This paper introduces a novel fractional differential and variational model that includes the terms of fusion and super-resolution, edge enhancement and noise suppression. In image fusion and super-resolution term, the structure tensor is employed to describe the geometry of all the input images. According to the fact that the fused image and the source inputs should have the same or similar structure tensor, the energy functional of the image fusion and super-resolution is established combining with the down-sampling operator. For edge enhancement, the bidirectional diffusion term is incorporated into the image fusion and super-resolution model to enhance the visualization of the fused image. In the noise suppression term, a new variational model is developed based on the fractional differential and fractional total variation. Thanks to the above three terms, the proposed model can realize the image fusion, super-resolution, and the edge information enhancement simultaneously. To search for the optimal solution, a gradient descent iteration scheme derived from the Euler-Lagrange equation of the proposed model is employed. The numerical results indicate that the proposed method is feasible and effective.

中文翻译：

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图像融合和超分辨率的分数微分和变分方法

本文介绍了一种新的分数微分和变分模型，其中包括融合和超分辨率、边缘增强和噪声抑制等项。在图像融合和超分辨率术语中，使用结构张量来描述所有输入图像的几何形状。根据融合图像与源输入应具有相同或相似结构的张量这一事实，结合下采样算子建立图像融合和超分辨率的能量泛函。对于边缘增强，双向扩散项被纳入图像融合和超分辨率模型，以增强融合图像的可视化。在噪声抑制项中，基于分数微分和分数总变分开发了一种新的变分模型。由于以上三项，所提出的模型可以同时实现图像融合、超分辨率和边缘信息增强。为了寻找最优解，采用了从所提出模型的欧拉-拉格朗日方程导出的梯度下降迭代方案。数值结果表明所提出的方法是可行和有效的。