A challenge: What is an edge?

Subject: A challenge: What is an edge?
From: Michelle T. Lin (michlin@cs.berkeley.edu)
Date: Tue Apr 04 2000 - 02:13:25 MET DST

Berkeley Initiative in Soft Computing (BISC)

To: BISC Group
From: L. A. Zadeh <zadeh@cs.berkeley.edu>

        Recently I attended a lecture in which the speaker touched
upon the problem of edge detection. This suggested to me the
question: what is an edge? In thinking about the answer, I realized
that the concept of edge is quite complex and how to define it
presents a real challenge. This is the spirit in which the following
should be read. Please note that what I have in mind are edges of
three dimensional objects and, more particularly, three-dimensional
cylindrical objects.

        There is an enormous literature on the problem of edge
detection. Professor S. K. Pal who is one of the foremost
authorities on pattern recognition and computer vision, sent to me a
list of his papers going back to 1983, in which edge is treated as a
fuzzy concept. Professor T. Pavlidis, former editor of PAMI sent me
a description of a system in which the concept of a fuzzy edge is
employed. My colleague, Jitendra Malik, a leading member of the
computer vision community, brought to my attention a book
"Geometry-Driven Diffusion in Computer Vision," in which a
mathematically sophisticated treatment of edge detection is presented.
So what is the problem in defining the concept of edge?

        The problem is that the concept of edge, along with the
concepts of causality, relevance, randomness, probability,
stationarity and many others, is an instance of what may be called an
amorphic concept. In essence, an amorphic concept has three facets:

        a) the crisp facet
        b) the fuzzy facet
        c) the amorphous facet.
        The crisp facet relates to instances where the meaning of the
concept is crisp and unequivocal. In the case of the concept of edge,
these are the instances where what is an edge is quite clear and

        The fuzzy facet relates to instances in which an edge is a
fuzzy set on the surface of the object. In many realistic situations
this is in fact the case. The problem. then, is to define the
membership function of the fuzzy set.

        The amorphous facet relates to instances in which an edge is a
fuzzy set but its membership function does not admit precisiation, and
there is no internal or external consensus on whether an edge exists
and, if it exists, how its membership function should be defined. In
other words, the membership function is a pseudo-function in the sense
in which I defined it in a recent abstract. In essence, a
pseudo-function is a function whose meaning cannot be defined in what
I call the Generalized Constraint Language (GCL) -- a language which
is maximally expressive.

        In this framework, a concept is crisp if its only facet is
crisp. A concept is fuzzy if has a crisp facet and a fuzzy facet.
And a concept is amorphic if it has all three facets.

        The problem with the concept of edge is that in many realistic
settings what comes into play is its amorphous facet. I plan to write
an article on amorphic concepts in which a number of examples will be

        The significance of the concept of amorphicity is that such
concepts do not lend themselves to mathematically precise definitions.
This explains why we do not have mathematically precise definitions of
such concepts as causality and randomness -- definitions which capture
our intuitive perception of what these concepts mean. The problem
with scientific theories is that in many theories amorphic concepts
are treated as if they were crisp.

        Your comments would be welcome. Please do not hesitate to
        With warm regards to all.
Lotfi A. Zadeh
Professor in the Graduate School and Director,
Berkeley Initiative in Soft Computing (BISC)
CS Division, Department of EECS
University of California
Berkeley, CA 94720-1776
Tel/office: (510) 642-4959 Fax/office: (510) 642-1712
Tel/home: (510) 526-2569 Fax/home: (510) 526-2433
email: zadeh@cs.berkeley.edu
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