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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.
