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Active contour model based on LIF model and optimal DoG operator energy for image segmentation
Journal Pre-proof Active contour model based on LIF model and optimal DoG operator energy for image segmentation Yaxi Duan, Taile Peng, Xianghui Qi
PII:
S0030-4026(19)31565-7
DOI:
https://doi.org/10.1016/j.ijleo.2019.163667
Reference:
IJLEO 163667
To appear in:
Optik
Received Date:
20 December 2018
Revised Date:
18 September 2019
Accepted Date:
16 October 2019
Please cite this article as: Duan Y, Peng T, Qi X, Active contour model based on LIF model and optimal DoG operator energy for image segmentation, Optik (2019), doi: https://doi.org/10.1016/j.ijleo.2019.163667
This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier.
Active contour model based on LIF model and optimal DoG operator energy for image segmentation Yaxi Duana,*, Taile Penga, and Xianghui Qib a School of Computer Science and Technology, Huaibei Normal University. Huaibei, 235000, China College of Computer and Communication, Lanzhou University of Technology, Lanzhou, 730050, China
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* Corresponding author. E-mail address: [email protected].
Abstract
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In order to solve the problem that the region-based active contour model is sensitive to the initial contour position, the convergence is poor, and the active contour model can not obtain good segmentation results when segmenting complex background images and severe intensity inhomogeneous images. In this paper, an active contour model which combines Local Image Fitting (LIF) and Difference of Gaussian (DoG) operator energy for image segmentation is proposed. Firstly, an optimal DoG operator is obtained by using the edge energy term, it can enhance the edge while smoothing inhomogeneous regions. Then, using the DoG energy term which is obtained in the first step and the LIF energy term to construct the total function energy terms. In this process, the regularization term is also established, it can control the smoothness of evolution curve, avoiding over-segmentation and re-initialization step. Finally, the variational method and gradient descent flow method are adopted to minimize the total energy functional for segmentation. Compared with other region-based active contour models, the experimental results show that the proposed method can achieve a better segmentation performance with less iterations and high calculation efficiency while segmenting the synthetic and real images with intensity inhomogeneity. Keywords: local image fitting; Difference of Gaussian; active contour model; image segmentation
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1. Introduction
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Image segmentation is an important research direction in the field of image processing and computer vision [1,2,3,4,5,6,7,8,9]. Since the active contour model [3] was proposed by Kass et al., the algorithm based on the active contour model has been widely used. The basic idea of the model is this, by using a continuous curve to represent the edges of the object and establishing an energy functional regarding continuous curve as the independent variable, so that the process of segmentation is transformed into calculating the minimum value of the energy functional. Generally, it can be achieved by calculating the Euler-Lagrange equation of function. When the energy reaches a minimum, the contour of the object appears. According to the different forms of contour expression, the active contour model can be generally divided into parametric active contour model and geometric active contour model. The parametric active contour model is affected by the location of initial contour and difficult to deal with the curve topological structure, limiting its further application. Compared with the parameter active contour model, the curve motion of the geometric active contour model is based on the geometric metric parameters rather than the expression parameters. Therefore, this model can handle the topological structure of curves better and enable the range of application wider. In general, we divide the geometric active contour model into two categories: edge-based active contour model [4,6,10, 11] and region-based active contour model [1,12,13,14,15,16,17,18]. For the edge-based active contour model, the most representative one is the geodesic active contour model [6], which detects the edge according to the size of the gradient. The basic idea of this model is to define an edge detector, in locations with large gradients, the curve stops evolving, so we get the contour of the object, but it is greatly affected by noise. Moreover, the model cannot get satisfactory segmentation
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results when segmenting the images with intensity inhomogeneity. However, the region-based active contour model mainly uses the level set method [15] to define internal energy and external energy, which drives the curve evolution. When the energy reaches the minimum, the part enclosed by the curve is the object contour, and this model is insensitive to noise. The most representative one is the CV model [1]. which performs well in segmenting the images with intensity homogeneity, but it cannot handle the images with intensity inhomogeneity very well. Li et al. proposed a Local Binary Fitting (LBF) model [13] to overcome the shortcomings of the CV model. This algorithm utilizes the local information of the image to smooth the inhomogeneous regions. but the model is easily influenced by the location of initial contour leading to fall into local minimum and has high computational complexity, noise also affects the model to a certain extent. Zhang et al. proposed a Local Image Fitting (LIF) model [17]. Compared with the LBF model, they have similar segmentation results, but this model takes less time. It also has same shortcomings as the LBF model. Then, a number of models based on local and global information have been proposed [12,16,18], but none of them have introduced edge energy term, which makes it difficult to achieve ideal segmentation results when segmenting images with weak edges.
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In view of the above problems. We use the combination of local image fitting (LIF) and the optimal Difference of Gaussian (DoG) energy in this paper. Firstly, the idea of DoG is applied to image segmentation by constructing edge energy term and using the optimal DoG operator to detect edges. Then, we combine the optimal DoG operator with the local energy term of the LIF model to construct the total energy terms. In addition, the introduction of length term and regular term makes it smoother in the evolutionary process. Moreover, it is not necessary to re-initialize the contour, which greatly improves the efficiency of its evolution. The experimental results show that the proposed model has good characteristics of speed, precision and robustness when segmenting different types of real images and synthetic images. The rest of the paper is organized as follows: in section 2, we briefly introduced and reviewed the CV model, the LBF model and the LIF model. The model and method proposed in this paper are described in detail in section 3. In section 4, the effectiveness of the proposed model is verified by experiments. The paper is summarized in section 5.
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2. Related Work 2.1. C-V Model
E cv (c1 , c2 , C ) 1
inside (C )
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The C-V model [1] is a region-based segmentation method, which is proposed by Chan and Vese. The basic idea of the model is to regard the image as the two parts of the homogeneous regions. By using level set method to calculate the intensity mean of internal and external contour, the energy functional would be minimized to construct the energy activity contour model. Assuming that Ω ⊂ R2 is a two-dimensional image space domain, I: Ω ⊂ R is an input image, and its minimized energy functional is given by (1). | u0 ( x, y) c1 |2 dxdy 2
outside (C )
| u0 ( x, y) c2 |2 dxdy Length.
(1)
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In (1), μ ≥ 0, λ1 and λ2 control the contributions of the internal and external energy terms. c1, c2 are the image intensities of the interior and exterior of contour C respectively. The first two terms are data fitting energy terms, and the third is length term.
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In order to facilitate the solution, a zero level set function ϕ(x, y) is introduced to replace the unknown contour C. If the point (x, y) is inside the curve C, ϕ(x, y) > 0, and if the point (x, y) is outside the curve C, ϕ(x, y) < 0. If the point (x, y) is on C, ϕ(x, y) = 0. Therefore, the level set function energy function can be expression is: E cv (c1 , c2 , ) 1 | u0 ( x, y) c1 |2 H ( ( x, y))dxdy 2 | u0 ( x, y) c2 |2 (1 H ( ( x, y)))dxdy
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( ( x, y)) | ( x, y) | dxdy
(2)
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In (2), Hε(ϕ(z)) and δε(ϕ(z)) are regularized Heaviside function and Dirac function respectively. And the corresponding relationships can be shown as formula (3) and formula (4). 1. if z 0 H ( z) 0. if z 0
(3)
Active contour model based on LIF model and optimal DoG operator energy for image segmentation
( z)
3
d ( H ( z )) dx
(4)
By calculating the Euler-Lagrange equation of the above energy functional Ecv in Equation (2), and by using the gradient descent flow method [19] to update the level set function ϕ(x, y) to obtain the following evolution equation: ( ) div( ) - 1 (u0 c1 ) 2 2 (u0 c2 ) 2 t | |
(5)
In (5), c1 and c2 are intensity averages inside and outside the evolving curve C respectively, which should be updated at each iteration. Their expressions are shown in formula (6) and (7).
u0 ( x, y) H ( ( x, y))dxdy
c2
(6)
H ( ( x, y))dxdy
u0 ( x, y )(1 H ( ( x, y )))dxdy
(1 H ( ( x, y )))dxdy
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(7)
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The main advantage of C-V model is that it can detect edge of objects whose boundaries are not based on gradient and can effectively extract the boundaries of objects. It is also not sensitive to noise. This model needs to re-initialize the contour at each time of evolution, which is time-consuming and cannot handle weak-edge images with intensity inhomogeneity. 2.2. LBF Model
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The LBF model [13] was proposed by Li et al., which is a region-based active contour model. By introducing the Gauss kernel function and using the local information of the image to achieve binary function fitting. The local region information is embedded in the variational level set function, which overcomes the limitations of the piecewise constant function model. As a result, the model can segment images with intensity inhomogeneity. The level set energy functional of this model is as follows: 2 2 ELBF ( f 1 , f 2 , ) 1 K ( x, y) | I ( y) f1 ( x) | H ( ( y))dy dx 2 K ( x, y) | I ( y) f 2 ( x) | (1 H ( ( y)))dy dx L( ( x)) P( ( x))
(8)
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In (8), the weight coefficients of λ1, λ2, υ, μ are positive parameters, f1 and f2 are two smooth functions, during the period of evolution performed by the curve of level set function, they represent the weighted average of the interior and exterior of the Gaussian window of the intensity image. The first two terms are data fitting energy terms. The third is the length term, which makes the evolution curve (C) smoother. The fourth is the penalty term [20], whose function is to make the level set function do not need to be re-initialized in the curve evolution process, and it can accelerate the speed of curve evolution, f1 and f2, L(ϕ) and P(ϕ) are show in formulas (9), (10), (11) and (12), respectively. f1
f2
K ( x) I ( x) H ( f ( x)) K ( x) H ( f ( x))
K ( x) I ( x)(1 H ( ( x))) K ( x) (1 H ( ( x)))
L( ) | H ( ( x, y)) | dxdy
P( )
1 (| H ( ( x, y)) | 1) 2 dxdy 2
(9)
(10)
(11)
(12)
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Where ∗ is the convolution operator in Equation (9) and Equation (10). The LBF model can segment the images with intensity inhomogeneity since it introduces the gaussian kernel function. However, this model requires initial contour more. On the one hand, selecting the inappropriate initial contour location would easily fall into the local minimum and cannot get the correct segmentation result. On the other hand, if the chosen variance σ of the Gaussian kernel is too large, it will generate the blurry edge and will not obtain the satisfied results similarly. Therefore, the model has some limitations for segmentation of images with complex background. 2.3. LIF Model
E LIF ( )
1 | I ( x) I LFI ( x) |2 dx 2
In (13), ILFI represents the local fitting image. The expression is as follows:
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I LFI m1 H ( ) m2 (1 H ( ))
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The LIF model [17] was proposed by Zhang et al. It is an improvement of the LBF model. For a given image I(x) in the image domain x∈Ω. Firstly, the local fitting information of the image is used to construct the local fitting energy term, leading to minimize the fitting energy to achieve segmentation. Secondly, after the end of each activity contour curve, Gauss filter is used to smooth the result. For this step would replace the length term and penalty term in the LBF model, it not only improves the efficiency of evolution, but also gets the similar result of LBF model. The energy function of the LIF model is expressed as follows:
In (14), the definitions of m1 and m2 are as follows:
(14)
(15)
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m1 mean( I ({x | ( x) 0} Wk ( x))) m2 mean( I ({x | ( x) 0} Wk ( x)))
(13)
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In (15), Wk (x) is a truncated Gaussian or constant window function. In this model, the truncated Gauss window function Kσ is used, the σ is the standard deviation and the width is 4K+1. ϕ is the zero level set of a Lipschitz function that represents the contour C. In the expression ILFI, Hε(ϕ) is used to regularize the Heaviside function. The Heaviside function and its derivative form Dirac function are shown as formula (16) and formula (17).:
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H ( z)
1 2 z 1 arctan 2
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( z)
1
z2 2
(16)
(17)
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By using the steepest descent method and the variational method to calculate the minimum energy functional ELIF(ϕ) in Equation (13) with respect to ϕ. Its gradient descent flow equation can be expressed by (18).
( I I LFI )(m1 m2 ) ( ) t
(18)
The LIF model performs Gaussian filtering on each evolutionary obtained image, which has same effect as adding the length term and the penalty term in the LBF model. However, the model is also sensitive to the location of initial contour, and it is not ideal to segment images with severe intensity inhomogeneity. 3. Proposed LIFDG Model
Active contour model based on LIF model and optimal DoG operator energy for image segmentation
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Motivated by the strengths of Difference of Gaussian and local region-based active contour models, we proposed an active contour model based on LIF model and optimal DoG operator energy. 3.1. Energy Functional of LIFDG Model The LIFDG model is composed of the local energy ELIF of the LIF model and the optimal DoG operator energy EDG, in addition, we have added regularized energy ER, which overcomes the shortcomings of the above models. The initial contour can be placed in any location of the image, which greatly improves the robustness of the model. The total energy functional is as follows: E LIFDG E LIF ( , m1 , m2 ) E DG ( ) E R
(19)
Using the gradient descent flow method, we minimize the energy functional ELIFDG in (19) with respect to ϕ to get the corresponding variational level set energy functional can be formulated as: (20)
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( )( I I LFI )(m1 m2 )+ ( ) D ( )div( ) ( div( )) t | | | |
Where β, λ, υ, μ are positive parameters. This evolution Equation (20) is applied in our model. 3.1.1. Introducing The Edge Energy Term
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Inspired by the literature [21], we introduced DoG operator fitting energy. The DoG operator is obtained by subtracting two filtered images. We define a Gaussian function with two standard deviations, σ1 and σ2, convolved with the original image I(x, y) respectively. The DoG operator for detecting the edges of the image will be obtained, and the function expression of the DoG operator is shown as formula (21).
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G 1 I ( x, y) G 2 I ( x, y) (G 1 G 2 ) I ( x, y) DoG I ( x, y)
(21)
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Because the DoG operator is a second-order difference operator, it is very sensitive to the gray change of the image. That is to say, there are some zero crossing points that are not the real edges of the image. It is mainly caused by the noise and the image with intensity inhomogeneity. Therefore, in order to solve these problems, we constructed the edge energy term, which can be written as: E DoG ( D) ((G 1 G 2 ) D 0)2 +(D (G 1 G 2 ) I ( x, y))2 dxdy
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(22)
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Where D represents the optimal DoG operator, the contour C is a closed curve, α is a positive weighting coefficient, which is used to balance first and second terms, and the greater the image noise, the greater the value of α. As for ω, it can preserve the object edges when ω is equal to 1. And when ω is larger than 1, it can enhance the object edges. In the homogeneous region of the image, D approaches zero, thus, the boundaries of image are enhanced. The first term is the data fitting term that measures the proximity between the optimal DoG operator and zero plane. And the second term is also the data fitting term, which is used to measure the approximation between the original DoG operator and the optimal DoG operator, at the edges of the image, they are close to each other, so that they can protect the edges very well. By minimizing the energy functional (22), we can obtain the corresponding Euler-Lagrange equation. the following iterative equation can be obtained by using the gradient descent flow method: D (G 1 G 2 ) D ( D (G 1 G 2 ) I ( x, y)) t
(23)
The stable solution of Equation (23) is the optimal DoG operator. In this model, we obtain it by iterating the original DoG operator 100 times. The adopted parameters and corresponding values are set as follows: α=3.5, ω=1, σ1=3.6, σ2=1.6,
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Δt=0.02. From the experimental analysis of Figure 1 in Section 4, under the same conditions, the best segmentation results would be achieved in shorter time and fewer iterations by using the optimal DoG operator. In addition, the regularization term ER is introduced in our model, the length term L(ϕ) and penalty term P(ϕ). They are consistent with Equation (11) and Equation (12), respectively. Here, we use zero level set function to represent the closed contour C. Therefore, the energy functional is proposed as follows: E DG ( ) D( ) L( ) P( ) 1 H ( )D( x, y) dxdy ( ) dxdy ( 1) 2 dxdy 2
(24)
Where λ, υ, μ are positive parameters. The first term is the edge energy term and the rest are regularized terms ER. H(ϕ) is Heaviside function, which is consistent with Equation (16). By Minimizing the energy functional EDG(ϕ) in Equation (24) with respect to ϕ, the corresponding gradient descent flow equation is obtained as follows:
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( ) D ( )div( ) ( div( )) t | | | |
(25)
In (25), λ, υ, μ are positive parameters. δε(ϕ) is Dirac function, which is consistent with Equation (17), and D is the optimal DoG operator, which satisfies the Equation (23)
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3.1.2. Introducing The Local Region Energy Term
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By minimizing the Equation (25), the boundary of the object can be extracted very quickly. Although this model that adopts the optimal DoG operator has achieved the goal of segmentation. However, the DoG operator is a global detection method. When segmenting images with severe intensity inhomogeneity, there will also be some degree of false segmentation or over-segmentation. This will be experimentally verified in section 4.
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The edge energy term has some advantages when dealing with images with weak edges. However, when segmenting images with severe intensity inhomogeneity and complex background, it fails to segment the object accurately, mainly due to the edge energy term that enhances the edges of complex background. Thus, in this paper, we add local energy term to segment the images with intensity inhomogeneity. For one thing, the model proposed in this paper can handle images with intensity inhomogeneity, for another, it can accurately extract the object boundary for images with weak edges. Through the analysis of the above models, we select the local energy term of the LIF model as local region energy term in this paper, see Equation (13), and its gradient descent flow equations is described as Equation (18). 3.2. Algorithm Steps
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Step 1. By calculating the Equation (23), the optimal D is obtained, and it is taken as the edge term of subsequent solution Step 2. For a given image I, initialize a contour ϕ0= 1 arbitrarily, we choose constant function as the initial contour. Step 3. According to Equation (11), (12), (14), and (15) calculate L, P, ILFI, m1 and m2. Step 4. The level set function is evolved according to gradient descent flow Equation (20). Step 5. If ϕi+1 satisfies the evolutionary stability condition, stop the evolution; otherwise, return to step 3.
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4. Experiment and Result Analysis In this section, to test the performance of our algorithm. we shall validate the proposed method on different types of complex images. The proposed method was coded by MATLAB R2016b (64 bits) on a computer with Intel(R) Core (TM) i5-3210M 2.50GHz CPU, 4 GB RAM, and Windows 10 (64 bits) operating system. In our experiments, unless otherwise specified, we usually use the following default setting of the parameters in our method: σ=3, ε=1.5, time step Δt=0.1, μ=2, υ=0.01×255². The values of β and λ are determined depending on the image. When calculating the optimal DoG operator, the parameters used in this paper are set as follows: α = 3.5, σ1 = 3.6, σ2 = 1.6, Δt = 0.02, iteration 100 times.
Active contour model based on LIF model and optimal DoG operator energy for image segmentation
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In order to quantitatively analyze and objectively evaluate the segmentation quality of different methods. We use the Jaccard similarity (JS) coefficient [14] and Dice similarity coefficient [22]to measure the accuracy of the image segmentation method.S1 and S2 represent the ground truth and the target area of the algorithm segmentation, respectively. N(∙) represents a collection of the number of pixels included. The JS and DSC are respectively expressed as follows.
JS N (S1 S2 ) N (S1 S2 )
(26)
DSC 2N (S1 S2 ) ( N (S1 ) N (S2 ))
(27)
The value of JS and DSC ranges from 0 to 1, with a higher value representing a more accurate segmentation result. 4.1. Verify The Effect of The Optimal DoG Operator
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Firstly, we analyze the comparison of image processing results using edge energy term when there is no local fitting energy term. For this reason, the DNA channel images are chosen to be segmented by adopting the original DoG operator and optimal DoG operator respectively. The initial level set function is set to a constant value function: ϕ0=1. The upper row represents the results obtained by using the original DoG operator, and the lower row represents the results obtained by using the optimal DoG operator. (a), (b) and (c) are the results of iterations 5 times, 10 times and 15 times respectively. From Figure 1, it can be seen intuitively that using the original DoG operator still failed to achieve a satisfactory segmentation effect when segmenting the DNA channel 15 times, while using the optimal DoG operator in the 10th iteration, the segmentation effect is already obvious. Therefore, using the optimal DoG operator has better effect on image segmentation and has a high efficiency for image segmentation. In the following experiments, we utilize the optimal DoG operator to segment images.
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(a) (b) (c) Figure 1. Comparison of original DoG operator and optimal DoG operator segmentation image results. (a) (Column 1): Segmentation results after 5 iterations, (b) (Column 2): Segmentation results after 10 iterations, (c) (Column 3): Segmentation results after 15 iterations
(a) (b) (c) (d) Figure 2. The result of segmentation of synthetic image. (a) Original synthetic image with initial contour (green line), (b) Image obtained with the optimal DoG operator,(c) Segmentation results with LIF model, (d) The result of the LIF model segmentation after adding the edge energy term
Secondly, we analyze the influence of edge energy term on LIF model. For this reason, we choose synthetic images with severe intensity inhomogeneity for experiments. The results are shown in Figure 2. As can be seen from Figure 2 (c), the curve still cannot evolve to the accurate object contour after evolved many times When segmenting an image using the
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LIF model. However, after adding the edge energy term in LIF model, it can be seen that the LIFDG model can generate reasonable segmentation results in Figure 2 (d). 4.2. Verify The Segmentation Effect of LIFDG Model for Different Types of Intensity Inhomogeneous Images.
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In order to test the segmentation performance of the LIFDG model for different images with intensity inhomogeneity. We choose 5 images with intensity inhomogeneity and weak edges. As shown in Figure 3. Figure 3 (a) is the original image with initial contour (green line), Figure 3 (d) is the final result image, and Figure 3 (b) and (c) are intermediate results of the curve evolution process. The first row shows the synthetic image with weak edges. In this paper, we use edge energy term, that is, the optimal DoG operator to detect edges. In order to achieve satisfactory segmentation results. The parameter settings are as follows: λ=−15，β=0.1, the number of iterations is 120 times, and the time used is 3.2860 seconds; The
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(a) (b) (c) (d) Figure 3. The segmentation results of LIFDG model. (a) (Column 1): The original images with initial contour (green line), (b) (Column 2) and (c) (Column 3) are intermediate results of the curve evolution process, (d) (Column 4): The final segmentation results (red line)
second and third row are all vessel images with intensity inhomogeneity, the former selected parameters are: λ=15, β=0.1, and the number of iterations is 76, the time-consuming is 4.2550 seconds. The parameters used by the latter are: λ=15, β=0.3, the number of iterations is 54, and the elapsed time is 2.3309 seconds; The fourth row is a natural image, the parameters used are λ=−5, β=0.01, the number of iterations is 78, and the time taken is 9.1027 seconds. The last row is an intensity inhomogeneous image with noise. The parameters we use are: α=8.5, λ=12, β=0.2, the number of iterations is 60 times, and the time used is 2.4309 seconds. In the experiment, we chose a positive number λ for images with brighter objects than the background (the second row, the third row and the last row). On the contrary (the first row and the forth row), the selected λ
Active contour model based on LIF model and optimal DoG operator energy for image segmentation
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is a negative number. The last row is an image with noise. When calculating the optimal DoG operator, we need to increase the value of α to reduce the influence of noise. 4.3. Verify The Robustness of The LIFDG Model to Intensity Inhomogeneity
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For the sake of verifying the robustness of this method to the initial contour. We choose two representative level set models, LBF model and LIF model, to compare with the LIFDG model in this paper. For comparison, we select a real vessel image for experimentation. Each model sets the optimal parameters according to the location of initial contour, and the segmentation results are shown in Figure 4. Among them, the green line in the first column (a) is the initial contour, and different locations of initial contour are manually initialized for the same images. The second column (b) is the result of segmentation using the LBF model. The third column (c) is the result of segmentation using the LIF model. And the fourth column (d) is the result of segmentation using our model. In this experiment, the adopted parameters are set as follows: σ=5, ε=1.5, time step Δt=0.1, μ=2, υ=0.001×2552, λ=15, β=0.3, and the number of iterations ranges from 50 to 60. The results of this experiment show that this model has no effect on the location of initial contour.
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(a) (b) (c) (d) Figure 4. Results of image segmentation with different locations of initial contour using LIFDG model, LIF model, and LBF model. (a) (Column 1): The original image with different locations of initial contour (green line), (b) (Column 2): The results of segmentation by using the LBF model, (c) (Column 3): The results of segmentation by using the LIF model, (d) (Column 4): The results of segmentation by using the LIFDG model
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However, when images are segmented by LBF and LIF model, the selection of initial contour location is crucial. If we choose inappropriately, it is easy to influence the experiment results and lead to mis-segmentation. Moreover, during each experiment, the parameters will change correspondingly with the location of initial contour. As for the same image, no matter where the initial contour location is, our model only need to set the parameters once. Therefore, compared with the other two models, the LIFDG model is insensitive to the initial contour location. In other words, the added edge energy optimal DoG operator term plays a significant role in image segmentation. By enhancing the edge of the objects and smoothing other regions, it not only reduces the iteration times of the model, but also makes it insensitive to the location of initial contour. It can effectively solve the problems of other region-based level set method. 4.4. Comparisons with The LBF Model, and The LIF Model
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Furthermore, in order to illustrate the superior performance of the proposed algorithm, we conduct experiments in two aspects. Firstly, we select two images with intensity inhomogeneity, synthetic image and real vessel image. When the segmentation results are approximately the same, we compare running time and number of iterations of the respective models. To explain the segmentation results more objective, we use the JS and DSC to measure the accuracy of the image segmentation method. In the experiment, the location of initial contour is fixed, and the optimal parameters of the corresponding model are adopted. Figure 5 and Figure 6 show experimental contrast images respectively. The parameter settings of each model are shown in Table 1. The number of iterations, the running time, the value of JS and DSC are shown in Table 2.
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(a) (b) (c) (d) Figure 5. The results are segmented by using LBF model, LIF model and LIFDG model (from left to right for each model iteration process). (a) (Column 1): The original images with initial contour (green line), (b) (Column 2) and (c) (Column 3) are intermediate results of the curve evolution process, (d) (Column 4): The final segmentation results
(a) (b) (c) (d) Figure 6. The results are segmented by using LBF model, LIF model and LIFDG model (from left to right for each model iteration process). (a) (Column 1): The original images with initial contour (green line), (b) (Column 2) and (c) (Column 3) are intermediate results of the curve evolution process, (d) (Column 4): The final segmentation results
Active contour model based on LIF model and optimal DoG operator energy for image segmentation
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As can be seen from Table 2, LIFDG model has shorter running time than the LBF and LIF model under the same conditions. For the second row in Figure 5, the LIF model was over-segmented due to the influence of the initial contour, and the result was not satisfactory. For the first and third rows of Figure 5, the number of iterations of the LIFDG model is about half less than that of the LBF model, but the results are similar. In the same way, as can be seen from the first row, second row, and third row of Figure 6, We get similar segmentation results. That is, the JS indicators and DSC indicators are roughly the same. But LIFDG model has shorter average running time and fewer iterations, which indicates that our model is more efficient. Secondly, in order to illustrate the superior performance of the method in this paper for segmentation of the images with intensity inhomogeneity. We choose 6 images for experiments. As shown in Figure 7. In the experiment, each model selects the optimal parameters. We compare our model with the LBF model and the LIF model respectively, and we use the running time, number of iterations , JS indicators and DSC indicators to evaluate the segmentation results, as shown in Table 3.
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We can intuitively see from the segmentation results of the different models in Figure 7. For the first row, the image is a synthetic image, it can be seen that the edge of it is very fuzzy, also contain noise and background, due to the influence of the initial contour position, the LBF model can't get a satisfactory segmentation result. It can be seen from the segmentation accuracy of Table 3, the LIF model and our model has similar coefficient of similarity, but our model number of iterations is about 1/3 times of LIF model, time is shorter. The second row is also a synthetic image, which has severe intensity inhomogeneity and weak edges. Since both the LBF model and the LIF model are based on regions, and they cannot handle images with weak edges. Therefore, they cannot obtain correct segmentation results. Due to the addition of edge energy term in our model, and edge energy term can enhance the edges. Thus, we can get satisfactory segmentation results. The third and fourth row have complex backgrounds, and the images with intensity inhomogeneity. Since LIFDG model has good edge fitting and detection capabilities, it also can eliminate the influence of noise. Therefore, our model has better segmentation results. However, both the LBF and LIF model are sensitive to the initial contour, and when dealing with complex background images, we need to adjust the variance (σ) largely enough to achieve the ability to eliminate background noise. If the variance (σ) is too large, the edge of the object will become more blurred and it's very hard to extract the object boundary. Due to the complex background noise, it is difficult to obtain correct segmentation results for the LIF model, and as the number of iterations increases, the LIF model has little effect on the segmentation results of the image. Therefore, when segmenting the two images, the number of iterations is respectively Set to 200. For the experiment of the third row of images, the parameters of each model are set as follows: For the LBF model: σ=5, ε=1, υ=0.05×255², μ=2; For the LIF model: σ11=20, σ12 =1, ε=1; For the LIFDG model: σ=10, υ=0.05×255², μ=2, ε=1.5, β=1, λ=10, and the time step of all models is Δt=0.1. As for the fourth row, the parameters of each model are set as follows: For the LBF model: σ=5, ε=1, υ=0.05×255², μ=1. For the LIF model: σ11=20, σ12 =1, ε=1; For the LIFDG model: σ=4, υ=0.05×255², μ=2, ε=1.5, β=1, λ=10, and the time step of all models is Δt=0.01. For the fifth row, the image is a synthetic image with noise. By subjectively analyzing the segmentation results of each model, the segmentation results of LIFDG model are more accurate than those of the other two models. When segmenting images with noisy, we set the parameter α=5 in our model. the LBF model and the LIF model need to enlarge the variance to reduce the influence of noise. For the last row, the variance parameters of the LBF and LIF model are set as follows: For the LBF model: σ=15; For the LIF model: σ11=23, σ12 =1. For the last row, It is a normalized and enhanced image, and the noise contained in the image is also correspondingly enhanced. It can be clearly seen that after the image is enhanced, the upper left and upper left corners of the image are very similar to the target, and therefore, it is difficult to segment. The LBF model, the LIF model and the model proposed in this paper are used to segment the image respectively. It can be seen from the experimental results that the model can obtain the correct segmentation result. Through objective data analysis, as shown in Table 3, the proposed model is also superior to the other two models.
Yaxi Duan, Taile Peng and Xianghui Qi
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(a) (b) (c) (d) Figure 7. The results are segmented by using LBF model, LIF model and LIFDG model. (a) (Column 1): The original images with initial contour (green line), (b) (Column 2): The results of segmentation by using the LBF model, (c) (Column 3): The results of segmentation by using the LIF model, (d) (Column 4): The results of segmentation by using the LIFDG model
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5. Conclusions This paper presents an active contour model, which combines LIF model and optimal DoG operator energy for image segmentation. Firstly, the optimal DoG operator is applied in the edge fitting energy term. Secondly, we combine the optimal DoG operator with the local energy term of the LIF model to achieve the purpose of segmenting the image. At the same time, due to the introduction of the edge energy term, the model is insensitive to the initial contour. And, when segmenting images with weak edges, we can also accurately extract the object boundary. Through the verification of multiple experiments, compared with other models, the experimental results show that LIFDG model is not only insensitive to initial contour, but also has high segmentation accuracy. It effectively reduces the running time and number of iterations.
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Acknowledgements This paper is jointly supported by Key Natural Science Research Projects in Colleges of Anhui China (Grant No. KJ2016A630). The authors would like to thank Dr. Chunming Li for sharing the code of the RSF model in website: http://www.engr.uconn.edu/cmli/. At the same time, I would also like to thank Dr. Kaihua Zhang for sharing the code of the LIF model in website: http://www4.comp.polyu.edu.hk/~cslzhang/papers.htm.
References 1. 2. 3. 4.
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Active contour model based on LIF model and optimal DoG operator energy for image segmentation
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22. Y Liu, C. He, Y. Wu, Zemin Ren. The L0-regularized discrete variational level set method for image
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segmentation[J]. Image and Vision Computing, 2018, 75: 32-43.
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Yaxi Duan, Taile Peng and Xianghui Qi
Biography Yaxi Duan is currently a teaching assistant from the school of computer science and technology in the Huaibei Normal University. His research interests include image segmentation. Peng Taile received his B.Sc degree in 1998 from Jilin University of Technology and M.Sc.degree in 2008 from Hefei University of Technology and Ph.D degree in 2016 from Shanghai University ;Now he is a professor in Huaibei Normal University. His main research interests include video understanding and machine learning,and etc.
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Xianghui Qi is a master student from the school of computer and communication in the Lanzhou University of Technology. His research interests include image segmentation.
Active contour model based on LIF model and optimal DoG operator energy for image segmentation
Table 1. Parameter settings of LBF model, LIF model and LIFDG model Parameters description Figure 5 σ 3 μ 0.003×255² Time step Δt 0.1 σ11 3 σ12 1 Time step Δt 0.01 σ 4 Weight coefficients β and λ β=0.1, λ= -5 Time step Δt 0.1
Model LBF model
LIF model
LIFDG model
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Figure 6 5 0.001×255² 0.1 5 1 0.011 5 β=1, λ=7 0.1
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Table 2. Comparison of LBF model, LIF model and LIFDG model on iteration numbers, running time ,JS indicator and DSC indicator Number of iterations (left to Model Images Running time JS indicator DSC indicator right except the first column) Row 1 in Figure 5 90 160 240 3.5684 0.9824 0.9911 LBF model Row 1 in Figure 6 40 80 120 2.5086 0.8730 0.9322 Row 2 in Figure 5 80 160 240 5.5223 0.8747 0.9332 LIF model Row 2 in Figure 6 90 180 264 7.7903 0.7818 0.8775 Row 3 in Figure 5 50 100 150 4.7740 0.9580 0.9786 LIFDG model Row 3 in Figure 6 20 40 56 2.2746 0.8548 0.9217
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Table 3. Comparison of LBF model, LIF model and LIFDG model on iteration numbers, running time and JS indicator and DSC indicator Model Evaluation parameters Row 1 Row 2 Row 3 Row 4 Row 5 Row 6 Number of iterations 210 500 200 464 63 200 2.8339 7.4561 9.0098 22.0585 7.1627 7.9402 Running time （s） LBF model JS indicator 0.4872 0.2292 0.9511 0.6976 0.9772 0.7169 DSC indicator 0.6552 0.3729 0.9749 0.8218 0.9885 0.8351 Number of iterations 130 316 200 200 34 42 Running time 3.7771 10.5623 136.2049 144.6047 7.1407 3.1063 LIF model JS indicator 0.9746 0.3355 0.3476 0.1257 0.9701 0.6755 DSC indicator 0.9871 0.5025 0.5159 0.2233 0.9848 0.8063 Number of iterations 38 42 70 128 22 140 Running time 1.5237 3.0816 14.2319 8.5226 1.2734 4.3308 LIFDG model JS indicator 0.9733 0.9654 0.9393 0.8727 0.9924 0.9916 DSC indicator 0.9865 0.9824 0.9687 0.9320 0.9962 0.9958
PII:
S0030-4026(19)31565-7
DOI:
https://doi.org/10.1016/j.ijleo.2019.163667
Reference:
IJLEO 163667
To appear in:
Optik
Received Date:
20 December 2018
Revised Date:
18 September 2019
Accepted Date:
16 October 2019
Please cite this article as: Duan Y, Peng T, Qi X, Active contour model based on LIF model and optimal DoG operator energy for image segmentation, Optik (2019), doi: https://doi.org/10.1016/j.ijleo.2019.163667
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Active contour model based on LIF model and optimal DoG operator energy for image segmentation Yaxi Duana,*, Taile Penga, and Xianghui Qib a School of Computer Science and Technology, Huaibei Normal University. Huaibei, 235000, China College of Computer and Communication, Lanzhou University of Technology, Lanzhou, 730050, China
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* Corresponding author. E-mail address: [email protected].
Abstract
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In order to solve the problem that the region-based active contour model is sensitive to the initial contour position, the convergence is poor, and the active contour model can not obtain good segmentation results when segmenting complex background images and severe intensity inhomogeneous images. In this paper, an active contour model which combines Local Image Fitting (LIF) and Difference of Gaussian (DoG) operator energy for image segmentation is proposed. Firstly, an optimal DoG operator is obtained by using the edge energy term, it can enhance the edge while smoothing inhomogeneous regions. Then, using the DoG energy term which is obtained in the first step and the LIF energy term to construct the total function energy terms. In this process, the regularization term is also established, it can control the smoothness of evolution curve, avoiding over-segmentation and re-initialization step. Finally, the variational method and gradient descent flow method are adopted to minimize the total energy functional for segmentation. Compared with other region-based active contour models, the experimental results show that the proposed method can achieve a better segmentation performance with less iterations and high calculation efficiency while segmenting the synthetic and real images with intensity inhomogeneity. Keywords: local image fitting; Difference of Gaussian; active contour model; image segmentation
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1. Introduction
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Image segmentation is an important research direction in the field of image processing and computer vision [1,2,3,4,5,6,7,8,9]. Since the active contour model [3] was proposed by Kass et al., the algorithm based on the active contour model has been widely used. The basic idea of the model is this, by using a continuous curve to represent the edges of the object and establishing an energy functional regarding continuous curve as the independent variable, so that the process of segmentation is transformed into calculating the minimum value of the energy functional. Generally, it can be achieved by calculating the Euler-Lagrange equation of function. When the energy reaches a minimum, the contour of the object appears. According to the different forms of contour expression, the active contour model can be generally divided into parametric active contour model and geometric active contour model. The parametric active contour model is affected by the location of initial contour and difficult to deal with the curve topological structure, limiting its further application. Compared with the parameter active contour model, the curve motion of the geometric active contour model is based on the geometric metric parameters rather than the expression parameters. Therefore, this model can handle the topological structure of curves better and enable the range of application wider. In general, we divide the geometric active contour model into two categories: edge-based active contour model [4,6,10, 11] and region-based active contour model [1,12,13,14,15,16,17,18]. For the edge-based active contour model, the most representative one is the geodesic active contour model [6], which detects the edge according to the size of the gradient. The basic idea of this model is to define an edge detector, in locations with large gradients, the curve stops evolving, so we get the contour of the object, but it is greatly affected by noise. Moreover, the model cannot get satisfactory segmentation
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Yaxi Duan, Taile Peng and Xianghui Qi
results when segmenting the images with intensity inhomogeneity. However, the region-based active contour model mainly uses the level set method [15] to define internal energy and external energy, which drives the curve evolution. When the energy reaches the minimum, the part enclosed by the curve is the object contour, and this model is insensitive to noise. The most representative one is the CV model [1]. which performs well in segmenting the images with intensity homogeneity, but it cannot handle the images with intensity inhomogeneity very well. Li et al. proposed a Local Binary Fitting (LBF) model [13] to overcome the shortcomings of the CV model. This algorithm utilizes the local information of the image to smooth the inhomogeneous regions. but the model is easily influenced by the location of initial contour leading to fall into local minimum and has high computational complexity, noise also affects the model to a certain extent. Zhang et al. proposed a Local Image Fitting (LIF) model [17]. Compared with the LBF model, they have similar segmentation results, but this model takes less time. It also has same shortcomings as the LBF model. Then, a number of models based on local and global information have been proposed [12,16,18], but none of them have introduced edge energy term, which makes it difficult to achieve ideal segmentation results when segmenting images with weak edges.
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In view of the above problems. We use the combination of local image fitting (LIF) and the optimal Difference of Gaussian (DoG) energy in this paper. Firstly, the idea of DoG is applied to image segmentation by constructing edge energy term and using the optimal DoG operator to detect edges. Then, we combine the optimal DoG operator with the local energy term of the LIF model to construct the total energy terms. In addition, the introduction of length term and regular term makes it smoother in the evolutionary process. Moreover, it is not necessary to re-initialize the contour, which greatly improves the efficiency of its evolution. The experimental results show that the proposed model has good characteristics of speed, precision and robustness when segmenting different types of real images and synthetic images. The rest of the paper is organized as follows: in section 2, we briefly introduced and reviewed the CV model, the LBF model and the LIF model. The model and method proposed in this paper are described in detail in section 3. In section 4, the effectiveness of the proposed model is verified by experiments. The paper is summarized in section 5.
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2. Related Work 2.1. C-V Model
E cv (c1 , c2 , C ) 1
inside (C )
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The C-V model [1] is a region-based segmentation method, which is proposed by Chan and Vese. The basic idea of the model is to regard the image as the two parts of the homogeneous regions. By using level set method to calculate the intensity mean of internal and external contour, the energy functional would be minimized to construct the energy activity contour model. Assuming that Ω ⊂ R2 is a two-dimensional image space domain, I: Ω ⊂ R is an input image, and its minimized energy functional is given by (1). | u0 ( x, y) c1 |2 dxdy 2
outside (C )
| u0 ( x, y) c2 |2 dxdy Length.
(1)
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In (1), μ ≥ 0, λ1 and λ2 control the contributions of the internal and external energy terms. c1, c2 are the image intensities of the interior and exterior of contour C respectively. The first two terms are data fitting energy terms, and the third is length term.
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In order to facilitate the solution, a zero level set function ϕ(x, y) is introduced to replace the unknown contour C. If the point (x, y) is inside the curve C, ϕ(x, y) > 0, and if the point (x, y) is outside the curve C, ϕ(x, y) < 0. If the point (x, y) is on C, ϕ(x, y) = 0. Therefore, the level set function energy function can be expression is: E cv (c1 , c2 , ) 1 | u0 ( x, y) c1 |2 H ( ( x, y))dxdy 2 | u0 ( x, y) c2 |2 (1 H ( ( x, y)))dxdy
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( ( x, y)) | ( x, y) | dxdy
(2)
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In (2), Hε(ϕ(z)) and δε(ϕ(z)) are regularized Heaviside function and Dirac function respectively. And the corresponding relationships can be shown as formula (3) and formula (4). 1. if z 0 H ( z) 0. if z 0
(3)
Active contour model based on LIF model and optimal DoG operator energy for image segmentation
( z)
3
d ( H ( z )) dx
(4)
By calculating the Euler-Lagrange equation of the above energy functional Ecv in Equation (2), and by using the gradient descent flow method [19] to update the level set function ϕ(x, y) to obtain the following evolution equation: ( ) div( ) - 1 (u0 c1 ) 2 2 (u0 c2 ) 2 t | |
(5)
In (5), c1 and c2 are intensity averages inside and outside the evolving curve C respectively, which should be updated at each iteration. Their expressions are shown in formula (6) and (7).
u0 ( x, y) H ( ( x, y))dxdy
c2
(6)
H ( ( x, y))dxdy
u0 ( x, y )(1 H ( ( x, y )))dxdy
(1 H ( ( x, y )))dxdy
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The main advantage of C-V model is that it can detect edge of objects whose boundaries are not based on gradient and can effectively extract the boundaries of objects. It is also not sensitive to noise. This model needs to re-initialize the contour at each time of evolution, which is time-consuming and cannot handle weak-edge images with intensity inhomogeneity. 2.2. LBF Model
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The LBF model [13] was proposed by Li et al., which is a region-based active contour model. By introducing the Gauss kernel function and using the local information of the image to achieve binary function fitting. The local region information is embedded in the variational level set function, which overcomes the limitations of the piecewise constant function model. As a result, the model can segment images with intensity inhomogeneity. The level set energy functional of this model is as follows: 2 2 ELBF ( f 1 , f 2 , ) 1 K ( x, y) | I ( y) f1 ( x) | H ( ( y))dy dx 2 K ( x, y) | I ( y) f 2 ( x) | (1 H ( ( y)))dy dx L( ( x)) P( ( x))
(8)
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In (8), the weight coefficients of λ1, λ2, υ, μ are positive parameters, f1 and f2 are two smooth functions, during the period of evolution performed by the curve of level set function, they represent the weighted average of the interior and exterior of the Gaussian window of the intensity image. The first two terms are data fitting energy terms. The third is the length term, which makes the evolution curve (C) smoother. The fourth is the penalty term [20], whose function is to make the level set function do not need to be re-initialized in the curve evolution process, and it can accelerate the speed of curve evolution, f1 and f2, L(ϕ) and P(ϕ) are show in formulas (9), (10), (11) and (12), respectively. f1
f2
K ( x) I ( x) H ( f ( x)) K ( x) H ( f ( x))
K ( x) I ( x)(1 H ( ( x))) K ( x) (1 H ( ( x)))
L( ) | H ( ( x, y)) | dxdy
P( )
1 (| H ( ( x, y)) | 1) 2 dxdy 2
(9)
(10)
(11)
(12)
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Yaxi Duan, Taile Peng and Xianghui Qi
Where ∗ is the convolution operator in Equation (9) and Equation (10). The LBF model can segment the images with intensity inhomogeneity since it introduces the gaussian kernel function. However, this model requires initial contour more. On the one hand, selecting the inappropriate initial contour location would easily fall into the local minimum and cannot get the correct segmentation result. On the other hand, if the chosen variance σ of the Gaussian kernel is too large, it will generate the blurry edge and will not obtain the satisfied results similarly. Therefore, the model has some limitations for segmentation of images with complex background. 2.3. LIF Model
E LIF ( )
1 | I ( x) I LFI ( x) |2 dx 2
In (13), ILFI represents the local fitting image. The expression is as follows:
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I LFI m1 H ( ) m2 (1 H ( ))
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The LIF model [17] was proposed by Zhang et al. It is an improvement of the LBF model. For a given image I(x) in the image domain x∈Ω. Firstly, the local fitting information of the image is used to construct the local fitting energy term, leading to minimize the fitting energy to achieve segmentation. Secondly, after the end of each activity contour curve, Gauss filter is used to smooth the result. For this step would replace the length term and penalty term in the LBF model, it not only improves the efficiency of evolution, but also gets the similar result of LBF model. The energy function of the LIF model is expressed as follows:
In (14), the definitions of m1 and m2 are as follows:
(14)
(15)
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m1 mean( I ({x | ( x) 0} Wk ( x))) m2 mean( I ({x | ( x) 0} Wk ( x)))
(13)
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In (15), Wk (x) is a truncated Gaussian or constant window function. In this model, the truncated Gauss window function Kσ is used, the σ is the standard deviation and the width is 4K+1. ϕ is the zero level set of a Lipschitz function that represents the contour C. In the expression ILFI, Hε(ϕ) is used to regularize the Heaviside function. The Heaviside function and its derivative form Dirac function are shown as formula (16) and formula (17).:
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( z)
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z2 2
(16)
(17)
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By using the steepest descent method and the variational method to calculate the minimum energy functional ELIF(ϕ) in Equation (13) with respect to ϕ. Its gradient descent flow equation can be expressed by (18).
( I I LFI )(m1 m2 ) ( ) t
(18)
The LIF model performs Gaussian filtering on each evolutionary obtained image, which has same effect as adding the length term and the penalty term in the LBF model. However, the model is also sensitive to the location of initial contour, and it is not ideal to segment images with severe intensity inhomogeneity. 3. Proposed LIFDG Model
Active contour model based on LIF model and optimal DoG operator energy for image segmentation
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Motivated by the strengths of Difference of Gaussian and local region-based active contour models, we proposed an active contour model based on LIF model and optimal DoG operator energy. 3.1. Energy Functional of LIFDG Model The LIFDG model is composed of the local energy ELIF of the LIF model and the optimal DoG operator energy EDG, in addition, we have added regularized energy ER, which overcomes the shortcomings of the above models. The initial contour can be placed in any location of the image, which greatly improves the robustness of the model. The total energy functional is as follows: E LIFDG E LIF ( , m1 , m2 ) E DG ( ) E R
(19)
Using the gradient descent flow method, we minimize the energy functional ELIFDG in (19) with respect to ϕ to get the corresponding variational level set energy functional can be formulated as: (20)
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( )( I I LFI )(m1 m2 )+ ( ) D ( )div( ) ( div( )) t | | | |
Where β, λ, υ, μ are positive parameters. This evolution Equation (20) is applied in our model. 3.1.1. Introducing The Edge Energy Term
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Inspired by the literature [21], we introduced DoG operator fitting energy. The DoG operator is obtained by subtracting two filtered images. We define a Gaussian function with two standard deviations, σ1 and σ2, convolved with the original image I(x, y) respectively. The DoG operator for detecting the edges of the image will be obtained, and the function expression of the DoG operator is shown as formula (21).
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G 1 I ( x, y) G 2 I ( x, y) (G 1 G 2 ) I ( x, y) DoG I ( x, y)
(21)
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Because the DoG operator is a second-order difference operator, it is very sensitive to the gray change of the image. That is to say, there are some zero crossing points that are not the real edges of the image. It is mainly caused by the noise and the image with intensity inhomogeneity. Therefore, in order to solve these problems, we constructed the edge energy term, which can be written as: E DoG ( D) ((G 1 G 2 ) D 0)2 +(D (G 1 G 2 ) I ( x, y))2 dxdy
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(22)
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Where D represents the optimal DoG operator, the contour C is a closed curve, α is a positive weighting coefficient, which is used to balance first and second terms, and the greater the image noise, the greater the value of α. As for ω, it can preserve the object edges when ω is equal to 1. And when ω is larger than 1, it can enhance the object edges. In the homogeneous region of the image, D approaches zero, thus, the boundaries of image are enhanced. The first term is the data fitting term that measures the proximity between the optimal DoG operator and zero plane. And the second term is also the data fitting term, which is used to measure the approximation between the original DoG operator and the optimal DoG operator, at the edges of the image, they are close to each other, so that they can protect the edges very well. By minimizing the energy functional (22), we can obtain the corresponding Euler-Lagrange equation. the following iterative equation can be obtained by using the gradient descent flow method: D (G 1 G 2 ) D ( D (G 1 G 2 ) I ( x, y)) t
(23)
The stable solution of Equation (23) is the optimal DoG operator. In this model, we obtain it by iterating the original DoG operator 100 times. The adopted parameters and corresponding values are set as follows: α=3.5, ω=1, σ1=3.6, σ2=1.6,
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Δt=0.02. From the experimental analysis of Figure 1 in Section 4, under the same conditions, the best segmentation results would be achieved in shorter time and fewer iterations by using the optimal DoG operator. In addition, the regularization term ER is introduced in our model, the length term L(ϕ) and penalty term P(ϕ). They are consistent with Equation (11) and Equation (12), respectively. Here, we use zero level set function to represent the closed contour C. Therefore, the energy functional is proposed as follows: E DG ( ) D( ) L( ) P( ) 1 H ( )D( x, y) dxdy ( ) dxdy ( 1) 2 dxdy 2
(24)
Where λ, υ, μ are positive parameters. The first term is the edge energy term and the rest are regularized terms ER. H(ϕ) is Heaviside function, which is consistent with Equation (16). By Minimizing the energy functional EDG(ϕ) in Equation (24) with respect to ϕ, the corresponding gradient descent flow equation is obtained as follows:
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( ) D ( )div( ) ( div( )) t | | | |
(25)
In (25), λ, υ, μ are positive parameters. δε(ϕ) is Dirac function, which is consistent with Equation (17), and D is the optimal DoG operator, which satisfies the Equation (23)
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3.1.2. Introducing The Local Region Energy Term
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By minimizing the Equation (25), the boundary of the object can be extracted very quickly. Although this model that adopts the optimal DoG operator has achieved the goal of segmentation. However, the DoG operator is a global detection method. When segmenting images with severe intensity inhomogeneity, there will also be some degree of false segmentation or over-segmentation. This will be experimentally verified in section 4.
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The edge energy term has some advantages when dealing with images with weak edges. However, when segmenting images with severe intensity inhomogeneity and complex background, it fails to segment the object accurately, mainly due to the edge energy term that enhances the edges of complex background. Thus, in this paper, we add local energy term to segment the images with intensity inhomogeneity. For one thing, the model proposed in this paper can handle images with intensity inhomogeneity, for another, it can accurately extract the object boundary for images with weak edges. Through the analysis of the above models, we select the local energy term of the LIF model as local region energy term in this paper, see Equation (13), and its gradient descent flow equations is described as Equation (18). 3.2. Algorithm Steps
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Step 1. By calculating the Equation (23), the optimal D is obtained, and it is taken as the edge term of subsequent solution Step 2. For a given image I, initialize a contour ϕ0= 1 arbitrarily, we choose constant function as the initial contour. Step 3. According to Equation (11), (12), (14), and (15) calculate L, P, ILFI, m1 and m2. Step 4. The level set function is evolved according to gradient descent flow Equation (20). Step 5. If ϕi+1 satisfies the evolutionary stability condition, stop the evolution; otherwise, return to step 3.
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4. Experiment and Result Analysis In this section, to test the performance of our algorithm. we shall validate the proposed method on different types of complex images. The proposed method was coded by MATLAB R2016b (64 bits) on a computer with Intel(R) Core (TM) i5-3210M 2.50GHz CPU, 4 GB RAM, and Windows 10 (64 bits) operating system. In our experiments, unless otherwise specified, we usually use the following default setting of the parameters in our method: σ=3, ε=1.5, time step Δt=0.1, μ=2, υ=0.01×255². The values of β and λ are determined depending on the image. When calculating the optimal DoG operator, the parameters used in this paper are set as follows: α = 3.5, σ1 = 3.6, σ2 = 1.6, Δt = 0.02, iteration 100 times.
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In order to quantitatively analyze and objectively evaluate the segmentation quality of different methods. We use the Jaccard similarity (JS) coefficient [14] and Dice similarity coefficient [22]to measure the accuracy of the image segmentation method.S1 and S2 represent the ground truth and the target area of the algorithm segmentation, respectively. N(∙) represents a collection of the number of pixels included. The JS and DSC are respectively expressed as follows.
JS N (S1 S2 ) N (S1 S2 )
(26)
DSC 2N (S1 S2 ) ( N (S1 ) N (S2 ))
(27)
The value of JS and DSC ranges from 0 to 1, with a higher value representing a more accurate segmentation result. 4.1. Verify The Effect of The Optimal DoG Operator
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Firstly, we analyze the comparison of image processing results using edge energy term when there is no local fitting energy term. For this reason, the DNA channel images are chosen to be segmented by adopting the original DoG operator and optimal DoG operator respectively. The initial level set function is set to a constant value function: ϕ0=1. The upper row represents the results obtained by using the original DoG operator, and the lower row represents the results obtained by using the optimal DoG operator. (a), (b) and (c) are the results of iterations 5 times, 10 times and 15 times respectively. From Figure 1, it can be seen intuitively that using the original DoG operator still failed to achieve a satisfactory segmentation effect when segmenting the DNA channel 15 times, while using the optimal DoG operator in the 10th iteration, the segmentation effect is already obvious. Therefore, using the optimal DoG operator has better effect on image segmentation and has a high efficiency for image segmentation. In the following experiments, we utilize the optimal DoG operator to segment images.
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(a) (b) (c) Figure 1. Comparison of original DoG operator and optimal DoG operator segmentation image results. (a) (Column 1): Segmentation results after 5 iterations, (b) (Column 2): Segmentation results after 10 iterations, (c) (Column 3): Segmentation results after 15 iterations
(a) (b) (c) (d) Figure 2. The result of segmentation of synthetic image. (a) Original synthetic image with initial contour (green line), (b) Image obtained with the optimal DoG operator,(c) Segmentation results with LIF model, (d) The result of the LIF model segmentation after adding the edge energy term
Secondly, we analyze the influence of edge energy term on LIF model. For this reason, we choose synthetic images with severe intensity inhomogeneity for experiments. The results are shown in Figure 2. As can be seen from Figure 2 (c), the curve still cannot evolve to the accurate object contour after evolved many times When segmenting an image using the
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LIF model. However, after adding the edge energy term in LIF model, it can be seen that the LIFDG model can generate reasonable segmentation results in Figure 2 (d). 4.2. Verify The Segmentation Effect of LIFDG Model for Different Types of Intensity Inhomogeneous Images.
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In order to test the segmentation performance of the LIFDG model for different images with intensity inhomogeneity. We choose 5 images with intensity inhomogeneity and weak edges. As shown in Figure 3. Figure 3 (a) is the original image with initial contour (green line), Figure 3 (d) is the final result image, and Figure 3 (b) and (c) are intermediate results of the curve evolution process. The first row shows the synthetic image with weak edges. In this paper, we use edge energy term, that is, the optimal DoG operator to detect edges. In order to achieve satisfactory segmentation results. The parameter settings are as follows: λ=−15，β=0.1, the number of iterations is 120 times, and the time used is 3.2860 seconds; The
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(a) (b) (c) (d) Figure 3. The segmentation results of LIFDG model. (a) (Column 1): The original images with initial contour (green line), (b) (Column 2) and (c) (Column 3) are intermediate results of the curve evolution process, (d) (Column 4): The final segmentation results (red line)
second and third row are all vessel images with intensity inhomogeneity, the former selected parameters are: λ=15, β=0.1, and the number of iterations is 76, the time-consuming is 4.2550 seconds. The parameters used by the latter are: λ=15, β=0.3, the number of iterations is 54, and the elapsed time is 2.3309 seconds; The fourth row is a natural image, the parameters used are λ=−5, β=0.01, the number of iterations is 78, and the time taken is 9.1027 seconds. The last row is an intensity inhomogeneous image with noise. The parameters we use are: α=8.5, λ=12, β=0.2, the number of iterations is 60 times, and the time used is 2.4309 seconds. In the experiment, we chose a positive number λ for images with brighter objects than the background (the second row, the third row and the last row). On the contrary (the first row and the forth row), the selected λ
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is a negative number. The last row is an image with noise. When calculating the optimal DoG operator, we need to increase the value of α to reduce the influence of noise. 4.3. Verify The Robustness of The LIFDG Model to Intensity Inhomogeneity
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For the sake of verifying the robustness of this method to the initial contour. We choose two representative level set models, LBF model and LIF model, to compare with the LIFDG model in this paper. For comparison, we select a real vessel image for experimentation. Each model sets the optimal parameters according to the location of initial contour, and the segmentation results are shown in Figure 4. Among them, the green line in the first column (a) is the initial contour, and different locations of initial contour are manually initialized for the same images. The second column (b) is the result of segmentation using the LBF model. The third column (c) is the result of segmentation using the LIF model. And the fourth column (d) is the result of segmentation using our model. In this experiment, the adopted parameters are set as follows: σ=5, ε=1.5, time step Δt=0.1, μ=2, υ=0.001×2552, λ=15, β=0.3, and the number of iterations ranges from 50 to 60. The results of this experiment show that this model has no effect on the location of initial contour.
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(a) (b) (c) (d) Figure 4. Results of image segmentation with different locations of initial contour using LIFDG model, LIF model, and LBF model. (a) (Column 1): The original image with different locations of initial contour (green line), (b) (Column 2): The results of segmentation by using the LBF model, (c) (Column 3): The results of segmentation by using the LIF model, (d) (Column 4): The results of segmentation by using the LIFDG model
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However, when images are segmented by LBF and LIF model, the selection of initial contour location is crucial. If we choose inappropriately, it is easy to influence the experiment results and lead to mis-segmentation. Moreover, during each experiment, the parameters will change correspondingly with the location of initial contour. As for the same image, no matter where the initial contour location is, our model only need to set the parameters once. Therefore, compared with the other two models, the LIFDG model is insensitive to the initial contour location. In other words, the added edge energy optimal DoG operator term plays a significant role in image segmentation. By enhancing the edge of the objects and smoothing other regions, it not only reduces the iteration times of the model, but also makes it insensitive to the location of initial contour. It can effectively solve the problems of other region-based level set method. 4.4. Comparisons with The LBF Model, and The LIF Model
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Furthermore, in order to illustrate the superior performance of the proposed algorithm, we conduct experiments in two aspects. Firstly, we select two images with intensity inhomogeneity, synthetic image and real vessel image. When the segmentation results are approximately the same, we compare running time and number of iterations of the respective models. To explain the segmentation results more objective, we use the JS and DSC to measure the accuracy of the image segmentation method. In the experiment, the location of initial contour is fixed, and the optimal parameters of the corresponding model are adopted. Figure 5 and Figure 6 show experimental contrast images respectively. The parameter settings of each model are shown in Table 1. The number of iterations, the running time, the value of JS and DSC are shown in Table 2.
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(a) (b) (c) (d) Figure 5. The results are segmented by using LBF model, LIF model and LIFDG model (from left to right for each model iteration process). (a) (Column 1): The original images with initial contour (green line), (b) (Column 2) and (c) (Column 3) are intermediate results of the curve evolution process, (d) (Column 4): The final segmentation results
(a) (b) (c) (d) Figure 6. The results are segmented by using LBF model, LIF model and LIFDG model (from left to right for each model iteration process). (a) (Column 1): The original images with initial contour (green line), (b) (Column 2) and (c) (Column 3) are intermediate results of the curve evolution process, (d) (Column 4): The final segmentation results
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As can be seen from Table 2, LIFDG model has shorter running time than the LBF and LIF model under the same conditions. For the second row in Figure 5, the LIF model was over-segmented due to the influence of the initial contour, and the result was not satisfactory. For the first and third rows of Figure 5, the number of iterations of the LIFDG model is about half less than that of the LBF model, but the results are similar. In the same way, as can be seen from the first row, second row, and third row of Figure 6, We get similar segmentation results. That is, the JS indicators and DSC indicators are roughly the same. But LIFDG model has shorter average running time and fewer iterations, which indicates that our model is more efficient. Secondly, in order to illustrate the superior performance of the method in this paper for segmentation of the images with intensity inhomogeneity. We choose 6 images for experiments. As shown in Figure 7. In the experiment, each model selects the optimal parameters. We compare our model with the LBF model and the LIF model respectively, and we use the running time, number of iterations , JS indicators and DSC indicators to evaluate the segmentation results, as shown in Table 3.
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We can intuitively see from the segmentation results of the different models in Figure 7. For the first row, the image is a synthetic image, it can be seen that the edge of it is very fuzzy, also contain noise and background, due to the influence of the initial contour position, the LBF model can't get a satisfactory segmentation result. It can be seen from the segmentation accuracy of Table 3, the LIF model and our model has similar coefficient of similarity, but our model number of iterations is about 1/3 times of LIF model, time is shorter. The second row is also a synthetic image, which has severe intensity inhomogeneity and weak edges. Since both the LBF model and the LIF model are based on regions, and they cannot handle images with weak edges. Therefore, they cannot obtain correct segmentation results. Due to the addition of edge energy term in our model, and edge energy term can enhance the edges. Thus, we can get satisfactory segmentation results. The third and fourth row have complex backgrounds, and the images with intensity inhomogeneity. Since LIFDG model has good edge fitting and detection capabilities, it also can eliminate the influence of noise. Therefore, our model has better segmentation results. However, both the LBF and LIF model are sensitive to the initial contour, and when dealing with complex background images, we need to adjust the variance (σ) largely enough to achieve the ability to eliminate background noise. If the variance (σ) is too large, the edge of the object will become more blurred and it's very hard to extract the object boundary. Due to the complex background noise, it is difficult to obtain correct segmentation results for the LIF model, and as the number of iterations increases, the LIF model has little effect on the segmentation results of the image. Therefore, when segmenting the two images, the number of iterations is respectively Set to 200. For the experiment of the third row of images, the parameters of each model are set as follows: For the LBF model: σ=5, ε=1, υ=0.05×255², μ=2; For the LIF model: σ11=20, σ12 =1, ε=1; For the LIFDG model: σ=10, υ=0.05×255², μ=2, ε=1.5, β=1, λ=10, and the time step of all models is Δt=0.1. As for the fourth row, the parameters of each model are set as follows: For the LBF model: σ=5, ε=1, υ=0.05×255², μ=1. For the LIF model: σ11=20, σ12 =1, ε=1; For the LIFDG model: σ=4, υ=0.05×255², μ=2, ε=1.5, β=1, λ=10, and the time step of all models is Δt=0.01. For the fifth row, the image is a synthetic image with noise. By subjectively analyzing the segmentation results of each model, the segmentation results of LIFDG model are more accurate than those of the other two models. When segmenting images with noisy, we set the parameter α=5 in our model. the LBF model and the LIF model need to enlarge the variance to reduce the influence of noise. For the last row, the variance parameters of the LBF and LIF model are set as follows: For the LBF model: σ=15; For the LIF model: σ11=23, σ12 =1. For the last row, It is a normalized and enhanced image, and the noise contained in the image is also correspondingly enhanced. It can be clearly seen that after the image is enhanced, the upper left and upper left corners of the image are very similar to the target, and therefore, it is difficult to segment. The LBF model, the LIF model and the model proposed in this paper are used to segment the image respectively. It can be seen from the experimental results that the model can obtain the correct segmentation result. Through objective data analysis, as shown in Table 3, the proposed model is also superior to the other two models.
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(a) (b) (c) (d) Figure 7. The results are segmented by using LBF model, LIF model and LIFDG model. (a) (Column 1): The original images with initial contour (green line), (b) (Column 2): The results of segmentation by using the LBF model, (c) (Column 3): The results of segmentation by using the LIF model, (d) (Column 4): The results of segmentation by using the LIFDG model
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5. Conclusions This paper presents an active contour model, which combines LIF model and optimal DoG operator energy for image segmentation. Firstly, the optimal DoG operator is applied in the edge fitting energy term. Secondly, we combine the optimal DoG operator with the local energy term of the LIF model to achieve the purpose of segmenting the image. At the same time, due to the introduction of the edge energy term, the model is insensitive to the initial contour. And, when segmenting images with weak edges, we can also accurately extract the object boundary. Through the verification of multiple experiments, compared with other models, the experimental results show that LIFDG model is not only insensitive to initial contour, but also has high segmentation accuracy. It effectively reduces the running time and number of iterations.
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Acknowledgements This paper is jointly supported by Key Natural Science Research Projects in Colleges of Anhui China (Grant No. KJ2016A630). The authors would like to thank Dr. Chunming Li for sharing the code of the RSF model in website: http://www.engr.uconn.edu/cmli/. At the same time, I would also like to thank Dr. Kaihua Zhang for sharing the code of the LIF model in website: http://www4.comp.polyu.edu.hk/~cslzhang/papers.htm.
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22. Y Liu, C. He, Y. Wu, Zemin Ren. The L0-regularized discrete variational level set method for image
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segmentation[J]. Image and Vision Computing, 2018, 75: 32-43.
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Biography Yaxi Duan is currently a teaching assistant from the school of computer science and technology in the Huaibei Normal University. His research interests include image segmentation. Peng Taile received his B.Sc degree in 1998 from Jilin University of Technology and M.Sc.degree in 2008 from Hefei University of Technology and Ph.D degree in 2016 from Shanghai University ;Now he is a professor in Huaibei Normal University. His main research interests include video understanding and machine learning,and etc.
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Xianghui Qi is a master student from the school of computer and communication in the Lanzhou University of Technology. His research interests include image segmentation.
Active contour model based on LIF model and optimal DoG operator energy for image segmentation
Table 1. Parameter settings of LBF model, LIF model and LIFDG model Parameters description Figure 5 σ 3 μ 0.003×255² Time step Δt 0.1 σ11 3 σ12 1 Time step Δt 0.01 σ 4 Weight coefficients β and λ β=0.1, λ= -5 Time step Δt 0.1
Model LBF model
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LIFDG model
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Figure 6 5 0.001×255² 0.1 5 1 0.011 5 β=1, λ=7 0.1
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Table 2. Comparison of LBF model, LIF model and LIFDG model on iteration numbers, running time ,JS indicator and DSC indicator Number of iterations (left to Model Images Running time JS indicator DSC indicator right except the first column) Row 1 in Figure 5 90 160 240 3.5684 0.9824 0.9911 LBF model Row 1 in Figure 6 40 80 120 2.5086 0.8730 0.9322 Row 2 in Figure 5 80 160 240 5.5223 0.8747 0.9332 LIF model Row 2 in Figure 6 90 180 264 7.7903 0.7818 0.8775 Row 3 in Figure 5 50 100 150 4.7740 0.9580 0.9786 LIFDG model Row 3 in Figure 6 20 40 56 2.2746 0.8548 0.9217
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Table 3. Comparison of LBF model, LIF model and LIFDG model on iteration numbers, running time and JS indicator and DSC indicator Model Evaluation parameters Row 1 Row 2 Row 3 Row 4 Row 5 Row 6 Number of iterations 210 500 200 464 63 200 2.8339 7.4561 9.0098 22.0585 7.1627 7.9402 Running time （s） LBF model JS indicator 0.4872 0.2292 0.9511 0.6976 0.9772 0.7169 DSC indicator 0.6552 0.3729 0.9749 0.8218 0.9885 0.8351 Number of iterations 130 316 200 200 34 42 Running time 3.7771 10.5623 136.2049 144.6047 7.1407 3.1063 LIF model JS indicator 0.9746 0.3355 0.3476 0.1257 0.9701 0.6755 DSC indicator 0.9871 0.5025 0.5159 0.2233 0.9848 0.8063 Number of iterations 38 42 70 128 22 140 Running time 1.5237 3.0816 14.2319 8.5226 1.2734 4.3308 LIFDG model JS indicator 0.9733 0.9654 0.9393 0.8727 0.9924 0.9916 DSC indicator 0.9865 0.9824 0.9687 0.9320 0.9962 0.9958