Generative Models Applied to Inverse Problems

我的研究是在深度学习和成像反问题的交叉点上进行的。解决反问题是通过(通常)已知的正演模型计算给出间接测量的未知物理量的任务。一个典型的例子是当成像技术被用于医学、工程、天文学和地球物理学时。有趣的逆问题几乎总是不适定的，在基于恢复图像进行决策的应用中，解决这一问题是至关重要的。具体地说，有时恢复的数据不足以准确地解决重建问题，并且必须在解决过程中结合关于对象的某种形式的先验知识。开发新的和改进现有的方法来处理这些案件是我的项目的目标。提出的方法使用深度学习方法来学习产生式模型。生成性模型隐含地模拟来自观测的数据的高维分布。然后，学习的分布可以作为求解反问题的先验，确保解是可行的。与手工制作的先验相比，学习的先验可以提供更具体的信息，同时保持对前向问题变化的灵活性。例如，我们可以学习一种生成模型，该模型可以生成所有可行的脑、心和肺CT图像。然后，从CT扫描仪接收低辐射剂量数据，我们在可行图像集合中搜索，以找到与数据最匹配的图像。尽管提供的数据量很少，但生成的图像仍将是高质量的。新的工程和/或物理科学研究的范围很广。找到由生成模型产生的最适合数据的图像所需的结果优化既是一个非线性且非凸的问题，并且将需要应用或开发最新技术。需要数值分析工具来限定误差和收敛。还需要开发与计算机科学家密切合作的数学方法，以检查和确保地面真实图像可以由生成模型产生。这将增加一个快速增长的研究领域，包括可变自动编码器和生成性对抗网络。

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.