00100	Neurophysiological studies suggest that cells in the mammalian visual
00200	cortex  extract  edges  from  retinal  data [].  
00300	However, edges between regions of uniform intensity are only one useful
00400	perceptual construct.
00500	 Often, surfaces are
00600	irregular on the scale of the image, such as hair, leaves,  feathers,
00700	foam; others are textured.  Gibson argues that textures and gradients
00800	of texture are important perceptual constructs [].   AI is  just  now
00900	coming  to  deal  with  textured scenes [].  Our perceptual goals are
01000	closely related to spatial interpretations in terms  of  objects  and
01100	their  spatial relations. Stereo and motion parallax depth perception
01200	provide information most closely related to these  perceptual  goals.
01300	It  is  clear  that  these abilities are important in perception and
01400	apparently are developed at birth [].  Perhaps other
01500	visual   learning   depends   on  the  structure  provided  by  depth
01600	perception.
01700	
01800	Though regions  of  uniform  intensity  and  their boundaries  are
01900	important  low  level  constructs,  it  is clear that the analysis of
02000	uniform regions is inadequate; we must be  able  to  find  boundaries
02100	between  textured  regions.    Natural scenes usually contain objects
02200	with well-defined surface boundaries.   The  retinal  projections  of
02300	such  scenes are patchworks of areas of obscuring objects.
02400	Some  edges
02500	correspond  to  boundaries between objects, some to interior edges of
02600	objects,  others  to  surface  markings,  others   to   shadows   and
02700	reflections. All signify features of potential interest, but there is
02800	no  direct  connection  between  intensity  boundaries  and   spatial
02900	interpretations (which are our perceptual goal). Thus, it is possible
03000	to find intensity discontinuities on a local basis, even though some edges with
03100	low  contrast  will  be  missed.    It  is  not  possible to find the
03200	"meaningful edges" on a local basis,  ie  to  make  a  "perfect  line
03300	drawing". Present techniques of findig intensity discontinuities
03400	require extensive calculation and are not really adequate.   We  feel
03500	that   computational   complexity   is  inherent.    We  expect  that
03600	performance approaching human visual acuity inherently requires great
03700	computation  cost (eg the retina has 1000 times as many cells as our
03800	tv images),  and that substantial economies in computation are
03900	possible only by sampling strategies which ignore much of the  image.
04000	  Thus, sampling coarsely allows
04100	detection of extended boundaries, from which many finer details  (but
04200	not all) can be determined.
04300	For most  problems  of  interest,  however,  those  reduced
04400	abilities  are  presumably  adequate.
04500	
04600	
04700	We prefer to discuss boundary finding (for regions  without  texture)
04800	in  terms  of computational components, local edge operators and edge
04900	organization procedures.  In the biological system,  the  local  edge
05000	operators are Hubel-Wiesel cells, defined over small disks (typically 
05100	1 degree in the monkey).  These
05200	cells are strongly directional [angular  resolution]  and  there  are
05300	cells  for  various  directions. 
05400	There are approximately [] of them.  A  computational equivalent is the
05500	Hueckel operator [], defined over a small disk.
05600	This operator is directional in that it calculates moments of the
05700	intensity function in a few directions and determines the direction of 
05800	best fit
05900	of a single edge on the disk.
06000	We can estimate the computational cost of sampling a picture coarsely
06100	with this operator.  We want to detect edges at all directions.
06200	We look for long features, thus we are
06300	content to sample at a few positions along edges,
06400	but at many positions perpendicular to an edge. 
06500	It is enough to sample the Hueckel operator at every other point
06600	along widely separated vertical and horizontal lines.
06700	Let us say we use 8 horizontal and 8 vertical lines.  At
06800	about 4msec per application, this amounts to 8 sec for a 256 by 256
06900	image, or 48 sec in three colors and stereo.
07000	
07100	
07200	Consider design criteria for a local edge operator.  It describes the
07300	light intensity surface over a small disk by a step function normal to
07400	a line through the disk.  We evaluate its utility by its sensitivity
07500	and computational cost for a specified error rate (eg sensitivity defined
07600	by the contrast required for 50% positive response at an edge for a threshold
07700	set for 5% false positives where there is no edge).  We can model light
07800	intensity at an edge by the edge signal with detector noise added.  Detector
07900	noise can be easily measured and modelled.  Surface specks and markings could
08000	be regarded as noise.
08100	Their contrast can be arbitrarily large, however, and to attempt to
08200	eliminate response to them by simply raising thresholds would lower
08300	sensitivity.  They are systematic phenomena at a local scale, but are
08400	not extended; the edge organization process (and texture organization)
08500	can deal with such specks.
08600	
08700	To what extent is improvement in sensitivity possible for edge operators?
08800	The Hueckel operator requires a threshold only 1.5 times that obtained
08900	by testing the difference of means on two sides of an edge for vertical
09000	and horizontal directions.  Thus, no great gain appears possible by
09100	improving operators.  Likewise, the minimum computation over an nxn window
09200	is some multiple k of the number of points, nxn, where k is at least two
09300	to account for horizontal and vertical moments to obtain directionality.
09400	Thus, for an 8x8 window, there are some multiple of 64 arithmetic operations
09500	at each application.  There does not seem to be room for dramatically less
09600	computation for the Hueckel operator.
09700	There is a tradeoff between sensitivity on the one hand, and computation
09800	cost and resolution on the other.  This suggests a range of sizes of
09805	operators from those for specks to those for extended edges.
09900	An important limitation of that operator is that it is sensitive primarily
10000	to the gradient of the intensity.  This means that in the presence of
10100	non-uniform illumination, predominantly from reflections, the operator
10200	returns false results over an entire surface.  The biological system
10300	is known to be insensitive to smooth gradients; lateral inhibition serves
10400	to eliminate these effects.  This technique has been used by Binford [],
10500	and elucidated by Horn and Marr [].
     

00100	Finding local edge fragments is local in the sense that the support
00200	of the calculation is a small disk.
00300	One stage of edge organization is also
00400	local in that the problem addressed is: find edge fragments which
00500	are nearby and have similar slopes.  We can imagine cells distributed
00505	over the visual field which compare the slopes of edge fragments within
00510	their receptive field.  
     

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.