基于深度卷积字典学习的亚像素分辨率衍射成像研究

Sub-pixel resolution diffraction imaging aims to recover high-resolution images from the observed low-resolution diffraction intensity patterns, and it has important applications for many scientific fields. Under the high noise levels and single observation scenarios, existing algorithms suffer from the poor imaging quality, the low resolution reconstruction and the slow imaging speed. To address these issues, this project intends to research on deep structured convolutional dictionary learning. Moreover, the inherent priors of the image are fused for sub-pixel resolution diffraction imaging. The concrete research contents are as follows. (1) The structure and condition constraints are incorporated into the deep convolutional dictionary. The deep structured convolutional dictionary learning models and methods in complex domain are studied based on the sparse representation theory of the complex-valued image and the deep convolutional sparse coding theory; (2) The non-local similarity, plug-and-play priors and deep image priors are applied for learning the optimal deep convolutional dictionary of the reconstructed image jointly or independently through a single noisy diffraction intensity pattern. Based on this adaptive sparse representation, the sub-pixel resolution diffraction imaging models that are robust to Poisson noise, Gaussian noise and Poisson-Gaussian hybrid noise are studied; (3) The fast methods of solving the diffraction imaging optimization problems by using the non-convex optimization theory, the deep unfolding principle and the epigraph principle are studied. The research achievements of this project will break through the diffraction-limited resolution and have important application value.

亚像素分辨率衍射成像旨在利用观测的低分辨率衍射强度图样重建高分辨率图像，它在多个科学领域具有重要应用。现有算法在噪声强度较大、单次观测等情况下，成像质量差、分辨率低、速度慢。为解决这些问题，本项目拟研究深度结构卷积字典学习，并结合图像固有先验进行亚像素分辨率衍射成像。具体研究内容包括:（1）以复图像稀疏表示和深度卷积稀疏编码理论为基础，对深度卷积字典施加结构与条件约束，研究复域深度结构卷积字典学习模型与方法；（2）通过单幅含噪衍射强度图样，联合或单独应用图像的非局部相似性、即插即用先验及深度图像先验辅助学习对重建图像本身最优表示的深度卷积字典。以这种自适应表示为基础，研究对泊松、高斯及泊松-高斯混合噪声鲁棒的亚像素分辨率衍射成像模型；（3）研究利用非凸优化理论、深度展开原理及镜像图原理快速求解衍射成像优化问题的方法。本项目的研究成果有望突破衍射极限分辨率限制，具有重要应用价值。

本项目首先研究了面向实/复图像的卷积字典学习及其深度展开方法。构建了卷积稀疏编码优化模型、综合卷积字典学习模型、对偶紧标架学习模型、双层紧标架学习模型等多种具有结构与条件约束的单层或双层卷积字典学习模型；研究了利用深度展开技术进行字典学习的方法，为研究利用深度展开的卷积字典学习技术进行亚像素分辨率衍射成像及图像重建奠定基础。. 其次，研究了对噪声鲁棒的亚像素分辨率衍射成像和图像重建方法。构建了基于线下训练好的深度卷积神经网络、对偶紧标架、双层紧标架、综合卷积字典的高斯去噪器，结合即插即用框架、镜像图方法和交替优化框架，联合或单独应用图像的非局部相似性、即插即用先验及深度先验研究了对噪声鲁棒的亚像素分辨率衍射成像和图像重建算法。. 再次，研究了基于深度展开卷积字典学习的亚像素或像素分辨率衍射成像方法。以图像在自适应卷积字典下的稀疏性为先验，研究了联合应用非局部相似性进行亚像素分辨率衍射成像的方法；研究了利用学习的卷积稀疏编码模型和卷积字典学习模型进行高质量衍射成像的算法；研究了基于对偶紧标架的收缩网络，并进行了高质量的衍射成像。. 最后，研究了针对亚像素分辨率衍射成像及图像重建优化问题的快速、有效求解方法。研究了通过变量分离法、交替优化方法将构建的亚像素分辨率衍射成像问题和图像重建转化为简单的子问题，并利用加速梯度投影法、阈值法、广义交替投影法对各个子问题进行了有效求解；研究了利用深度展开原理将迭代求解方法展开成深度神经网络的快速成像方法；研究了利用Pytorch并行计算构架实现各个步骤的大规模复杂优化问题的并行处理。. 项目共发表学术论文10篇，其中SCI收录论文7篇（包括二区论文5篇），EI收录论文2篇（CCF B类会议论文及CCF C类会议论文各1篇），核心论文1篇（燕山大学学报）。授权发明专利1项，软件著作权2项。培养青年教师1人，博士研究生1人，硕士研究生4人，完成了研究目标。