This work describes a Bayesian framework for reconstructing the boundaries that represent targeted features in an image, as well as the regularity (i.e., roughness vs. smoothness) of these boundaries.This regularity often carries crucial information in many inverse problem applications, e.g., for identifying malignant tissues in medical imaging. We represent the boundary as a radial function and characterize the regularity of this function by means of its fractional differentiability. We propose a hierarchical Bayesian formulation which, simultaneously, estimates the function and its regularity, and in addition we quantify the uncertainties in the estimates. Numerical results suggest that the proposed method is a reliable approach for estimating and characterizing object boundaries in imaging applications, as illustrated with examples from X-ray CT and image inpainting. We also show that our method is robust under various noise types, noise levels, and incomplete data.

翻译：本文描述了一个贝叶斯框架，用于重建图像中表征目标特征的边界，以及这些边界的正则性（即粗糙度与平滑度）。在许多逆问题应用中，这种正则性往往承载关键信息，例如在医学成像中识别恶性组织。我们将边界表示为径向函数，并利用该函数的分数阶可微性表征其正则性。我们提出了一种分层贝叶斯公式，能够同时估计函数及其正则性，此外还量化了估计中的不确定性。数值结果表明，所提方法是一种在成像应用中估计和表征目标边界的可靠方法，通过X射线CT和图像修复的实例进行了说明。我们还展示了该方法在多种噪声类型、噪声水平及不完整数据条件下的鲁棒性。