On the localization performance measure and optimal edge detection
Tagare, H.D.; de Figueiredo, R.J.P.
Multidimensional Signal Processing Workshop, 1989., Sixth
Volume , Issue , 6-8 Sep 1989 Page(s):114 -
Digital Object Identifier 10.1109/MDSP.1989.97065
Summary:Summary form only given, as follows. Two measures have been suggested in the literature to characterize the localization performance of an edge detector. The first one was proposed by Abdou and Pratt (1979) and the second one by Canny (1986). The limitations of these localization measures are shown. The former is heuristic, while Canny's measure has not been formulated correctly. The problem of localization of an edge detector has been formulated with the help of the theory of zero-crossings of stochastic processes. The measure of localization of the edge detector proposed is the extent to which it causes the density of maxima to be reduced as we get farther away from the true edge. It is possible to show that with this localization measure and a constraint of the width of the filter, the optimal linear filter for edge detection is the derivative of a Gaussian
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Edge detection.
A notion of gauge coordinates which has been adopted in the
computer vision community is to express image descriptors
in terms of local directional derivatives defined from
certain preferred coordinate systems.
At any image point, introduce a local (u, v)-system
such that the v-direction is parallel to the gradient
direction 
,
and introduce directional derivative operators
along these directions by
Then, we can define an edge point as a point for which
the gradient assumes a local maximum in the gradient direction,
and restate this edge definition as
where 
and 
denote second- and third-order
directional derivatives in the v-direction.
After expansion to Cartesian coordinates and simplification,
this edge definition assumes the form
Interpolating for zero-crossings of 
within
the sign-constraints of 
gives a
straightforward method for sub-pixel edge detection.
Figure 5(a) shows the result of applying
this edge detector to an image of an arm at scale levels
t = 1.0, 16.0 and 256.0.
Observe how qualitatively different types of edge curves are extracted
at the different scales.
A characteristic behaviour is that most of the sharp edge structures
corresponding to object boundaries
give rise to edge curves at both fine and coarse scales.
Moreover, the number of spurious edges due to noise is
much larger at fine scales,
whereas the localization of the edges can be poor at coarse scales.
Notably, the shadow of the arm can only be extracted as
a connected curve at a coarse scale.
This example constitutes one illustration of the need for including
image operators at coarse scales when extracting general classes of
image structures from real-world data.
Figure 5: Edges and bright ridges detected at scale levels t = 1.0, 16.0
and 256.0, respectively.
Tony Lindeberg
Tue Jul 1 14:57:47 MET DST 1997