Canny算子：一种边缘检测的计算方法
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收藏 6.98MB PDF 举报 "这篇论文是John Canny在1986年发表于IEEETRANSACTIONSONPATTERNANALYSISANDMACHINEINTELLIGENCE上的《A Computational Approach to Edge Detection》，文中详细介绍了Canny边缘检测算法。Canny算子是一种高效且广泛使用的边缘检测方法，它基于对边缘检测目标的全面定义，包括检测、定位以及避免多响应等问题。"
在Canny的边缘检测算法中，首先，他提出了一种理想的边缘检测器应满足的一系列精确而明确的目标。这些目标旨在确保边缘检测器能够在不预先假设解决方案形式的情况下，准确地识别图像中的边缘。检测目标是指算法能正确识别边缘的存在，而定位目标则是指算法能准确确定边缘的位置。
Canny算法的核心包括以下几个关键步骤：
1. 预处理：使用高斯滤波器进行平滑，以消除图像噪声。Canny发现一阶高斯导数核在边缘检测中表现出色，因为它可以有效地平滑图像并保留边缘信息。
2. 计算梯度幅度和方向：通过应用一阶导数算子（如Sobel或Prewitt算子）来估计图像的梯度幅度和方向。这一步骤有助于确定边缘候选点，因为边缘通常对应于图像梯度的显著变化。
3. 非最大值抑制：这是Canny算法的关键特性之一。根据边缘定义为在梯度方向上具有最大梯度值的点，非最大值抑制过程会消除那些在梯度方向上不是局部最大值的点，从而减少假阳性边缘。
4. 双阈值检测：设置两个阈值，低阈值用于检测弱边缘，高阈值用于检测强边缘。连接低于高阈值的相邻像素，形成连续的边缘，而低于低阈值但高于高阈值的点则可能被连接到更强的边缘，或者如果它们孤立，则会被丢弃。
5. 边缘跟踪：通过边缘连接和去除孤立点，以得到连贯的边缘。
在Canny的分析中，针对阶梯边缘的特性，他发现检测性能与定位性能之间存在一种自然的不确定性原理。这意味着无法同时达到最佳的检测和定位效果，而必须在两者之间做出权衡。基于这个原则，Canny导出了一个在特定尺度下最优的运算符形状，即所谓的Canny边缘检测算子。
Canny边缘检测算法通过一系列数学和计算方法，成功地平衡了边缘检测的准确性和效率，使其成为计算机视觉和图像处理领域中的经典算法。尽管现代技术已经发展出许多其他边缘检测方法，Canny算子仍然是许多实际应用中的首选工具。
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: Harbin Institute of Technology. Downloaded on March 16,2010 at 09:30:29 EDT from IEEE Xplore. Restrictions apply.
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