Summary:
The mechanical model of the magnetorheological damper is the foundation of the structure control of the magnetorheological damper. Aiming at the Bingham magnetorheological damper mechanical model, the shear strain rate is fractionally differentiated, a fractional order Bingham magnetorheological damper mechanical model is proposed, and this model is introduced into the image denoising to derive a fraction Order Bingham image denoising model, and define a fractional order edge detection operator and numerical solution, and finally used for denoising synthetic image and Lena image. MATLAB experiment results show that the algorithm can not only keep the image target edge, but also achieve image denoising.
0 Preface
Magnetorheological damper is a kind of vibration damping control device with a wide application prospect manufactured by the rapid and reversible rheological characteristics of magnetorheological fluid under strong magnetic field [1]. Due to the nonlinearity of the magnetorheological damper, the establishment of a more accurate dynamic model of the damper is one of the key factors for developing and using the damper to obtain a good control effect [2]. The hydromechanical properties of magnetorheological fluids are affected by many factors such as external magnetic fields. The constitutive relationship is complex, and it is difficult to establish an accurate mathematical model. At present, there is no unified magnetorheological damper mechanical model [3].
The Bingham model is a commonly used plastic hydrodynamic model of magnetorheological dampers [4]. The Herschel-Bulkley model is an improved nonlinear Bingham model with a power-law constitutive relationship, and its integer-order derivative model requires the introduction of multiple derivative terms and material parameters [5]. Fractional derivatives can well describe complex nonlinear natural and social phenomena with power laws [6]. To this end, this paper combines the fractional differential with the Bingham model to obtain the fractional Bingham model, describing the dynamic process of the magnetorheological damper.
Image denoising is an important step in image preprocessing [7], and image denoising based on partial differential equations (PDE) is a research hotspot in image analysis and processing [8]. In this paper, a fractional order Bingham PDE image denoising model is derived from the fractional order Bingham model, and a fractional order edge detection operator is defined for image denoising. The test results prove the effectiveness of the algorithm.
1 Fractional calculus and Bingham model
1.1 Definition of fractional calculus
Fractional calculus is an important branch of mathematical analysis. At present, there are many definition expressions of fractional calculus [9], among which there are three classic time domain expressions, including Grümwald-Letnikov, Riemann-Liouville and Capotu definitions , The three definitions are equivalent under certain conditions. The definition of fractional calculus is:
In the formula, v is the fractional order, G represents the definition of GL, the subscripts a and t represent the lower and upper bounds of the integral formula, and a is the initial value of time t.
1.2 Bingham model
The Bingham model is one of the mechanical models commonly used in magnetorheological dampers. The relationship between stress and strain is:
1.3 Herschel-Bulkley model
The Herschel-Bulkley model is an improved Bingham model that can explain the shear thinning and shear thickening behavior of magnetorheological fluids after yielding. The model expression is:
2 Fractional Bingham magnetorheological damper mechanical model
The Herschel-Bulkley model is an integer derivative model. Equation (3) has power-law characteristics. Its stress and strain responses depend on time and strain rate, are related to load and deformation history, and have memory. Compared with the Bingham model, it introduces multiple derivative terms and material parameters to describe the dynamic characteristics of the magnetorheological fluid more accurately. For the power rate characteristics of the model, Gemant [4] suggested that the fractional order time derivative be introduced into the constitutive relationship of the material, which can accurately describe the dynamic characteristics of a large number of complex memory materials with fewer material parameters. . Therefore, the fractional order of equation (1) is introduced into equation (2), and a mechanical model of fractional order Bingham magnetorheological damper is proposed, as shown in equation (4).
3 Fractional Bingham image denoising model
3.1 Image denoising model
Let u(0)=u0 be the initial noisy image, and obtain the PDE image denoising iterative model:
3.2 Numerical solution
The numerical solution uses the finite difference method. Think of the image as a vector [1×MN], the vector element ui is the pixel gray value, i ∈ [1,..., MN], h represents the discrete grid size, tk=τk is the discrete time, where k is a positive integer, τ is the time step. Differentiate the image in the time domain:
The fractional order differential mask operator [11], the Tainsi operator, is obtained, as shown in Figure 1. Using the Tainsi operator to perform fractional order differential processing on equation (11), a fractional order edge detection operator is obtained.
4 Test results
4.1 Performance evaluation
There are two evaluation methods used to evaluate the image denoising results [12-13], namely the signal-to-noise ratio (SNR) and the average structural similarity (MSSIM). SNR image denoising evaluation uses equation (17):
In the formula, M is the number of local windows, and X and Y are reference standards and denoised images.
4.2 Results
De-noise the synthetic image and compare the processing results of the image denoising algorithm with different orders (0≤v≤1), as shown in Figure 2. The size of Figure 2(a) is 150×150; the SNR of the noise image Figure 2(b) is 7.966 5 and the noise σ=10; Figure 2(c)～(f) are the fractional order of 0, 0.3, 0.7, 1. Image denoising result with 40 iterations. It can be seen from the figure that when the order is 0.7, the image denoising effect is the best, both maintaining the edges and removing the noise. In equation (6), v1 and v2 take different values, and the fractional order Bingham image denoising model (7) is used to denoise Figure 2(b) to obtain SNR and MSSIM values, as shown in Table 1. When v1=v2=0.7, the SNR is the largest; when v1=v2=0.5, the MSSIM is the largest.
Denoise the Lena image, and compare the processing results of the image denoising algorithm with different orders (0≤v≤1), as shown in Figure 3. Figure 3(a) has a size of 256×256, the noise image 3(b) has an SNR of 12.653 1, and noise σ=10. Figure 3(c) to (f) show fractional orders of 0, 0.3, 0.7, and 1, respectively. , The image denoising result with 10 iterations. It can be seen from the figure that when the order is 0.3, the image denoising effect is the best, both maintaining the edges and removing the noise. In formula (6), v1 and v2 take different values, and use the fractional Bingham image denoising model (7) to denoise Figure 3(b) to obtain SNR and MSSIM values, as shown in Table 2. When v1=0.2 and v2=0.3, the SNR is the largest; when v1=v2=0.4, the MSSIM is the largest.
4.3 Algorithm analysis
Extract the pixel value of the 75th line in Figure 2, as shown in Figure 4, the figure includes the noise-free pixel value, the pixel value of the noise-containing image, and the pixel value of the denoising image of different orders (v=0, 0.7, 1). It can be seen from the figure that in the flat area, the pixel value is closer to the true value when v = 0.7, the pixel value oscillation amplitude is the largest when v = 1.0, and the denoising effect is the worst; in the edge area, the edge retention effect is when v = 1.0 The best, v=0.7, v=0 is the worst; comprehensive comparison, when v=0.7, the fractional Bingham mechanical model works best. It not only maintains the high-frequency edge information of the image, but also suppresses the low-frequency area noise. The phenomenon is consistent with the numerical performance of Table 1. In Table 1 and Table 2, the peak positions of SNR and MSSIM are close, and the numerical change trend of SNR and MSSIM is consistent, which proves that choosing the appropriate fractional order can indeed improve the image denoising effect of the Bingham mechanical model.
5 Conclusion
Image denoising is one of the most basic problems in image processing. It is an important goal of image denoising to achieve denoising while maintaining high-frequency information at the edge of the image. In this paper, using the Bingham magnetorheological damper mechanical model combined with the principle of fractional calculus, a fractional order magnetorheological damper mechanical image denoising model is proposed. MATLAB test results show that:
(1) According to the mechanical model of Bingham magnetorheological damper, it is effective to derive the fractional order Bingham image denoising model by fractional processing the shear strain rate.
(2) The defined fractional-order edge detection operator and fractional-order mask operator (Tainsi operator) are feasible, can effectively detect image edges, and have good anti-noise performance.
(3) By selecting the appropriate fractional order, the image denoising effect can be optimized, which can not only maintain the image edge information, but also achieve good image denoising.
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