基于深度生成先验的贝叶斯医学图像重建及不确定性量化

面向医学图像重建的高精度不确定性量化方法是当前计算数学领域的研究热点。在仅有少量标注数据的情况下，针对传统贝叶斯方法不能精确度量数据和模型不确定性的关键难题，本项目拟提出一种基于深度生成先验的贝叶斯图像重构方法。该方法首先发展一种TV先验和深度生成对抗网络先验结合的混合先验，以提高图像先验的分层特征表示能力。然后，分别使用直接代入法和限制条件法把混合先验嵌入贝叶斯推断框架，发展结合交替方向乘子法和Crank-Nicolson郎之万方法的抽样算法，实现高维非凸非线性后验分布的高效快速采样。最后，在CT/PET医学影像问题中验证算法的有效性。预期的研究成果将丰富医学图像反问题的不确定性量化方法，推动智能化影像检测技术的发展，具有重要的科学意义和实际应用价值。

The efficient schemes for quantifying the uncertainty of medical image reconstruction and analysis are currently hot research areas in modern computational mathematics. Medical images with limited labeled data are highly challenging for traditional Bayesian inference with model-driven prior due to the randomness of observations and uncertainty of the model, this project attempts to systematically design some efficient and stable Bayesian inference methods with deep generative prior for reconstructing medical images under uncertainty. We first propose a model-driven generative adversarial network prior on combing the total variation(TV) prior and generative adversarial network(GAN) prior to highly exploit hierarchical feature representations, and then the hybrid GAN prior is plugged in Bayesian rule in indirect and direct ways. Second, we provide a Markov chain Monte Carlo algorithm, based on the Crank-Nicolson Langevin method and alternating direction method of multipliers(ADMM), to sample from high dimensional non-convex posterior distribution. Finally, we apply the proposed method to a CT/PET image reconstruction problem that is difficult due to the uncertainty of data and model. The outcome of the project shall enrich the uncertainty quantification methods of medical imaging inverse problems and promote the development of the intelligent imaging diagnostic system, and thus be of great both theoretical and practical significance.