We derive a new pointwise characterization of the subdifferential of the total variation (TV) functional. It involves a full trace operator which maps certain Lq - vectorfields to integrable functions with respect to the total variation measure of the derivative of a bounded variation function. This full trace operator extents a notion of normal trace, frequently used, for example, to characterize the total variation flow.
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Bredies et al. (2016) studied this question.
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