Question:
This paper proposes an advanced image segmentation algorithm that combines active contour model, superpixel segmentation, and deep object detection algorithm. This is effective because:
A. These three methods are highly adaptable.
B. This combination can solve the problems existing in current active contour models.
C. Previous active contour models has weak noise resistance and boundary localization capabilities.
D. Repeated experiments have proven that this combination is effective.

Answer: B

More in depth, we propose an ACM combined with fuzzy superpixel centers (FSCs) and nonlinear diffusion filter (NDF) to solve the manual contour initialization problem of ACM, transforming the object detection algorithm into instance segmentation. First, the object detection algorithm locates the area of interest, while the original image is divided into superpixels. Then, the superpixels of interest are extracted, and their joint boundary is obtained as the initial contour of ACM. Furthermore, the FCM clustering based on regularization entropy on superpixels is applied, while the clustered regional information is integrated into the energy function of ACM to achieve an optimized iterative process. In addition, a module based on a nonlinear diffusion function is designed to enhance the boundary information of the target object and suppress background interference. The proposed model, called FSC&NDF, achieves high-precision segmentation compared with traditional and deep learning algorithms. The main innovations are as follows.
1) The object detection algorithm and superpixels work together to extract the rough contour of key objects, avoiding manual contour initialization of ACM.
2) A new energy description method is proposed, which incorporates the information of superpixels extracted by FSCs and optimizes the iteration direction.
3) NDF module is designed to enhance boundary gradient and reduce background noise, further improving the efficiency of contour evolution.

IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 74, 2025
5035013
An Active Contour Model Based on Fuzzy
Superpixel Centers and Nonlinear Diﬀusion
Filter for Instance Segmentation
Yiyang Chen , Member, IEEE, Fuzheng Zhang , Guina Wang , Guirong Weng ,
and Daniele Fontanelli , Fellow, IEEE
Abstract—The signiﬁcant weaknesses of the active contour
model (ACM) are the manual setting of contour and the inability
to process images with complex information, which limits its
eﬃciency and application scope. In this article, an ACM, called
FSC&NDF, is combined with fuzzy superpixel centers (FSCs) and
nonlinear diﬀusion ﬁlter (NDF) to solve the above two problems
simultaneously. YOLOv9 is adopted to locate the superpixels of
interest; the joint boundaries of these superpixels are set as the
initial contour, which is close to the morphological features of the
target. Improved fuzzy superpixel clustering is applied to extract
image features and yield superpixel centers, and the clusters
are integrated into the main body of the energy function, NDF
module further enhances boundary positioning and suppresses
noise. In addition, the proposed connection mechanism makes
it possible to convert object detection to instance segmentation.
Experimental results show that FSC&NDF overcomes the limi-
tations of previous ACMs in all aspects and its FPS, AP, AP50,
and APM are higher than mainstream deep learning algorithms.
The platform experiment based on the telecentric lens further
proves the practicality of FSC&NDF.
Index Terms—Active contour, automatic contour generation,
fuzzy superpixel centers (FSCs), image segmentation, nonlinear
diﬀusion ﬁlter (NDF).
I. INTRODUCTION
A
VARIETY
of
image
segmentation
algorithms
are
designed to solve diﬀerent image problems, includ-
ing particle swarm optimization, wavelet analysis, genetic
algorithm, watershed transformation (WT), active contour evo-
lution, and deep learning model. These algorithms improve
the eﬃciency of segmentation to a certain extent and are
accepted in a wide range of industrial applications [1],
[2], [3]. Active contour model (ACM) is a comprehen-
sive image segmentation method, which is frequently used
in natural image segmentation [4], [5], [6], medical image
Received 11 October 2024; revised 18 January 2025; accepted 16 May
2025. Date of publication 2 June 2025; date of current version 10 June
2025. This work was supported in part by the National Natural Science
Foundation of China under Grant 62103293 and in part by the Natural Science
Foundation of Jiangsu Province under Grant BK20210709. The Associate
Editor coordinating the review process was Dr. Pritam Paral. (Corresponding
author: Guina Wang.)
Yiyang
Chen,
Fuzheng
Zhang,
Guina
Wang,
and
Guirong
Weng
are with the School of Mechanical and Electrical Engineering, Soo-
chow University, Suzhou 215137, China (e-mail: yychen90@suda.edu.cn;
fzzhang1128@stu.suda.edu.cn; wangguina@suda.edu.cn; wgr@suda.edu.cn).
Daniele Fontanelli is with the Department of Industrial Engineering, Uni-
versity of Trento, 38123 Trento, Italy (e-mail: daniele.fontanelli@unitn.it).
Digital Object Identiﬁer 10.1109/TIM.2025.3573369
processing [7], [8], [9], remote sensing [10], [11], [12].
The geometric active contour (GAC) model [13] is a clas-
sic ACM which is also known as the level set method.
The contour is regarded as the intersection of a continuous
3-D surface and the 2-D image, and the core energy function
is constructed based on this surface, which is an implicit
expression of the 2-D contour. According to the Euler equation
and calculus of variations [14], the derivative of the energy
function with respect to the surface equal to the inverse of
the derivative of the surface with respect to time, which is a
brachistochrone problem. Therefore, the evolution of the 2-D
contour is transformed into the iteration of the 3-D surface,
which is equivalent to solve low-dimensional problems with
high-dimensional functions. The drawback is that the initial
state of the contour is decisive. While the initial contour is set
manually, inappropriate ones can lead to undersegmentation.
The bias ﬁeld theory and clustering method are applied to
reform the original image in [15] and [16], which extracts
the grayscale information of each position. Since the clus-
tering method is sensitive to the original image, the results
may be severely distorted by high-level noise interference.
Another mainstream method is to convert image ﬁtting into
distribution approximation [17], [18]. Classic divergence such
as Jeﬀreys divergence and K–L divergence are suitable to
describe the grayscale distribution of the image, providing
a reasonable channel for quantiﬁcation of image texture and
regional similarity. When the number of targets in the image
increases signiﬁcantly, it is diﬃcult to accurately describe the
distribution of each target. Fuzzy c-means (FCMs) clustering
is combined with diﬀerent modules of ACM to yield promis-
ing results [19], [20]. By clustering the preprocessed image,
key information are obtained and integrated into the energy
function. However, each iteration requires repeated clustering
operations, which increases the time cost and computational
overhead.
Generating a suitable initial contour for ACM depends on
the feature information of the image. Superpixel segmentation
is a commonly used preprocessing method in image processing
tasks, which uses the feature information of the image, such
as texture, color, and brightness, to group the pixels of the
original image, and each of these groups is called a superpixel.
Based on this, the basic information of the image is classiﬁed
and simpliﬁed, providing a guarantee for subsequent accurate
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IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 74, 2025
segmentation. Simple linear iterative clustering (SLIC) [21]
is one of the most well-known algorithms, which implements
clustering by calculating the color distance and spatial distance
between pixels. Its classiﬁcation results are presented as regu-
lar matrices, and for irregular targets, the generalization results
are not ideal. Kernel function similarity measurement is added
in LSC [22] to optimize the comprehensive distance between
pixels and achieve high memory eﬃciency. Adaptive morpho-
logical reconstruction (AMR) is applied in [23] to screen seed
regions from multiple scales and obtain all boundaries that
may contain target information.
Superpixels target the entire image domain, to extract
superpixel subblocks in key areas, it is necessary to locate
these areas in advance. Object detection algorithms are tai-
lored for this purpose, which are based on deep learning
and have gained widespread attention due to their excellent
target positioning capabilities. They can be divided into two
categories, one is to set numerous candidate boxes on the
image and ﬁlter them to obtain the ﬁnal detection box [24],
[25], [26], [27], and the other is to predict the center point of
the target object, and obtain the four vertices of the detection
box through the regression of the center point [28], [29].
Generally, the detection box holds the complete target area,
so a novel point is that the rough contour of the target object
can be acquired by extracting the superpixels within the box.
With the support of ACM, the rough contour evolves toward
the ground truth, so that instance segmentation is realized.
This is a win-win design: the object detection algorithm is
converted into instance segmentation without adding complex
segmentation modules, and the initial contour of ACM is set
automatically.
More in depth, we propose an ACM combined with fuzzy
superpixel centers (FSCs) and nonlinear diﬀusion ﬁlter (NDF)
to solve the manual contour initialization problem of ACM,
transforming the object detection algorithm into instance seg-
mentation. First, the object detection algorithm locates the area
of interest, while the original image is divided into superpixels.
Then, the superpixels of interest are extracted, and their joint
boundary is obtained as the initial contour of ACM. Further-
more, the FCM clustering based on regularization entropy on
superpixels is applied, while the clustered regional information
is integrated into the energy function of ACM to achieve an
optimized iterative process. In addition, a module based on a
nonlinear diﬀusion function is designed to enhance the bound-
ary information of the target object and suppress background
interference. The proposed model, called FSC&NDF, achieves
high-precision segmentation compared with traditional and
deep learning algorithms. The main innovations are as follows.
1) The object detection algorithm and superpixels work
together to extract the rough contour of key objects,
avoiding manual contour initialization of ACM.
2) A new energy description method is proposed, which
incorporates the information of superpixels extracted by
FSCs and optimizes the iteration direction.
3) NDF module is designed to enhance boundary gradient
and reduce background noise, further improving the
eﬃciency of contour evolution.
II. MOTIVATIONS
Previous ACMs rely on manually drawn initial contours
to achieve a stable segmentation process, which limits the
application scenarios of the model. These models are short
of feature information extraction, resulting in the inability
to handle complex images with multiple targets and scenes.
In addition, their antinoise and boundary positioning capabil-
ities need to be further improved.
A. Necessity of Automatic Contour Generation
Although the segmentation eﬀect of ACM is commendable,
it is deeply aﬀected by the state of the initial contour. Three
models, RSF [15], GADF [30], and OLPFI [31], are adopted
to present the problem. As shown in Fig. 1, the initial contours
are all set manually. When the contours present the state
of Case 1, the three models yield ideal segmentation eﬀects
on single-target images, while they are not up to par with
multitarget images. When the contours change to the state
of Case 2, all segmentation results become worse, with over-
segmentation and under-segmentation appearing. Taking RSF
model as an example, the structure of its energy equation is
similar to that of most ACMs, which is
E =
2
X
i=1
ci
Z
Ωi
Z
Kλ (I (p) −fi (q))2 Mi (p) dp

dq.
(1)
Notice that i takes values between 1 and 2, which means that
the model treats the entire image domain as Ω1 inside the
contour and Ω2 outside the contour, I(p)−f1(q) and I(p)−f2(q)
represent the energy of these two regions, respectively. This
is equivalent to ignoring all other contours when one contour
evolves. Therefore, when processing multitarget images, it is
necessary to manually draw a contour for each target, and the
same mechanism applies to GADF and OLPFI. Technically
speaking, the location and shape of the initial contour are not
speciﬁed by the model, meaning that the initial contour can
be set arbitrarily in the image domain.
Let the target area set be ΩT, and the intersection of the
closed contour and the target area be ΩIS ; then, there are
three cases: ΩIS = ∅, ΩIS ⊂ΩT, and ΩIS = ΩT. When ΩIS =
∅, Kλ(I(p) −f1(q))2 has no relationship with the grayscale
information of the ground-truth target, while the probability
distribution it obeys may be similar to the ground-truth target,
which may lead to model misjudgment such as falling into
the local optimum and failing to overcome the noise. When
ΩIS ⊂ΩT, Kλ(I(p) −f1(q))2 is partially related to the ground-
truth information, hence the contained edge gradient makes
the transformation of the contour close to the real boundary.
However, the existence of uncertainties may interfere with
the evolution direction of the contour, inappropriate learning
rate may force the contour to cross the real boundary and
enter inside the target, causing missegmentation. Finally, when
ΩIS = ΩT, the initial contour, and the ground-truth boundary
completely overlap; therefore, Kλ(I(p) −f1(q))2 and the real
target area follow the same distribution. This fact reveals an
important shortcoming of the existing model: the setting of the
prior contour is not only labor-intensive but also meaningless
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CHEN et al.: ACM BASED ON FSCs AND NONLINEAR DIFFUSION FILTER FOR INSTANCE SEGMENTATION
5035013
Fig. 1. Segmentation problems caused by manual calibration of the initial contour. The green line is the initial contour, and the red line is the ﬁnal contour.
Row 1: results of single target image under diﬀerent initial contours. Row 2: results of multitarget images under diﬀerent initial contours.
Fig. 2. Boundary positioning results of LIF and SIRE. The green line is the
initial contour, and the red line is the ﬁnal contour. Row 1: polygon target.
Row 2: multiple targets and complex scenes.
when the contour is set extremely close to the ground-truth
boundary.
B. Boundary Positioning and Denoising
A common phenomenon is that ACM works well when
dealing with simple targets, once the target or scene becomes
intricate, i.e., the targets are polygonal or various types of
objects appear in the scene, the segmentation eﬀect becomes
poor. The key to solving this problem is to improve boundary
positioning capability. SIRE [32] introduces the Softsign func-
tion to enhance the boundary information; the corresponding
iteration function is as follows:
8
ˆˆ<
ˆˆ:
∂φ
∂t = −ν (∂S 1 (φ)/∂φ) Softsign ((d1 (p) −d2 (p)) /U)
S 1 (x) = 1/
1 + e−x
Softsign (x) = x/(1 + |x|)
(2)
where ν is a constant, S 1 is the Sigmoid function, φ is the level
set function, and d1 and d2 represent the energy description
of the region inside and outside the contour, respectively.
U is the regularization term, that is,
U =
v
u
u
t 1
P
X
p∈Ω
I (p) −¯I
2.
(3)
P is the total number of pixels in the image and ¯I is the
average gray value of the image. Since the gradient near the
boundary is large, the gradient is further ampliﬁed by (2),
and the smaller grayscale approaches 0, thereby enhancing the
boundary gradient. In Fig. 2, LIF [33] is chosen to compare
Fig. 3.
Noise segmentation results of SIRE and AFOD. The green line is
the initial contour, and the red line is the ﬁnal contour. Gaussian noise with
a variance of 0.01 is applied.
with SIRE, it is diﬃcult for LIF to segment images with
complex shapes, while SIRE suﬀers when multiple types of
objects appear in the scene.
AFOD [34] proposes an ACM based on adaptive fractional
order diﬀerentiation, which quantiﬁes the intensity frequency
of all image information. Its level set iteration function is
∂φ
∂t = −H′ (φ)

b1
Z
Ω
Kλ (I −( f1 + f2) /2) dp
+b2
Z
Ω
Kλ
DθI −(d1 + d2) /2

dp

(4)
where H is the Heaviside function, b1 and b2 are adjustable
parameters, Ωrefers to the whole image domain, Kλ is the
kernel function, f1 and f2 are ﬁtting intensities of I inside and
outside the contour, d1 and d2 are ﬁtting intensities of DθI
inside and outside the contour, Dθ means fractional diﬀerential
operation, and θ is deﬁned as follows:
θ (x) =
|∇I (x)|
max
x∈Ω{|∇I (x)|}
(5)
where ∇is the gradient operator. In the case of noise inter-
ference, each domain is given a speciﬁc diﬀerential process,
which improves the positioning eﬀect of the target object.
Fig. 3 shows the comparison between SIRE and AFOD in
dealing with noisy images. For single-target images, noise
is overcome by AFOD, when faced with complex images,
the segmentation eﬀect is still poor. Due to the discussed
limitations, YOLOv9 combined with AMR-WT is adopted in
this work to solve the initial contour problem, and instance
segmentation based on YOLOv9 is achieved.
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IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 74, 2025
III. METHODOLOGY
This
section
elaborates
on
the
theoretical
basis
of
FSC&NDF framework. The feature information of superpix-
els, such as texture, regional color, and shape, is merged by
FSCs, boosting the target positioning eﬀect. In addition, the
boundary gradient is enhanced and the noise is suppressed
by the proposed NDF module, leading to faster segmentation
speed and greater boundary processing capability.
A. Construction of NDF Module
Diﬀusion is a common physical process that balances the
concentration of a substance without changing its mass. For
images, pixel grayscale can be regarded as concentration, and
the diﬀusion of grayscale implies gradient change. By using
gradient information to drive the local grayscale to diﬀuse
toward the target boundary, while keeping the overall grayscale
stable, diﬀusion ﬁltering is designed. Let Ω⊂R2 be the
grayscale image, and each pixel in the image is denoted
by p, i.e., p ∈Ω. Inspired by Fick’s law, a relational expression
between the potential function Ψ, ﬂux Φ, and diﬀusivity g are
given as follows:
∇(Ψ (|∇u|)) = Φ (∇u) = g
|∇u|2
|∇u|
(6)
where u(p) is the grayscale image, ∇u = [∂ux, ∂uy] is the
gradient of each point in the image, and |∇u| = (∂ux
2+∂uy
2)1/2
is the modulus of the gradient. The rightmost expression
of (6) is a variant of Fick’s law, and g determines the
strength of diﬀusion. To protect the original information of
the image, the value of Φ must remain stable, as described in
the following proposition.
Proposition 1: The relationship between g and ∇u should
meet the following requirements.
1) g(|∇u|2) is a decreasing function with |∇u| as the inde-
pendent variable, satisfying 0 < g(|∇u|2) < 1.
2)
lim
|∇u|→∞g(|∇u|2) = 0.
3)
lim
|∇u|→0 g(|∇u|2) = 1.
Proof: The detailed proof is described in [35].
In this sense, when the gradient is rather small, the diﬀusion
rate approaches the maximum value of 1, and grayscale speeds
up its movement and crosses the nonboundary area. When the
gradient is rather large, the diﬀusion rate decreases, indicating
that it approaches the boundary. Thus, the value of g(|∇u|)|∇u|
is kept at a reasonable level, retaining uniform grayscale.
Based on the above requirements and references in [35], this
article proposes the following diﬀusion coeﬃcient function:
g (s) =
2
1 + es
(7)
that, in light of (6), yields the following ﬂux function:
Φ (s) =
2s
1 + es2
(8)
and the potential function
Ψ (s) = s2 −ln

1 + es2
/2

(9)
Fig. 4. Schematic of the three basic functions.
the function graphs are shown in Fig. 4. The role of Ψ is to
constrain the image gradient which ﬂuctuates in a large range
to a small interval so that it is convenient to ﬁx parameters to
control the change of image information. For instance, when
|∇u| is too large, Ψ(|∇u|) < 0.7 to prevent ﬂuctuations in
the iterative process, when |∇u| is small, Ψ(|∇u|) →0+ to
eliminate minor disturbances.
In simple terms, (9) is derived by s to obtain (8), to ensure
that (9) is close to 0 when s →0, ln(2) is added at the end of
the equation to control its sign change. As can be seen from
Fig. 4, the diﬀusion coeﬃcient equation g(s) satisﬁes the three
conditions mentioned above, and the ﬂux function Φ(s) does
reﬂect the change rate of the potential function Ψ(s) with s.
Next, the potential function Ψ and the image gradient |∇u|
are combined to form a local energy function as follows:
EPG (x) ≈
Z
Ω
Ψ (|∇u (x)|)dx.
(10)
EPG contains the normalized gradient information of the entire
image domain, whose optimization process is equivalent to
ﬁnd an optimal grayscale distribution that minimizes the sum
of the normalized gradients. In fact, minimizing the normal-
ized gradients increases the contrast between large gradients
and small gradients, thereby enhancing the foreground and
suppressing the background. In the variational method, the
standard way to solve (10) is to ﬁnd a steady-state solution to
the gradient ﬂow equation, i.e.,
−∂EPG
∂u
= ∂u
∂t = div
g
|∇u|2
∇u

(11)
where g(|∇u|2) is the diﬀusion coeﬃcient function as in (11).
Since the image gradient reﬂects the image characteristics,
g(|∇u|2) adjusts the diﬀusion coeﬃcient through |∇u|2 to reach
the excellent performance of edge protection, which reduces
the noise and maintains the image edge information. If (11)
is employed directly to iterate the initial contour, the edge
information, and noise may be uniﬁed into a ﬂat area, making
it unsuitable for contour to distinguish boundaries and noise.
Therefore, a threshold should be set in the image iteration
process to ﬁne-tune the gap between the target edge and noise.
Thus, (11) is modiﬁed as follows:
∂uPG
∂t
= div
g
|∇u|2 /η2
∇u

(12)
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CHEN et al.: ACM BASED ON FSCs AND NONLINEAR DIFFUSION FILTER FOR INSTANCE SEGMENTATION
5035013
Fig. 5. Eﬀect of NDF module. Row 1: original image with Gaussian noise,
and the processing eﬀect under m = 10, m = 20, and m = 30. Rows 2 and 3:
color histograms in diﬀerent states.
Fig. 6. Eﬀects of ﬁve superpixel segmentation models.
where η = √std2(I(x)) is the edge threshold and std2 repre-
sents the calculation of standard deviation. Its role is to reduce
the energy ﬂuctuation range while maintaining the signiﬁcance
of the edge and ﬂat area, which better ensures the distinction
between boundaries and noise.
Letting g(|∇u|2/η2) be g∗, and the solution of (12) is
approximated by the discrete equation
um+1 = um + ∆t1 (div (g∗· |∇um|))
(13)
where m is the current iteration number, ∆t1 is the iteration step
size, which is a constant, and g∗is updated in time with um.
Depending on m, the degree of grayscale diﬀusion is diﬀerent,
the grayscale distribution under diﬀerent m is shown in Fig. 5.
As m increases, the number of pixels with larger grayscale
values remains stable, while those with grayscale values close
to 0 increase signiﬁcantly.
B. Contour Generation Based on YOLOv9 and AMR-WT
Morphological reconstruction (MR) is frequently used in
WT algorithms such as GMMSP [36] and MMGR-WT [37],
while adaptive MR (AMR) further improves the segmentation
eﬀectiveness and yields AMR-WT [23]. Traditional superpixel
segmentation requires prespecifying the number of superpix-
els, which may lead to the individual characteristics loss of
each target. However, AMR is able to automatically ﬁlter out
meaningless regions in the original superpixels, so that the
shape and size of each superpixel is diﬀerent, and it is closer to
the appearance of the real subregion in the image, as shown in
Fig. 6. The area of interest can be fused from these superpixels,
so the key task is to reasonably select the target superpixels.
This article adopts YOLOv9 [25] to locate the area of
interest, calculates the intersection between the detection box
and each superpixel, and retains the complete superpixels
within the box. Then, the nonpublic boundaries of these
superpixels are connected to form the initial contour. More
in depth, denote the output of AMR-WT as S, having its size
equal to the original image. Let the ith superpixel region be S i,
the mask of the original image be MI, and the region within
the deformed detection box be S b, the regional correlation
equation is deﬁned as follows:
MI = 0,
if and only if (S i ∩S b = S i) ∧(S i ∪S b = S b)
MI = 1
(14)
the regional boundary with a value of 1 in MI is the initial
contour of a single target. This depicted process determines
the initial value of the level set. The generation of the initial
contour is illustrated in Fig. 7.
C. Energy Function Combined With FSCs
The initial contour C is speciﬁed in the image domain Ω,
which divides the image into two nonoverlapping regions, with
the initial energy function deﬁned as follows:
EI (C) = λ1
Z
outside(C)

I (x) −I f
1 (x)

dx
+ λ2
Z
inside(C)

I (x) −I f
2 (x)

dx
(15)
in which (I(x) −I f
1 (x)) and (I(x) −I f
2 (x)) describe the image
information outside and inside C, while I f
1 and I f
2 are the ﬁtting
functions of the image to be deﬁned.
Usually, the contour C is implicit in the level set function
φ. To embed the presence of φ, the Heaviside function Hε(φ)
[15] is introduced into (15) to have
8
ˆˆˆˆˆˆˆˆˆ<
ˆˆˆˆˆˆˆˆˆ:
EI (φ) = λ1
Z
Ω

I (x) −I f
1 (x)

Hε (φ) dx
+ λ2
Z
Ω

I (x) −I f
2 (x)

(1 −Hε (φ)) dx
Hε (φ) = 1
2

1 + 2
πarc tan
φ
ε

Hε
′ (φ) = δε (φ) = 1
π
ε
ε2 + φ2 .
(16)
Similar to the approach followed for (16), the same method to
ﬁnd the minimum value of (16) is computed with its partial
diﬀerential equation, i.e.,
−∂EI
∂φ = ∂φ
∂t = −λ1

I (x) −I f
1 (x)

δε (φ)
+ λ2

I (x) −I f
2 (x)

δε (φ) .
(17)
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Fig. 7. Contour generation process.
To simplify the model computation and the parameter control,
let λ1 = λ2 = λ, i.e., the parameter λ is a single signed
constant. Then, (17) is rewritten as follows:
∂φ
∂t = λ

I f
1 (x) −I f
2 (x)

δε (φ) .
(18)
In (15), I(x) −I f
1 (x) and I(x) −I f
2 (x) represent the high
frequency components outside and inside the contour C,
respectively. While superpixels contain global information of
the entire image domain, the classiﬁcation of these superpixels
is considered to obtain approximate features inside and outside
the contour. Wu et al. [37], this article performs an optimized
FCM clustering on all superpixels, with the number of clusters
being 2, representing the two regions inside and outside the
contour, respectively. Its objective function is
J =
NS
X
r=1
2
X
k=1
S rbkr ∥Xr −ck∥2+
NS
X
r=1
2
X
k=1
S rbkr · ln
 bkr
1 −ξk

(19)
where NS is the number of superpixels, S r represents each
superpixel region, bkr is the membership of the rth super-
pixel region with respect to the kth cluster center, which
satisﬁes P2
k=1 bkr = 1, ck refers to the kth cluster center, and,
ﬁnally, 1 −ξk is the regularization coeﬃcient, which refers to
the prior probability of Xr belonging to ck, with the constraint
P2
k=1 (1 −ξk) = 1. The second term of (19) is the regularized
entropy, which computes the correlation between each super-
pixel and the other superpixels. The closer the predicted bkr is
to the prior probability, the higher the correlation between the
two superpixels is; hence, a higher weight is associated during
the clustering. In this way, the similarity of diﬀerent regions is
taken into account during clustering, making the classiﬁcation
results more eﬀective. Xr is the representative element of each
superpixel, which is deﬁned as follows:
Xr = arg min
„ ƒ‚ …
xri
8
<
:
N′
X
j=q,q,i
ˇˇxri −xr j
ˇˇ
9
=
; ,
i = 1, 2, . . . , N′
(20)
having xri as the current pixel in the rth superpixel region,
xr j denoting all the pixels in the region except the current pixel
and N′ is the number of pixels in each region. Hence, Xr is
the pixel in the rth superpixel region that has the smallest L1
norm among all the pixels. Since regions with lots of feature
information are clustered, variations in membership during the
iterative process are unstable. Thus, the second term in (19)
aims to control the change amplitude of the membership bkr,
which is called regularization entropy.
Proposition 2: According to the Lagrange multiplier
method, the update equations of ck, bkr, and ξk are
ck =
PNS
r=1 S rbkrXr
PNS
r=1 S rbkr
(21)
bkr =
(1 −ξk) exp
ξk −∥Xr −ck∥2
P2
k=1 (1 −ξk) exp
ξk −∥Xr −ck∥2
(22)
ξk = 1 −
PNS
r=1 S rbkr
PNS
r=1 S r
.
(23)
Proof: See the Appendix.
The clustered superpixel regions obtained by (19) are
recorded as S c1 and S c2. The intersection ratio of the
two regions with the area inside the contour is then com-
puted, which is denoted by IOUc1 and IOUc2, respectively.
If IOUc1 > IOUc2, S c1 is embedded in I f
2 ; otherwise, S c1 is
embedded in I f
1 , and the other area is assigned to the remaining
ﬁtting function. The clustered regions embedded in I f
1 and I f
2
are recorded as IS 1 and IS 1, respectively. Therefore, the ﬁtting
functions are constructed as follows:
8
ˆ<
ˆ:
I f
1 = I + IS 1 −Iσ1
I f
2 = I + IS 2 −Iσ2
Iσi = I ∗Gσi,
i = 1, 2
(24)
where IS 1 is the clustered superpixel region that belongs
outside the contour, IS 2 is that belongs inside the contour, and
Gσi is the Gaussian kernel with size ωi × ωi. Then, (18) is
rewritten as follows:
∂φ
∂t = λ
IS 1 −IS 2 −I ∗
Gσ1 −Gσ2

δε (φ)
(25)
which is the original state of the global energy function.
D. Evolution of the Final Iterative Equation
A positive peak and a negative valley appear on both sides
of a zero point when the target boundary is second-order
diﬀerentiated, which detects the sudden change in the gray
level of the image. Therefore, the model has a strong ability
to locate target edges, the downside of which is that it is more
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5035013
sensitive to noise. Due to the edge enhancement characteristic
of the NDF module, a combination with the energy function
is foreseen. Let e(x, t = 0) = IS 1 −IS 2 −I ∗(Gσ1 −Gσ2) replace
u with e in (13) for iteration. Record e after n iterations as
e(x, t = n), and then, by substituting it into (25), the ﬁnal
iterative equation
∂φ
∂t = λ · e (x, t = n) · δε (φ)
(26)
is obtained. Under the action of the diﬀusion coeﬃcient
function, the target boundary information is enhanced, while
the information are strengthened due to the presence of η.
In (26), e(x, t
=
n) is the data driven term, and its
size is highly dependent on the gray value of the image.
To improve the parameter robustness, an energy uniform
function is deﬁned as follows:
8
ˆˆˆˆ<
ˆˆˆˆ:
ρ (x) = 2
πsign (x) · arc tan
x2
sign (x) =
8
ˆ<
ˆ:
1,
x > 0
0,
x = 0
−1,
x < 0
(27)
and (26) is further rewritten as follows:
∂φFSC&NDF
∂t
= λρ (e (x, t = n) /τ) · δε (φ)
(28)
its corresponding discretization equation is
φn+1 = φn + ∆t (λρ (e (x, t = n) /τ) · δε (φ))
(29)
where τ = 2η. Let the initial level set function be
φ0 =
8
ˆ<
ˆ:
−c0,
x ∈Ωc −∂Ωc
0,
x ∈∂Ωc
c0,
x ∈Ω−Ωc
(30)
where c0 is a constant, Ωis the image domain, ∂Ωc is the target
boundary, and Ωc is the target domain without its boundary.
Obviously, the value range of e is constrained between 0
and 1, which means that the value of λ can be ﬁne-tuned within
a wider range. The fault tolerance rate of λ is improved and
a satisfactory iteration eﬀect is achieved.
Since the level set function becomes bumpier as the number
of iterations increases, this article deﬁnes a new regularization
term as follows:
φR = ρ (ks · φn) ,
ks = 11
(31)
where
ρ
is
the
energy
uniform
function
in
(27)
and
φn represents the level set function after the iteration process
is completed. In addition, ks is the slope improvement factor,
which is set to 11 empirically for stability. The level set
function retains the properties of the signed distance function
while being diﬀerentiable.
For the energy minimization, the contour C derives some
redundant curves under the background interference, which
aﬀects the ﬁnal segmentation result. To account for this
detrimental eﬀect, we construct a curve trimming function as
follows:
φT (xc) = 1
N′′
N′′
X
i=1
φR (xi)
(32)
Algorithm 1 Implementation of FSC&NDF
Input: original image I, c0, φ0, m, ∆t1, ε, λ, ω1, ω2, σ1, σ1,
∆t, N, γ.
Output: φT and target contours.
1: Default settings: c0 = ε = ∆t1 = ∆t = 1, m = 30, λ = 10,
ω1 = ω2 = 15, σ1 = 0.5, σ2 = 3.5, N = 50, and γ = 7.
2: Input the image into YOLOv9 and AMR-WT to obtain
detection boxes and superpixels.
3: Initialize φ0 by (30).
4: Compute IS 1 and IS 2 by ((21) - (23)).
5: Use IS 1, IS 2, I (x), Gσ1, and Gσ2 to compute e (x, t = 0),
execute (13) to calculate e (x, t = n).
6: Compute ρ (e (x, t = n) /τ) by (27).
7: for i from 1to N do
8: Calculate δε (φ).
9: Update φi using (29).
10: if
ˇˇφi+1 −φiˇˇ ≤0.001 Break iteration; end
11: Regularize φi+1 and obtain φR by (31).
12: Trimming φR and obtain φT by (32).
13: end
14: return φT and the zero level set.
Fig. 8. Eﬀects of curve trimming function. Row 1: contours before and after
trimming. Row 2: φ before and after trimming.
a truncated window of size γ × γ is utilized to traverse
each point in the image, and N′′ = γ × γ. Notice that (32)
plays the role of the convolution kernel, which performs mean
ﬁltering on the entire image domain. The grayscale value of
each pixel is obtained by averaging the grayscale values of
the surrounding pixels. In this way, the positions where the
grayscale value is more prominent, such as in the presence
of noise and blur, can be ﬁltered out. For predicted target
boundaries, curve trimming reduces the ﬂuctuation of φ, so
that the parts on both sides of the ground truth boundary move
closer to it at the same time. As can be noted from Fig. 8,
the predicted contour is closer to the ground-truth boundary
after trimming. In addition, φ becomes smoother, which is
similar to the initial binary state, providing a prerequisite for
the next iteration. Algorithm 1 depicts the overall described
solution.
IV. MODEL EXPERIMENTS
In this section, the segmentation ability of FSC&NDF is
veriﬁed based on the COCO benchmark. Comparative experi-
1Registered trademark.
2Trademarked.
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TABLE I
METRICS VALUES AND SEGMENTATION TIME T (S) FOR NATURAL IMAGES. A∼REFERS TO THE AVERAGE VALUE OF EACH METRIC
Fig. 9. Comparison results with ACMs. FSC&NDF achieves better segmen-
tation eﬀects on simple and complex objects.
ments of the model with conventional and noisy images from
previous ACMs are conducted. Comparisons with Mask R-
CNN [24], BlendMask [38], PolarMask [39], DeepSnake [40],
E2EC [41], and SAM [42] is discussed. Ablation and platform
experiments are also conducted. For experiments on traditional
ACMs and robustness, we adopt dice similarity coeﬃcient
(DSC), Jaccard similarity coeﬃcient JSC), and conﬁdence
similarity coeﬃcient (CSC) as evaluation metrics, which are
deﬁned as follows:
DSC = 2 (|Ma| ∩|Mb|)
|Ma| + |Mb|
(33)
JSC = |Ma| ∩|Mb|
|Ma| ∪|Mb|
(34)
CSC = 3 −|Ma| + |Mb|
|Ma| ∩|Mb|
(35)
where Ma represents the segmented area and Mb represents
the ground-truth area. For experiments with deep learning
models, the standard AP, AP50, AP75, APS , APM, and APL are
Fig. 10. Noise segmentation results. Gaussian noise with a variance of 0.05
is applied, which is a common strong noise.
used to evaluate the results. All experiments are conducted on
Python 3.8 with PyTorch 1.13.0 on an Intel1 core2 i9-13900KF
3.00-GHz NVIDIA RTX 4090 workstation. The parameters are
ﬁxed according to Algorithm 1.
A. Comparative Experiments With Previous ACMs
Four recently proposed ACMs, LKLD [18], APFJD [17],
GADF [30], and SIRE [32], are adopted to conduct com-
parative experiments against the proposed FSC&NDF. The
segmentation results on natural images are shown in Fig. 9,
LKLD falls into local optimal solutions, so that many noise
points appear when segmenting the target area. For the other
three models, there are evident under-segmentation and over-
segmentation. For image 1, only FSC&NDF recognizes the
leg contour, for image 3, the result of APFJD is the closest
to the proposed model, and the other models have large
deviations. The contrast between the target and the background
in image 5 is low, and the grayscale information is complex,
thus signiﬁcant errors are noticeable, while FSC&NDF still
accurately segments the target. The metrics values of each
model are given in Table I: all the values of FSC&NDF
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5035013
TABLE II
METRICS VALUES AND SEGMENTATION TIME T (S) FOR NOISY IMAGES. A∼REFERS TO THE AVERAGE VALUE OF EACH METRIC
are at the forefront, with the only exception of image 8,
APFJD has a higher accuracy than FSC&NDF. However, its
segmentation time 1.13 lags behind FSC&NDF by 0.09, and
the accuracy of FSC&NDF remains acceptable. Moreover,
FSC&NDF shows the highest average DSC, JSC, CSC, and
the shortest segmentation time T. In general, the proposed
model has the best segmentation performance on most images.
Although the performance of previous ACMs on a few images
is similar to that of the proposed model, its greatest advantage
lies in the automatic contour initialization, which is a unique
feature of the proposed solution.
The antinoise experimental results are given in Fig. 10,
where strong Gaussian noise is applied to these images. Notice
how the noise has diﬀerent impacts on diﬀerent methods:
LKLD has errors when segmenting images 1–3; APFJD has
over-segmentation when dealing with image 3; the same
problem appears when AFOD and SIRE segment images 2
and 3. For images 4 and 5, all the algorithms have diﬃculty
in processing low-contrast details, except for FSC&NDF. This
fact is collaborated by the metrics in Table II: FSC&NDF
outperforms all the other models in each metric, having all
the DSC, JSC, and CSC values constantly above 0.85, while
its average segmentation time is shortened to 0.14, which is
deemed satisfactory for the application at hand. The DoG
operator is designed to suppress the noise, while, due to
the presence of Gaussian convolutions, the real boundary
information is also smoothed. With the help of the NDF
module, the grayscale value diﬀuses to the boundary; hence,
the total grayscale value of the image remains unchanged,
thereby achieving boundary enhancement and reducing the
background interference. In addition, the precise description
of the key area by the FSCs enhances the contour guidance
ability: indeed, the force component that drives the contour in
the right direction becomes larger, thus enabling noise crossing
at a faster speed.
B. Comparative Experiments With Deep Learning Models
The segmentation ability of FSC&NDF is comparable to
other instance segmentation algorithms, the comparison results
are shown in Fig. 11. Images 2 and 3 are relatively easy
to segment, and the segmentation results of the six models
are ideal. Images 1 and 5 are challenging, with multiple
overlapping objects: FSC&NDF successfully segments all the
objects and has comparable results compared to all the other
methods. As a large-vision model, SAM shows excellent
performance except for image 2, with missegmentation of
Fig. 11. Qualitative results of several models on the COCO dataset. Row 1:
Mask R-CNN. Row 2: BlendMask. Row 3: PolarMask. Row 4: DeepSnake.
Row 5: E2EC. Row 6: SAM. Row 7: FSC&NDF.
TABLE III
RESULTS ON THE COCO DATASET. APVAL REFERS TO AP VALUE OF THE
VALIDATION SET. THE REMAINING METRICS VALUES ARE
OBTAINED FROM THE TESTING SET
background information. Table III lists the comparison of
the metric values of all models, where FPS refers to the
number of images that the model can process per second.
FSC&NDF performs better than the other models except
SAM in APval and AP50 and performs the best in FPS, AP,
and APM. Under the transformer architecture, which is based
on a large-pretraining dataset, the segmentation accuracy
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TABLE IV
RESULTS OF DIFFERENT SUPERPIXEL SEGMENTATION
METHODS ON THE COCO DATASET
of SAM is higher than the proposed model. Nevertheless,
FSC&NDF has a simpler model architecture, fewer param-
eters, and application simplicity. Although SAM has higher
segmentation accuracy at diﬀerent scales, FSC&NDF has
greater FPS, AP, and APM, which shows that the proposed
model has faster segmentation speed and better segmentation
eﬀect for medium objects. As aforementioned, the superiority
of SAM is based on a large dataset, hence, when there is a
limited available set of data, it is hard for SAM to achieve
ideal results. In general, SAM is superior when the task is
diﬃcult and there is suﬃcient data, otherwise, FSC&NDF is
the correct choice to make.
C. Ablation Experiments
1) Eﬀects of Diﬀerent Superpixel Segmentation Methods:
Since superpixels are crucial components of the proposed
model, experiments based on superpixels with diﬀerent shapes
are reported. The proposed solution thus implicitly demon-
strates that it is compatible with diﬀerent morphological
superpixels. SLIC [21], LSC [22], GMMSP [36], and MMGR-
WT [37] are adopted, whose results are qualitatively depicted
in Fig. 6. Results based on these four state-of-the-art super-
pixel segmentation methods are quantiﬁed and reported in
Table IV. The eﬀectiveness based on diﬀerent choices is
acceptable, which indicates that alternative superpixel gen-
eration processes can be adopted. Notice that the overall
performance of the proposed model with AMR-WT attains
the best results.
2) Eﬀects of Diﬀerent Regional Representative Elements:
In order to empirically prove the superiority of (20) in com-
puting the representative feature of each superpixel region
denoted by Xr-L1, experiments based on diﬀerent feature com-
putational methods are conducted. Namely, Xr-L1 is compared
against mean, variance, standard deviation (std), entropy, and
(20) based on L2 norm (Xr-L2), that is,
Xr = arg min
„ ƒ‚ …
xri
8
<
:
N′
X
j=q,q,i
xri −xr j
2
9
=
; ,
i = 1, 2, . . . , N′
(36)
where the meaning of the variables is the same as of (20).
From Table V, which reports the quantitative comparison,
the segmentation eﬀects based on linear feature extraction
operators are better than those based on nonlinear operators,
which is due to the magnifying characteristics of the nonlinear
operator, e.g., small (or large) distances become more extreme
after being processed by such operators, thus leading to an
TABLE V
RESULTS OF DIFFERENT SUPERPIXEL FEATURE CALCULATION
METHODS ON THE COCO DATASET
accuracy loss. Instead, among all the linear operators, Xr-
L1 retains more information on the distance, so the extracted
feature is more stable.
3) Eﬀects of Diﬀerent Parameter Values: The solution
parameters reported in Algorithm 1 have been chosen by ﬁne-
tuning on the experimental data. More precisely, when the
values of m and N increase to a certain extent, (13) and (29)
remain convergent, so the impact of a further increase on the
segmentation accuracy becomes negligible. Instead, the impact
of diﬀerent parameter choices for λ, ω1, ω2, σ1, σ2, and γ is
more involved and asks for experimental ﬁne tuning. As shown
in Table VI (for λ, ω1, and ω2) and Table VII (for σ1, σ2, and
γ). The segmentation accuracy under diﬀerent ﬁgures of merit
(namely, DSC, JSC, and CSC) reaches the maximum around
the reported choices.
D. Complexity Analysis
To further analyze the proposed approach, the complexity
of the overall solution is reported, with diﬀerent metric val-
ues according to the diﬀerent compared models. For ACMs,
the standard complexity representation O( f(n)) is adopted
(Table VIII); for deep learning models, giga ﬂoating point
operations per second (GFLOPs) and parameters (Params) are
computed to quantify time and space complexity (Table IX).
In addition, the inference time of all models is also recorded,
and all statistics are yielded based on the COCO benchmark.
Note that the proposed model is compared with previous
ACMs without the contour generation module (w/o CG), and
it is compared with deep learning models with the contour
generation module (w/ CG), for the sake of fairness. The
time complexity of GADF and SIRE is higher than that
of the other three models, with inference speed faster than
LKLD and APFJD. LKLD, APFJD, and the proposed model
have the same complexity, while the proposed model is the
fastest among all models. In Table IX, the metrics values of
FSC&NDF are calculated based on YOLOv9, which can be
replaced by other models. The point is that with a lightweight
model like YOLOv9 as the front-end network, our method is
able to handle most of the tasks. Notice that the inference time
of FSC&NDF ranks among the best, while the complexity is
the lowest.
E. Validation Based on Experimental Platform
The proposed FSC&NDF can be deployed on hardware plat-
forms to assist industrial production, Fig. 12 shows our gantry
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5035013
TABLE VI
RESULTS OF DIFFERENT λ, ω1, AND ω2 ON THE COCO DATASET
TABLE VII
RESULTS OF DIFFERENT σ1, σ2, AND γ ON THE COCO DATASET
TABLE VIII
COMPARISON OF ALGORITHM COMPLEXITY WITH ACMS
TABLE IX
COMPARISON OF ALGORITHM COMPLEXITY WITH
DEEP LEARNING MODELS
Fig. 12. Laser processing platform based on gantry robot.
robot for laser processing. The platform combines a CMOS
camera and a telecentric lens to achieve target segmentation
and positioning. First, the camera is moved directly above
Fig. 13. Platform segmentation results. Row 1: chips segmentation. Row 2:
fabric defect segmentation.
the processing plane manually. At this time, the processing
plane displays the source pattern to be engraved on the
material. The digital image is acquired by the camera, which
is processed by the FSC&NDF to obtain the pattern contour.
Then, the pixel coordinates of the contour are converted to real
world coordinates using the intrinsic and extrinsic calibration
matrices of the camera. The position information of this
coordinate are transmitted to the gantry robot to guide the
laser to the starting position, where the original material is
correctly placed. Finally, the laser moves along the coordinates
of each contour vertex, which engraves the target pattern on
diﬀerent materials. To verify the practicality of the algorithm,
two materials are selected for real-time segmentation under
the telecentric lens. In Fig. 13, Row 1 shows the segmentation
results of the chip, and Row 2 shows the segmentation results
of the fabric defect. It is obvious that the proposed model
performs well at the microscopic level.
V. CONCLUSION AND FUTURE WORK
In this article, an instance segmentation algorithm, called
FSC&NDF, combined with YOLOv9 and advanced ACM
is proposed. The novel innovations are: the combination of
YOLOv9 and AMR-WT realizes initial contour positioning;
the NDF module is designed to enhance the target information
and suppress the background noise; the superpixel clustering
boosts the feature information extraction. FSC&NDF frees tra-
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ditional ACMs from manual contour initialization and shows
segmentation eﬀects that are not inferior to mainstream deep
learning methods. AMR-WT acts as a glue to make YOLOv9
and ACM work together harmoniously. The proposed NDF
module and superpixel fuzzy clustering improve the noise
resistance eﬀect and feature capture ability.
In the future work, we will extend the applicability of
the approach to various object detection algorithms so as to
solve segmentation tasks in diﬀerent ﬁelds, i.e., transferability.
Instead, to increase its eﬀectiveness, the algorithm will be
updated to improve the evolutionary ability of ACMs together
with their information description on the inner and outer
contour areas, which can be modiﬁed to better guide the
contour iteration. Finally, we are planning to use the feature
map output by a CNN as the input to the energy function,
leading to the improvement of its feature analysis ability.
APPENDIX
PROOF OF PROPOSITION 2
From the constraints, the Lagrange function of (19) is
L =
NS
X
r=1
2
X
k=1
S rbkr ∥Xr −ck∥2+
NS
X
r=1
2
X
k=1
S rbkr · ln
 bkr
1 −ξk

+
NS
X
r=1
αr

1 −
2
X
k=1
bkr
!
+
NS
X
r=1
βr

1 −
2
X
k=1
(1 −ξk)
!
s.t.
2
X
k=1
bkr = 1,
2
X
k=1
(1 −ξk) = 1
where αr and βr are the Lagrange multipliers. Then, the partial
derivative of each variable is taken, and we have
∂L
∂ck
=
NS
X
r=1
2 · S rbkr ∥Xr −ck∥= 0
the solution of the above equation yields
ck =
PNS
r=1 S rbkrXr
PNS
r=1 S rbkr
.
The next PDE is
∂L
∂bkr
= S r

∥Xr −ck∥2 +

ln
 bkr
1 −ξk

+ 1 −ξk

.
−αr = 0.
By shifting the terms, there exists
bkr = (1 −ξk) exp
αr
S r
+ ξk −1 −∥Xr −ck∥2

and substitute the above equation into P2
k=1 bkr = 1
bkr =
(1 −ξk) exp
ξk −∥Xr −ck∥2
P2
k=1
(1 −ξk) exp
ξk −∥Xr −ck∥2.
The ﬁnal PDE is
∂L
∂ξk
=
NS
X
r=1
 S rbkr
1 −ξk
+ βr

= 0
since P2
k=1 (1 −ξk) = 1, which gives rise to
ξk = 1 −
PNS
r=1 S rbkr
PNS
r=1 S r
.
REFERENCES
[1]
H. Bi, L. Xu, X. Cao, Y. Xue, and Z. Xu, “Polarimetric SAR image
semantic segmentation with 3D discrete wavelet transform and Markov
random ﬁeld,” IEEE Trans. Image Process., vol. 29, pp. 6601–6614,
2020.
[2]
B. Chen et al., “Attention-guided and noise-resistant learning for robust
medical image segmentation,” IEEE Trans. Instrum. Meas., vol. 73,
pp. 1–13, 2024.
[3]
S. Minaee, Y. Y. Boykov, F. Porikli, A. J. Plaza, N. Kehtarnavaz, and
D. Terzopoulos, “Image segmentation using deep learning: A survey,”
IEEE Trans. Pattern Anal. Mach. Intell., vol. 44, no. 7, pp. 3523–3542,
Jul. 2021.
[4]
B. Wang, D. Tao, R. Dong, Y. Tang, and X. Gao, “A contour co-
tracking method for image pairs,” IEEE Trans. Image Process., vol. 30,
pp. 5402–5412, 2021.
[5]
G. Wang, F. Zhang, Y. Chen, G. Weng, and H. Chen, “An active contour
model based on local pre-piecewise ﬁtting bias corrections for fast and
accurate segmentation,” IEEE Trans. Instrum. Meas., vol. 72, pp. 1–13,
2023.
[6]
H. Li, J. Song, and L. Dai, “Superpixel segmentation based on feature
fusion and boundary constraint for ferrograph image segmentation,”
IEEE Trans. Instrum. Meas., vol. 72, pp. 1–11, 2023.
[7]
Z. Liu, X. Bai, H. Liu, and Y. Zhang, “Multiple-surface-approximation-
based
FCM with interval
memberships for
bias
correction
and
segmentation of brain MRI,” IEEE Trans. Fuzzy Syst., vol. 28, no. 9,
pp. 2093–2106, Sep. 2020.
[8]
C. J. J. Sheela and G. Suganthi, “Brain tumor segmentation with
radius contraction and expansion based initial contour detection for
active contour model,” Multimedia Tools Appl., vol. 79, nos. 33–34,
pp. 23793–23819, Sep. 2020.
[9]
S. Zhao et al., “Attractive deep morphology-aware active contour
network for vertebral body contour extraction with extensions to het-
erogeneous and semi-supervised scenarios,” Med. Image Anal., vol. 89,
Oct. 2023, Art. no. 102906.
[10] H. Qi, X. Kong, L. Cheng, J. Hu, and J. Gu, “Addressing ﬁne-
grained lake water body extraction: A hybrid approach combining vision
transformer and geodesic active contour,” IEEE Trans. Geosci. Remote
Sens., vol. 62, 2024, Art. no. 4204614.
[11] P. Polewski, J. Shelton, W. Yao, and M. Heurich, “Instance segmen-
tation of fallen trees in aerial color infrared imagery using active
multi-contour evolution with fully convolutional network-based intensity
priors,” ISPRS J. Photogramm. Remote Sens., vol. 178, pp. 297–313,
Aug. 2021.
[12] Y. Chen, P. Ge, G. Wang, G. Weng, and H. Chen, “An overview of
intelligent image segmentation using active contour models,” Intell.
Robot., vol. 3, no. 1, pp. 23–55, 2023.
[13] V. Caselles, F. Catt´e, T. Coll, and F. Dibos, “A geometric model for
active contours in image processing,” Numerische Math., vol. 66, no. 1,
pp. 1–31, Dec. 1993.
[14] M. Kass, A. Witkin, and D. Terzopoulos, “Snakes: Active con-
tour
models,”
Int.
J.
Comput.
Vis.,
vol. 1,
no. 4,
pp. 321–331,
Jan. 1988.
[15] C. Li, C.-Y. Kao, J. C. Gore, and Z. Ding, “Minimization of region-
scalable ﬁtting energy for image segmentation,” IEEE Trans. Image
Process., vol. 17, no. 10, pp. 1940–1949, Oct. 2008.
[16] G. Wang, Z. Li, G. Weng, and Y. Chen, “An optimized denoised
bias
correction
model
with
local
pre-ﬁtting
function
for
weak
boundary image segmentation,” Signal Process., vol. 220, Jul. 2024,
Art. no. 109448.
[17] P. Ge, Y. Chen, G. Wang, and G. Weng, “An active contour model
driven by adaptive local pre-ﬁtting energy function based on Jeﬀreys
divergence for image segmentation,” Expert Syst. Appl., vol. 210, Dec.
2022, Art. no. 118493.
[18] C. Yang, G. Weng, and Y. Chen, “Active contour model based on local
Kullback–Leibler divergence for fast image segmentation,” Eng. Appl.
Artif. Intell., vol. 123, Aug. 2023, Art. no. 106472.
[19] R. Jin and G. Weng, “Active contours driven by adaptive functions and
fuzzy c-means energy for fast image segmentation,” Signal Process.,
vol. 163, pp. 1–10, Oct. 2019.
Authorized licensed use limited to: Soochow University. Downloaded on June 12,2025 at 12:45:00 UTC from IEEE Xplore.  Restrictions apply.
CHEN et al.: ACM BASED ON FSCs AND NONLINEAR DIFFUSION FILTER FOR INSTANCE SEGMENTATION
5035013
[20] G. Liu, Y. Jiang, B. Chang, and D. Liu, “Superpixel-based active contour
model via a local similarity factor and saliency,” Measurement, vol. 188,
Jan. 2022, Art. no. 110442.
[21] R. Achanta, A. Shaji, K. Smith, A. Lucchi, P. Fua, and S. S¨usstrunk,
“SLIC superpixels compared to state-of-the-art superpixel methods,”
IEEE Trans. Pattern Anal. Mach. Intell., vol. 34, no. 11, pp. 2274–2282,
Nov. 2012.
[22] Z. Li and J. Chen, “Superpixel segmentation using linear spectral
clustering,” in Proc. IEEE Conf. Comput. Vis. Pattern Recognit. (CVPR),
Jun. 2015, pp. 1356–1363.
[23] T. Lei, X. Jia, T. Liu, S. Liu, H. Meng, and A. K. Nandi, “Adaptive
morphological
reconstruction
for
seeded
image
segmentation,”
IEEE
Trans.
Image
Process.,
vol. 28,
no. 11,
pp. 5510–5523,
Nov. 2019.
[24] K.
He,
G.
Gkioxari,
P.
Doll´ar,
and
R.
Girshick,
“Mask
R-
CNN,” in Proc. IEEE Int. Conf. Comput. Vis. (ICCV), Oct. 2017,
pp. 2980–2988.
[25] C.-Y. Wang, I.-H. Yeh, and H.-Y. M. Liao, “YOLOv9: Learning what
you want to learn using programmable gradient information,” in Proc.
Eur. Conf. Comput. Vis., Oct. 2024, pp. 1–21.
[26] W. Liu et al., “SSD: Single shot MultiBox detector,” in Proc. Eur. Conf.
Comput. Vis., Jan. 2016, pp. 21–37.
[27] Z. Liu et al., “Swin transformer: Hierarchical vision transformer using
shifted windows,” in Proc. IEEE/CVF Int. Conf. Comput. Vis. (ICCV),
Oct. 2021, pp. 9992–10002.
[28] H. Law and J. Deng, “CornerNet: Detecting objects as paired
keypoints,”
Int.
J.
Comput.
Vis.,
vol. 128,
no. 3,
pp. 642–656,
Mar. 2020.
[29] K. Duan, S. Bai, L. Xie, H. Qi, Q. Huang, and Q. Tian, “CenterNet:
Keypoint triplets for object detection,” in Proc. IEEE/CVF Int. Conf.
Comput. Vis. (ICCV), Oct. 2019, pp. 6568–6577.
[30] D. Chen, J. Spencer, J.-M. Mirebeau, K. Chen, M. Shu, and
L. D. Cohen, “A generalized asymmetric dual-front model for active
contours and image segmentation,” IEEE Trans. Image Process., vol. 30,
pp. 5056–5071, 2021.
[31] X. Yan and G. Weng, “Hybrid active contour model driven by optimized
local pre-ﬁtting image energy for fast image segmentation,” Appl. Math.
Model., vol. 101, pp. 586–599, Jan. 2022.
[32] B. Dong, G. Weng, Q. Bu, Z. Zhu, and J. Ni, “An active contour model
based on shadow image and reﬂection edge for image segmentation,”
Expert Syst. Appl., vol. 238, Mar. 2024, Art. no. 122330.
[33] K. Zhang, H. Song, and L. Zhang, “Active contours driven by local
image ﬁtting energy,” Pattern Recognit., vol. 43, no. 4, pp. 1199–1206,
Apr. 2010.
[34] M.-M.
Li
and
B.-Z.
Li,
“A
novel
active
contour
model
for
noisy
image
segmentation
based
on
adaptive
fractional
order
diﬀerentiation,” IEEE Trans. Image Process., vol. 29, pp. 9520–9531,
2020.
[35] P. Perona and J. Malik, “Scale-space and edge detection using
anisotropic diﬀusion,” IEEE Trans. Pattern Anal. Mach. Intell., vol. 12,
no. 7, pp. 629–639, Jul. 1990.
[36] Z. Ban, J. Liu, and L. Cao, “Superpixel segmentation using Gaus-
sian mixture model,” IEEE Trans. Image Process., vol. 27, no. 8,
pp. 4105–4117, Aug. 2018.
[37] C. Wu, L. Zhang, H. Zhang, and H. Yan, “Improved superpixel-
based
fast
fuzzy
C-means
clustering
for
image
segmentation,”
in
Proc.
IEEE
Int.
Conf.
Image
Process.
(ICIP),
Sep.
2019,
pp. 1455–1459.
[38] H. Chen, K. Sun, Z. Tian, C. Shen, Y. Huang, and Y. Yan, “BlendMask:
Top-down meets bottom-up for instance segmentation,” in Proc.
IEEE/CVF Conf. Comput. Vis. Pattern Recognit. (CVPR), Jun. 2020,
pp. 8570–8578.
[39] E. Xie et al., “PolarMask: Single shot instance segmentation with polar
representation,” in Proc. IEEE/CVF Conf. Comput. Vis. Pattern Recognit.
(CVPR), Jun. 2020, pp. 12190–12199.
[40] S. Peng, W. Jiang, H. Pi, X. Li, H. Bao, and X. Zhou, “Deep snake
for real-time instance segmentation,” in Proc. IEEE/CVF Conf. Comput.
Vis. Pattern Recognit. (CVPR), Jun. 2020, pp. 8530–8539.
[41] T. Zhang, S. Wei, and S. Ji, “E2EC: An end-to-end contour-based
method for high-quality high-speed instance segmentation,” in Proc.
IEEE/CVF Conf. Comput. Vis. Pattern Recognit. (CVPR), Jun. 2022,
pp. 4433–4442.
[42] A. Kirillov et al., “Segment anything,” in Proc. IEEE/CVF Int. Conf.
Comput. Vis., Oct. 2023, pp. 4015–4026.
Yiyang Chen (Member, IEEE) received the M.Eng.
degree from the Imperial College London, Lon-
don, U.K., in 2013, and the Ph.D. degree from
the University of Southampton, Southampton, U.K.,
in 2017.
After that, he worked as a Research Fellow in
control systems from 2017 to 2018 and in traﬃc
signal control from 2018 to 2020 with the University
of Southampton. He joined the School of Mechani-
cal and Electrical Engineering, Soochow University,
Suzhou, China, in 2020, as an Associate Professor.
He has published several articles in top control conferences and journals.
His research interests include iterative learning control, optimization, artiﬁcial
intelligence, image processing, and robotic systems.
Fuzheng Zhang received the B.S. degree in vehicle
engineering from Shanghai University of Engi-
neering Science, Shanghai, China, in 2022. He is
currently pursuing the M.S. degree in image pro-
cessing with the School of Mechanical and Electrical
Engineering, Soochow University, Suzhou, China.
His research interests include image segmentation,
artiﬁcial intelligence, deep learning, and LLM.
Guina Wang received the M.S. and Ph.D. degrees
from Nanjing University of Aeronautics and Astro-
nautics,
Nanjing,
China,
in
2007
and
2021,
respectively.
She is currently an Associate Professor with the
School of Mechanical and Electric Engineering,
Soochow University, Suzhou, China. Her current
research interests include optical sensing technology
and image processing techniques.
Guirong Weng received the M.S. degree from Nan-
jing University of Science and Technology, Nanjing,
China, in 1985.
He is currently a Professor with the School of
Mechanical and Electric Engineering, Soochow Uni-
versity, Suzhou, China, and an Executive Director of
Jiangsu Province Automation Society. His research
interests include image processing and biological
information processing.
Daniele Fontanelli (Fellow, IEEE) received the
Ph.D. degree in automation, robotics, and bioengi-
neering from the University of Pisa, Pisa, Italy,
in 2006.
He is the Co-Founder of Polytec Intralogistics Srl
and IDRA Labs. He is the PI of the EU Project
MAGICIAN-iMmersive leArninG for ImperfeCtion
detectIon and repAir through human-robot interac-
tioN, and he was the Co-Founder and the PI of the
EIT-Digital international master’s on “Autonomous
Systems” from 2017 to 2023. He is currently a
Full Professor with the Department of Industrial Engineering, Univer-
sity of Trento, Trento, Italy. He has authored or co-authored more than
250 scientiﬁc articles in peer-reviewed top journals and conference proceed-
ings. His research interests include distributed and real-time estimation and
control, localization algorithms, synchrophasor estimation, clock synchro-
nization algorithms, resource aware control, wheeled mobile robots, service
robotics, and human-robot interaction and estimation.
Dr. Fontanelli is a member of the IMEKO TC17-Measurement in Robotics.
He is a Senior Area Editor of IEEE TRANSACTIONS ON INSTRUMENTATION
AND MEASUREMENT and an Associate Editor of IEEE ROBOTICS AND
AUTOMATION LETTERS.
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