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：本文描述了一种贝叶斯框架，用于重建图像中目标特征的边界以及这些边界的规则性（即粗糙度与光滑度）。在许多逆问题应用中，这种规则性往往承载着关键信息，例如医学影像中恶性组织的识别。我们将边界表示为径向函数，并通过其分数阶可微性来刻画该函数的规则性。我们提出了一种分层贝叶斯公式，能够同时估计函数及其规则性，并量化估计中的不确定性。数值结果表明，所提方法是成像应用中估计和表征物体边界的可靠方法，X射线CT和图像修复实例验证了其有效性。我们还证明，该方法在多种噪声类型、噪声水平及数据不完整情况下均具有稳健性。