Level Set Evolution Without Re-initialization: A New Variational Formulation
Chunming Li
1
, Chenyang Xu
2
, Changfeng Gui
3
, and Martin D. Fox
1
1
Department of Electrical and
2
Department of Imaging
3
Department of Mathematics
Computer Engineering and Visualization University of Connecticut
University of Connecticut Siemens Corporate Research Storrs, CT 06269, USA
Storrs, CT 06269, USA Princeton, NJ 08540, USA gui@math.uconn.edu
{cmli,fox}@engr.uconn.edu chenyang.xu@siemens.com
Abstract
In this paper, we present a new variational formulation
for geometric active contours that forces the level set func-
tion to be close to a signed distance function, and therefore
completely eliminates the need of the costly re-initialization
procedure. Our variational formulation consists of an in-
ternal energy term that penalizes the deviation of the level
set function from a signed distance function, and an exter-
nal energy term that drives the motion of the zero level set
toward the desired image features, such as object bound-
aries. The resulting evolution of the level set function is
the gradient flow that minimizes the overall energy func-
tional. The proposed variational level set formulation has
three main advantages over the traditional level set formu-
lations. First, a significantly larger time step can be used for
numerically solving the evolution partial differential equa-
tion, and therefore s peeds up the curve evolution. Second,
the level set function can be initialized with general func-
tions that are more efficient to construct and easier to use
in practice than the widely used signed distance function.
Third, the level set evolution in our formulation can be
easily implemented by simple finite difference scheme and
is computationally more efficient. The proposed algorithm
has been applied to both simulated and real images with
promising results.
1. Introduction
In recent years, a large body of work on geometric ac-
tive contours, i.e., active contours implemented via level set
methods, has been proposed to address a wide range of im-
age segmentation problems in image processing and com-
puter vision (cf. [3, 5, 7]). Level set methods were first in-
troduced by Osher and Sethian [11] for capturing moving
fronts. Active contours were introduced by Kass, Witkins,
and Terzopoulos [1] for segmenting objects in images using
dynamic curves. The existing active contour models can be
broadly classified as either parametric active contour mod-
els or geometric active contour models according to their
representation and implementation. In particular, the para-
metric active contours [1, 2] are represented explicitly as
parameterized curves in a Lagrangian framework, while the
geometric active contours [5–7] are represented implicitly
as level sets of a two-dimensional function that evolves in
an Eulerian framework.
Geometric active contours are independently introduced
by Caselles et al. [5] and Malladi et al. [7], respectively.
These models are based on curve evolution theory [10] and
level set method [17]. The basic idea is to represent con-
tours as the zero level set of an implicit function defined in a
higher dimension, usually referred as the level set function,
and to evolve the level set function according to a partial dif-
ferential equation (PDE). This approach presents several ad-
vantages [4] over the traditional parametric active contours.
First, the contours represented by the level set function may
break or merge naturally during the evolution, and the topo-
logical changes are thus automatically handled. Second, the
level set function always remains a function on a fixed grid,
which allows efficient numerical schemes.
Early geometric active contour models (cf. [5–7]) are
typically derived using a Lagrangian formulation that yields
a certain evolution PDE of a parametrized curve. This PDE
is then converted to an evolution PDE for a level set func-
tion using the related Eulerian formulation from level set
methods. As an alternative, the evolution PDE of the level
set function can be directly derived from the problem of
minimizing a certain energy functional defined on the level
set function. This type of variational methods are known as
variational level set methods [8, 9, 14].
Compared with pure PDE driven level set methods,
the variational level set methods are more convenient and
natural for incorporating additional information, such as
region-based information [8] and shape-prior information
[9], into energy functionals that are directly formulated in
the level set domain, and therefore produce more robust re-
sults. For examples, Chan and Vese [8] proposed an ac-
Proceedings of the 2005 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR’05)
1063-6919/05 $20.00 © 2005 IEEE
