A bounded vector field $p$ satisfying this equation immediately provides an upper bound for the star-norm of $f$. Hierarchical constructions developed by Eitan Tadmor in connection with the Bourgain-Brezis problem give a constructive way of producing such bounded solutions. However, obtaining sharp and computationally useful estimates of the star-norm of a general image remains an interesting open problem.

 

I will use this question as a starting point for a related line of work based on multiscale total-variation methods. Hierarchical $(BV,L^2)$ decompositions lead, in a continuous-scale limit, to total-variation flow. For weighted TV flow, the accumulated flow parameter gives an explicit upper bound on the weighted star-norm of the residual and, after finite-time extinction, on the star-norm of the image itself. Introducing spatially varying weights also makes it possible to suppress noise while reducing diffusion across significant edges.

 

I will then discuss how these ideas arise in electron backscatter diffraction (EBSD), where images encode crystallographic orientations of polycrystalline materials. In this setting, denoising and restoration must account for crystallographic symmetry while preserving grain boundaries, whose geometry carries important physical information. I will describe our use of total-variation and weighted total-variation methods for restoring EBSD orientation maps and extracting their geometric structure.

 

I will conclude with several open questions, including sharper and more computable estimates of the star-norm, connections between hierarchical bounded solutions and TV-based multiscale representations, and variational methods that better preserve the geometry of crystallographic images. These problems suggest several directions in which techniques from nonlinear analysis, geometric analysis, and image processing may interact.