We consider the problem of blob detection for uncertain images, such as images that have to be inferred from noisy measurements. Extending recent work motivated by astronomical applications, we propose an approach that represents the uncertainty in the position and size of a blob by a region in a three-dimensional scale space. Motivated by classic tube methods such as the taut-string algorithm, these regions are obtained from level sets of the minimizer of a total variation functional within a high-dimensional tube. The resulting non-smooth optimization problem is challenging to solve, and we compare various numerical approaches for its solution and relate them to the literature on constrained total variation denoising. Finally, the proposed methodology is illustrated on numerical experiments for deconvolution and models related to astrophysics, where it is demonstrated that it allows to represent the uncertainty in the detected blobs in a precise and physically interpretable way.

翻译：我们研究了不确定图像中的斑点检测问题，例如需要从噪声测量中推断的图像。受天文学应用启发的最新工作延伸，我们提出了一种方法，通过三维尺度空间中的区域来表示斑点位置和大小的不确定性。受经典管状方法（如紧弦算法）的启发，这些区域通过在高维管状约束下求解总变差泛函最小化问题的水平集获得。由此产生的非光滑优化问题求解难度较高，我们比较了多种数值方法并关联了约束总变差去噪的相关文献。最后，该方法在去卷积和天体物理相关模型的数值实验中得到了验证，结果表明它能以精确且具有物理可解释性的方式表示检测到的斑点的不确定性。