629
IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE,
VOL.
12.
NO.
7.
JULY
1990
Scale-Space and Edge Detection Using Anisotropic
Diffusion
PIETRO PERONA
AND
JITENDRA MALIK
Abstracf-The scale-space technique introduced by Witkin involves
generating coarser resolution images by convolving the original image
with a Gaussian kernel. This approach has a major drawback: it is
difficult to obtain accurately the locations of the “semantically mean-
ingful” edges at coarse scales. In this paper we suggest a new definition
of scale-space, and introduce a class of algorithms that realize it using
a diffusion process. The diffusion coefficient is chosen to vary spatially
in such a way as to encourage intraregion smoothing in preference to
interregion smoothing. It is shown that the “no new maxima should be
generated at coarse scales” property of conventional scale space is pre-
served. As the region boundaries in
our
approach remain sharp, we
obtain a high quality edge detector which successfully exploits global
information. Experimental results are shown on a number of images.
The algorithm involves elementary, local operations replicated over the
image making parallel hardware implementations feasible.
Zndex
Terms-Adaptive filtering, analog
VLSI,
edge detection, edge
enhancement, nonlinear diffusion, nonlinear filtering, parallel algo-
rithm, scale-space.
1.
INTRODUCTION
HE importance of multiscale descriptions of images
T
has been recognized from the early days of computer
vision, e.g., Rosenfeld and Thurston [20].
A
clean for-
malism for this problem is the idea of scale-space filtering
introduced by Witkin [21] and further developed in Koen-
derink [ll], Babaud, Duda, and Witkin [l], Yuille and
Poggio [22], and Hummel
[71,
[SI.
The essential idea
of
this approach is quite simple:
embed the original image in a family of derived images
I(x,
y,
t)
obtained by convolving the original image
Io(x,
y)
with a Gaussian kernel
G(x,
y;
t)
of variance
t:
Z(X,
Y,
f)
=
41(x,
y)
*
G(x,
y;
f).
(1)
Larger values of
t,
the scale-space parameter, corre-
spond to images at coarser resolutions. See Fig.
1.
As pointed out by Koenderink
[
111 and Hummel [7],
this one parameter family of derived images may equiv-
alently be viewed as the solution of the heat conduction,
or diffusion, equation
I,
=
AZ
=
(Zxx
+
IJy)
(2)
Manuscript received May 15, 1989; revised February 12, 1990. Rec-
ommended for acceptance by R.
J.
Woodham.
This
work was supported
by the Semiconductor Research Corporation under Grant 82-11-008
to
P.
Perona, by an
IBM
faculty development award and a National Science
Foundation PYI award to
J.
Malik, and by the Defense Advanced Research
Projects Agency under Contract N00039-88-C-0292.
The authors are with the Department of Electrical Engineering and
Computer Science, University of Califomia, Berkeley, CA 94720.
IEEE
Log
Number 90361 10.
Fig. 1.
A
family of l-D signals
I(x,
t)
obtained by convolving the original
one (bottom) with Gaussian kernels whose variance increases from bot-
tom
to
top (adapted from Witkin [21]).
with the initial condition
I(x,
y,
0)
=
Zo(x,
y),
the orig-
inal image.
Koenderink motivates the diffusion equation formula-
tion by stating two criteria.
I)
Causality:
Any feature at a coarse level of resolu-
tion is required
to
possess a (not necessarily unique)
“cause” at a finer level of resolution although the reverse
need not be true. In other words, no spurious detail should
be generated when the resolution is diminished.
2)
Homogeneity and
Isotropy:
The blurring is required
to be space invariant.
These criteria lead naturally to the diffusion equation
formulation. It may be noted that the second criterion is
only stated for the sake of simplicity. We will have more
to say on this later. In fact the major theme of this paper
is to replace this criterion by something more useful.
It should also be noted that the causality criterion does
not force uniquely the choice
of
a Gaussian to do the blur-
ring, though it is perhaps the simplest. Hummel [7] has
made the important observation that a version of the max-
imum principle from the theory
of
parabolic differential
equations is equivalent to causality. We will discuss this
further in Section IV-A.
This paper is organized
as
follows: Section I1 critiques
the standard scale space paradigm and presents an addi-
tional set of criteria for obtaining ‘‘semantically meaning-
ful” multiple scale descriptions. In Section I11 we show
that by allowing the diffusion coefficient to vary, one can
satisfy these criteria. In Section IV-A the maximum prin-
ciple is reviewed and used to show how the causality cri-
terion is still satisfied by our scheme. In Section V some
0162-8828/90/0700-0629$01
.OO
0
1990
IEEE
630
IEEE
TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE,
VOL.
12.
NO.
7.
JULY
1990
experimental results are presented. In Section
VI
we com-
pare our scheme with other edge detection schemes. Sec-
tion
VI1
presents some concluding remarks.
11.
WEAKNESSES
OF
THE
STANDARD SCALE-SPACE
PARADIGM
We now examine the adequacy of the standard scale-
space paradigm for vision tasks which need “semanti-
cally meaningful” multiple scale descriptions. Surfaces
in nature usually have
a
hierarchical organization com-
posed of a small discrete number of levels
[
131.
At the
finest level,
a
tree is composed of leaves with an intricate
structure of veins. At the next level, each leaf is replaced
by a single region, and at the highest level there is
a
single
blob corresponding to the treetop. There is
a
natural range
of resolutions (intervals of the scale-space parameter) cor-
responding to each of these levels
of
description. Fur-
thermore at each level of description, the regions (leaves,
treetops, or forests) have well-defined boundaries.
In the standard scale-space paradigm the true location
of
a
boundary at
a
coarse scale
is
not directly available in
the coarse scale image. This can be seen clearly in the
1-D example in Fig.
2.
The locations
of
the edges at the
coarse scales are shifted from their true locations. In
2-D
images there is the additional problem that edge junc-
tions, which contain much of the spatial information of
the edge drawing, are destroyed. The only way to obtain
the true location of the edges that have been detected at
a
coarse scale is by tracking across the scale space to their
position in the original image. This technique proves to
be complicated and expensive
[SI.
The reason for this spatial distortion is quite obvious-
Gaussian blurring does not “respect” the natural bound-
aries of objects. Suppose we have the picture of
a
treetop
with the sky
as
background. The Gaussian blurring pro-
cess would result in the green of the leaves getting
“mixed” with the blue of the sky, long before the treetop
emerges
as a
feature (after the leaves have been blurred
together). Fig.
3
shows
a
sequence
of
coarser images ob-
tained by Gaussian blurring which illustrates this phe-
nomenon. It may
also
be noted that the region boundaries
are generally quite diffuse instead of being sharp.
With this
as
motivation, we enunciate
[18]
the criteria
which we believe any candidate paradigm for generating
multiscale “semantically meaningful” descriptions of
images must satisfy.
I)
Causaliry:
As pointed out by Witkin and Koender-
ink. a scale-space representation should have the property
that no spurious detail should be generated passing from
finer to coarser scales.
2)
Immediate Localization:
At each resolution, the re-
gion boundaries should be sharp and coincide with the
semantically meaningful boundaries at that resolution.
3) Piecewise Smoothing:
At all scales, intraregion
smoothing should occur preferentially over interregion
smoothing. In the tree example mentioned earlier, the leaf
regions should be collapsed to
a
treetop
before
being
merged with the sky background.
Fig.
2.
Position
of
the edges (zeros
of
the Laplacian with respect
to
x)
through the linear scale space
of
Fig.
1
(adapted from Witkin
[21]).
Fig.
3.
Scale-space (scale parameter increasing
from
top to bottom, and
from left to right) produced by isotropic linear diffusion
(0.
2.
4,
8,
16.
32
iterations
of
a discrete
8
nearest-neighbor implementation. Compare
to Fig.
12.
111.
ANISOTROPIC DIFFUSION
There is
a
simple way of modifying the linear scale-
space paradigm to achieve the objectives that we have put
forth in the previous section. In the diffusion equation
framework of looking at scale-space, the diffusion coef-
ficient
c
is assumed to be
a
constant independent
of
the
space location. There is no fundamental reason why this
must be
so.
To quote Koenderink
[
1
1,
p.
3641,
“
. .
.
I do
not permit space variant blurring. Clearly this is not es-
sential to the issue, but
it
simplifies the analysis greatly.”
We will show how
a
suitable choice of
c(x,
y,
t)
will
enable
us
to satisfy the second and third criteria listed in
the previous section. Furthermore this can be done with-
out sacrificing the causality criterion.
Consider the anisotropic diffusion equation
I,
=
div (c(x,
y,
t)Vl)
=
c(x,
y.
r)Al
+
Vc
VI
(3)
where we indicate with
div
the divergence operator, and
with
V
and
A
respectively the gradient and Laplacian op-
erators, with respect to the space variables. It reduces
to the isotropic heat diffusion equation
I,
=
cAZ
if
c(x,
y,
t)
is a constant. Suppose that at the time (scale)
t,
we knew the locations of the region boundaries appro-
priate for that scale. We would want to encourage
smoothing
within
a
region
in
preference to smoothing
across
the boundaries. This could be achieved by setting
the conduction coefficient to be 1 in the interior of each
region and
0
at the boundaries. The blurring would then
take place separately in each region with no interaction
between regions. The region boundaries would remain
sharp.
Of course. we do
not
know in advance the region
boundaries at each scale (if we did the problem would
already have been solved!). What can be computed is
a
PERONA AND MALIK: SCALE-SPACE AND EDGE DETECTION
63
1
S
Fig.
4.
The qualitative shape
of
the nonlinearity
g
(
.
).
current best estimate of the location of the boundaries
(edges) appropriate to that scale.
Let
E(x,
y,
t)
be such an estimate: a vector-valued
function defined on the image which ideally should have
the following properties:
1)
E(x,
y,
t)
=
0
in the interior of each region.
2)
E(x,
y,
t)
=
Ke(x,
y,
t)
at each edge point, where
e
is a unit vector normal to the edge at the point, and
K
is the local contrast (difference in the image intensities on
the left and right) of the edge.
Note that the word
edge
as used above has not been
formally defined-we mean here the perceptual subjective
notion of an edge as a region boundary. A completely
satisfactory formal definition is likely to be part of the
solution, rather than the problem definition!
If an estimate
E(x,
y,
t)
is available, the conduction
coefficient
c(x,
y,
t)
can be chosen to be a function
c
=
g
(
)I
E
11
)
of the magnitude of
E.
According to the previ-
ously stated strategy
g(
)
has to be a nonnegative
monotonically decreasing function with
g(0)
=
1
(see
Fig.
4).
This way the diffusion process will mainly take
place in the interior of regions, and it will not affect the
region boundaries where the magnitude of
E
is large.
It is intuitive that the success of the diffusion process
in satisfying the three scale-space goals
of
Section I1 will
greatly depend on how accurate the estimate
E
is as a
“guess” of the position of the edges. Accuracy though is
computationally expensive and requires complicated al-
gorithms. We are able to show that fortunately the sim-
plest estimate of the edge positions, the gradient of the
brightness function, i.e.,
E(x,
y,
t)
=
VZ(x,
y,
t),
gives
excellent results.
There are many possible choices for
g
(
),
the most ob-
vious being a binary valued function. In the next section
we show that in case we use the edge estimate
E(x,
y,
t)
=
VZ(x,
y,
t)
the choice of
g(
)
is
restricted to a subclass
of the monotonically decreasing functions.
IV.
PROPERTIES
OF
ANISOTROPIC
DIFFUSION
We first establish that anisotropic diffusion satisfies the
causality criterion of Section I1 by recalling a general re-
sult of the partial differential equation theory, the maxi-
mum principle. In Section IV-B we show that a diffusion
in which the conduction coefficient is chosen locally as a
function of the magnitude of the gradient
of
the brightness
function, i.e.,
will not only preserve, but also sharpen, the brightness
edges if the function
g
(
.
)
is
chosen properly.
A.
The Maximum Principle
The causality criterion requires that no new features are
introduced in the image in passing from fine to coarse
scales in the scale-space. If we identify “features” in the
images with “blobs” of the brightness function
Z(x,
y,
t)
for different values of the scale parameter
t,
then the birth
of
a new “blob” would imply the creation of either a
maximum or
a
minimum which would have to belong
either to the interior or the top face
Z(x,
y,
tf)
of
the scale
space
(q
is the coarsest scale
of
the scale-space). There-
fore the causality criterion can be established by showing
that all maxima and minima in the scale-space belong to
the original image.
The diffusion equation
(3)
is a special case of a more
general class of elliptic equations that satisfy a maximum
principle. The principle states that all the maxima of the
solution of the equation in space and time belong to the
initial condition (the original image), and to the bound-
aries of the domain of interest provided that the conduc-
tion coefficient is positive. In our case, since we use adi-
abatic boundary conditions, the maximum principle is
even stronger: the maxima only belong to the original im-
age.
A
proof of the principle may be found in
[17];
to
make the paper self-contained we provide an easy proof
in the Appendix, where the adiabatic boundary case is also
treated, and weaker hypotheses on the conduction coeffi-
cient are used. A discrete version of the maximum prin-
ciple is proposed in Section
V.
B.
Edge Enhancement
With conventional low-pass filtering and linear diffu-
sion the price paid for eliminating the noise, and for per-
forming scale space, is the blurring of edges. This causes
their detection and localization to be difficult. An analysis
of this problem is presented in
[4].
Edge enhancement and reconstruction of blurry images
can be achieved by high-pass filtering or running the dif-
fusion equation backwards in time. This is an ill-posed
problem, and gives rise to numerically unstable compu-
tational methods, unless the problem is appropriately con-
strained or reformulated
[9].
We will show here that if the conduction coefficient is
chosen to be an appropriate function of the image gradient
we can make the anisotropic diffusion enhance edges while
runningfonvard in time, thus enjoying the stability of dif-
fusions which is guaranteed by the maximum principle.
We model an edge as a step function convolved with a
Gaussian. Without
loss
of generality, assume that the edge
is aligned with the
y
axis.
The expression for the divergence operator simplifies
to
a
ax
div
(c(x,
y,
t)VZ)
=
-
(~(x,
Y,
t)&).
632
IEEE TRANSACTIONS
ON
PATTERN ANALYSIS AND MACHINE INTELLIGENCE.
VOL.
12,
NO.
7.
JULY
1990
K
S
Fig.
6.
A
choice
of
the function
4
(
.
)
that leads
to
edge enhancement.
negative. This may be a source of concern since it is
known that constant-coefficient diffusions running back-
wards are unstable and amplify noise generating ripples.
In our case this concern is unwarranted: the maximum
principle guarantees that ripples are not produced. Exper-
quickly shrink, and the process
keeps stable.
imentally one observes that the
areas where
4r
(1,)
<
O
V
-T-
Fig.
5.
(TOP
to bottom)
A
mollified step edge and its
1st.
2nd, and 3rd
derivatives.
V.
EXPERIMENTAL RESULTS
Our anisotropic diffusion, scale-space, and edge detec-
tion ideas were tested using a simple numerical scheme
that
is
described
in
this
section.
Equation
(3)
can be discretized on a square lattice, with
brightness values associated to the vertices, and conduc-
We choose
c
to be a function of the gradient of
I:
c
(x,
g(zx)
.
I,
Then the
1-D
version of the diffusion equation
(3)
be-
y,
t)
=
g(z,(x,
y,
t))
as
in
(4).
Let
4(z,)
denote the flux
c
I,.
comes
(5)
a
ax
4
=
-
4(Ix)
=
4r(4)
4,.
We are interested in looking at the variation in time
of
the slope
of
the edge:
a/at(Z,).
If
c(
)
>
0
the function
I(
)
is smooth, and the order
of
differentiation may be
inverted:
I:,
+
4'
*
L,.
(6)
=
4rr
.
Suppose the edge is oriented in such a way that
I,
>
0.
At the point of inflection
I,,
=
0,
and
I,,,
<<
0
since the
point of inflection corresponds to the point with maximum
slope (see Fig.
5).
Then in a neighborhood of the point
of inflection
a/at(Z,)
has sign opposite to
4r(Z,).
If
4r
(I,)
>
0
the slope of the edge will decrease with time;
if, on the contrary
4'
(I,)
<
0
the slope will increase with
time.
Notice that this increase in slope cannot be caused by a
scaling of the edge, because this would violate the max-
imum principle. The edge becomes sharper.
There are several possible choices of
4
(
.
),
for exam-
ple,
g(Z,)
=
C/(
1
+
(I,/K)'+")
with
a
>
0
(see Fig.
6).
Then there exists a certain threshold value related to
K,
and
a,
below which
4(
-
)
is monotonically increasing,
and beyond which
4
( )
is monotonically decreasing, giv-
ing the desirable result of blurring small discontinuities
and
sharpening edges. Notice also that in a neighborhood
of the steepest region of an edge the diffusion may be
thought of as running "backwards" since
+'(I,)
in
(5)
is
tion coefficients to the arcs (see Fig.
7).
A 4-nearest-
neighbors discretization of the Laplacian operator can be
used:
I!:,;'
=
r:,j
+
h[CN VNI
+
cs
VsI
+
CE
.
VEI
+
cw
oWl]:,j
(7)
where
0
I
X
I
1
/4 for the numerical scheme to be sta-
ble,
N,
S,
E,
Ware the mnemonic subscripts for North,
South, East, West, the superscript and subscripts on the
square bracket are applied to all the terms it encloses, and
the symbol
V
(not to be confused with V, which we use
for the gradient operator) indicates nearest-neighbor dif-
ferences:
VNI;,j
E
Ii-
1.j
-
1j.j
VS1i.j
E
Il+~,j
-
';,I
VEZj,j
=
Zi,j+I
-
z;,j
vU/z;,j
=
I;,j-l
-
z;,j.
(8)
The conduction coefficients are updated at every itera-
tion as a function
of
the brightness gradient (4):
(9)
PERONA AND MALIK: SCALE-SPACE AND EDGE DETECTION
633
Fig.
7.
The structure
of
the discrete computational scheme for simulating
the diffusion equation (see Fig.
8
for a physical implementation). The
brightness values
I,,,
are associated with the nodes of a lattice, the con-
duction coefficients
c
to
the arcs. One node
of
the lattice and its four
North, East, West, and South neighbors are shown.
Fig.
8.
The structure of a network realizing the implementation
of
aniso-
tropic diffusion described in Section
V,
and more in detail in
[19].
The
charge on the capacitor at each node of the network represents the bright-
ness
of
the image at a pixel. Linear resistors produce isotropic linear
diffusion. Resistors with an
I-V
characteristic as in Fig.
6
produce an-
isotropic diffusion.
The value of the gradient can be computed on different
neighborhood structures achieving different compromises
between accuracy and locality. The simplest choice con-
sists in approximating the norm of the gradient at each arc
location with the absolute value of its projection along the
direction of the arc:
cb,l
=
g(
1
vN1:,~l)
CktJ
=
g(
I
VsI:,Jl)
ck,
=
8(
1
VEZ,JI)
clv,,
=
s(l
VwC,,I).
(10)
This scheme is not the exact discretization of
(3),
but
of similar diffusion equation in which the conduction ten-
sor is diagonal with entries
g
(
I
I,
\
)
and
g
(
I
Iy
1
)
instead
of
g
(
11
VI
11
)
and
g
(
)I
VI
(1
).
This discretization scheme
preserves the property of the continuous equation
(3)
that
the total amount of brightness in the image is preserved.
Additionally the “flux” of brightness through each arc of
the lattice only depends on the values of the brightness at
the two nodes defining it, which makes the scheme a nat-
ural choice for analog
VLSI
implementations
[19].
See
Fig.
8.
Less crude approximations of the gradient yielded
perceptually similar results at the price of increased com-
putational complexity.
It is possible to verify that, whatever the choice of the
approximation of the gradient, the discretized scheme still
satisfies the maximum (and minimum) principle provided
that the function
g
is bounded between
0
and
1.
We can in fact show this directly from
(7),
using the
facts
X
E
[0,
1/41,
and
c
E
[0,
11,
and defining
(IM):,
5
max
{
(I,
IN,
IS,
ZE,
Zw):,,},
and
(Im)f,j
min
{
(1,
IN,
Is,
IE,
Iw):,,
}
,
the maximum and minimum of the neigh-
bors of
Z,,,
at iteration
t.
We can prove that
(LJ,
5
C,?
5
(Id:,,
(11)
i.e., no (local) maxima and minima are possible in the
interior of the discretized scale-space. In fact:
]f+l
‘J
=
I:,,
+
X[CN
*
VNI
+
CS
VSI
+
CE
*
VEI
+
CW
*
VwI]:,,
=
If,,(
-
A(cN
+
CS +
CE
+
cW):,,)
+
X(C,
.
IN
+
CS
*
IS
+
CE
IE
+
cW.
Zw):,,
5
IM:J(l
-
X(cN
+
CS
+
CE
+
cW):,J)
and, similarly:
+
XIm:,J(~~
+
CS
+
CE
+
CW):,~
=
Im:,.
(13)
The numerical scheme used to obtain the pictures in this
paper is the one given by equations
(7),
(8),
and
(lo),
using the original image as the initial condition, and adi-
abatic boundary conditions, i.e., setting the conduction
coefficient to zero at the boundaries of the image.
A
con-
stant value for the conduction coefficient
c
(i.e.,
g
(
*
)
=
1)
leads to Gaussian blurring (see Fig.
3).
Different functions were used for
g
(
-
)
giving percep-
tually similar results. The images in this paper were ob-
tained using
g(vI)
=
&mI/K)~)
(Fig.
9),
and
(Figs.
12-14).
The scale-spaces generated by these two
functions are different: the first privileges high-contrast
edges over low-contrast ones, the second privileges wide
regions over smaller ones.
The constant
K
was fixed either by hand at some fixed
value (see Figs.
9-14),
or using the “noise estimator”
described by Canny
[4]:
a histogram of the absolute val-
ues of the gradient throughout the image was computed,