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Image denoising model based on improved fractional calculus mathematical equation

Pubblicato online: 15 Jul 2022
Volume & Edizione: AHEAD OF PRINT
Pagine: -
Ricevuto: 12 Feb 2022
Accettato: 11 Apr 2022
Dettagli della rivista
License
Formato
Rivista
eISSN
2444-8656
Prima pubblicazione
01 Jan 2016
Frequenza di pubblicazione
2 volte all'anno
Lingue
Inglese
Preface

With the rapid development of artificial intelligence technology, digital image has become an important source of information and knowledge, penetrating into all aspects of people's life. Computer vision extends people's ability to obtain and process information to machines, and image quality has become the most critical factor restricting the follow-up image processing system. In the real environment, due to the influence of equipment, system and other factors, most images will have varying degrees of noise, which will have a great adverse impact on the subsequent processing work. The problem of image denoising is to retain the effective feature information in the original image as much as possible and remove the noise contained in the image, so as to obtain a high-quality image. At present, the image denoising method based on PDE model is widely studied, but it has the disadvantages of low efficiency and difficult to remove multiplicative noise. Based on the PDE denoising model, an improved image denoising model based on fractional calculus equation is proposed to realize the rapid removal of image multiplicative noise.

Image denoising model based on integer order partial differential
Image noise classification

In the process of image generation, image noise is generated due to the influence of equipment circuit design, manufacturing process, external electromagnetic interference, temperature change and other factors.

According to the relationship between noisy image f, original image u and noise n, noise is divided into additive noise and multiplicative noise.[1]

Additive noise: F = u + n, indicating the additive and independent relationship between the signal of the original image and the noise signal. In the real scene, the category of additive noise is encountered most. Since the noise is independent of the original image, the noise will exist whether there is an input image or not. The typical additive noise is Gaussian noise, and the probability density function is as follows: p(x)=12πσe[(xμ)22σ2] p\left( x \right) = {1 \over {\sqrt {2\pi } \sigma }}{e^{\left[ {{{{{\left( {x - \mu } \right)}^2}} \over {2{\sigma ^2}}}} \right]}} Among μ represents the expected value of X, σ Represents variance.

Multiplicative noise: F = u * n, indicating that the signal of the original image is multiplied by the noise signal and has a close relationship. When there is no image input, the noise does not exist, and the multiplicative noise is coupled to the input image[2]. The typical multiplicative noise is Rayleigh noise, and the probability division function is as follows: p(x)=2b(xa)e[(xa)2b](xa) p\left( x \right) = {2 \over b}\left( {x - a} \right){e^{\left[ {{{{{\left( {x - a} \right)}^2}} \over b}} \right]}}\,\left( {x \ge a} \right) Where: when x < a, P (x) = 0, the expected value of X is μ=α+πb/4 \mu = \alpha + \sqrt {\pi b/4} , variance is σ2 = b(4 − π)/4.

image denoising model based on PDE

In 1990, the nonlinear diffusion P-M model for additive noise removal proposed by Perona and Malik needs to introduce the gradient information of the image into the diffusion coefficient in order to maintain the edge while denoising.[3] However, due to the negative characteristic of the diffusion coefficient in a certain direction, the model is required to be regular in space and time. Its model is: ut=div(11+x2/α2u) {u_t} = div\left( {{1 \over {1 + {x^2}/{\alpha ^2}}}\nabla u} \right) Where: Among them, a is a constant greater than 0 and ∇ is the gradient.

P-M model can only solve the problem of additive noise, and it is inefficient and can not deal with the problem of multiplicative noise. The image noise is removed by variational method, and the problem of multiplicative noise is solved. The solution of variational evolution equation is transformed into the solution of partial differential equation PDE model. The model mainly considers the gradient information, curvature information and diffusion direction of the image, and can remove the multiplicative noise of the image. Later, the gray value is introduced into the model to remove the multiplicative noise. Its model is: ut=div(2|u|αsupxu(x)+|u|α×u(1+|u|2)1β2) {u_t} = div\left( {{{2{{\left| u \right|}^\alpha }} \over {{{\sup }_x}\,u\left( x \right) + {{\left| u \right|}^\alpha }}} \times {{\nabla u} \over {{{\left( {1 + {{\left| {\nabla u} \right|}^2}} \right)}^{{{1 - \beta } \over 2}}}}}} \right) Where: a>0, β∈[0,1]. The model combines the gradient value and gray value of the image and introduces it into the diffusion coefficient to improve the denoising effect of the image.[4] The above model is an integer order PDE denoising model, which can not be used for noise removal in texture images.

Improved fractional calculus image denoising model
Fractional calculus

In order to overcome the shortcomings of integer order calculus PDE model, the processing of image denoising is shown in mathematical expression (formula 7) by combining the basic transformation theory of Fourier series (formula 5) with fractional order variational model (formula 6). f(x)=a0+n=1(ancosnπxL+bnsinnπxL) f\left( x \right) = {a_0} + \sum\limits_{n = 1}^\infty {\left( {{a_n}\cos {{n\pi x} \over L} + {b_n}\sin {{n\pi x} \over L}} \right)} minΩu(uf)2dxdy+Ωu|u|dxdy \min \int_\Omega ^u {{{\left( {u - f} \right)}^2}dxdy + \int_\Omega ^u {\left| {\nabla u} \right|dxdy} } |ω|γe[γπi3floor(ω)]f(ω),γR {\left| \omega \right|^\gamma }{e^{\left[ {{{\gamma \pi i} \over 3}floor\left( \omega \right)} \right]}}f\left( \omega \right),\,\gamma \, \in \,R Among them, ω is the angular frequency, γ is γ differential operator, floor () stands for rounding down the value.

According to the formula, with the enhancement of fractional differentiation, the edge and texture features of the image are retained, which proves that fractional integration can eliminate image noise.[5]

Fractional calculus PDE model

When the second-order PDE model processes the image, it will produce “ladder effect”, so that the denoising effect of the image is not obvious. Y-K model, by introducing Laplace operator, solves the problem that the image edge cannot be preserved. However, when eliminating the noise in the smooth area, the image will have spots. Moreover, due to the large number of iterations, the denoising process of the whole image takes a long time. The mathematical expression of Y-K model is as follows: ut=Δ(e(sk)2Δu) {{\partial u} \over {\partial t}} = - \Delta \left( {{e^{ - {{\left( {{s \over k}} \right)}^2}}}\Delta u} \right) Where, Δ is Laplace operator.

Improved fourth-order PDE model

When removing the multiplicative noise of texture image, this paper introduces the fractional Fick law, substitutes the nonlinear diffusion coefficient into the conservation law equation, and realizes the removal of multiplicative noise by establishing the correlation space fractional diffusion equation model.[6] The nonlinear diffusion coefficient is composed of texture detection operator with fractional derivative and gray value detection operator. The mathematical model is as follows: Dx±αu(x,y,z)=1τ(mα)(±x)mxdu(a,y,z)(ax)α+1mda D_{x \pm }^\alpha u\left( {x,y,z} \right) = {1 \over {\tau \left( {m - \alpha } \right)}}{\left( { \pm {\partial \over {\partial x}}} \right)^m}\int_x^d {{{u\left( {a,\,y,\,z} \right)} \over {{{\left( {a - x} \right)}^{\alpha + 1 - m}}}}da} Dy±αu(x,y,z)=1τ(mα)(±y)mxdu(x,b,z)(by)α+1mdb D_{y \pm }^\alpha u\left( {x,y,z} \right) = {1 \over {\tau \left( {m - \alpha } \right)}}{\left( { \pm {\partial \over {\partial y}}} \right)^m}\int_x^d {{{u\left( {x,\,b,\,z} \right)} \over {{{\left( {b - y} \right)}^{\alpha + 1 - m}}}}db}

In the fractional order operation, under the condition that the effect of image denoising is not affected, only the diffusion flow of the formula needs to be considered without defining the adjoint operator, which can effectively avoid complex operation and improve the efficiency of operation. To analyze the fractional derivative, you only need to α Taking different values, we can solve various models.

In order to solve the shortcomings of the traditional PDE model, aiming at the speckle phenomenon when eliminating the noise in the smooth area, we use the smooth term |sm|2+β2 \sqrt {{{\left| {sm} \right|}^2} + {\beta ^2}} instead of the non smooth term |sm|, the edge structure can be kept clear in the process of image denoising.[7] However, when solving, the traditional algorithm is very cumbersome and the convergence speed is slow. In order to improve the operation speed, the alternative direction method of multipliers (ADMM) algorithm is used. The idea of the algorithm is to first introduce auxiliary variables to replace the variables that are difficult to solve in the original problem, and then convert the target problem into an unconstrained extreme value problem for solution. In the process of solving, The unconstrained problem is decomposed into several solvable subproblems, which are solved alternately until the results are obtained.

The constrained problem is transformed into an unconstrained problem, and the smooth term is adopted. The Lagrange function obtained is as follows: g(u)=αH(sm|sm|2+β2)+γ1T(du)+ρ1du2+γ2T(zu)+ρ2zu2 g\left( u \right) = \alpha H \bullet \left( {{{sm} \over {\sqrt {{{\left| {sm} \right|}^2} + {\beta ^2}} }}} \right) + \gamma _1^T\left( {d - \nabla u} \right) + {\rho _1}{\left\| {d - \nabla u} \right\|^2} + \gamma _2^T\left( {z - u} \right) + {\rho _2}{\left\| {z - u} \right\|^2} Among them, γ1 and γ2 is the operator of Lagrange function, ρ1 and ρ2 is the penalty parameter, and the required value is greater than 0.

Simulation experiment and result analysis

Under the windows10 system platform, Intel i5-5200u CPU and 16G memory are configured. Matlab R12 is used to complete the image denoising simulation experiment for two 256 * 256 scene and map images with different textures under unknown noise level. The original images of the two pictures are shown below.

Figure 1

original image

Visual analysis

P-M model, PDE model and improved PDE model are used to denoise the image, and the advantages and disadvantages of several models are compared. First, determine the parameters a, γ1, γ2, ρ1 and ρ2. The optimal range value of relevant parameters such as 2 is set through numerical experiments a= 3, γ1= 0.71, γ2= 0.52, ρ1 = 0.3 and ρ2=0.5. Set the iteration times of the three models to 1000, denoise the images respectively, and visually analyze and compare the images obtained by the three models. As shown in Image 2 and 3.

Figure 2

denoising effect of different models on map image

Figure 3

denoising effect of different algorithms on texture image

According to the visual observation results of the image, the denoising effect of P-M model is the worst, and the image denoising effect of the improved fractional PDE model is significantly better than other models, showing the best image quality.

quantitative analysis

The evaluation of image denoising quality mainly includes fidelity and intelligibility. Fidelity can be judged by visual observation, and it can be observed by the naked eye. There may be some subjective factors in the process of evaluation. Some small differences need to be reflected by specific values. Peak signal to noise ratio (PSNR) is the most widely used quantitative evaluation index of image denoising quality at present. It can more intuitively measure the deviation between the test image and the reference image through specific values, so as to give the observer a clearer reference index. In addition, the low efficiency of different image denoising models has also been criticized to varying degrees in the industry. Comparing the running time of different models can reflect the advantages and disadvantages of different models to some extent.

Assuming that the calculation formula of PSNR of reference image X and test image y with size m * n is: PSNR(x,y)=10lg(2552m×nxy2) PSNR\left( {x,\,y} \right) = 10\;{\rm{lg}}\left( {{{{{255}^2}m \times n} \over {{{\left\| {x - y} \right\|}^2}}}} \right)

The image denoising of the algorithm is analyzed quantitatively. The PSNR value and running time of image denoising of the three models are shown in Table 1 and Table 2 respectively.

PSNR values of different models after denoising

P-M model PDE model Improved PDE model
imageA 19.81 20.25 27.64
imageB 19.73 20.37 28.58

running time of different models for denoising (s)

P-M model PDE model Improved PDE model
imageA 252.78 332.65 178.98
imageB 232.63 317.59 167.92

It can be seen from table 1 that the PSNR value of the improved PDE model after processing the two images is higher than that of the other two comparison models. The image processing efficiency of the image denoising model is low. The three models inevitably encounter the same problem, which can not meet the needs of real-time calculation of the algorithm. The image processing time is as long as a few minutes, but through comparison, the operation efficiency of the improved PDE model is much better than the other two models.

Conclusions

In this paper, an improved image denoising model based on fractional calculus mathematical equation is proposed. Through experimental comparison, the new model is obviously better than other models. The new model is still unable to deal with the problem of MR image denoising in low field and its operation efficiency is relatively low. In addition, the new image denoising algorithm based on neural network has also achieved good results, which can be combined with the improved PDE model to further optimize the model.

Figure 1

original image
original image

Figure 2

denoising effect of different models on map image
denoising effect of different models on map image

Figure 3

denoising effect of different algorithms on texture image
denoising effect of different algorithms on texture image

PSNR values of different models after denoising

P-M model PDE model Improved PDE model
imageA 19.81 20.25 27.64
imageB 19.73 20.37 28.58

running time of different models for denoising (s)

P-M model PDE model Improved PDE model
imageA 252.78 332.65 178.98
imageB 232.63 317.59 167.92

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