基于Lennard-Jones势场的图像分割几何活动轮廓模型
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本研究论文探讨了"几何活动轮廓的Lennard-Jones力场"这一主题，主要关注于将经典的Lennard-Jones势能模型应用于图像分割领域中的几何活动轮廓（GAC）方法。Lennard-Jones力场通常用于描述原子间的相互作用，但在本文中，作者创新性地将这种理论应用到计算机视觉中的图像处理任务，尤其是无边缘地图依赖的图像分割。
传统上，GAC模型依赖于预先计算出的边缘信息，以引导轮廓沿着图像的边界或特征进行演化。然而，这篇论文提出了一种新的GAC模型，它不再受限于这样的预处理步骤。作者将每个像素视为一个"粒子"，设计了一个基于Lennard-Jones力的交互模型，通过模拟像素间的吸引力和排斥力来驱动轮廓的动态变化。这种设计使得模型能够自适应地捕捉图像内部的结构和纹理信息，而不仅仅是边缘。
为了增强模型的灵活性，论文引入了一个距离规整参数，允许用户调整力的强度，从而适应不同场景下的需求。此外，作者还定义了一个能量函数，将Lennard-Jones势能与GAC的形态学优化结合起来，以形成一个完整的能量优化框架。这种结合使得模型能够在保持形状变化的同时，兼顾全局的结构相似性和局部的细节一致性。
论文的核心贡献在于提出了一种新颖的GAC求解策略，它在不依赖于额外特征提取的情况下，利用Lennard-Jones力场实现对图像内容的有效分割。这种方法可能在处理复杂图像、物体分割以及自适应边缘检测等领域展现出优越性能，因为它能更好地反映像素间的内在关系，并在一定程度上克服了传统GAC模型对于边缘依赖的局限性。
该研究不仅提升了图像分割技术的理论基础，也为实际应用提供了强大的工具，特别是在那些边缘不明显或者需要考虑像素间复杂相互作用的任务中。通过将物理世界的分子间作用力理论融入到计算机视觉的图像处理中，论文展示了跨学科知识在解决实际问题中的潜在价值。
According to the inverse-square law in physics, we choose
m ¼ 2, n ¼ 1, and
s
¼ 1 for simplicity in the paper.
We take the parameter c as a zero-nonzero switch,
which controls two different characteristics of the force
field. When c ¼ 0, only the attractive part takes effect. In
this case, the vector flows bi-directionally converge to the
boundaries. If ca0, the combination of the attractive
effect and the repulsive effect produces equilibrium
adjacent to the boundary with a distance. In this situation,
the morphological dilation effect is achieved by the
equilibrium positions. We will use these two different
characteristics for different applications (see details in
Section 5). In the following, we take two toy examples, the
line edge and the step edge, to show these characteristics
of the L-J force intuitively.
3.1.1. Case of c ¼ 0
This case means that we only use the attractive term in
(7),
F
Attr
¼exp
D
2
ðe
i
; e
j
Þ
k
!
1
ðjrjþ
d
Þ
2
r
jrj
: ð8Þ
Fig. 2(a) shows the attraction performance of L-J force
field on the line edge. The top of Fig. 2(a) is the vector
flows of the line edge map, and the amplitude response of
L-J force field is shown at the bottom. The sign of
amplitude response curve represents the direction of
vector flows: the negative stands for direction from right
to left, and the positive for the opposite direction. In
Fig. 2(a), the vector flows bi-directionally converge to the
line edge, i.e., the L-J force forms a head-to-head vector
flows across the line edge. The performance of L-J force
field on the step edge is illustrated in Fig. 2(b). The vector
flows and the amplitude responses of the step edge are
similar to those of the line edge in Fig. 2(a). It implies that
the L-J force field not only forms the vector flows bi-
directionally converging to the boundaries in the line-like
edge map, but also it can achieve the same effect to the
step-like edge map or piecewise-content image.
2
3.1.2. Case of ca0
The case of ca0 means both the attractive force and
the repulsive force are taken into account. Fig. 2(c)
illustrates the performance of L-J force field on the line
edge map in this case, where the top shows the vector
flows of the line edge and the bottom is the amplitude
response curve. Across the line edge, the vector flows
converge to the equilibrium positions at both sides of the
line edge with a distance to the line edge. The equili-
briums correspond to the zero-crossing points in the
amplitude response curve. This phenomenon can be used
to form the morphological dilation effect. The L-J force
field of the step edge is shown in Fig. 2(d), and it is similar
to Fig. 2(c): for either the step edge or the line edge the L-J
force field can achieve the same morphological dilation
effect.
3.2. L-J force driven GAC
Our purpose is to design a GAC model based on the L-J
force field. An intuitive way is to replace F
Img
in (1) with
the L-J force F
LJ
img
, i.e., C
t
¼ð
k
ðCÞþb þ F
LJ
img
NÞN, where
k
and b are same as in (1). However, the constant
propagation expansion term b usually cannot respond
properly to the local situation. At the boundary positions
whose responses are weaker than b, i.e., the weak edges,
the evolving curve will be divergent. If we drop out the
balloon force or set it with a small value improperly, the
curve evolution will show the phenomenon of conglutina-
tion at those positions where F N 0, or where the
amplitude of the local force is too weak to drive C. To deal
with the above issues, we use an adaptive propagation
expansion term b according to the local force field [11]:
b
ðsÞ¼sign
ðFðsÞNÞgðjFðsÞNjÞ; ð9Þ
where sign
ðxÞ is a sign function to indicate the local
direction change (inflation or deflation) of the curve:
sign
ðxÞ¼1 when xZ
b
, otherwise sign
ðxÞ¼1.
b
is a
constant to balance the expansion and the contraction
effect. And gðÞ is a monotonically decreasing function that
adaptively controls the curve evolution according to the
local vector field. Therefore the proposed L-J force driven
GAC model is given as
C
t
¼ð
k
ðCÞþb
ðCÞþF
LJ
img
NÞN: ð10Þ
U shape is often used to test the performance of the
GAC model for the special case with deep concave shape.
In [9], GVF showed good property for U shape. Fig. 3 shows
the results of U shape by the proposed method and GVF. It
can be found that the proposed L-J force driven GAC deals
with the U shape as well as GVF.
Fig. 4 shows the performances with different
expansion term schemes. Fig. 4(c) is the result with
b ¼ 0. The curve cannot go into the deep narrow gaps,
because the vectors near the boundaries are nearly
perpendicular to the boundaries and the width of the
gap is very narrow so that there are no vectors digging
into the depth of the gap. Fig. 4(d) shows the result using a
constant expansion term ðb ¼ 0:3Þ. Since b is relatively
larger than some edge responses, some weak edges of
matches are leaked; a constant expansion term is not
suitable for the situation where the edge responses are
non-uniform. Fig. 4(b) shows the result using the local
adaptive expansion term. The whole boundaries of
matches are well segmented.
4. Analysis of vector flow field
In this section, we first discuss how to evaluate the
force field in GAC model quantitatively. Then we will give
a comparison between the L-J force field and GVF. At last,
we will prove that the electrostatic model [13,20] is
actually a special case of the L-J force field model.
4.1. Two quantitative criteria for the force field evaluation
To our best knowledge, all the previous works
evaluated the force field used in GAC by observing the
ARTICLE IN PRESS
2
By selecting proper DðÞ for feature set F , image can be
decomposed into approximately ‘‘stepwise’’ components in the sense
of homogeneity of metric.
Please cite this article as: Z. Li, et al., Lennard-Jones force field for geometric active contour, Signal Process. (2009),
doi:10.1016/j.sigpro.2009.10.008
Z. Li et al. / Signal Processing ] (]] ]]) ]]]–]]]4
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