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In view of the problem that the local active contour model is difficult to achieve image segmentation accurately and quickly, an improved image segmentation method based on Local Image Fitting (LIF) is proposed. Firstly, the local median is used as the fitting center of the curve to enhance the robustness of the model to noise. Secondly, a minimized Laplacian of gaussian energy (Log) term is introduced, and the Log operator is used to smooth the image and enhance the edges of the image. Finally, the minimized Log energy term is combined with the LIF, which together drives the curve to the boundary. Experimental results show that the Precision rate, Recall rate and Dice Similarity Coefficient of this model are closest to 1. Compared with other main region-based models, the image segmentation accuracy of this method is significantly higher than that of other algorithms, which improves the anti-noise performance and image segmentation speed.

Image segmentation is still a challenging work at present; it has great significance for the understanding and analysis of images. Active contour model is a very successful image segmentation method, which is divided into parametric active contour model [

Currently, the Piecewise Constant (PC) model [

On the basis of the above models, the researchers put forward many improved models. Liu et al. [

Based on LIF model, an improved active contour model is proposed in this paper. First, we replace the mean value of the LIF model with the median intensity of the local area as the fitting center to enhance the robustness to noise. Secondly, the energy equation about the LOG operator is established to minimize it, which can enhance the edge information while smoothing the uniform region. Finally, the minimized Log energy term is combined with the LIF term to drive the curve to the boundary. The experimental results show that the segmentation accuracy and speed of the proposed model are significantly improved and the segmentation effect is good.

Let Ω ⊂ R 2 , for a given image I ( x , y ) : Ω → R , The closed contour C is constructed by the level set function ϕ : Ω → R ,

In order to make full use of local image information, Li et al. [

E R S F ( f 1 , f 2 , ϕ ) = λ 1 ∫ [ ∫ K σ ( x − y ) | I − f 1 | 2 H ε ( ϕ ( y ) ) d y ] d x + λ 2 ∫ [ ∫ K σ ( x − y ) | I − f 2 | 2 ( 1 − H ε ( ϕ ( y ) ) ) d y ] d x (1)

In Equation (1), K σ is the Gaussian kernel function, σ controls the size of the kernel function.

K σ = 1 ( 2 π ) n / 2 σ n e − | x 2 | / 2 σ 2 (2)

f 1 , f 2 is the mean intensity inside and outside the locally approximated curve, and the expression is as follows:

f 1 ( x ) = K σ ( x ) ∗ [ H ε ( ϕ ) I 0 ( x ) ] K σ ( x ) ∗ H ε ( ϕ ) f 2 ( x ) = K σ ( x ) ∗ [ ( 1 − H ε ( ϕ ) ) I 0 ( x ) ] K σ ( x ) ∗ ( 1 − H ε ( ϕ ) ) (3)

H ε ( ϕ ) is the smooth approximation function of Heaviside function H ( ϕ ) , δ ε ( ϕ ) is Dirac function, ε is a constant.

H ε ( ϕ ) = 1 2 [ 1 + 2 π arctan ( ϕ ε ) ] δ ε ( ϕ ) = 1 π ⋅ ε ε 2 + ϕ 2 (4)

In general, λ 1 = λ 2 = 1 , the curve evolution equation obtained through curve evolution theory is as follows:

∂ ϕ ∂ t = − δ ε ( ϕ ) K σ ∗ ( | I − f 1 ( x ) | 2 − | I − f 2 ( x ) | 2 ) (5)

Since the kernel function K σ , f 1 , f 2 is the weighted average of the intensity of the image in the Gaussian window inside and outside the curve, the high-precision segmentation of the image with uneven intensity can be achieved. However, if the initial contour is not set properly, the evolution rate of the curve may drop sharply and eventually fall into the local minimum. This means that different positions of the initial contour may lead to different segmentation results, and inappropriate initial contour may lead to segmentation failure.

Zhang et al. [

E L I F ( m 1 , m 2 , ϕ ) = 1 2 ∫ Ω | I 0 − I L F I | 2 d x (6)

I L F I is defined as follows：

I L F I = m 1 H ε ( ϕ ) + m 2 ( 1 − H ε ( ϕ ) ) (7)

where m 1 ( x ) , m 2 ( x ) is calculated as follows:

{ m 1 ( x ) = a v e r a g e ( I 0 ( x ) ∈ ( { x ∈ Ω | ϕ < 0 } ∩ W k ( x ) ) ) m 2 ( x ) = a v e r a g e ( I 0 ( x ) ∈ ( { x ∈ Ω | ϕ > 0 } ∩ W k ( x ) ) ) (8)

Finally, its evolution equation is as follows:

∂ ϕ ∂ t = δ ε ( ϕ ) ( I − I L F I ) ( m 1 − m 2 ) (9)

In LIF model, m 1 ( x ) and m 2 ( x ) can be viewed as the average of the image strength in the window. Thus, m 1 ( x ) and m 2 ( x ) are the same as f 1 ( x ) and f 2 ( x ) in the RSF model. By introducing local image information, LIF model can segment images with uneven intensity, calculating only two reels at a time, which is half of that of RSF model, so the calculation cost is lower. However, with the introduction of localization information, the model becomes more sensitive to the location of the initial contour and is prone to fall into the local minimum.

Ding et al. [

E L o g ( L ( x , y ) ) = ∫ Ω g ( | ∇ I | ) × L 2 + ( 1 − g ( | ∇ I | ) ) × ( L − C × Δ ( G σ ∗ I ) ) 2 d x d y (10)

where C is a constant, Δ ( G σ ∗ I ) is Laplacian of gaussian, defined as Equation (11).

Δ ( G σ ∗ I ) = ( ∂ 2 G σ ( x , y ) ∂ x 2 + ∂ 2 G σ ( x , y ) ∂ y 2 ) ∗ I (11)

E L o g is the energy function of the Log of image, when E L o g declines, g ( | ∇ I | ) × L 2 will drive L to 0, which helps to smooth the regions. When approaching the target edge, ( 1 − g ( | ∇ I | ) ) × ( L − C × Δ ( G σ ∗ I ) ) 2 will drive L to close to Δ ( G σ ∗ I ) , when C > 1, it can enhance the target edges.

Its evolution equation is as follows:

∂ L ∂ t = g ( | ∇ I | ) L − ( 1 − g ( | ∇ I | ) ) ∗ ( L − C ∗ L ) (12)

In the LIF model, the fitting center is represented by the gray mean of images inside and outside the evolution curve. This method has poor robustness to noise points, and when there are many noise points in the image, the gray mean cannot accurately reflect the change of gray level of the image. In order to solve this problem, this paper uses the median value of grayscale of images inside and outside the evolution curve instead. When there are noise points in the image, the median value can guarantee the accuracy of the fitting center and better reflect the gray level change of the image area. The fitting center can be defined as:

{ f 1 ( x ) = m e d ( I 0 ( x ) ∈ ( { x ∈ Ω | ϕ < 0 } ∩ W k ( x ) ) ) f 2 ( x ) = m e d ( I 0 ( x ) ∈ ( { x ∈ Ω | ϕ > 0 } ∩ W k ( x ) ) ) (13)

In Equation (13), med() is the median operator. Because the noise point belongs to the mutation pixel point, it will have a great influence on the result when calculating with the mean value. However, the calculation of median does not need to calculate the specific values of all pixel points, so the median is more robust to noise points.

In

Next, we propose the following model:

E = ω ∫ Ω L O H ε ( ϕ ) d x + 1 2 ∫ Ω | I − I n | 2 d x (14)

I n = f 1 H ε ( ϕ ) + f 2 ( 1 − H ε ( ϕ ) ) (15)

where f 1 , f 2 are calculated in Equation (13), L O is calculated by Equation (16).

L n + 1 = L n + Δ t 1 ( g ( | ∇ I | ) L n − ( 1 − g ( | ∇ I | ) ) ∗ ( L n − C ∗ L n ) ) (16)

In order to make the curve evolve more stably and quickly, the length constraint term L ( ϕ ) and distance regularization term P ( ϕ ) are introduced into the model, L ( ϕ ) keeps the curve smooth, and P ( ϕ ) keeps the steady evolution of the level set function without the need for re-initialization.

L ( ϕ ) = μ ∫ Ω δ ε ( ϕ ) | ∇ ϕ | d x d y (17)

P ( ϕ ) = ∫ Ω 1 2 ( | ∇ ϕ | − 1 ) 2 d x d y (18)

The final evolution equation is as follows:

∂ ϕ ∂ t = δ ε ( ϕ ) ( I − I n ) ( f 1 − f 2 ) + ω δ ε ( ϕ ) ∗ L O + u δ ε ( ϕ ) d i v ( ∇ ϕ | ∇ ϕ | ) + v ( ∇ 2 ϕ − d i v ( ∇ ϕ | ∇ ϕ | ) ) (19)

In this paper, the finite difference method is used to solve the differential Equation (19), the central difference method is used to approximate the partial derivatives in the spatial domain, and the forward difference method is used to approximate the partial derivatives in the time domain. The discrete form of Equation (19) is:

ϕ i , j n + 1 = ϕ i , j n + Δ t ( A i , j + B i , j + C i , j ) (20)

where A i , j , B i , j and C i , j are calculated as follows:

{ A i , j = δ ε ( ϕ ) ∗ ( [ I − f 1 H ε ( ϕ ) − f 2 ( 1 − H ε ( ϕ ) ) ] ( f 1 − f 2 ) ) B i , j = ω δ ε ( ϕ ) L O C i , j = u δ ε ( ϕ ) ∇ ( ∇ ϕ i , j | ∇ ϕ i , j | ) + v ( ∇ 2 ϕ i , j − d i v ( ∇ ϕ i , j | ∇ ϕ i , j | ) ) (21)

The segmentation steps of the above model can be summarized as the following steps:

Step 1: Initialize ϕ ( x , y , t = 0 ) = 0 , and set parameters.

Step 2: Calculate L O according to Equation (16), Calculate f 1 , f 2 according to Equation (13).

Step 3: Update ϕ according to Equation (20).

Step 4: Judge whether the curve evolution is stable. If so, stop iteration and update the contour to obtain segmentation results. If you are not satisfied, go to Step 2.

In addition, we used precision rate P, recall rate R and DICE Similarity Coefficient DSC [

{ P = N ( S g ∩ S m ) N ( S m ) R = N ( S g ∩ S m ) N ( S g ) D S C = 2 N ( S g ∩ S m ) N ( S g ) ∪ N ( S m ) (22)

where S g represents the ground truth, and S m represents the area where the model solves. The closer the value of P is to 1, the better the image split. At P = 1 , it is stated that the solved area is the same as the real area, and the split result is the best. The same is true for R and DSC, where only approximately close to 1 represents the better segmentation effect.

In order to verify the validity of this algorithm, this section gives the model mentioned in this article and the image segmentation experimental results of this model. In this model, ϕ 0 is set to a small constant function ϕ 0 = c 0 . When calculating L O , Δ t 1 = 0.05 , C = 5 , number of iterations n = 100 . When evolving the curve ϕ : c 0 = 1 ; u = 0.001 × 255 × 255 ; v = 2 ; ε = 3 ; Δ t = 0.1 , ω = 15 . The window size of K σ is k = 3 , and the standard deviation is σ = 2 . These parameters can be adjusted for different images.

The four original images in

In order to objectively evaluate the segmentation quality of each model, we present the ground truth of four images, as shown in

Image | iterations | P | R | DSC |
---|---|---|---|---|

A | 10 | 0.9984 | 0.9579 | 0.9777 |

B | 6 | 0.9870 | 0.9881 | 0.9875 |

C | 15 | 0.9874 | 0.9989 | 0.9931 |

D | 20 | 0.9159 | 0.9813 | 0.9475 |

Image | RSF | LIF | LOGF | Our |
---|---|---|---|---|

A | 0.9212 | 0.8476 | 0.8335 | 0.3853 |

B | 0.7864 | 0.7338 | 0.6238 | 0.3255 |

C | 0.8442 | 0.8755 | 0.5432 | 0.5645 |

D | 0.9275 | 0.6834 | 0.8532 | 0.5853 |

In conclusion, the proposed model can better balance the effect of image segmentation and the efficiency of image segmentation, and compared with the other three models, the proposed model can accurately perform image segmentation while meeting the requirements of image segmentation efficiency. According to the final segmentation result, the required running time and the number of iterations, it can be seen that the model in this paper has good segmentation effect and unique advantages in the accuracy and speed of the algorithm.

In view of the traditional active contour, model in the zone of partial information is sensitive to initial contour, segmentation and segmentation accuracy rate needs to be improved; this paper puts forward an improved local area active contouring model, which uses the median fitting center to improve the robustness of noise, at the same time, combined with the optimization of the Log function, enhances the image edge information. Compared with RSF, LIF and LOGF, the methods of segmentation accuracy, segmentation effect, calculation speed and noise resistance are better.

The authors declare no conflicts of interest regarding the publication of this paper.

Chen, W.Q., Liu, C.J. and Pan, B. (2021) An Improved Active Contour Model Based on Local Information. Open Access Library Journal, 8: e7187. https://doi.org/10.4236/oalib.1107187