Functional Analysis and Parallel Domain Decomposition for the TV-Stokes Model

TL;DR

This paper provides a rigorous mathematical foundation for the TV-Stokes image denoising model, analyzes its structure, identifies inconsistencies, and introduces a parallel domain decomposition method for efficient large-scale implementation.

Contribution

It establishes the first functional-analytic framework for TV-Stokes, corrects its formulation, and develops a parallelizable domain decomposition algorithm.

Findings

Mathematically consistent reformulation of TV-Stokes

Existence of orthogonal projection onto divergence-free space

Effective parallel domain decomposition method

Abstract

The TV-Stokes model is a two-step variational method for image denoising that combines the estimation of a divergence-free tangent field with total variation regularization in the first step and then uses that to reconstruct the image in the second step. Although effective in practice, its mathematical structure and potential for parallelization have remained unexplored. In this work, we establish a rigorous functional-analytic foundation for the TV-Stokes model. We formulate both steps in appropriate infinite-dimensional function spaces, derive their dual formulations, and analyze the compatibility and mathematical consistency of the coupled system. In particular, we identify analytical inconsistencies in the original formulation and demonstrate how an alternative model resolves them. We also examine the orthogonal projection onto the divergence-free subspace, proving its existence in a continuous setting and establishing consistency with its discrete counterpart. Building on this theoretical framework, we develop the first domain decomposition method for TV-Stokes by applying overlapping Schwarz-type iterations to the duals of both steps. Although the divergence-free constraint gives rise to a global projection operator in the continuous model, we show that it becomes locally computable in the discrete setting. This insight enables a fully parallelizable algorithm suitable for large-scale image processing in memory-constrained environments. Numerical experiments demonstrate the correctness of the domain decomposition approach and its usability in parallel image reconstruction.