目的 针对传统总变分方法在去除泊松噪声时容易出现“阶梯效应”和图像边缘模糊的问题，提出了一种基于分数阶变分的自适应去泊松噪声新模型。方法 首先新模型在分析了泊松噪声分布特点的基础上导出了非凸自适应正则项，它能够根据图像不同区域的特点自适应地调节正则项系数，以达到保持图像边缘的目的。然后，新模型利用分数阶离散微分向量能够结合更多图像信息的特点，将正则项中的一阶离散微分向量替换为分数阶离散微分向量，以此来达到抑制“阶梯效应”的目的。对于新模型的求解，结合交替迭代法和加权原始-对偶法提出了一种高效的数值解法。结果 新模型明显优于传统总变分去泊松噪声模型，在有效抑制“阶梯效应”的同时图像边缘也得到了较好地保护，以经典的Peppers图片为例，新模型相比于传统模型，峰值信噪比（PSNR）由28.98 dB提高到了30.24 dB，图像结构相似度（SSIM）由0.77提高到了0.87。另外，所提的数值解法具有收敛速度快、复杂度低的特点，收敛时间从偏微分方程、Chambolle投影等传统数值解法的0.5 s与0.1 s缩短至0.056 s。结论 实验结果表明，所提模型与数值解法的可行性，模型与数值解法在主要客观评价指标和图像视觉效果方面均优于传统的变分去泊松噪声模型，且模型与数值解法具有较好的普适性。但是模型中分数阶的阶次选取有待进一步优化。

Objective In the traditional total variation method, the removal of Poisson noise causes the “staircase effect” and image edge blurring in the plain area of the image. In reality, laser radar, satellite remote sensing, and medical imaging CT are based on the system of light quantum counting. The interference in image acquisition is basically subject to the Poisson distribution of quantum noise. How to effectively suppress the “staircase effect” and protect the edge of the image have thus become variational Poisson noise problems. To solve these problems, a new adaptive fractional-variation Poisson noise denoising model based on the traditional variational method is proposed.Methods The new model performs adaptive non-convex regularization based on the analysis of the Poisson noise distribution characteristics. Compared with the original model, the new non-convex regularization model can adjust the regularization coefficient adaptively according to the characteristics of regularization to different regions of the image; the image edge is therefore maintained. Regularization in the traditional total variational model involves the first-order discrete differential vector. Owing to the function characteristic of the bounded variation of the first-order differential vector, the “staircase effect” is easily caused in the plain area during denoising. To suppress the “staircase effect”, the new model uses the fractional discrete differential vector to combine the characteristics of image information and replaces the first-order discrete differential vector with the fractional discrete vector in the regularization. Given that the regularization of the new model is non-convex and the discrete differential is the fractional order differential vector, the traditional partial differential equation and the Chambolle projection algorithm cannot quickly and effectively obtain the numerical solution for the new model. For this reason, a more efficient numerical solution is proposed by combining the iterative method and the weighted primitive-dual method.Result Numerical results show that the new model is superior to the traditional total variational Poisson noise denoising model. The edge of the image is well protected, and the “staircase effect” is effectively suppressed. With the two classic images of Peppers and Lena as an example, the peak signal-to-noise ratio (PSNR) of the new model in the Peppers image is increased from 28.98 to 30.24 compared with the traditional model. The signal-to-noise ratio (SNR) is increased from 15.01 to 16.31, image structure similarity (SSIM) is increased from 0.77 to 0.87, and the mean square error of the image (MSE) is decreased from 82.24 to 61.52. In the Lena image, the PSNR of the new method is increased from 29.08 to 29.62, SNR is increased from 14.55 to 15.08, SSIM is increased from 0.78 to 0.83, and the MSE of the image is reduced from 80.37 to 70.97 compared with the traditional model. In addition, the numerical solution proposed in this study exhibits rapid convergence and low complexity compared with the traditional numerical solution. Similarly, with the classic Lena image as an example, the convergence time of the algorithm is reduced to 0.056 seconds compared with the convergence time of the partial differential equation, Chambolle projection, and other traditional numerical solutions (e.g., 0.5 and 0.1 seconds).Conclusion Experimental results reveal the feasibility of the numerical method and model proposed in this study. The model and numerical solution are superior to traditional variants in terms of PSNR, SNR, MSE, and image visual effect. Numerical experimentation on several representative images shows that the new model and numerical solution possess good universality. The new model effectively suppresses the “staircase effect” and enhances the edge information of the image. It can effectively remove noise in radar and medical images. The fractional order in the new model is a fixed value. Obviously, different regions of the image use different orders of the fractional order to exert different effects. The effect of using a fixed value of the fractional order for the entire image could be subject to further improvement.