Closed-Form Approximation of the Total Variation Proximal Operator

Chandler, Edward P.; Shoushtari, Shirin; Wohlberg, Brendt Egon; Kamilov, Ulugbek S.

doi:10.1109/TCI.2025.3603689

Closed-Form Approximation of the Total Variation Proximal Operator

Journal Article · Thu Aug 28 00:00:00 EDT 2025 · IEEE Transactions on Computational Imaging

DOI:https://doi.org/10.1109/TCI.2025.3603689· OSTI ID:2999945

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Washington Univ., St. Louis, MO (United States)
Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)

Total variation (TV) is a widely used function for regularizing imaging inverse problems that is particularly appropriate for images whose underlying structure is piecewise constant. TV regularized optimization problems are typically solved using proximal methods, but the way in which they are applied is constrained by the absence of a closed-form expression for the proximal operator of the TV function. A closed-form approximation of the TV proximal operator has previously been proposed, but its accuracy was not theoretically explored in detail. Here, we address this gap by making several new theoretical contributions, proving that the approximation leads to a proximal operator of some convex function, it is equivalent to a gradient descent step on a smoothed version of TV, and that its error can be fully characterized and controlled with its scaling parameter. We experimentally validate our theoretical results on image denoising and sparse-view computed tomography (CT) image reconstruction.

View Accepted Manuscript (DOE)

Research Organization:: Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)

Sponsoring Organization:: USDOE Laboratory Directed Research and Development (LDRD) Program; USDOE National Nuclear Security Administration (NNSA); USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR)

Grant/Contract Number:: 89233218CNA000001

OSTI ID:: 2999945

Alternate ID(s):: OSTI ID: 2999956

Report Number(s):: LA-UR--24-32988; LA-UR--24-33356; 10.1109/TCI.2025.3603689

Journal Information:: IEEE Transactions on Computational Imaging, Journal Name: IEEE Transactions on Computational Imaging Vol. 11; ISSN 2334-0118; ISSN 2573-0436; ISSN 2333-9403

Publisher:: Institute of Electrical and Electronics Engineers (IEEE)Copyright Statement

Country of Publication:: United States

Language:: English

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Closed-Form Approximation of the Total Variation Proximal Operator

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