快速离散分数傅里叶变换及其在图像处理中的应用

Digital image can vividly present real-world scene during image capture and it has become an essential tool to express and convey information. However, due to the variety of image acquisition equipment and transmission channel, the fine features of an image will loss or distortion. Traditional methods based on Fourier transform analyze an image in the frequency domain and cannot dose a fine job of describing the texture details. We need to study new theory and method to supplement and perfect current methods. .The fractional Fourier transform (FRFT) can reflect all of the image texture spectrums from spatial domain to frequency domain. Therefore it can describe image fine features. This fact enables us to achieve better image processing effect by using the FRFT. At present, however, the efficient discrete algorithm for the FRFT is absent and the variation rule for the fractional spectrum is not known. This project purposes to push the application of the FRFT to the field of image processing. We will study the fast discrete algorithm for the FRFT and establish the reality-preserving theoretical framework for multiple-parameter fractional transform. Based on the framework model, several new fractional transforms will be constructed. And then, we will analyze the change rule of digital image in different fractional domains and develop the response model of different image textures to fractional Fourier transform. On this basis, an efficient color image encryption method will be proposed in multiple fractional domains..This project will improve the efficiency of the discrete algorithm for the FRFT and provide new methods and tools for digital image security. Our research is significant to enrich the theoretical system of modern digital image processing.

数字图像能够生动的反应信息采集时的场景，是人们表达和传递消息的主要手段。但因采集设备和传输通道的多样性，常使图像的精细特征丢失或扭曲。传统基于傅里叶变换的方法仅能在频域分析，无法更好的完成对图像细节的描述，亟需研究新理论新方法对此进行补充完善。.分数傅里叶变换（FRFT）可以反映图像从空域渐变到频域的所有特征，能实现对精细纹理的描述，有望达到更好的处理效果。但目前尚缺乏对FRFT的高效离散化算法，图像的分数域谱变化机理尚不清楚。本项目以促进FRFT在图像处理的应用为目的，研究对FRFT的快速离散化算法，提出保实多参数分数变换的框架，构建多种新型保实分数变换，进而，分析图像在分数域的特征变化规律，建立图像不同纹理对FRFT的响应模型，并基于此，提出一种多分数域的彩图加密方法。.本项目将提高FRFT的离散化效率，为图像安全领域提供新方法、新工具，对丰富现代数字图像处理的理论体系具有重要意义。

数字图像能够生动的反应信息采集时的场景，是人们表达和传递消息的主要手段。但因采集设备和传输通道的多样性，常使图像的精细特征丢失或扭曲。传统基于傅里叶变换的方法仅能在频域分析，无法更好的完成对图像细节的描述，亟需研究新理论新方法对此进行补充完善。分数傅里叶变换可以反映图像从空域渐变到频域的所有特征，能实现对精细纹理的描述，有望达到更好的图像处理效果。本项目自2018年实施以来，首先分析了特征分解型和采样型离散分数傅里叶变换的特点，建立了两种类型之间的等价关系，为不同应用中如何选择合适的离散化算法提供了理论指导；然后提出了保实多参数分数Hartley变换和保实多参数分数角变换，并深入分析了其性质特征；进而根据图像不同纹理对分数傅里叶变换的不同响应，结合混沌理论提出了两种保实多参数分数域的彩图加密算法。本项目的研究成果能够为图像加密领域提供新方法、新工具，能够进一步丰富和发展现代数字图像处理的理论体系。