IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 19, NO. 12, DECEMBER 2010 3243
Distance Regularized Level Set Evolution and Its
Application to Image Segmentation
Chunming Li, Chenyang Xu, Senior Member, IEEE, Changfeng Gui, and Martin D. Fox, Member, IEEE
Abstract—Level set methods have been widely used in image
processing and computer vision. In conventional level set formula-
tions, the level set function typically develops irregularities during
its evolution, which may cause numerical errors and eventually de-
stroy the stability of the evolution. Therefore, a numerical remedy,
called reinitialization, is typically applied to periodically replace
the degraded level set function with a signed distance function.
However, the practice of reinitialization not only raises serious
problems as when and how it should be performed, but also affects
numerical accuracy in an undesirable way. This paper proposes
a new variational level set formulation in which the regularity of
the level set function is intrinsically maintained during the level
set evolution. The level set evolution is derived as the gradient flow
that minimizes an energy functional with a distance regularization
term and an external energy that drives the motion of the zero
level set toward desired locations. The distance regularization
term is defined with a potential function such that the derived
level set evolution has a unique forward-and-backward (FAB)
diffusion effect, which is able to maintain a desired shape of the
level set function, particularly a signed distance profile near the
zero level set. This yields a new type of level set evolution called
distance regularized level set evolution (DRLSE). The distance
regularization effect eliminates the need for reinitialization and
thereby avoids its induced numerical errors. In contrast to com-
plicated implementations of conventional level set formulations, a
simpler and more efficient finite difference scheme can be used to
implement the DRLSE formulation. DRLSE also allows the use of
more general and efficient initialization of the level set function.
In its numerical implementation, relatively large time steps can
be used in the finite difference scheme to reduce the number
of iterations, while ensuring sufficient numerical accuracy. To
demonstrate the effectiveness of the DRLSE formulation, we apply
it to an edge-based active contour model for image segmentation,
and provide a simple narrowband implementation to greatly
reduce computational cost.
Index Terms—Forward and backward diffusion, image segmen-
tation, level set method, narrowband, reinitialization.
I. INTRODUCTION
T
HE LEVEL set method for capturing dynamic interfaces
and shapes was introduced by Osher and Sethian [1] in
1987. The basic idea of the level set method is to represent a
Manuscript received September 06, 2007; revised October 12, 2009; accepted
October 12, 2009. Date of publication August 26, 2010; date of current version
November 17, 2010. The associate editor coordinating the review of this manu-
script and approving it for publication was Prof. Scott T. Acton.
C. Li is with Institute of Imaging Science, Vanderbilt University, Nashville,
TN 37232 USA (e-mail: lchunming@gmail.com).
C. Xu is with Technology to Business Center, Siemens Corporate Research,
Berkeley, CA 94704 USA (e-mail: chenyang.xu@siemens.com).
C. Gui is with Department of Mathematics, University of Connecticut, Storrs,
CT 06269 USA (e-mail: gui@math.uconn.edu).
M. D. Fox is with Department of Electrical and Computer Engineering, Uni-
versity of Connecticut, Storrs, CT 06269 USA (e-mail: fox@engr.uconn.edu).
Digital Object Identifier 10.1109/TIP.2010.2069690
contour as the zero level set of a higher dimensional function,
called a level set function (LSF), and formulate the motion of
the contour as the evolution of the level set function. Some key
ideas in the level set method were proposed earlier by Dervieux
and Thomasset [2], [3] in the late 1970s, but their work did
not draw much attention. It was only after the work by Osher
and Sethian in [1], the level set method became well known
and since then has had far-reaching impact in various applica-
tions, such as computational geometry, fluid dynamics, image
processing, and computer vision.
In image processing and computer vision applications, the
level set method was introduced independently by Caselles
et al.
[4] and Malladi et al. [5] in the context of active contour (or
snake) models [6] for image segmentation. Early active contour
models are formulated in terms of a dynamic parametric contour
with a spatial parameter in [0,1],
which parameterizes the points in the contour, and a temporal
variable
. The curve evolution can be expressed as
(1)
where
is the speed function that controls the motion of the
contour, and
is the inward normal vector to the curve .
The curve evolution in (1) in terms of a parameterized con-
tour can be converted to a level set formulation by embedding
the dynamic contour
as the zero level set of a time de-
pendent LSF
. Assuming that the embedding LSF
takes negative values inside the zero level contour and positive
values outside, the inward normal vector can be expressed as
, where is the gradient operator. Then, the
curve evolution equation (1) is converted to the following par-
tial differential equation (PDE):
(2)
which is referred to as a level set evolution equation. The ac-
tive contour model given in a level set formulation is called an
implicit active contour or geometric active contour model. For
specific active contours in parametric and their corresponding
level set formulations, the readers are referred to [7].
A desirable advantage of level set methods is that they can
represent contours of complex topology and are able to handle
topological changes, such as splitting and merging, in a natural
and efficient way, which is not allowed in parametric active con-
tour models [6], [8], [9] unless extra indirect procedures are in-
troduced in the implementations. Another desirable feature of
level set methods is that numerical computations can be per-
formed on a fixed Cartesian grid without having to parameterize
the points on a contour as in parametric active contour models.
1057-7149/$26.00 © 2010 IEEE
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