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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY, VOL. 23, NO. 5, MAY 2013 883

Generalized Gradient Vector Flow for Snakes: New

Observations, Analysis, and Improvement

Lunming Qin, Ce Zhu, Senior Member, IEEE, Yao Zhao, Senior Member, IEEE, Huihui Bai, and Huawei Tian

Abstract—Snakes, or active contours, have been widely used

in image processing applications. An external force for snakes

called gradient vector ﬂow (GVF) attempts to address traditional

snake problems of initialization sensitivity and poor convergence

to concavities, while generalized GVF (GGVF) aims to improve

GVF snake convergence to long and thin indentations (LTIs).

In this paper, we ﬁnd and show that both GVF and GGVF

snakes essentially yield the same performance in capturing LTIs

of odd widths, and generally neither can converge to even-width

LTIs. Based on a thorough investigation of the GVF and GGVF

ﬁelds within the LTI during their iterative processes, we identify

the crux of the convergence problem, and accordingly propose

a novel external force termed as component-normalized GGVF

(CN-GGVF) to eliminate the problem. CN-GGVF is obtained by

normalizing each component of initial GGVF vectors with respect

to its own magnitude. Experimental results and comparisons

against GGVF snakes show that the proposed CN-GGVF snakes

can capture LTIs regardless of odd or even widths with a

remarkably faster convergence speed, while preserving other

desirable properties of GGVF snakes with lower computational

complexity in vector normalization.

Index Terms—Active contour models, convergence, external

force, gradient vector ﬂow (GVF), snakes.

I. Introduction

INCE SNAKES, or active contours, were ﬁrst proposed

by Kass et al. [1] in 1987, they have become one of the

most active and successful research areas in computer vision

and image processing applications [2]. Active contours are

curves deﬁned within an image domain that deform under the

inﬂuence of internal and external forces to conform to desired

features (like edges). Internal forces deﬁned within the curve

itself are determined by the geometric properties of the curve,

Manuscript received April 7, 2012; revised September 3, 2012; accepted

October 17, 2012. Date of publication January 24, 2013; date of current ver-

sion May 1, 2013. This work was supported in part by the 973 Program, under

Grant 2012CB316400; the National Natural Science Funds for Distinguished

Young Scholar, under Grant 61025013; the National NSF of China, under

Grant 61210006, Grant 61202240, Grant 60903066, Grant 61272051, and

Grant 61201393. This paper was recommended by Associate Editor H. Wang.

L. Qin, H. Bai, and H. Tian are with the Institute of Information Science,

Beijing Jiaotong University, Beijing Key Laboratory of Advanced Infor-

mation Science and Network Technology, Beijing 100044, China (e-mail:

lunming.qin@hotmail.com; hhbai@bjtu.edu.cn; hwtian@live.cn).

C. Zhu is with the School of Electronic Engineering, University of Elec-

tronic Science and Technology of China, Chengdu 611731, China (e-mail:

eczhu@uestc.edu.cn).

Y. Zhao is with the Institute of Information Science, Beijing Jiaotong

University, and with the State Key Laboratory of Rail Trafﬁc Control and

Safety, Beijing 100044, China (e-mail: yzhao@bjtu.edu.cn).

Color versions of one or more of the ﬁgures in this paper are available

online at http://ieeexplore.ieee.org.

Digital Object Identiﬁer 10.1109/TCSVT.2013.2242554

while external forces are derived from the image data, which

have been the most discussed issue in active contour models.

An energy function associated with the curve is deﬁned, so

that the problem of ﬁnding desired features is converted into

an energy minimization process. Due to its high efﬁciency,

the snake model has found many applications, including

edge detection [1], shape recovery [3], [4], object tracking

[5]–[8], and image segmentation [9]–[12]. However, there are

two key shortcomings with the traditional snake. First, the

initial contour must be close enough to the desired features,

otherwise the snake will likely converge to a wrong result.

Second, it is difﬁcult for the snake to move into boundary

concavities [13], [14].

An external force for snakes, called gradient vector ﬂow

(GVF) [15], was introduced to largely overcome the two

limitations of the traditional snake. GVF is computed as a

diffusion of the gradient vectors of a gray-level or binary

edge map derived from an image. Since its publication about a

decade ago, GVF has been widely used and adapted to various

models and problems, e.g., segmentation [16]–[19], tracking

[6], [8], registration [20], and skeletonization [21]. Addition-

ally, automatic initialization methods of the GVF snake can

be found in [22] and [23]. Efﬁcient numerical schemes are

applied to speed up the GVF computation [24], [25].

Although GVF has been widely used and improved in vari-

ous models, it still exhibits some defects, such as the difﬁculty

in forcing a snake into long and thin indentations (LTIs) as

well as noise sensitivity. In [26], Xu and Prince proposed a

generalized version of GVF (generalized GVF or GGVF) by

introducing two spatially varying weighting functions, which

has been reported to improve GVF convergence to LTIs as well

as robustness to noise. In [27], the GVF in the normal direction

(NGVF) snake is proposed, which can enter into LTIs at a

faster convergence speed. Furthermore, the normally biased

GVF (NBGVF) snake is claimed to perform well on weak edge

preserving and noise robustness while maintaining the desir-

able properties of capturing LTIs [28]. In [29], by adding a new

external force, an improved GVF active contour is presented

to detect LTIs. Besides these general snakes, there are also

some speciﬁc active contours aiming to segment the narrow

structures, like blood vessels in medical data or roads in aerial

images [30], [31]. In spite of the various improvements for

GVF/GGVF, the underlying mechanism for the difﬁculty in

capturing LTIs has still not been fully understood.

In this paper, we revisit the conventional edge-based

parametric GVF and GGVF snakes to present some new

1051–8215/$31.00

 2013 IEEE

884 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY, VOL. 23, NO. 5, MAY 2013

observations and the associated analysis, and then identify

the intrinsic difﬁculties with respect to GVF and GGVF

convergence to LTIs. Based on the theoretical analysis we

propose an improved snake model. Speciﬁcally, our main

contributions are summarized as follows.

1) New observations on GVF and GGVF snakes are given

to show that the two models essentially yield the same

performance in capturing LTIs of odd widths, and gener-

ally neither can converge to even-width LTIs. The obser-

vations are different from the opinion in [26] that GGVF

snakes outperform GVF snakes in the convergence to

LTIs.

2) The force characteristics within LTIs are theoretically

investigated, based on which, we identify two intrinsic

difﬁculties for GVF and GGVF snakes in convergence to

LTIs, namely, the obliteration and noise problems. More

details on the theoretical analysis and the two problems

can be found in Sections III and IV, respectively.

3) To cope with the obliteration problem, a simple yet

effective external force called component-normalized

GGVF (CN-GGVF) is developed, which can efﬁciently

move the snake into even-width LTIs, with details in

Section V.

The remainder of the paper is organized as follows.

Section II reviews the traditional snake, GVF and GGVF

snakes, and presents some new observations. Section III

theoretically examines the force characteristics within LTIs

and Section IV identiﬁes the two intrinsic difﬁculties in

convergence to LTIs, i.e., the obliteration and noise problems,

in GVF and GGVF. In Section V, a novel external force termed

CN-GGVF for snakes is proposed. Furthermore, performance

of the CN-GGVF snake is examined in Section VI and the

conclusion is given in Section VII.

II. Background and New Observations

A. Traditional Snake Model

The traditional snake is represented by a parametric curve

c(s)=[x(s),y(s)], s ∈ [0, 1], which deforms through the

spatial domain of an image to minimize the energy function [1]

E(c(s))=





(α





(s)



+β





(s)



)+E

ext

(c(s))



ds (1)

where α and β are weighting parameters controlling the

snake’s tension and rigidity, respectively, and c



(s) and c



(s)

are the ﬁrst and second derivatives of c(s) with respect to s.

The ﬁrst two terms within the above integral stand for the

internal energy. The external energy E

ext

(c(s)) derived from

the image data takes on its smaller values at the desired

features. Representative external energy functions for a gray-

level image I(x, y) for seeking step edges are given as [1]

(1)

ext

(x, y)=−

∇I(x, y)

(2)

ext

(x, y)=−

∇[G

(x, y) ∗ I(x, y)]

(3)

where G

(x, y) is a 2-D Gaussian function with standard

deviation σ, ∗ denotes linear convolution, and ∇ is the gradient

operator. Typical external energy functions for a line-drawing

(black on white) [32] are

(3)

ext

(x, y)=I(x, y)(4)

(4)

ext

(x, y)=G

(x, y) ∗ I(x, y). (5)

To minimize E(c(s)), the snake must satisfy the Euler–

Lagrange equation

αc



(s) − βc



(s) −∇E

ext

(c(s)) = 0 (6)

which can be viewed as a force balance equation

int

(c(s)) + F

ext

(c(s)) = 0 (7)

where F

int

(c(s)) = αc



(s) − βc



(s) is the internal force

shrinking and smoothing the contour, and the external force

ext

(c(s)) = −∇E

ext

(c(s)) pulls the snake to the desired

features.

To ﬁnd a solution to (6), the active contour c(s) has to evolve

dynamically as a function of time t, i.e., c(s, t). By making the

partial derivative of c(s, t) with respect to t, i.e., c

(s, t), equal

to the left-hand side of (6), the dynamic snake function is

given as

(s, t)=αc



(s, t) − βc



(s, t) −∇E

ext

(c(s, t)). (8)

The solution c(s) of (6) is obtained when the steady-state

solution c(s, t) of (8) is reached (c

(s, t) = 0) from an initial

contour c(s, 0). A numerical solution of c(s, t) on a discrete

grid can be obtained by discretizing (8) and solving the

discrete equation iteratively [1], [33].

B. GVF Snake Model

Xu and Prince proposed gradient vector ﬂow (GVF) [15]

as an external force for active contours to largely overcome

the two key shortcomings of the traditional snake. The force

balance equation (7) is used as a starting point for designing

a GVF snake. The external force term −∇E

ext

(c(s)) in (7) is

replaced with a GVF ﬁeld v(x, y)=[u(x, y),v(x, y)] deﬁned

as the equilibrium solution of the following vector partial

differential equation:

(

x, y, t

)

= μ∇

(

x, y, t

)

−

∇f



(

x, y, t

)

−∇f



(9)

where v

(

x, y, t

)

is the partial derivative of v(x, y, t) with

respect to t, and ∇

= ∂



∂x

+ ∂



∂y

denotes the Laplacian

operator. f is an edge map derived from the image and deﬁned

to have larger values at the desired features, which is typically

the additive inverse of an external energy function as given in

(2)–(5). μ controls the smoothness degree of the GVF ﬁeld

and should be set according to the noise level in the image

(larger μ for more noise).

C. GGVF Snake Model

1) GGVF Field: By introducing two spatially varying

weighting functions into the GVF formulation, Xu and Prince

proposed an external force referred to as generalized gradient

vector ﬂow (GGVF) [26]. As a generalization of GVF, GGVF

was reported to improve contour convergence to LTIs and

QIN et al.: GENERALIZED GRADIENT VECTOR FLOW FOR SNAKES 885

robustness to noise, which is deﬁned as the equilibrium

solution of the following partial differential equation:

(x, y, t)=g(|∇f |)∇

v(x, y, t) − h(|∇f |)[v(x, y, t) −∇f ]

(10)

where

∇f

)=exp(−

∇f



k) (11)

∇f

)=1− exp(−

∇f



k). (12)

The parameter k regulates to some extent the tradeoff between

the ﬁrst term (known as smoothing term) and the second term

(known as data term) in the right-hand side of (10) and it

should be set according to the amount of noise in the image

(larger k for higher noise levels).

2) Numerical Implementation: As in [26], the partial

differential equation (10) specifying the GGVF ﬁeld can be

implemented using an explicit ﬁnite difference scheme. In

(10), |∇f | can be calculated using any digital image gradient

operator, e.g., simple central differences employed in this

paper. Since the data term in (10) is deﬁned to encourage the

GGVF ﬁeld to be close to the edge map gradient computed

from the image, its weighting coefﬁcient h(|∇f |) must be

greater than or equal to 0. According to (12), the parameter k

must be greater than 0.

To set up the iterative solution, let the spatial sample

intervals be x and y and the time step for each iteration be

t. The partial derivative v

(x, y, t) can then be approximated

as [v(x, y, t + t) − v(x, y, t)]/t. On a discrete grid, the

Laplacian term can be approximated by

∇

(

x, y, t

)

xy

A ∗ v

(

x, y, t

)

(13)

where A =



010

1 −41

010



is the Laplacian matrix.

Substituting the above approximations into (10) gives an

iterative solution to the GGVF ﬁeld as follows:

v(x, y, t + t)= v(x, y, t)+

t

xy

g(|∇f |) A ∗ v(x, y, t)

−th(|∇f |)[v(x, y, t) −∇f ]. (14)

According to the GGVF stability constraint t



(

xy

)

≤



4 [26], t must be not larger than 1



4 by making x =

y = 1. To obtain the maximum smoothness of the GGVF

ﬁeld, we set t =1



4. Accordingly, (14) can be rewritten as

v(x, y, t + t)= v(x, y, t)+

g(|∇f |) A ∗ v(x, y, t)

−

h(|∇f |)[v(x, y, t) −∇f ]. (15)

The desired external force ﬁeld v(x, y, t) is iteratively calcu-

lated from the initial vector ﬁeld v(x, y, 0) = ∇f , i.e., the

gradient of the edge map. Typically, the iteration number

(denoted as n) required to calculate the GGVF ﬁeld or the

above GVF ﬁeld for an n

×n

-pixel image is

√

× n

[15].

After the iterative process, external forces in actual imple-

mentations are normalized to unit vectors to make the snake

evolve at a constant speed. Typically, the initial GGVF vectors

v(x, y)=[u(x, y),v(x, y)] are normalized with respect to their

magnitudes using

vn−ggvf

(x, y)=



v(x, y)/|v(x, y)|, |v(x, y)| =0

[

0, 0

]

, |v(x, y)| =0

(16)

which is referred to as vector-based normalization.

Fig. 1. Performance of GVF and GGVF snakes in detecting an odd-width

LTI. The initial and resulting snakes are indicated by cyan dashed and solid

lines, respectively. The external force ﬁelds in the top and bottom rows are

properly and over smoothed, respectively. First column: GVF snakes. Second

column: GVF ﬁelds within the LTI. Third column: GGVF snakes. Fourth

column: GGVF ﬁelds within the LTI. (a) GVF snake. (b) Properly smoothed.

(g) GGVF snake. (h) Over-smoothed.

D. New Observations on GVF and GGVF Snakes

It was reported in [26] that GGVF snakes outperform GVF

snakes in the convergence to LTIs. GVF’s difﬁculty of forcing

a snake into an LTI is caused by excessive smoothing of

the ﬁeld near the LTI boundary, governed by the smoothing

coefﬁcient μ in (9). When a much smaller μ is used, the GVF

snake can also achieve a good convergence to the LTI similar

to that of GGVF, but at the expense of an order of magnitude

higher computational complexity than that of GGVF [26].

However, to our observation, GVF and GGVF snakes yield

almost the same performance in convergence to LTIs. Speciﬁ-

cally, when the LTI width is of an odd number of pixels, both

snakes can converge completely to the LTI if their external

force ﬁelds are appropriately smoothed

, as reported in [26],

whereas the computational cost required for the GVF ﬁeld is

almost the same as that for the GGVF ﬁeld, which is opposed

to the opinion in [26]. However, if their vector ﬁelds are over-

smoothed

, not only the GVF snake but also the GGVF snake

fails to capture the LTI. On the other hand, when the LTI width

is an even number of pixels, in general, neither the GVF snake

nor the GGVF snake is able to detect the LTI no matter how

their external force ﬁelds are smoothed. The following will

further elaborate our observations.

Fig. 1 shows the performance of the GVF and GGVF snakes

in capturing an odd-width LTI. The original image in Fig. 1

is a line-drawing of a square-shaped object (shown in black)

with the odd-width LTI. All resulting snakes indicated by cyan

solid lines are derived from a same initialization indicated by

a cyan dashed line. All ﬁelds within the LTI are sampled by

a factor of 3 in the y direction. We ﬁnd that at almost the

same computational cost, both the GVF snake in Fig. 1(a)

The vector ﬁeld is considered to be appropriately smoothed when k in (10)

is less than a selected value to be discussed in Section IV-B.

The vector ﬁeld is considered to be over-smoothed when k is greater than

the selected value.

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