Digital model creation and image meticulous processing based on variational partial differential equation

First, variational expression. Assuming it has been clear that the average value of Gaussian white noise N μ = 0 meets this condition, then combined with the analysis of maximum likelihood criterion, an approximation of I can be obtained by using the computational least squares problem. The specific formula is shown as follows: minI∈Ω||Inoisy−I||2dΩ \mathop {\min}\limits_I {\in _\Omega}||{I_{noisy}} - I|{|^2}d\Omega

In the above formula, ||I_noisy − I||² represents a measure of image difference during denoising, as shown below: ||Inoisy−I||2 ||{I_{noisy}} - I|{|^2}

Since n represents the Gaussian white noise with the mean μ = 0 and the standard deviation of σ, all points in a radius sphere with I noisy as the centre of the circle can be approximated by I, in other words, they may all be solutions of the above formula. Therefore, in order to obtain more valuable solutions, the related problems need to be normalised. If the smoothing property of the image can be measured by using the normalised term R (I), and the corresponding I becomes more positive with the subsequent decline, then the optimal solution of I can be obtained according to the level set of R.

Second, divergence operator expression. The divergence operator model framework proposed by Weickert regards the PDE image denoising problem as a chemical concentration diffusion process following certain physical laws. In this way, the variational PDE denoising model is sorted into a more general framework based on divergence operator. The specific formula is shown as follows: ∂I∂t=div(D∇I) {{\partial I} \over {\partial t}} = {\rm{div}}(D\nabla I)

In the above formula, ∇I stands for the gradient field of the image, which corresponds to the concentration gradient of the chemical concentration diffusion, and D refers to the diffusion tensor field constructed based on the diffusion tensor. At this time, the diffusion behaviour is combined with the following formula, and the corresponding D needs to be designed, so as to obtain the following formula: ∂I∂t=div(D∇I) {{\partial I} \over {\partial t}} = {\rm{div}}(D\nabla I)

By comparison with the above formula, it can be seen that Φ functional expression is a special case of divergence expression, and the diffusion tensor corresponding to it is: D=ϕ′(||∇I||)||∇I||Id D = {{\phi '(||\nabla I||)} \over {||\nabla I||}}Id

In the above formula, Id represents the identity matrix of 2×2, and the following formula can also be regarded as a special case of the divergence expression, and the corresponding diffusion heat formula is:

Heat diffusion equation: ∂I∂t=ΔI=div(∇I) {{\partial I} \over {\partial t}} = \Delta I = {\rm{div}}(\nabla I)

Thirdly, trace operator expression. Combined with the above research and analysis on thermal diffusion, P-M model, Φ functional expression and directional smoothing, the expression form of trace operator can be seen as follows: ∂I∂t=c1Iuu+c2Ivv \matrix{{{{\partial I} \over {\partial t}}} \hfill & {= {c_1}{I_{uu}} + {c_2}{I_{vv}}} \hfill\cr}

In the above formula, u,v ∈ R² and ⊥ v,c₁,c₂ ≥ 0, Iuu and Ivv represent the second directional derivative of image I in the direction u and v, respectively. Combined with the expression of the second order directional derivative, the expression formula of trace operator of its diffusion tensor can be clearly expressed as: ∂I∂t=c1Iuu+c2Ivv=c1trace(HuuT)+c2trace(HvvT)=trace[H(c1uuT+c2vvT)]=trace(HT) \matrix{{{{\partial I} \over {\partial t}}} \hfill & {= {c_1}{I_{uu}} + {c_2}{I_{vv}}} \hfill\cr{} \hfill & {= {c_1}trace(Hu{u^T}) + {c_2}trace(Hv{v^T})} \hfill\cr{} \hfill & {= trace[H({c_1}u{u^T} + {c_2}v{v^T})]} \hfill\cr{} \hfill & {= trace(HT)} \hfill\cr}

In order to remove the noise effectively and retain the edge as much as possible, it should be processed in accordance with the direction of the relevant gradient and the smoothing intensity of the direction of the isoluminance line. In other words, the characteristic value of the diffusion tensor T, λ₁,λ₂ should be adjusted autonomically according to the characteristics of the region in which it is located. When processing different image features, Tschumperle uses μ1*+μ2* \sqrt {\mu _1^* + \mu _2^*} to distinguish them. In this case, λ₁,λ₂ represents the μ1*+μ2* \sqrt {\mu _1^* + \mu _2^*} function. In this process, the following design principles need to be observed:

First, the isotropic smoothing and effective denoising is implemented in the non-edge region, and the specific formula is as follows: ∂I∂t≈ΔI=trace(H)namely: limμ1*+μ2*→0T=Id {{\partial I} \over {\partial t}} \approx \Delta I = trace(H)\;\;{\rm{namely}}:\mathop {\lim}\limits_{\sqrt {\mu _1^* + \mu _2^*}\to 0} T = Id

And should be reflected in: λ₁,λ₂ limμ1*+μ2*→0λ2=limμ1*+μ2*→0λ2=1 li{m_{\sqrt {\mu _1^* + \mu _2^*}\to 0}}{\lambda _2} = li{m_{\sqrt {\mu _1^* + \mu _2^*}\to 0}}{\lambda _2} = 1

Second, the anisotropic smoothing process is implemented in the edge region, and the edge region is preserved as far as possible while the noise is removed. At this time, smoothing should be processed with the direction of ξ^*of isoluminance line. The specific formula is as follows: ∂I∂t≈λ2ξ*ξ*T=trace(λ2ξ*ξ*TH) {{\partial I} \over {\partial t}} \approx {\lambda _2}{\xi ^*}{\xi ^{*T}} = trace({\lambda _2}{\xi ^*}{\xi ^{*T}}H)

Also available: limμ1*+μ2*→∞λ1λ2=0 li{m_{\sqrt {\mu _1^* + \mu _2^*}\to \infty}}{{{\lambda _1}} \over {{\lambda _2}}} = 0

The choices proposed by Tschumperle that meet the above requirements are: {λ1=11+μ1*+μ2*λ2=11+μ1*+μ2* \left\{{\matrix{{{\lambda _1} = {1 \over {1 + \mu _1^* + \mu _2^*}}}\cr{{\lambda _2} = {1 \over {\sqrt {1 + \mu _1^* + \mu _2^*}}}}\cr}} \right.

By referring the processing formula of the edge region above to the calculation formula of the trace operator’s diffusion tensor, the expression formula of T can be obtained: T=11+μ1*+μ2*η*η*T+11+μ1*+μ2*ξ*ξ*T T = {1 \over {1 + \mu _1^* + \mu _2^*}}{\eta ^*}{\eta ^{*T}} + {1 \over {\sqrt {1 + \mu _1^* + \mu _2^*}}}{\xi ^*}{\xi ^{*T}}

In other words, under a ||∇I||, the effect of the above formula is approximately linear isotropic diffusion.

Second, the repair model based on divergence operator expression. Although using the variational method to deal with and analyse the denoising model can provide a global explanation for the denoising process, part of the flexibility of model design will be lost, and an efficient repair model cannot be obtained. Therefore, it is necessary to use more local expression forms to deal with the problem, namely discrete operator and trace operator. Taking the curvature-driven (CDD) repair model as an example, D is regarded as the area waiting to be repaired, and E is regarded as the outer neighbourhood of the area waiting to be repaired. Under the condition of no noise analysis, the TV repair model can be simplified as: min∈E∪D||∇I||dxdy \min {\in _{E \cup D}}||\nabla I||dxdy

The actual equation of the most rapid descent is: ∂I∂t=div(1||∇I||∇I) {{\partial I} \over {\partial t}} = {\rm{div}}\left({{1 \over {||\nabla I||}}\nabla I} \right)

The Fast Curvature Driven (FCDD) repair model improves the actual computing speed on the basis of the original model, and the specific form is shown as follows: ∂I∂t=div(g(|k|)∇I) {{\partial I} \over {\partial t}} = {\rm{div}}(g(|k|)\nabla I)

Thirdly, the repair model based on trace operator expression. The repair model with divergence operator as the core mainly regards the repair process as a process of information diffusion to the damaged area. The core of the application of this model is that the diffusion characteristic of each point in the damaged area is the corresponding diffusion tensor. Since the diffusion tensor contained in the circuit operator cannot fully and intuitively present the actual diffusion situation, the reference trace operator should be written to express it. Generally speaking, trace operator expression will push information into the damaged area one by one in a smooth way. It is same as the expression form of divergence operator, and its design core is also a diffusion tensor. Taking the self-aware repair model as an example, it is assumed that I0 represents the original image, D represents the area waiting to be repaired, and DC refers to the complement of D, that is, the clear area without damage in the image. In this case, the image repair model based on trace operator expression is: {∂I∂t=trace(TH),X∈DI=I0,X∈Dc \left\{{\matrix{{{{\partial I} \over {\partial t}} = trace(TH),X \in D}\cr{I = {I^0},X \in {D^c}}\cr}} \right.

In the above formula, H represents the Hessian matrix and T represents the diffusion tensor, which can intuitively present the smooth behaviour of some regions of the model.

The key difficulty of using trace operator expression to design image repair model lies in diffusion tensor T, which is related to the corresponding feature value λ₁,λ₂ as follows: T=λ1uuT+λ2vvT T = {\lambda _1}u{u^T} + {\lambda _2}v{v^T}

In the curvature-driven directional 1DLaplaicans repair model, the final solution can be calculated using the classical time-stepping method, which can be divided into the following points: (1)

The central difference method is used to analyse the first-order spatial derivatives Ix, Iy and the second-order spatial derivatives Ixx, Iyy and Ixy;

In the above formula, Δ T represents the length of the time distribution, and I represents the value of the waiting repair point at the time of N Δ T.

Repeat the above steps until the repair work is completed [5, 6].