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Snakes, Shapes, and Gradient Vector Flow
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有关GVF梯度矢量流的相关文献,该方法可用于图像的分割、边缘检测等方面的研究。
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IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 7, NO. 3, MARCH 1998 359
Snakes, Shapes, and Gradient Vector Flow
Chenyang Xu, Student Member, IEEE, and Jerry L. Prince, Senior Member, IEEE
Abstract—Snakes, or active contours, are used extensively in
computer vision and image processing applications, particularly
to locate object boundaries. Problems associated with initializa-
tion and poor convergence to boundary concavities, however,
have limited their utility. This paper presents a new external force
for active contours, largely solving both problems. This external
force, which we call gradient vector flow (GVF), is computed
as a diffusion of the gradient vectors of a gray-level or binary
edge map derived from the image. It differs fundamentally from
traditional snake external forces in that it cannot be written as the
negative gradient of a potential function, and the corresponding
snake is formulated directly from a force balance condition rather
than a variational formulation. Using several two-dimensional
(2-D) examples and one three-dimensional (3-D) example, we
show that GVF has a large capture range and is able to move
snakes into boundary concavities.
Index Terms—Active contour models, deformable surface mod-
els, edge detection, gradient vector flow, image segmentation,
shape representation and recovery, snakes.
I. INTRODUCTION
S
NAKES [1], or active contours, are curves defined within
an image domain that can move under the influence of
internal forces coming from within the curve itself and external
forces computed from the image data. The internal and external
forces are defined so that the snake will conform to an object
boundary or other desired features within an image. Snakes
are widely used in many applications, including edge detection
[1], shape modeling [2], [3], segmentation [4], [5], and motion
tracking [4], [6].
There are two general types of active contour models
in the literature today: parametric active contours [1] and
geometric active contours [7]–[9]. In this paper, we focus on
parametric active contours, although we expect our results
to have applications in geometric active contours as well.
Parametric active contours synthesize parametric curves within
an image domain and allow them to move toward desired
features, usually edges. Typically, the curves are drawn toward
the edges by potential forces, which are defined to be the
negative gradient of a potential function. Additional forces,
such as pressure forces [10], together with the potential forces
comprise the external forces. There are also internal forces
designed to hold the curve together (elasticity forces) and to
keep it from bending too much (bending forces).
Manuscript received November 1, 1996; revised March 17, 1997. This work
was supported by NSF Presidential Faculty Fellow Award MIP93-50336. The
associate editor coordinating the review of this manuscript and approving it
for publication was Dr. Guillermo Sapiro.
The authors are with the Image Analysis and Communications Laboratory,
Department of Electrical and Computer Engineering, The Johns Hopkins
University, Baltimore, MD 21218 USA (e-mail: prince@jhu.edu).
Publisher Item Identifier S 1057-7149(98)01745-X.
There are two key difficulties with parametric active contour
algorithms. First, the initial contour must, in general, be
close to the true boundary or else it will likely converge
to the wrong result. Several methods have been proposed to
address this problem including multiresolution methods [11],
pressure forces [10], and distance potentials [12]. The basic
idea is to increase the capture range of the external force
fields and to guide the contour toward the desired boundary.
The second problem is that active contours have difficulties
progressing into boundary concavities [13], [14]. There is no
satisfactory solution to this problem, although pressure forces
[10], control points [13], domain-adaptivity [15], directional
attractions [14], and the use of solenoidal fields [16] have
been proposed. However, most of the methods proposed to
address these problems solve only one problem while creating
new difficulties. For example, multiresolution methods have
addressed the issue of capture range, but specifying how
the snake should move across different resolutions remains
problematic. Another example is that of pressure forces, which
can push an active contour into boundary concavities, but
cannot be too strong or “weak” edges will be overwhelmed
[17]. Pressure forces must also be initialized to push out or
push in, a condition that mandates careful initialization.
In this paper, we present a new class of external forces
for active contour models that addresses both problems listed
above. These fields, which we call gradient vector flow (GVF)
fields, are dense vector fields derived from images by mini-
mizing a certain energy functional in a variational framework.
The minimization is achieved by solving a pair of decoupled
linear partial differential equations that diffuses the gradient
vectors of a gray-level or binary edge map computed from the
image. We call the active contour that uses the GVF field as its
external force a GVF snake. The GVF snake is distinguished
from nearly all previous snake formulations in that its external
forces cannot be written as the negative gradient of a potential
function. Because of this, it cannot be formulated using
the standard energy minimization framework; instead, it is
specified directly from a force balance condition.
Particular advantages of the GVF snake over a traditional
snake are its insensitivity to initialization and its ability to
move into boundary concavities. As we show in this paper,
its initializations can be inside, outside, or across the object’s
boundary. Unlike pressure forces, the GVF snake does not
need prior knowledge about whether to shrink or expand
toward the boundary. The GVF snake also has a large capture
range, which means that, barring interference from other
objects, it can be initialized far away from the boundary. This
increased capture range is achieved through a diffusion process
that does not blur the edges themselves, so multiresolution
1057–7149/98$10.00 1998 IEEE
360 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 7, NO. 3, MARCH 1998
methods are not needed. The external force model that is
closest in spirit to GVF is the distance potential forces of
Cohen and Cohen [12]. Like GVF, these forces originate from
an edge map of the image and can provide a large capture
range. We show, however, that unlike GVF, distance potential
forces cannot move a snake into boundary concavities. We
believe that this is a property of all conservative forces that
characterize nearly all snake external forces, and that exploring
nonconservative external forces, such as GVF, is an important
direction for future research in active contour models.
We note that part of the work reported in this paper has
appeared in the conference paper [18].
II. B
ACKGROUND
A. Parametric Snake Model
A traditional snake is a curve
,
, that moves through the spatial domain of an image to
minimize the energy functional
(1)
where
and are weighting parameters that control the
snake’s tension and rigidity, respectively, and
and
denote the first and second derivatives of with respect to
. The external energy function is derived from the image
so that it takes on its smaller values at the features of interest,
such as boundaries. Given a gray-level image
, viewed
as a function of continuous position variables
, typical
external energies designed to lead an active contour toward
step edges [1] are
(2)
(3)
where
is a two-dimensional Gaussian function with
standard deviation
and is the gradient operator. If the
image is a line drawing (black on white), then appropriate
external energies include [10]:
(4)
(5)
It is easy to see from these definitions that larger
’s will cause
the boundaries to become blurry. Such large
’s are often
necessary, however, in order to increase the capture range of
the active contour.
A snake that minimizes
must satisfy the Euler equation
(6)
This can be viewed as a force balance equation
(7)
where
and . The
internal force
discourages stretching and bending while
the external potential force
pulls the snake toward the
desired image edges.
To find a solution to (6), the snake is made dynamic by
treating
as function of time as well as —i.e., .
Then, the partial derivative of
with respect to is then set
equal to the left hand side of (6) as follows:
(8)
When the solution
stabilizes, the term vanishes
and we achieve a solution of (6). A numerical solution to
(8) can be found by discretizing the equation and solving the
discrete system iteratively (cf., [1]). We note that most snake
implementations use either a parameter which multiplies
in order to control the temporal step-size, or a parameter to
multiply
, which permits separate control of the external
force strength. In this paper, we normalize the external forces
so that the maximum magnitude is equal to one, and use a unit
temporal step-size for all the experiments.
B. Behavior of Traditional Snakes
An example of the behavior of a traditional snake is shown
in Fig. 1. Fig. 1(a) shows a 64
64-pixel line-drawing of a
U-shaped object (shown in gray) having a boundary concavity
at the top. It also shows a sequence of curves (in black)
depicting the iterative progression of a traditional snake (
, ) initialized outside the object but within the
capture range of the potential force field. The potential force
field
where pixel is shown in
Fig. 1(b). We note that the final solution in Fig. 1(a) solves
the Euler equations of the snake formulation, but remains split
across the concave region.
The reason for the poor convergence of this snake is
revealed in Fig. 1(c), where a close-up of the external force
field within the boundary concavity is shown. Although the
external forces correctly point toward the object boundary,
within the boundary concavity the forces point horizontally
in opposite directions. Therefore, the active contour is pulled
apart toward each of the “fingers” of the U-shape, but not
made to progress downward into the concavity. There is no
choice of
and that will correct this problem.
Another key problem with traditional snake formulations,
the problem of limited capture range, can be understood by
examining Fig. 1(b). In this figure, we see that the magnitude
of the external forces die out quite rapidly away from the
object boundary. Increasing
in (5) will increase this range,
but the boundary localization will become less accurate and
distinct, ultimately obliterating the concavity itself when
becomes too large.
Cohen and Cohen [12] proposed an external force model
that significantly increases the capture range of a traditional
snake. These external forces are the negative gradient of a
potential function that is computed using a Euclidean (or
chamfer) distance map. We refer to these forces as distance po-
tential forces to distinguish them from the traditional potential
forces defined in Section II-A. Fig. 2 shows the performance
of a snake using distance potential forces. Fig. 2(a) shows
both the U-shaped object (in gray) and a sequence of contours
(in black) depicting the progression of the snake from its
initialization far from the object to its final configuration. The
XU AND PRINCE: GRADIENT VECTOR FLOW 361
(a) (b) (c)
Fig. 1. (a) Convergence of a snake using (b) traditional potential forces, and (c) shown close-up within the boundary concavity.
(a) (b) (c)
Fig. 2. (a) Convergence of a snake using (b) distance potential forces, and (c) shown close-up within the boundary concavity.
distance potential forces shown in Fig. 2(b) have vectors with
large magnitudes far away from the object, explaining why the
capture range is large for this external force model.
As shown in Fig. 2(a), this snake also fails to converge to
the boundary concavity. This can be explained by inspecting
the magnified portion of the distance potential forces shown in
Fig. 2(c). We see that, like traditional potential forces, these
forces also point horizontally in opposite directions, which
pulls the snake apart but not downward into the boundary
concavity. We note that Cohen and Cohen’s modification to
the basic distance potential forces, which applies a nonlinear
transformation to the distance map [12], does not change the
direction of the forces, only their magnitudes. Therefore, the
problem of convergence to boundary concavities is not solved
by distance potential forces.
C. Generalized Force Balance Equations
The snake solutions shown in Figs. 1(a) and 2(a) both satisfy
the Euler equations (6) for their respective energy model.
Accordingly, the poor final configurations can be attributed
to convergence to a local minimum of the objective function
(1). Several researchers have sought solutions to this problem
by formulating snakes directly from a force balance equation
in which the standard external force
is replaced by a more
general external force
as follows:
(9)
The choice of
can have a profound impact on both
the implementation and the behavior of a snake. Broadly
speaking, the external forces
can be divided into two
classes: static and dynamic. Static forces are those that are
computed from the image data, and do not change as the snake
progresses. Standard snake potential forces are static external
forces. Dynamic forces are those that change as the snake
deforms.
Several types of dynamic external forces have been invented
to try to improve upon the standard snake potential forces. For
example, the forces used in multiresolution snakes [11] and
the pressure forces used in balloons [10] are dynamic external
forces. The use of multiresolution schemes and pressure forces,
however, adds complexity to a snake’s implementation and
unpredictability to its performance. For example, pressure
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