Biomedical Imaging Group, École Polytechnique fédérale de Lausanne, Lausanne CH–1015, Switzerland.
IEEE Trans Image Process. 2013 Jul;22(7):2699-710. doi: 10.1109/TIP.2013.2255305. Epub 2013 Mar 28.
We present a novel statistically-based discretization paradigm and derive a class of maximum a posteriori (MAP) estimators for solving ill-conditioned linear inverse problems. We are guided by the theory of sparse stochastic processes, which specifies continuous-domain signals as solutions of linear stochastic differential equations. Accordingly, we show that the class of admissible priors for the discretized version of the signal is confined to the family of infinitely divisible distributions. Our estimators not only cover the well-studied methods of Tikhonov and l1-type regularizations as particular cases, but also open the door to a broader class of sparsity-promoting regularization schemes that are typically nonconvex. We provide an algorithm that handles the corresponding nonconvex problems and illustrate the use of our formalism by applying it to deconvolution, magnetic resonance imaging, and X-ray tomographic reconstruction problems. Finally, we compare the performance of estimators associated with models of increasing sparsity.
我们提出了一种新的基于统计的离散化范例，并推导出一类最大后验（MAP）估计器，用于解决病态线性反问题。我们受到稀疏随机过程理论的指导，该理论将连续域信号指定为线性随机微分方程的解。相应地，我们表明，信号的离散化版本的可容许先验类被限制在可分分布族内。我们的估计器不仅涵盖了作为特例的 Tikhonov 和 l1 型正则化等广为人知的方法，而且还为通常非凸的更广泛的稀疏促进正则化方案开辟了道路。我们提供了一种处理相应非凸问题的算法，并通过将其应用于反卷积、磁共振成像和 X 射线断层重建问题来说明我们形式主义的使用。最后，我们比较了与越来越稀疏的模型相关的估计器的性能。
