COMMENT ⊗   VALID 00003 PAGES
C REC  PAGE   DESCRIPTION
C00001 00001
C00002 00002		OPTIMAL CURVES OF LENGTH N
C00009 00003	1. Extract local descriptors:
C00010 ENDMK
C⊗;
	OPTIMAL CURVES OF LENGTH N


The situation in low level vision is similar to that in speech understanding
systems: the current highest priority is for complete systems which use context
and mobilize all available information.  However, those systems as implemented
appear to be ungeneralizable.  The quality of low level descriptors limits
system effectiveness as much as any other link.

THE GOAL: WHAT WILL WE BE ABLE TO DO IF SUCCESSFUL THAT WE COULD NOT DO NOW:
Obtain boundaries of region discontinuities ranging in size from salt and pepper
microtexture to large uniform regions, in computation time a small multiple
of that required for the Hueckel operator, with sensitivity equal to that of
humans.
HOW:
a. Theory of an optimal curve procedure
b. Estimate computational complexity of the procedure
c. Implement if warranted by a. and b. 

TIME SCALE: one student for 6 months (1/4 man year).

When humans look at a TV image, they see edges which are missed by Hueckel,
Binford-Horn, or other edge operators.  Although some would say that we know
what we are looking for, this is not solely a matter of context, as some
context-free experiments show.  We propose an analysis of the limitations of
edge operators and an extension of technique.

Why are edge operators insensitive?  Because they threshold on a very local
context.  It is a simple statistical result that the limit of sensitivity of
an operator is determined by the support of the operator.  When the edge
operator is part of a system, the limit of sensitivity of the system is
determined by the support of that threshold process.  For example, one system
procedure might be to look for candidates for an edge, and then verify.  Only
those candidates which show up on the limited support will be found.
It is a myth that we can find edges with  small contrast just by changing
thresholds on the operator.  An operator is useful only if it provides few
enough false positives that the rest of the system can function.  Raising
thresholds from 3 std dev to 2 std dev increases noise a factor of 7.  
Decreasing thresholds only renders the operator unselective (remember, in the
limit of small threshold, the operator becomes a no-op).

What other defects do they have?  Edge operators lack resolution to deal with
small scale texture, etc.

How to improve?  There is not much hope for improving local operators.  They
are near the statistical limit for operators with that support.  Binford has
shown that the Hueckel operator threshold is 1.5 times the optimal sensitivity
for that support.  
To increase the sensitivity, increase the support.  However, a bigger support
for a Hueckel opeator means less utility, because more curves are possible
within the support, ie fewer discs are well described by a single straight
edge within the disc.  Then determine optimal curves within a bigger support.
To determine all curves is a combinatorially explosive process.  Several
implementations have been made of optimal curve search methods (Martelli,
Chien and Fu).  Their results are far from applicable.  The cure

DECREASE THE NUMBER OF POSSIBLE CURVES BY LIMITING TO SMOOTH CURVES;
DECREASE COMPUTATION BY CASCADING SUMS
Their results don't apply to finding all curves in an image.
The revised problem: FIND ALL LOCALLY OPTIMAL CURVES OF LENGTHS 1≤n≤N
based on contrast.
The solution:
Describe the intensity surface over support n.  Find curves smooth over
support n by thresholding over support n.  However, keep the descriptors
at this stage to combine to form descriptions of the intensity surface over
support 2n by cascading.  Cascade to get support 2n,4n,8n, etc.  At each
stage, threshold to determine curves smooth over that support.  Thus,
calculate the Hueckel descriptors over a disc.  Find the descriptor over
support 2n by combining the results of two discs.
EXPECTED RESULTS
The sensitivity of support n is about linear with n; statistical is sqrt(n)
if pts are independent; continuity limits number of possibilities.
Cost of computation is proportional to log n; this may be optimistic

1. Extract local descriptors:
  Fourier components over a disk; Hueckel operator
   to describe the intensity surface in a way which is adequate for
   simple configurations (a single edge)
2. Descriptors for extended neighborhoods
3. Search on a reduced grid:
  take the local descriptors on 2x2 centers