Total Variation Denoising (TVD) is a mathematical technique used in image processing and signal processing to remove noise from images while preserving important features such as edges. The underlying idea of TVD is to minimize the total variation of an image, which is a measure of its smoothness, while still attempting to fit the observed noisy data.
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For on a bounded domain, total variation denoising has a unique minimizer. A bounded minimizing sequence has a weakly convergent subsequence; variation is a supremum of weakly continuous test-function pairings and the fidelity has weak lower semicontinuity. The strictly convex fidelity proves uniqueness. Bounded-variation contraction under clipping proves that data range bounds pass to the minimizer.