# Construction of Robot Computer Image Segmentation Model Based on Partial Differential Equation

Authors: Quan Zhang
Journal: Journal of Sensors
 (2022)

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## Abstract

The application of partial differential equation in robot computer image smoothing and image segmentation based on curve evolution is studied to achieve the breakthrough and innovation of classical methods. First of all, the image is interpolated and enlarged, usually with double primary interpolation and double secondary interpolation. The robot computer image obtained by interpolation has a better effect on the smooth part of the image, but the edge becomes blurred. Then, the model is used to enhance the amplified image and remove the noise introduced during interpolation. In the process of PDE image segmentation, using the level set method, natural continuation cannot guarantee that the embedded function is always a signed distance function in the evolution process. The results show that compared with the classical fourth-order model, this model can well retain the details of the image, smooth bevels, and edges of the image. The proposed new model is better than the second-order model and the classical fourth-order model both in PSNR value and MAE value and in visual effect. Secondly, the model based on piecewise smooth Mumford-Shah not only expresses the original image better but also solves the problem of gradual image segmentation with gray value. Therefore, the robot computer image segmentation model based on partial differential equation can segment the image better.


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## Body

## 1. Introduction

Image segmentation, as its name suggests, generally refers to the technology and process of extracting meaningful or interesting features from images. With the help of image segmentation and other technologies, the features in the image are extracted, which makes image analysis and understanding possible (Figure1). It is a key step from image processing to image analysis. Using partial differential equation for image segmentation has strong theoretical support and is consistent with the classical image theory. Level set method is a method to solve the curve evolution. It implicitly expresses the closed curve of the plane; that is, the evolution curve is expressed as a partial differential equation with embedded function, which avoids the tracking and parameterization of the curve in the evolution process. Therefore, the level set method is relatively easy to deal with the changes of curve topology, with high calculation accuracy and stable algorithm [1].Figure 1
Image segmentation model.In the development of image processing, mathematics always plays an important role and permeates all branches of image processing. In the 1960s and 1970s, linear processing methods represented by Fourier analysis occupied the whole field of digital image processing. With the help of stochastic process theory, people have established image model. The frame of image coding is established by probability theory and information theory. Linear filter (Wiener filter, Kalman filter) provides a strong theoretical support for lower level image processing; FFT is widely used in all branches of image processing. These mathematical tools greatly promote the development and application of image processing. Since the 1980s, nonlinear science has gradually penetrated into image processing methods, and many novel mathematical tools have been introduced into the field of image processing, such as the application of wavelet and multiscale analysis in image segmentation, image filtering, edge detection, feature extraction, and analysis: the application of fractal in image coding and texture recognition [2]. In particular, image processing methods based on partial differential equations have become a continuous hotspot in recent years. Starting with the analysis of the mechanism of image noise reduction, combined with mathematical tools such as variational method, functional analysis, differential geometry, and projective geometry, it establishes an axiom system related to partial differential equations. It has been widely used in image segmentation, image smoothing, image reconstruction, image recognition, remote sensing image processing, image analysis, edge detection, image interpolation, medical image processing, motion analysis, and so on. In many directions of image processing, image segmentation is one of the important research hotspots.This paper mainly studies the application of partial differential equation in image smoothing based on anisotropic diffusion and image segmentation based on curve evolution, trying to achieve a breakthrough and innovation of classical methods by deeply analyzing image features and using appropriate mathematical tools to obtain a more appropriate description of the image. Some new methods and ideas are proposed in this paper. Experiments and theoretical analysis have proved that the new method has obvious advantages over the classical method. It is an important supplement to the curve evolution and anisotropic diffusion theory and can be applied to all aspects of image processing engineering [3].


## 2. Literature Review

Nakamura and others believe that image segmentation is to segment the regions of interest in the image. Image segmentation is the basis of target recognition, tracking, and object-based coding. Therefore, image denoising has a very important application in the field of image processing. However, image segmentation is a very challenging task [4]. Zhao and others found that the storage and transmission of digital images are the main application direction of image compression. People’s demand for high compression ratio of image and high quality of restored image is endless. The combination of partial differential equation and wavelet decomposition can effectively eliminate the Gibbs phenomenon at the edge in the process of wavelet decomposition, which provides a new idea for high-quality image compression [5]. Weng and others applied discrete wavelet, wavelet packet, and wavelet frame decomposition to texture analysis and proposed methods for multiscale texture feature extraction, texture classification, and texture segmentation in wavelet or wavelet packet transform domain [6]. Liu and others found that image segmentation algorithms based on partial differential equations are mainly divided into two categories: image edge-based segmentation model and image region-based segmentation model. Snake model, geodesic active contour (GAC) model, and gradient vector flow (GVF) model are edge-based segmentation models. Snake model is a variational model, which is divided into internal force and external force. Internal force is a contraction force, which minimizes the arc length of the contour and reduces the oscillation of the arc length. The external force is to control the contour on the edge of the image, so as to achieve the effect of segmentation [7]. Tu and others found that the representatives of region-based segmentation model are M-S model and C-V model, which approach the original image in the form of segmented plane, so as to achieve the purpose of segmentation [8]. Ren and others found that the storage and transmission of digital images are the main application direction of image compression. People’s demand for high compression ratio of image and high quality of restored image is endless. The combination of partial differential equation and wavelet decomposition can effectively eliminate the Gibbs phenomenon at the edge in the process of wavelet decomposition, which provides a new idea for high-quality image compression [9]. Ma and others believe that for the numerical solution of the segmentation model, the commonly used methods are the level set method and the variational level set square, which represent the contour through the zero level set of an embedded function. The embedding distance of the gradient function should meet the requirements of 1. However, with the evolution of the embedded function, the gradient modulus will deviate from 1, so it is reinitialized during the evolution of the embedded function. Reinitialization not only reduces the computational efficiency but also destroys the evolution of embedded functions. Therefore, a distance regularized level set evolution (DRLSE) method is generated [10].


## 3. Research Methods

### 3.1. Generation of Fourth-Order Denoising Model

In essence, the second-order model will evolve the image towards the segmented plane, so most of the second-order models have the “ladder” effect; that is, the smooth region in the image produces false boundaries, which is visually unpleasant and may make the computer vision system mistakenly recognize some edges. The fourth-order partial differential equation denoising model can avoid the “ladder” effect [11].The fourth-order partial differential equation denoising model can avoid the “ladder” effect because it approximates the original image through piecewise inclined plane. However, the classical fourth-order models start with variation. By improving the energy functional of the second-order model, the fourth-order partial differential equation is derived by using the gradient descent flow method.


### 3.2. A Kind of Fourth-Order Denoising Model Based on Image Features

Generally, an image can be divided into three types of regions: smooth region, edge region, and texture region. When the image is enlarged by the usual interpolation method, the edge of the image will be blurred [12]. The algorithm proposed in this section is as follows: firstly, the image is interpolated and enlarged to the desired size. Usually, double primary interpolation and double secondary interpolation are used. The smooth part of the image obtained by interpolation is better, but the edge becomes blurred. Then, use the model in this paper to enhance the enlarged image and remove the noise introduced during interpolation. The specific process is shown in Figure 2.Figure 2
Image resolution enhancement algorithm flow.Based on the properties of the fourth-order partial differential equation and making full use of the characteristic information of the image, this paper selects the edge detection operator as the diffusion coefficient of the fourth-order equation [13] and directly constructs a new fourth-order partial differential equation. The constructed fourth-order partial differential equation is
(1)∂u∂t=−Φ1uxxuxxxx−Φ2uyyuyyyy,x,y∈Ω,t∈0,T,∂u∂n⟶=0,Φ1uxxuxxxn1+Φ2uyyuyyyn2=0,x,y∈∂Ω,t∈0,T,u0,x,y=fx,y,x,y∈Ω.Here,n⟶=n1,n2 represents the normal vector outside the unit on the boundary ∂Ω of the solution region, and the diffusion coefficient Φi, i=1,2, is a function related to ∇u, ux, and uy, and Φi=Φi∇u,ux,uy is the edge detection function. The boundary conditions are selected as zero derivative in the first-order external normal direction and zero derivative in the third-order external normal direction [14].


### 3.3. Level Set Method

In the process of PDE image segmentation, using level set method, natural continuation cannot ensure that the embedded function always remains a signed distance function in the evolution process. Therefore, the traditional approach is to reinitialize the embedded function into a signed distance function in the evolution process of the embedded function. This greatly reduces the efficiency of the segmentation algorithm. In order to avoid reinitialization, a regular term is added to the energy functional of the embedded function to keep the embedded function as a distance function as much as possible in the evolution process [15].The geodesic active contour (GAC) model minimizes the following “weighted” arc length, as shown in(2)LRC=∫0LCg∇uCsds.Minimize the gradient downflow corresponding to the above equation as(3)∂C∂t=gCκN−∇g·NN.N is the normal direction of curve evolution. Different from the previous parametric active contour model, the biggest feature of this kind of model is that it can automatically deal with the changes of topology.The level set method transforms the original evolution of the curve into a partial differential equation of the level set function. In this way, the segmentation problem is transformed from the original differential geometry problem into the problem of solving partial differential equations. There are many existing results for the numerical solution of partial differential equations, and a better scheme can be selected for the discretization of the model [16].A curve is represented by the level set of high-dimensional function, as shown in(4)Ct=x,yϕx,y=k.The original curve evolution problem is replaced by the evolution problem of two-dimensional function. The time-varying curveC can be expressed as the level set of ϕx,y,t in the time-varying function, that is,
(5)Ct=x,yϕx,y,t=k,when curveCt is as follows:
(6)∂C∂t=V.During evolution, the embedded functionBx,y will evolve into
(7)∂ϕ∂t=β∇ϕ.Here,β=V∗N is the normal component of the velocity of motion. The above formula is the basic equation of curve evolution level set method.The choice of embedding functionϕ is not unique. The most commonly used is to let ϕx,y represent the signed distance function from point x,y on the plane to curve C, that is,
(8)ϕx,y=dx,y,C,∀x,y∈Cout,−dx,y,C,∀x,y∈Cin.The distance function satisfies(9)∇ϕ=1.This means that the change of embedded function is neither slow nor rapid, which meets the requirements of natural continuation. Assuming that the given curve motion equation is valid for all level sets of embedded functions, this natural continuation method is usually selected. However, with the evolution of the embedding function, the embedding function will gradually deviate from the distance function, and the deviation from the distance function will lead to the instability of numerical calculation [16, 17].The level set method has the following advantages:(1)
More than two edges can be detected, that is, with topological variability(2)
It is easy to express curvature and unit normal vectorN by embedded function(3)
The numerical calculation can be carried out on the standard grid, which is a parameter-free method(4)
It can be used in high-dimensional cases. For example, the surface can be expressed as a zero level set of volume functionIf the level set method given above is applied, the evolution equation corresponding to the embedding function of GAC model is(10)∂ϕ∂t&=gκ−∇g·N∇ϕ=∇ϕdivg∇ϕ∇ϕ.This is a Hamilton-Jacobi equation, which needs to be solved by the upwind scheme. Zhao and Chan proposed a variational level set method in 1996. The variational level set directly starts from the energy functional and introduces the Heaviside function to replace the functional about the contour with the functional of the embedded function. The definition of Heaviside function is shown in(11)Hz=1,z≥0,0,z<0.The derivative of Heaviside functionH in the sense of distribution is Dirac function δ. In the actual calculation process, it is necessary to regularize the Heaviside function for numerical solution. Hz is defined as the regularized Heaviside function, and it satisfies
(12)Hεz⟶ε⟶0Hz.Two commonly used regularized Heaviside functions are given below, in the form of(13)Hεz=1,z>ε,0,z<−ε,121+zε+1πsinπzε,z≤ε,(14)Hεz=121+2πarctanzε.Both regularization functions are odd symmetric.Hεz is continuous and differentiable. Its derivative is defined as
(15)δεz=dHεzdz.δεz here can be regarded as the regularization of δ. For the corresponding derivative δε, when the embedded function ϕ is used as the independent variable of function H, there is
(16)δεϕ=dHεϕdϕ,∇Hεϕ=δεϕ∇ϕ.By introducing the Heaviside function, the line integral is transformed into an area integral, as shown in(17)∮CgCds=∬Ωg∇Hεϕx,ydxdy=∬Ωδεϕ∇ϕgdxdy.The gradient downflow is obtained as(18)∂ϕ∂t=δεϕdivg∇ϕ∇ϕ.Through the above analysis and derivation, we can find that the type of PDE obtained by variational level set method is different from that obtained by level set method. The former is a parabolic equation and the latter is a hyperbolic equation. The difficulty in the process of numerical solution is different. Therefore, the variational level set method is a more widely used method [18, 19].


### 3.4. Numerical Scheme of Fourth-Order Regularized Level Set Method

In this paper, the numerical discretization of the fourth-order regularized level set method is given. The half-point center difference scheme is adopted. Based on the explicit scheme of the original segmentation model, the discretization of the fourth-order regular term is added. For the fourth-order regularization termf, the central difference explicit scheme is adopted, as shown in
(19)Fi,jn=−DxxΦ1φi,jnαεφi,jnDxxφi,jn+DyyΦ2φi,jnβεφi,jnDyyφi,jn.


## 3.1. Generation of Fourth-Order Denoising Model

In essence, the second-order model will evolve the image towards the segmented plane, so most of the second-order models have the “ladder” effect; that is, the smooth region in the image produces false boundaries, which is visually unpleasant and may make the computer vision system mistakenly recognize some edges. The fourth-order partial differential equation denoising model can avoid the “ladder” effect [11].The fourth-order partial differential equation denoising model can avoid the “ladder” effect because it approximates the original image through piecewise inclined plane. However, the classical fourth-order models start with variation. By improving the energy functional of the second-order model, the fourth-order partial differential equation is derived by using the gradient descent flow method.


## 3.2. A Kind of Fourth-Order Denoising Model Based on Image Features

Generally, an image can be divided into three types of regions: smooth region, edge region, and texture region. When the image is enlarged by the usual interpolation method, the edge of the image will be blurred [12]. The algorithm proposed in this section is as follows: firstly, the image is interpolated and enlarged to the desired size. Usually, double primary interpolation and double secondary interpolation are used. The smooth part of the image obtained by interpolation is better, but the edge becomes blurred. Then, use the model in this paper to enhance the enlarged image and remove the noise introduced during interpolation. The specific process is shown in Figure 2.Figure 2
Image resolution enhancement algorithm flow.Based on the properties of the fourth-order partial differential equation and making full use of the characteristic information of the image, this paper selects the edge detection operator as the diffusion coefficient of the fourth-order equation [13] and directly constructs a new fourth-order partial differential equation. The constructed fourth-order partial differential equation is
(1)∂u∂t=−Φ1uxxuxxxx−Φ2uyyuyyyy,x,y∈Ω,t∈0,T,∂u∂n⟶=0,Φ1uxxuxxxn1+Φ2uyyuyyyn2=0,x,y∈∂Ω,t∈0,T,u0,x,y=fx,y,x,y∈Ω.Here,n⟶=n1,n2 represents the normal vector outside the unit on the boundary ∂Ω of the solution region, and the diffusion coefficient Φi, i=1,2, is a function related to ∇u, ux, and uy, and Φi=Φi∇u,ux,uy is the edge detection function. The boundary conditions are selected as zero derivative in the first-order external normal direction and zero derivative in the third-order external normal direction [14].


## 3.3. Level Set Method

In the process of PDE image segmentation, using level set method, natural continuation cannot ensure that the embedded function always remains a signed distance function in the evolution process. Therefore, the traditional approach is to reinitialize the embedded function into a signed distance function in the evolution process of the embedded function. This greatly reduces the efficiency of the segmentation algorithm. In order to avoid reinitialization, a regular term is added to the energy functional of the embedded function to keep the embedded function as a distance function as much as possible in the evolution process [15].The geodesic active contour (GAC) model minimizes the following “weighted” arc length, as shown in(2)LRC=∫0LCg∇uCsds.Minimize the gradient downflow corresponding to the above equation as(3)∂C∂t=gCκN−∇g·NN.N is the normal direction of curve evolution. Different from the previous parametric active contour model, the biggest feature of this kind of model is that it can automatically deal with the changes of topology.The level set method transforms the original evolution of the curve into a partial differential equation of the level set function. In this way, the segmentation problem is transformed from the original differential geometry problem into the problem of solving partial differential equations. There are many existing results for the numerical solution of partial differential equations, and a better scheme can be selected for the discretization of the model [16].A curve is represented by the level set of high-dimensional function, as shown in(4)Ct=x,yϕx,y=k.The original curve evolution problem is replaced by the evolution problem of two-dimensional function. The time-varying curveC can be expressed as the level set of ϕx,y,t in the time-varying function, that is,
(5)Ct=x,yϕx,y,t=k,when curveCt is as follows:
(6)∂C∂t=V.During evolution, the embedded functionBx,y will evolve into
(7)∂ϕ∂t=β∇ϕ.Here,β=V∗N is the normal component of the velocity of motion. The above formula is the basic equation of curve evolution level set method.The choice of embedding functionϕ is not unique. The most commonly used is to let ϕx,y represent the signed distance function from point x,y on the plane to curve C, that is,
(8)ϕx,y=dx,y,C,∀x,y∈Cout,−dx,y,C,∀x,y∈Cin.The distance function satisfies(9)∇ϕ=1.This means that the change of embedded function is neither slow nor rapid, which meets the requirements of natural continuation. Assuming that the given curve motion equation is valid for all level sets of embedded functions, this natural continuation method is usually selected. However, with the evolution of the embedding function, the embedding function will gradually deviate from the distance function, and the deviation from the distance function will lead to the instability of numerical calculation [16, 17].The level set method has the following advantages:(1)
More than two edges can be detected, that is, with topological variability(2)
It is easy to express curvature and unit normal vectorN by embedded function(3)
The numerical calculation can be carried out on the standard grid, which is a parameter-free method(4)
It can be used in high-dimensional cases. For example, the surface can be expressed as a zero level set of volume functionIf the level set method given above is applied, the evolution equation corresponding to the embedding function of GAC model is(10)∂ϕ∂t&=gκ−∇g·N∇ϕ=∇ϕdivg∇ϕ∇ϕ.This is a Hamilton-Jacobi equation, which needs to be solved by the upwind scheme. Zhao and Chan proposed a variational level set method in 1996. The variational level set directly starts from the energy functional and introduces the Heaviside function to replace the functional about the contour with the functional of the embedded function. The definition of Heaviside function is shown in(11)Hz=1,z≥0,0,z<0.The derivative of Heaviside functionH in the sense of distribution is Dirac function δ. In the actual calculation process, it is necessary to regularize the Heaviside function for numerical solution. Hz is defined as the regularized Heaviside function, and it satisfies
(12)Hεz⟶ε⟶0Hz.Two commonly used regularized Heaviside functions are given below, in the form of(13)Hεz=1,z>ε,0,z<−ε,121+zε+1πsinπzε,z≤ε,(14)Hεz=121+2πarctanzε.Both regularization functions are odd symmetric.Hεz is continuous and differentiable. Its derivative is defined as
(15)δεz=dHεzdz.δεz here can be regarded as the regularization of δ. For the corresponding derivative δε, when the embedded function ϕ is used as the independent variable of function H, there is
(16)δεϕ=dHεϕdϕ,∇Hεϕ=δεϕ∇ϕ.By introducing the Heaviside function, the line integral is transformed into an area integral, as shown in(17)∮CgCds=∬Ωg∇Hεϕx,ydxdy=∬Ωδεϕ∇ϕgdxdy.The gradient downflow is obtained as(18)∂ϕ∂t=δεϕdivg∇ϕ∇ϕ.Through the above analysis and derivation, we can find that the type of PDE obtained by variational level set method is different from that obtained by level set method. The former is a parabolic equation and the latter is a hyperbolic equation. The difficulty in the process of numerical solution is different. Therefore, the variational level set method is a more widely used method [18, 19].


## 3.4. Numerical Scheme of Fourth-Order Regularized Level Set Method

In this paper, the numerical discretization of the fourth-order regularized level set method is given. The half-point center difference scheme is adopted. Based on the explicit scheme of the original segmentation model, the discretization of the fourth-order regular term is added. For the fourth-order regularization termf, the central difference explicit scheme is adopted, as shown in
(19)Fi,jn=−DxxΦ1φi,jnαεφi,jnDxxφi,jn+DyyΦ2φi,jnβεφi,jnDyyφi,jn.


## 4. Result Analysis

### 4.1. Numerical Experiment and Result Analysis of Fourth-Order Denoising Model

This paper applies the algorithm proposed in this paper to denoise different noisy images and compares the experimental results with the classical second-order model (TV, regularized PM) and the classical fourth-order model (Y-K, LLT). Experiments were carried out on three different images: GZC01.bmp: segmented bevel image (128×128 pixels), lena.bmp (512×512 pixels), and pepper.bmp (256×256 pixels). The standard deviation is 10, 20, and 30. The numerical format proposed above is realized by MATLAB, and the experimental results are shown in Table 1.Table 1
Experimental PSNR, MAE, and CPU time.
σPSNRMAECPU time (s)102030102030102030Segmented bevel image GZC01.bmp (128×128 pixels)Second-order model28.2022.1318.707.8615.9423.67TV41.4237.6334.051.482.613.460.771.342.32Regularized PM40.9536.6733.251.402.453.660.260.290.27New model47.6342.2337.560.711.302.004.384.848.83lena.bmp (512×512 pixels)Second-order model28.1322.1318.707.9715.9223.74LLT34.2831.0329.253.675.086.1820.0555.6097.76Y-K32.9929.7928.114.075.616.7524.69101.17242.22New model34.4231.4029.643.534.765.7241.63109.33191.08pepper.bmp (256×256 pixels)Second-order model28.1822.1618.767.9415.8423.45LLT33.9630.0827.873.785.707.273.389.8318.89Y-K32.4928.6226.324.246.378.505.9419.1542.47New model34.4630.7628.563.455.146.519.7118.5147.49The denoising ratio of absolute noise to noise and the denoising ratio of absolute noise to noise (noise, noise, noise, noise) of smooth image are obtained.For the noise-free smooth imageu0 and the smooth image u obtained by denoising, the indicators to measure the denoising effect are peak signal to noise ratio (PSNR) and mean absolute-deviation error (MAE).Compared with the classical fourth-order model, it shows that this model can well retain the detailed information of the image, the smooth slope, and the edge of the image. The new model proposed in this paper is better than the second-order model and the classical fourth-order model in terms of PSNR value, MAE value, and visual effect [20].From the data in the above table, it can be found that the PSNR value of the new model is much larger than that of other models, indicating that the fourth-order denoising model proposed in this paper can better retain the structure information of the image [21].


### 4.2. Segmentation Experiment of Fourth-Order Regularized Level Set Method

Figure3 quantitatively analyzes and compares the edge positioning accuracy and noise resistance of the edge detection method, Canny edge detection method, and log edge detection method proposed in this paper for a synthetic image [22].Figure 3
Comparison of image edge positioning accuracy of different detection methods.This paper presents the experimental results of several groups of segmentation models, as shown in Figures4 and 5. The initial conditions are selected as binary functions [23].Figure 4
Segmentation accuracy.Figure 5
Time spent in segmentation.The model is based on piecewise smooth Mumford-Shah, which not only better expresses the original image but also solves the problem of gray value progressive image segmentation. Secondly, the new method has the characteristics of global optimization and has no special requirements for the position of the initial contour line. Even if it is far away from the edge of the initial closed contour line, it can be segmented accurately. Third, this method does not rely on edge information. For fuzzy or discrete edges, it can still obtain ideal segmentation effect [24, 25]. This experiment also reviews the typical methods of vector image segmentation based on active contour and extends the method in this paper to segment vector image. The new vector image segmentation method maintains many advantages of this method in gray image segmentation and can find some information that is difficult to obtain in single channel image from vector image.


## 4.1. Numerical Experiment and Result Analysis of Fourth-Order Denoising Model

This paper applies the algorithm proposed in this paper to denoise different noisy images and compares the experimental results with the classical second-order model (TV, regularized PM) and the classical fourth-order model (Y-K, LLT). Experiments were carried out on three different images: GZC01.bmp: segmented bevel image (128×128 pixels), lena.bmp (512×512 pixels), and pepper.bmp (256×256 pixels). The standard deviation is 10, 20, and 30. The numerical format proposed above is realized by MATLAB, and the experimental results are shown in Table 1.Table 1
Experimental PSNR, MAE, and CPU time.
σPSNRMAECPU time (s)102030102030102030Segmented bevel image GZC01.bmp (128×128 pixels)Second-order model28.2022.1318.707.8615.9423.67TV41.4237.6334.051.482.613.460.771.342.32Regularized PM40.9536.6733.251.402.453.660.260.290.27New model47.6342.2337.560.711.302.004.384.848.83lena.bmp (512×512 pixels)Second-order model28.1322.1318.707.9715.9223.74LLT34.2831.0329.253.675.086.1820.0555.6097.76Y-K32.9929.7928.114.075.616.7524.69101.17242.22New model34.4231.4029.643.534.765.7241.63109.33191.08pepper.bmp (256×256 pixels)Second-order model28.1822.1618.767.9415.8423.45LLT33.9630.0827.873.785.707.273.389.8318.89Y-K32.4928.6226.324.246.378.505.9419.1542.47New model34.4630.7628.563.455.146.519.7118.5147.49The denoising ratio of absolute noise to noise and the denoising ratio of absolute noise to noise (noise, noise, noise, noise) of smooth image are obtained.For the noise-free smooth imageu0 and the smooth image u obtained by denoising, the indicators to measure the denoising effect are peak signal to noise ratio (PSNR) and mean absolute-deviation error (MAE).Compared with the classical fourth-order model, it shows that this model can well retain the detailed information of the image, the smooth slope, and the edge of the image. The new model proposed in this paper is better than the second-order model and the classical fourth-order model in terms of PSNR value, MAE value, and visual effect [20].From the data in the above table, it can be found that the PSNR value of the new model is much larger than that of other models, indicating that the fourth-order denoising model proposed in this paper can better retain the structure information of the image [21].


## 4.2. Segmentation Experiment of Fourth-Order Regularized Level Set Method

Figure3 quantitatively analyzes and compares the edge positioning accuracy and noise resistance of the edge detection method, Canny edge detection method, and log edge detection method proposed in this paper for a synthetic image [22].Figure 3
Comparison of image edge positioning accuracy of different detection methods.This paper presents the experimental results of several groups of segmentation models, as shown in Figures4 and 5. The initial conditions are selected as binary functions [23].Figure 4
Segmentation accuracy.Figure 5
Time spent in segmentation.The model is based on piecewise smooth Mumford-Shah, which not only better expresses the original image but also solves the problem of gray value progressive image segmentation. Secondly, the new method has the characteristics of global optimization and has no special requirements for the position of the initial contour line. Even if it is far away from the edge of the initial closed contour line, it can be segmented accurately. Third, this method does not rely on edge information. For fuzzy or discrete edges, it can still obtain ideal segmentation effect [24, 25]. This experiment also reviews the typical methods of vector image segmentation based on active contour and extends the method in this paper to segment vector image. The new vector image segmentation method maintains many advantages of this method in gray image segmentation and can find some information that is difficult to obtain in single channel image from vector image.


## 5. Conclusion

Based on the modulus of directional curvature, this paper proposes a functional to describe the smoothness of image and deduces a fourth-order partial differential equation (PDE) image denoising model. The processing result is piecewise linear image (including linear image), and there is a step in the gradient at the edge. Taking advantage of this feature, this paper proposes a new geodesic active contour (new GAC) model, which improves the contour extraction performance of traditional GAC and is much faster. It is worth paying full attention to the new derivation of new GAC model. The region-based active contour model has certain robustness to noise, but the processing of strong noise image is still limited. The noise reduction model in this paper also greatly improves the segmentation effect of region based active contour model.In short, variational methods and partial differential equations are widely used in image processing. They are a new and developing discipline. Many image processing problems can be expressed by variational models and differential equations. The research work can be deepened from two aspects: the improvement of model form and the efficiency and accuracy of algorithm implementation. We need to make more efforts to continuously improve the processing effect of model and improve the execution efficiency and accuracy of algorithm implementation.



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*Source: 6216423-2022-07-18.xml*