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# The Practice System of Physics and Electronics Courses in Higher Vocational Colleges Based on Fractional Differential Equations

###### Accepté: 23 Apr 2022
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Introduction

Electronic image denoising is an important research topic in image processing. It is also an important step in image preprocessing. Image noise is mainly divided into additive and multiplicative [1]. At present, the removal of additive Gaussian noise is relatively mature. For example, the total variation minimization method can remove noise and preserve the edges of the image well. Some scholars have given corresponding algorithms for removing multiplicative noise. The noise contained in the image is signal-dependent and follows a Poisson distribution. The photoelectron noise caused in the photoelectric conversion process of the image sensor has such a density distribution function [2]. Some scholars have studied the suppression of image Poisson noise by the variational method. They attributed the removal of Poisson noise to the evolution of a second-order partial differential equation to a stable state [3]. Some scholars have overcome this shortcoming with fourth-order partial differential equations. Some scholars use the alternate projection method to reduce the amount of calculation.

Recently, some new integro-differential equation methods have been applied in image denoising. Some scholars have proposed a new integral-differential equation. This equation does a good job of removing noise and decomposing images into cartoon and texture components. Some scholars have proposed partial differential equations with fractional time and spatial derivatives [4]. However, its essence is still integro-differential equations. Some scholars have proposed integro-differential equations with time integration. This equation has certain advantages over the traditional partial differential equation method in removing Gaussian noise. In this paper, a new multi-scale hierarchical image representation model based on Poisson noise is established based on the classical minimum total variation (TV) model. Then we introduce continuous-time variables to get new integro-differential equations with time integrals. The new integro-differential equation is an inverse scale-space method. The method evolves from the zero image to the original noisy image. The paper discusses some properties of integro-differential equations and obtains the energy decomposition theorem [5]. Numerical experiments show that the proposed algorithm is superior to the classical TV and fourth-order differential equation algorithms.

TV model based on Poisson noise

The classical TV model is obtained from minimizing the energy functional of equation (1). $ETV(u)=∫Ω|∇u|dΩ+β∫Ω(f−u)2dΩ$ {E_{TV}}\left( u \right) = \int_\Omega {\left| {\nabla u} \right|d\Omega + \beta \int_\Omega {{{\left( {f - u} \right)}^2}\,d\Omega } } where β is the regularization parameter. The first term in formula (1) is called the regularization term. The second item is called the loyalty item. The TV model can remove Gaussian noise well and preserve the edges of the image well [6]. Some scholars first analyze the extreme value problem of image denoising from the perspective of Bayesian estimation and then construct a variational model of Poisson distribution characteristics. Assume that the image additive noise follows a Poisson distribution: $Pμ(n)=e−μμnn!,n≥0$ {P_\mu }\left( n \right) = {{{e^{ - \mu }}{\mu ^n}} \over {n!}},\,n \ge 0

Both mean and variance are μ. We observe image f and estimate original noise-free image μ. We analyze this problem using maximum a posteriori estimation [7]. $∑i(u(xi)−f(xi)logu(xi))+λ∫Ω|∇u|dΩ$ \sum\limits_i {\left( {u\left( {{x_i}} \right) - f\left( {{x_i}} \right)\log \,u\left( {{x_i}} \right)} \right) + \lambda \int_\Omega {\left| {\nabla u} \right|d\Omega } }

We introduce a continuous variable x, so equation (7) can be regarded as the discretization of the following functional: $E(u)=∫Ω(u−flogu)dΩ+∫Ωλ|∇u|dΩ$ E\left( u \right) = \int_\Omega {\left( {u - f\,\log \,u} \right)d\Omega + } \int_\Omega {\lambda \left| {\nabla u} \right|d\Omega }

Functional (4) is defined in uBV and satisfies log uL1 (Ω). Its minimization solution is essentially different from the functional formula (1). The minimization solution of functional formula (4) cannot keep the average gray value of the image [8]. So we get the following properties:

Theorem 1

Any given λ has $∫ΩudΩ=∫ΩfdΩ−‖u‖BV$ \int_\Omega {ud\Omega = } \int_\Omega {fd\Omega - {{\left\| u \right\|}_{BV}}} . where $‖u‖BV=∫Ω|∇u|dΩ$ {\left\| u \right\|_{BV}} = \int_\Omega {\left| {\nabla u} \right|d\Omega } is the BV semi-norm. Prove that the Euler-Lagrange equation of equation (4) is represented as $f+λudiv(∇u|∇u|)=u$ f + \lambda udiv\left( {{{\nabla u} \over {\left| {\nabla u} \right|}}} \right) = u .

Integrate both ends and obtain $∫ΩfdΩ+λ∫Ωudiv(∇u|∇u|)dΩ=∫ΩfdΩ−λ(∇u|∇u|,∇u)=∫ΩfdΩ−λ‖u‖BV=∫ΩudΩ$ \matrix{\int_\Omega {fd\Omega + \lambda \int_\Omega {udiv\left( {{{\nabla u} \over {\left| {\nabla u} \right|}}} \right)d\Omega } } \hfill\cr { = \int_\Omega {fd\Omega - \lambda \left( {{{\nabla u} \over {\left| {\nabla u} \right|}},\nabla u} \right)} } \hfill \cr { = \int_\Omega {fd\Omega - \lambda {{\left\| u \right\|}_{BV}} = \int_\Omega {ud\Omega } } } \hfill \cr } by Green's formula. The proof is complete.

New integro-differential equations

The minimal solution of functional (4) is the denoised image. We rewrite functional (4) into the following equation: $E(u)=∫Ω|∇u|dΩ+λ∫Ω(u−flogu)dΩ$ E\left( u \right) = \int_\Omega {\left| {\nabla u} \right|d\Omega + \lambda } \int_\Omega {\left( {u - f\,\log \,u} \right)d\Omega }

We can use functional (5) to solve the minimization of uλ for very small λ. This equation contains only the main contours of the image. As λ gets bigger and bigger, uλ contains more and more image details [9]. If f is decomposed at the initial scale λ0, we have $uλ0=argminυ=u−flogu{‖u‖BV+λ0∫ΩυdΩ}$ {u_{{\lambda _0}}} = \mathop {\arg \min}\limits_{\upsilon = u - f\log u} \left\{ {{{\left\| u \right\|}_{BV}} + {\lambda _0}\int_\Omega {\upsilon d\Omega } } \right\}

Where the residual image is υλ0 = fuλ0. It still contains some image details. So υλ0 can still be decomposed at scale λ1: $uλ1=argminυ=u−uλ0logu{|u|BV+λ1∫ΩυdΩ}$ {u_{{\lambda _1}}} = \mathop {\arg \min}\limits_{\upsilon = u - {u_{{\lambda _0}}}\log u} \left\{ {{{\left| u \right|}_{BV}} + {\lambda _1}\int_\Omega {\upsilon d\Omega } } \right\}

Where the residual image is υλ1 = υλ0υλ1. If the fixed scale parameter λ is replaced by a series of varying scale parameters λ1 < λ2 < ⋯, then a new multi-scale hierarchical image representation is obtained. We solve the minimal solution of the following series of functionals: $uλj=argminυ=u−uλjlogu{|u|BV+λ1∫ΩυdΩ}$ {u_{{\lambda _j}}} = \mathop {\arg \min}\limits_{\upsilon = u - {u_{{\lambda _j}}}\log u} \left\{ {{{\left| u \right|}_{BV}} + {\lambda _1}\int_\Omega {\upsilon d\Omega } } \right\}

Where the residual image is υλj = υλj−1uλj. We can obtain new integro-differential equations from this relationship [10]. The article introduces successive time steps τ and re-solves uλj in units of τ. In this way, the following equation can be obtained. $uλj=argminυ=τu−uλj−1logτu{|u|BV+λj∫ΩυdΩ}$ {u_{{\lambda _j}}} = \mathop {\arg \min}\limits_{\upsilon = \tau u - {u_{{\lambda _{{j^{ - 1}}}}}}\log \tau u} \left\{ {{{\left| u \right|}_{BV}} + {\lambda _j}\int_\Omega {\upsilon d\Omega } } \right\}

Among them, the residual image is υλj = υλj−1τuλj, so after the N + 1 step, the following multi-scale layered image is obtained. We denote it as f = τuλ0 + υλ0 = τuλ0 + τuλ0 + τuλ1 + υλ1 = ⋯ = τuλ0 + τuλ1 + ⋯ + τuλN + υλn. which is $f=∑k=0Nτuλk+υλN$ f = \sum\limits_{k = 0}^N {\tau {u_{{\lambda _k}}}} + {\upsilon _{{\lambda _N}}}

The N+1 step of multi-scale image representation is expressed as follows: $uλN=argminυ=τu−υλN−1logτu{|u|BV+λN∫ΩυdΩ}$ {u_{{\lambda _N}}} = \mathop {\arg \min}\limits_{\upsilon = \tau u - {\upsilon _{{\lambda _{N - 1}}}}\log \tau u} \left\{ {{{\left| u \right|}_{BV}} + {\lambda _N}\int_\Omega {\upsilon d\Omega } } \right\}

Where the residual image is υλN = υλN−1υλN. Its Euler-Lagrange equation is $τuλN−υλN−1=1λNuλNdiv(∇uλN|∇uλN|)$ \tau {u_{{\lambda _N}}} - {\upsilon _{{\lambda _{N - 1}}}} = {1 \over {{\lambda _N}}}{u_{{\lambda _N}}}\,div\left( {{{\nabla {u_{{\lambda _N}}}} \over {\left| {\nabla {u_{{\lambda _N}}}} \right|}}} \right)

We have the expression $υλN−1=f−∑k=0N−1τuλk$ {\upsilon _{{\lambda _{N - 1}}}} = f - \sum\nolimits_{k = 0}^{N - 1} {\tau {u_{{\lambda _k}}}} from equation (10). We substitute it into equation (12) to get $∑k=0Nτuλk−f=1λNuλNdiv(∇uλN|∇uλN|)$ \sum\limits_{k = 0}^N {\tau {u_{{\lambda _k}}} - f = {1 \over {{\lambda _N}}}{u_{{\lambda _N}}}\,div\left( {{{\nabla {u_{{\lambda _N}}}} \over {\left| {\nabla {u_{{\lambda _N}}}} \right|}}} \right)}

Let τ → 0 get the new integro-differential equation (14): $∫0tu(x,y,s)ds−f(x,y)=1λ(t)u(x,y,t)div(∇u(x,y,t)|∇u(x,y,t)|)$ \int_0^t {u\left( {x,\,y,\,s} \right)ds - f\left( {x,\,y} \right)} = {1 \over {\lambda \left( t \right)}}u\left( {x\,,y,\,t} \right)div\left( {{{\nabla u\left( {x\,,y,\,t} \right)} \over {\left| {\nabla u\left( {x\,,y,\,t} \right)} \right|}}} \right)

Where the initial condition is u(x, y, 0) = 0, and the boundary condition is $∂u∂n|∂Ω=0(t≥0)$ {{\partial u} \over {\partial n}}\left| {_{\partial \Omega } = 0\left( {t \ge 0} \right)} \right. . The scaling function λ(t) is monotonically increasing. In this paper, the image with Poisson noise is denoised by the integro-differential equation (14). The solution $U(x,y,t)=∫0tu(x,y,s)ds$ U\left( {x,\,y,\,t} \right) = \int_0^t {u\left( {x,\,y,s} \right)} ds converges to the initial noisy image f when t → ∞. So the function family {U(x, y, t)}t≥0 can be regarded as the inverse scale-space representation of. t is the inverse scale parameter. Therefore we choose an appropriate time parameter t, U(x, y, t). V(x, y, t) = f(x, y) − U(x, y, t) is the residual image [11]. The new integro-differential equation has the following properties:

Theorem 2

We can obtain the following relation $∫ΩUdΩ=∫ΩfdΩ−1λ(t)‖u‖BV$ \int_\Omega {Ud\Omega = \int_\Omega {fd\Omega - {1 \over {\lambda \left( t \right)}}{{\left\| u \right\|}_{BV}}} } according to the integro-differential equation (14).

Theorem 3

We can obtain the energy decomposition formula: $4∫0t1λ(s)∫Ωu(x,y,s)|∇u(x,y,s)|dΩds+‖V(x,y,t)‖L22=‖f‖|L22$ 4\int_0^t {{1 \over {\lambda \left( s \right)}}\int_\Omega {u\left( {x,\,y,\,s} \right)\left| {\nabla u\left( {x,\,y,\,s} \right)} \right|d\Omega ds + \left\| {V\left( {x,\,y,\,t} \right)} \right\|_{{L^2}}^2 = \left. {\left\| f \right\|} \right|_{{L^2}}^2} } according to the integro-differential equation (14). $∫Ω(U(⋅,t)−f)ddt(U(⋅,t)−f)dΩ=12ddf∫Ω(U(⋅,t)−f)(U(⋅,t)−f)dΩ=∫Ω1λ(t)div(∇u(⋅,t)|∇u(⋅,t)|)u2(⋅,t)dΩ$ \matrix{ {\int_\Omega {\left( {U\left( { \cdot ,t} \right) - f} \right){d \over {dt}}\left( {U\left( { \cdot ,t} \right) - f} \right)d\Omega } } \hfill \cr { = {1 \over 2}{d \over {df}}\int_\Omega {\left( {U\left( { \cdot ,t} \right) - f} \right)\,\left( {U\left( { \cdot ,t} \right) - f} \right)d\Omega } } \hfill \cr { = \int_\Omega {{1 \over {\lambda \left( t \right)}}div\left( {{{\nabla u\left( { \cdot ,t} \right)} \over {\left| {\nabla u\left( { \cdot ,t} \right)} \right|}}} \right){u^2}\,\left( { \cdot ,t} \right)d\Omega } } \hfill \cr }

Next, we perform time integration on both sides of equation (15) on [0, t] to get $12∫Ω(U(⋅,t)−f)(U(⋅,t)−f)dΩ−12∫Ωf⋅fdΩ∫st∫Ω1λ(s)div(∇u(⋅,s)|∇u(⋅,s)|)u2(⋅,s)dΩds=‖U(⋅,t)−f‖L22−‖f‖L22=∫s=0t2λ(s)(div(∇u(⋅,s)|∇u(⋅,s)|)u2(⋅,s)ds)$ \matrix{ {{1 \over 2}\int_\Omega {\left( {U\left( { \cdot ,t} \right) - f} \right)\,\left( {U\left( { \cdot ,t} \right) - f} \right)d\Omega - {1 \over 2}\int_\Omega {f \cdot fd\Omega } } } \hfill \cr {\int_s^t {\int_\Omega {{1 \over {\lambda \left( s \right)}}div\left( {{{\nabla u\left( { \cdot ,s} \right)} \over {\left| {\nabla u\left( { \cdot ,s} \right)} \right|}}} \right){u^2}\,\left( { \cdot ,s} \right)d\Omega ds} } } \hfill \cr { = \left\| {U\left( { \cdot ,t} \right) - f} \right\|_{{L^2}}^2 - \left\| f \right\|_{{L^2}}^2} \hfill \cr { = \int_{s = 0}^t {{2 \over {\lambda \left( s \right)}}\left( {div\left( {{{\nabla u\left( { \cdot ,s} \right)} \over {\left| {\nabla u\left( { \cdot ,s} \right)} \right|}}} \right){u^2}\,\left( { \cdot ,s} \right)ds} \right)} } \hfill \cr }

We get the following equation from Green's formula: $=−∫s=0t2λ(s)(∇u(⋅,s)|∇u(⋅,s)|u2(⋅,s))ds=−∫s=0t2λ(s)∫Ω2u(⋅,s)|∇u(⋅,s)|dΩds$ \matrix{ { = - \int_{s = 0}^t {{2 \over {\lambda \left( s \right)}}\left( {{{\nabla u\left( { \cdot ,s} \right)} \over {\left| {\nabla u\left( { \cdot ,s} \right)} \right|}}{u^2}\,\left( { \cdot ,s} \right)} \right)ds} } \hfill \cr { = - \int_{s = 0}^t {{2 \over {\lambda \left( s \right)}}\int_\Omega {2u\,\left( { \cdot ,s} \right)\left| {\nabla u\left( { \cdot ,s} \right)} \right|d\Omega ds} } } \hfill \cr }

Our decomposition Theorem 3 gives the relationship between the residual images V and f.

There are further results if λ(t) satisfies a certain condition [12].

Theorem 4

Assume a noisy image fBV. From the integrodifferential equation (14) we can conclude that if the scaling function λ(t) increases fast enough to form a Hadamard sequence, $limt→∞λ(t/2)λ(t)=0$ \mathop {\lim }\limits_{t \to \infty } {{\lambda \left( {t/2} \right)} \over {\lambda \left( t \right)}} = 0 , then there is $∫Ωf(x,y)dΩ=∫Ω∫0∞u(x,y,s)dsdΩ$ \int_\Omega {f\left( {x,y} \right)d\Omega = \int_\Omega {\int_0^\infty {u\left( {x,y,s} \right)dsd\Omega } } } .

Proof

The residual image V(·, t) can be written as $V(⋅,t)=f−∫s=0t/2u(⋅,s)ds−∫s=t/2tu(⋅,s)ds$ V\left( { \cdot ,t} \right) = f - \int_{s = 0}^{t/2} {u\left( { \cdot ,s} \right)ds - \int_{s = t/2}^t {u\left( { \cdot ,s} \right)ds} } . So there is inequality $1λ(t)‖V(⋅,t)‖BV≤1λ(t)‖f‖BV+λ(t/2)λ(t)∫s=0t/21λ(s)⋅‖u(⋅,s)‖BV+∫s=t/2t1λ(s)‖u(⋅,s)‖BVds$ \matrix{ {{1 \over {\lambda \left( t \right)}}{{\left\| {V\left( { \cdot ,t} \right)} \right\|}_{BV}} \le {1 \over {\lambda \left( t \right)}}{{\left\| f \right\|}_{BV}} + {{\lambda \left( {t/2} \right)} \over {\lambda \left( t \right)}}\int_{s = 0}^{t/2} {{1 \over {\lambda \left( s \right)}}} } \hfill \cr { \cdot {{\left\| {u\left( { \cdot ,s} \right)} \right\|}_{BV}} + \int_{s = t/2}^t {{1 \over {\lambda \left( s \right)}}{{\left\| {u\left( { \cdot ,s} \right)} \right\|}_{BV}}ds} } \hfill \cr }

Because when t → ∞ is λ(t)→∞, the first term on the right-hand side of the above inequality tends to 0. By hypothesis and theorem 3, it is found that the second and third terms also tend to 0. This way we get $limt→∞1λ(t)‖V(⋅,t)‖BV=0$ \mathop {\lim }\limits_{t \to \infty } {1 \over {\lambda \left( t \right)}}{\left\| {V\left( { \cdot ,t} \right)} \right\|_{BV}} = 0 . Because of ||u(·, t) ||BV ≤|| V(·, t)||BV, we have $∫ΩV(⋅,t)dΩ=1λ(t)‖u(⋅,t)‖BV≤1λ(t)‖V(⋅,t)‖BV$ \int_\Omega {V\left( { \cdot ,t} \right)d\Omega = {1 \over {\lambda \left( t \right)}}{{\left\| {u\left( { \cdot ,t} \right)} \right\|}_{BV}} \le {1 \over {\lambda \left( t \right)}}{{\left\| {V\left( { \cdot ,t} \right)} \right\|}_{BV}}} according to Theorem 2. So $limt→∞∫ΩV(⋅,t)dΩ=0$ \mathop {\lim }\limits_{t \to \infty } \int_\Omega {V\left( { \cdot ,t} \right)d\Omega } = 0 .

Theorem 4 states that with the increase of time, the average gray value of the residual image converges to zero.

Numerical Algorithms and Experiments

We first assume that the initial discrete image f is of size m × m. It is the sampling of consecutive images at consistent sampling intervals [13]. The sampling point is (0, 0), ⋯, u(x, y) = u(xΔx, yΔy). where x, y = 0, ⋯, m − 1. We choose the spatial sampling interval Δx and Δy as Δx = Δy = 1. We make sure the denominator is not zero. We replace |∇u| with $|∇u|2+ε$ \sqrt {{{\left| {\nabla u} \right|}^2} + \varepsilon } in equation (14). where ɛ is a small positive number. Jacobian iteration produces the sequence ωk+1Wn+1un+1 Δt when k → ∞. In the actual calculation, if $‖ωk+1−ωk‖22≤0.01$ \left\| {{\omega ^{k + 1}} - {\omega ^k}} \right\|_2^2 \le 0.01 , then stop the iteration. Next update image U: $Un+1=Un+Wn+1$ {U^{n + 1}} = {U^n} + {W^{n + 1}}

This paper selects the quadratic function λ(t) = 0.25 × t2 as the scaling function. Where the time step is taken as Δt = 1. Other types of monotonically increasing functions can also be used as scaling functions. According to numerical experiments, we found that the quadratic scaling function has fewer iterations and can achieve a good denoising effect. Figure 1 presents the evolution results at 4 scales [14]. The solution U of equation (15) contains only the blurred outline of the “Lenna” image. The solution U is already very close to the original noisy image f when t = 110. This shows that the new integro-differential equation is an inverse scale space method. As t and λ(t) increase, more and more details are added to the solution U.

The second experiment added Poisson noise to the standard 512512′ images of “Lenna,” “Barbara”, “FishingBoat”, and “Pepper.” We use the signal-to-noise ratio (SNR) to measure the denoising effect and take Δt = 0.5 it as the time step. We choose an appropriate stop time t so that the signal-to-noise ratio reaches its maximum value. In Table 1, the signal-to-noise ratios of 3 different algorithms for denoising these 4 images. We can see that our algorithm achieves the highest signal-to-noise ratio [15]. Figure 2 shows the comparison of the denoising effects of the three algorithms on the “Lenna” image. We can see that the second-order differential equation method produces a block effect. The fourth-order differential equation method and the method in this paper have no block effect. However, our method preserves image details the best. This is mainly because the algorithm in this paper is an inverse scale space algorithm. The inverse scale space algorithm can better preserve the details of the image. Figure 3 shows the partial enlargement of the image after denoising by the three algorithms. We can see that the second-order differential equation algorithm has an obvious block effect. The image obtained by the algorithm in this paper has the best visual effect.

Comparison of signal-to-noise ratios of three algorithms (dB)

The original image Lenna Barbara fishing boat Pepper
Noisy image 4.13 4.18 4.71 4.38
Second order equation 11.65 8.86 10.78 13.05
Fourth order equation 11.84 10.07 10.81 13.11
Algorithm 13.35 10.44 11.14 13.65
Conclusion

We construct a new integro-differential equation denoising algorithm for Poisson-distributed noise in images. At the same time, we give a numerical solution method. The method in this paper has the following characteristics: (1) This paper firstly establishes a multi-scale hierarchical image representation model with a series of scale parameters. Then we serialize it to get new integro-differential equations with scaling functions. (2) The new integro-differential equation method is an inverse scale space method. It evolves the constant image to the original one, so an appropriate stopping time should be chosen to achieve a good denoising effect. Numerical experiments also prove that the proposed algorithm has a higher signal-to-noise ratio and visual effects than the classical TV and fourth-order differential equation algorithms.

#### Comparison of signal-to-noise ratios of three algorithms (dB)

The original image Lenna Barbara fishing boat Pepper
Noisy image 4.13 4.18 4.71 4.38
Second order equation 11.65 8.86 10.78 13.05
Fourth order equation 11.84 10.07 10.81 13.11
Algorithm 13.35 10.44 11.14 13.65

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