MICHAILOVICH et al.: IMAGE SEGMENTATION USING ACTIVE CONTOURS 2789
last few decades [31]. This methodology is based on the uti-
lization of deformable contours which conform to various ob-
ject shapes. The contour deformation is typically driven by a
gradient flow stemming from minimization of an energy func-
tional, which can be dependent on either local (e.g., gradient
field) or global (e.g., mean intensity) properties of the image.
In the latter case, the resulting active contours are referred to as
region-based, and this is the group of methods to which the ap-
proach reported in this paper belongs.
The paper is organized as follows. Section II briefly revises
some fundamental aspects of the level-set framework and
introduces the Bhattacharyya energy functional, as well as the
related gradient flow. The statistical assumptions underpinning
the proposed approach are discussed in Section III. Some
possible choices of the discriminative features are summarized
here, as well. Section IV deals with the problem of automati-
cally controlling the smoothness of empirical distributions and
describes a simple method to perform this control. Section V
discusses a number of possible ways of extending the results
of the preceding sections to multiobject scenarios. Several
key results of our experimental study are demonstrated in
Section VI, while Section VII provides examples of practical
cases in which the proposed methodology can be advantageous
over some existing techniques. Section VIII finalizes the paper
with a discussion and conclusions.
II. B
HATTACHARYYA
FLOW
A. Level-Set Representation of Active Contours
In order to facilitate the discussion, we confine the derivations
below to the case of two classes (i.e., when the problem to be
dealt with is that of segmenting an object of interest out of its
background), followed by describing some possible ways to ex-
tent the proposed methodology to multiobject scenarios.
In the two-class case, the segmentation problem is reduced to
the problem of partitioning the domain of definition
of
an image
(with ) into two mutually exclusive and
complementary subsets
and . These subsets can be rep-
resented by their respective characteristic functions
and ,
which can, in turn, be defined by means of a level-set function
in the following manner. Let be the Heaviside
function defined in the standard way as
if
if
(2)
Then, one can define
and
, with .
Given a level-set function
, its zero level set
is used to implicitly represent a curve—active con-
tour—embedded into
. For the sake of concreteness, we asso-
ciate the subset
with the support of the object of interest,
while
is associated with the support of corresponding back-
ground. In this case, the objective of active-contour-based image
segmentation is, given an initialization
, to construct a
convergent sequence of level-set functions
(with
) such that the zero level-set of coin-
cides with the boundary of the object of interest.
The above sequence of level-set functions can be con-
veniently constructed using the variational framework [32].
Specifically, this sequence can be defined by means of a gra-
dient flow that minimizes the value of a properly defined cost
functional [31]. In the case of the present study the latter is
derived in the following way. First, the image to be segmented
is transformed into a vector-valued image of its local
features
.
2
Note that the feature image
ascribes to
every pixel of
an -tuple of its associated features, and,
hence, it can be formally represented as a map from
to .
Subsequently, given a level-set function
, the following
two quantities are computed:
(3)
and
(4)
where
, and and are two scalar-valued
functions with either compact or effectively compact supports
(e.g., Gaussian densities). Provided that the kernels
and
are normalized to have unit integrals with respect to the
feature vector
, viz. , the
functions
and given by (3) and (4)
are nothing else but kernel-based estimates of the probability
density functions (pdfs) of the image features observed over the
subdomains
and , respectively [33], [34].
The core idea of the preset approach is quite intuitive and it is
based on the assumption that, for a properly selected subset of
image features, the “overlap” between the informational con-
tents of the object and of the background has to be minimal.
In other words, if one thinks of the active contour as of a dis-
criminator that separates the image pixels into two subsets, then
the optimal contour should minimize the mutual information be-
tween these subsets. It is worthwhile noting that, for the case at
hand, minimizing the mutual information is equivalent to maxi-
mizing the KL divergence between the pdfs associated with the
“inside” and “outside” subsets of pixels. For the reasons dis-
cussed below, however, instead of the divergence, we propose to
maximize the Bhattacharyya distance between the pdfs. Specif-
ically, the optimal active contour
is defined as
(5)
where
(6)
2
In the section that follows, we will elaborate on some possible choices of
the feature space. Meanwhile, in order to clarify the meaning standing behind
of this operation, suffice it to note that
J
could be, for example, the image
I
(
x
)
itself or the vector-valued image of its partial derivatives.