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.

The classical TV model is obtained from minimizing the energy functional of equation (1).

Both mean and variance are

We introduce a continuous variable

Functional (4) is defined in ^{1} (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:

Any given

Integrate both ends and obtain

The minimal solution of functional (4) is the denoised image. We rewrite functional (4) into the following equation:

We can use functional (5) to solve the minimization of _{λ}_{λ}_{0}, we have

Where the residual image is _{λ0} = _{λ0}. It still contains some image details. So _{λ0} can still be decomposed at scale _{1}:

Where the residual image is _{λ1} = _{λ0} − _{λ1}. If the fixed scale parameter _{1} < _{2} < ⋯, then a new multi-scale hierarchical image representation is obtained. We solve the minimal solution of the following series of functionals:

Where the residual image is _{λj}_{λj−1} − _{λj}_{λj}

Among them, the residual image is _{λj}_{λj−1} − _{λj}_{λ0} + _{λ0} = _{λ0} + _{λ0} + _{λ1} + _{λ1} = ⋯ = _{λ0} + _{λ1} + ⋯ + _{λN}_{λn}

The

Where the residual image is _{λN}_{λN−1} − _{λN}

We have the expression

Let

Where the initial condition is _{t≥0} can be regarded as the inverse scale-space representation of.

We can obtain the following relation

We can obtain the energy decomposition formula:

Next, we perform time integration on both sides of equation (15) on [0,

We get the following equation from Green's formula:

Our decomposition Theorem 3 gives the relationship between the residual images

There are further results if

Assume a noisy image

The residual image

Because when _{BV} ≤|| _{BV}, we have

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

We first assume that the initial discrete image ^{k}^{+1} → ^{n}^{+1} ≡ ^{n}^{+1} Δ

This paper selects the quadratic function ^{2} as the scaling function. Where the time step is taken as Δ

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 Δ

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 |

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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