Generative Models Applied to Inverse Problems

我的研究工作在深度学习和成像逆问题的交叉点。求解逆问题是通过（通常）已知的正向模型计算给定间接测量的未知物理量的任务。一个典型的例子是成像技术用于医学、工程、天文学和电子物理学。有趣的逆问题几乎总是不适定的，解决这一问题在决策是基于恢复的图像的应用中至关重要。特别是，有时恢复的数据不足以准确地解决重建问题，并且必须将关于对象的某种形式的先验知识并入解决方案过程中。开发新的和改进现有的方法来解决这些问题是我的项目的目标。所提出的方法使用深度学习方法来学习生成模型。生成模型隐式地对来自观测的数据的高维分布进行建模。然后，学习的分布可以用作解决逆问题时的先验，确保解决方案是可行的。学习的先验知识可以提供比手工制作的先验知识更具体的信息，同时对正向问题的变化保持灵活性。例如，我们可以学习一个生成模型，它可以生成所有可行的大脑心脏和肺部CT图像。然后从CT扫描仪接收低辐射剂量数据，我们搜索通过可行的图像集，以找到最适合的图像数据。尽管所提供的数据量很小，但所产生的图像将是高质量的。新的工程和/或物理科学研究的范围很广。找到最适合数据的生成模型产生的图像所需的最终优化是非线性和非凸问题，并且将需要应用或开发最先进的技术。需要数值分析工具来限制误差和收敛。还需要开发与计算机科学家密切合作的数学方法，以检查和确保生成模型可以生成地面实况图像。这将增加一个快速增长的研究领域，包括变分自编码器和生成对抗网络。

My research works at the intersection of deep learning and inverse problems in imaging. Solving an inverse problem is the task of computing an unknown physical quantity given indirect measurements via a (usually) known forward model. A typical example is when imaging technologies are used in medicine, engineering, astronomy and geophysics. Interesting Inverse Problems are nearly always ill-posed and addressing this is critical in applications where decision making is based on the recovered image. In particular, sometimes the data recovered is not sufficient to solve the reconstruction problem accurately and some form of prior knowledge about the object must be incorporated in the solution process. Developing new and improving existing approaches to these cases is the aim of my project. The proposed approaches uses deep learning methods to learn a generative model. Generative models implicitly model high-dimensional distributions of data from observations. The learnt distribution can then be used as a prior when solving the inverse problem, ensuring solutions are feasible. Learnt priors could provide more specific information than that of a hand-crafted prior while remaining flexible to changes in the forward problem. For example, we could learn a generative model that can produce all feasible brain heart and lung CT images. Then receiving low radiation dose data from the CT scanner, we search through the set of feasible images to find the image that best fits the data. The resulting image will be high quality despite the small amount of provided data. There is wide scope for novel engineering and/or physical sciences research. The resulting optimisation required for finding the image produced by the generative model that best fits the data is both a non-linear and non-convex problem and will require application or development of state of the art techniques. Numerical analysis tools are needed to bound errors and convergence. Mathematical approaches working closely with computer scientists also need to be developed for checking and ensuring that the ground truth images can be produced by the generative model. This will add to a fast growing research area including variational autoencoders and generative adversarial networks.