图像处理：边缘检测的计算方法
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"A Computational Approach to Edge Detection" 是一篇由 John Canny 在1986年发表在 IEEE Transactions on Pattern Analysis and Machine Intelligence 上的论文，主要探讨了图像处理中的边缘检测问题，采用了一种计算方法来优化边缘检测的效果。
这篇论文提出了一种计算方法来解决边缘检测的问题，其成功之处在于定义了一套全面的目标来确定边缘点的计算。这些目标需精确到足以界定边缘检测器的理想行为，同时尽量减少对解决方案形式的假设。Canny 提出，边缘检测应该包括检测（detection）和定位（localization）两个关键标准。
检测标准是指在图像中正确识别出边缘的存在，而定位标准则要求准确地确定边缘的位置。为了实现这两个目标，Canny 将它们数学化表示为操作器脉冲响应上的函数。此外，他还引入了第三个标准，确保检测器对单一边缘只产生一次响应，避免多边检测和假阳性。
论文中，Canny 通过数值优化技术设计了针对多种常见图像特征的检测器，特别是针对阶跃边缘（step edges）。在分析阶跃边缘时，他发现检测性能和定位性能之间存在一个自然的不确定性原理。这意味着在追求高精度的同时，可能会影响边缘位置的精确度，反之亦然。
基于这个不确定性原则，Canny 得出了一个在不同尺度下都具有最优性能的操作器形状，即著名的 Canny 边缘检测算法。这个算法至今仍被广泛使用，它通过多尺度分析和非最大值抑制等步骤，有效地平衡了检测率和假阳性率，提供了高质量的边缘检测结果。
总结来说，这篇论文对边缘检测的理论和实践做出了重大贡献，Canny 的方法不仅提供了边缘检测的理论框架，还催生了一个高效且广泛应用的边缘检测算法，对图像处理和计算机视觉领域产生了深远的影响。
IEEE
TRANSACTIONS
ON
PATTERN
ANALYSIS
AND
MACHINE
INTELLIGENCE,
VOL.
PAMI-8,
NO.
6,
NOVEMBER
1986
lows.
From
the
Schwarz
inequality
for
integrals
we
can
show
that
SNR
(3)
is
bounded
above
by
n-I
G
|t;
G2(x)
dx
w
and
localization
(9)
by
I
w
no
1
81|G'2(x)dCX.
Both
bounds
are
attained,
and
the
product
of
SNR
and
localization
is
maximized
when
f(x)
=
G(
-
x)
in
[-
W].
Thus,
according
to
the
first
two
criteria,
the
optimal
detector
for
step
edges
is
a
truncated
step,
or
difference
of
boxes
operator.
The
difference
of
boxes
was
used
by
Rosenfeld
and
Thurston
[25],
and
in
conjunction
with
lat-
eral
inhibition
by
Herskovits
and
Binford
[11].
However
it
has
a
very
high
bandwidth
and
tends
to
exhibit
many
maxima
in
its
response
to
noisy
step
edges,
which
is
a
serious
problem
when
the
imaging
system
adds
noise
or
when
the
image
itself
contains
textured
regions.
These
ex-
tra
edges
should
be
considered
erroneous
according
to
the
first
of
our
criteria.
However,
the
analytic
form
of
this
criterion
was
derived
from
the
response
at
a
single
point
(the
center
of
the
edge)
and
did
not
consider
the
interac-
tion
of
the
responses
at
several
nearby
points.
If
we
ex-
amine
the
output
of
a
difference
of
boxes
edge
detector
we
find
that
the
response
to
a
noisy
step
is
a
roughly
tri-
angular
peak
with
numerous
sharp
maxima
in
the
vicinity
of
the
edge
(see
Fig.
1).
These
maxima
are
so
close
together
that
it
is
not
pos-
sible
to
select
one
as
the
response
to
the
step
while
iden-
tifying
the
others
as
noise.
We
need
to
add
to
our
criteria
the
requirement
that
the
function
f
will
not
have
"too
many"
responses
to
a
single
step
edge
in
the
vicinity
of
the
step.
We
need
to
limit
the
number
of
peaks
in
the
response
so
that
there
will
be
a
low
probability
of
declar-
ing
more
than
one
edge.
Ideally,
we
would
like
to
make
the
distance
between
peaks
in
the
noise
response
approx-
imate
the
width
of
the
response
of
the
operator
to
a
single
step.
This
width
will
be
some
fraction
of
the
operator
width
W.
In
order
to
express
this
as
a
functional
constraint
on
f,
we
need
to
obtain
an
expression
for
the
distance
between
adjacent
noise
peaks.
We
first
note
that
the
mean
distance
between
adjacent
maxima
in
the
output
is
twice
the
dis-
tance
between
adjacent
zero-crossings
in
the
derivative
of
the
operator
output.
Then
we
make
use
of
a
result
due
to
Rice
[24]
that
the
average
distance
between
zero-cross-
ings
of
the
response
of
a
function
g
to
Gaussian
noise
is
(
R(O)
112
where
R(i)
is
the
autocorrelation
function
of
g.
In
our
case
we
are
looking
for
the
mean
zero-crossing
spacing
for
the
function
f'.
Now
since
R(O)
=
g2(x)
dx
and
R
"(0)
=
-
g'2(x)
dx
-
00
b
z
r
00
the
mean
distance
between
zero-crossings
off'
will
be
+
OD
~
)1/2
'2
(x)
dx
x,,(f)
=
r
(+
f
tO
t2(x)
dx
\
oo
(12)
The
distance
between
adjacent
maxima
in
the
noise
re-
sponse
of
f,
denoted
Xmax,
will
be
twice
xzc.
We
set
this
distance
to
be
some
fraction
k
of
the
operator
width.
Xmax(f)
=
2x,,(f)
=
kW.
(13)
This
is
a
natural
form
for
the
constraint
because
the
re-
sponse
of
the
filter
will
be
concentrated
in
a
region
of
width
2
W,
and
the
expected
number
of
noise
maxima
in
this
region
is
Nn
where
2W
2
N
=
=
-m
k
Xmax
k
(14)
Fixing
k
fixes
the
number
of
noise
maxima
that
could
lead
to
a
false
response.
We
remark
here
that
the
intermaximum
spacing
(12)
scales
with
the
operator
width.
That
is,
we
first
define
an
operator
f,
which
is
the
result
of
stretching
f
by
a
factor
of
w,
fw(x)
=
f(xlw).
Then
after
substituting
into
(12)
we
find
that
the
intermaximum
spacing
for
f,
is
x,,(
fj)
=
wxzc(f
).
Therefore,
if
a
function
f
satisfies
the
multiple
response
constraint
(13)
for
fixed
k,
then
the
function
f,
will
also
satisfy
it,
assuming
W
scales
with
w.
For
any
fixed
k,
the
multiple
response
criterion
is
invariant
with
respect
to
spatial
scaling
of
f.
III.
FINDING
OPTIMAL
DETECTORS
BY
NUMERICAL
OPTIMIZATION
In
general
it
will
be
difficult
(or
impossible)
to
find
a
closed
form
for
the
functionfwhich
maximizes
(10)
sub-
ject
to
the
multiple
response
constraint.
Even
when
G
has
a
particularly
simple
form
(e.g.,
it
is
a
step
edge),
the
form
off
may
be
complicated.
However,
if
we
are
given
a
candidate
function
f,
evaluation
of
(10)
and
(12)
is
straightforward.
In
particular,
if
the
function
f
is
repre-
sented
by
a
discrete
time
sequence,
evaluation
of
(10)
requires
only
the
computation
of
four
inner
products
between
sequences.
This
suggests
that
numerical
optimi-
zation
can
be
done
directly
on
the
sampled
operator
im-
pulse
response.
The
output
will
not
be
an
analytic
form
for
the
operator,
but
an
implementation
of
a
detector
for
the
edge
of
inter-
est
will
require
discrete
point-spread
functions
anyway.
It
is
also
possible
to
include
additional
constraints
by
using
a
penalty
method
[15].
In
this
scheme,
the
constrained
optimization
is
reduced
to
one,
or
possibly
several,
un-
constrained
optimizations.
For
each
constraint
we
define
a
penalty
function
which
has
a
nonzero
value
when
one
682
Authorized licensed use limited to: Imperial College London. Downloaded on June 07,2010 at 19:36:46 UTC from IEEE Xplore. Restrictions apply.
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