Uncertainty Quantification in Image Segmentation Using the Ambrosio–Tortorelli Approximation of the Mumford–Shah Energy

The quantification of uncertainties in image segmentation based on the Mumford–Shah model is studied. The aim is to address the error propagation of noise and other error types in the original image to the restoration result and especially the reconstructed edges (sharp image contrasts). Analytically, we rely on the Ambrosio–Tortorelli approximation and discuss the existence of measurable selections of its solutions as well as sampling-based methods and the limitations of other popular methods. Numerical examples illustrate the theoretical findings.

Complex extreme measurable selections

We give a characterization of complex extreme measurable selections for a suitable set-valued map. We use this result to obtain necessary and sufficient conditions for a function to be a complex extreme point of the closed unit ball of Lp (ω, Σ, ν X), where (ω, σ, ν) is any positive, complete measure space, X is a separable complex Banach space, and 0 < p < ∞.