  This work proposes the variable exponent Lebesgue modular as a replacement
for the 1-norm in total variation (TV) regularization. It allows the exponent
to vary with spatial location and thus enables users to locally select whether
to preserve edges or smooth intensity variations. In contrast to earlier work
using TV-like methods with variable exponents, the exponent function is here
computed offline as a fixed parameter of the final optimization problem,
resulting in a convex goal functional. The obtained formulas for the convex
conjugate and the proximal operators are simple in structure and can be
evaluated very efficiently, an important property for practical usability.
Numerical results with variable $L^p$ TV prior in denoising and tomography
problems on synthetic data compare favorably to total generalized variation
(TGV) and TV.
