	local: find edge fragments which
00500	are nearby and have similar slopes.
00600	Little is known from physiology about mechanisms for the organization
00700	of edge information from Hubel-Wiesel cells.  One conjecture is that
00800	their output goes directly to form a Fourier transform.  We argue later
00900	against the utility of that conjecture.  There are many models from
01000	machine perception.  One of these is the boundary of a connected region.
01100	Another is the edge-following technique; the edge at 
01200	
01300	Local edge operators are useful for curves which are locally straight
01400	over the disk of the operator.  For large disks, there are few situations
01500	for  which a step function across a straight line adequately describes the
01600	intensity surface.  There are many more possibilities, for several lines,
01700	for curves, or for small textural features.  One approach is to find
01800	optimal curves which maximize some local contrast function (eg the gradient
01900	along the curve).  That approach is very expensive in computation because
02000	of the enormous number of possibilities, all enumerations of adjacent points,
02100	but by neatly enumerating combinatorics and by use of contiuity, the expected
02200	shape of the curve, and heuristic search, these techniques have been made
02300	interesting for situations such as analysis of x-rays, for which the shape
02400	of the curve is known in advance.
02500	
02600	Combinatorics can be minimized by a two step process of local edge hypothesis
02700	by thresholding output of a local edge operator followed by edge verification
02800	organization.  We assume that such stages follow the Hubel-Wiesel operators
02900	in biological systems.  There are two global techniques which work well for
03000	straight lines: clustering of edge fragments in the space of line parameters
03100	[] (angle and minimum distance); local clustering of edge fragments projected
03200	along a variety of directions.  For curves, no economical equivalent has been
03300	used.
03400	
03500	Other techniques are local; the simplest of these is the region growing
03600	form of edge organization.  Clearly, region interior is complementary to 
03700	region exterior; the boundary of the region interior is an edge, linked
03800	without any notion of smoothness, which might bridge gaps.  The aim of
03900	edge organization is to link edge fragments which are nearby and have
04000	similar slopes.  This may be done in an edge following mode [] in which
04100	the edge operator is used to track along edges, or in a raster mode,
04200	which has a parallel equivalent, for which the edge operator is applied
04300	in a raster scan; edge fragments are linked to one of a number of nearby
04400	unterminated edges [].
     

00100	It appears to be difficult to
00200	extend the idea of a region uniform in intensity to regions uniform
00300	in the spatial distribution of features.
00400	We feel that these
00500	difficulties are related in part to the inherent computation complexity
00600	associated with two dimensional geometry.