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The Ambrosio–Tortorelli approximation scheme with weighted underlying metric is investigated. It is shown that it Γ-converges to a Mumford–Shah image segmentation functional depending on the weightω⁢d⁢x{\omega\,dx}, where ω is a special function of bounded variation, and on its values at the jumps.

Higher order Ambrosio–Tortorelli scheme with non-negative spatially dependent parameters