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On the ROF Model in Rectilinear Anisotropy: Piecewise Constant Approximation and Universal Minimality
Clemens Kirisits1,2, Eric Setterqvist3
1Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria.
Applied Mathematics and Optimization
|August 6, 2026
Summary
This study analyzes the Rudin-Osher-Fatemi (ROF) functional, proving convergence rates for its discrete minimizers. These findings advance image processing and inverse problems by refining error bounds for the ROF model.
Area of Science:
- Image processing
- Inverse problems
- Variational methods
Background:
- The Rudin-Osher-Fatemi (ROF) functional is a cornerstone in image denoising and restoration.
- Understanding the convergence properties of its discrete approximations is crucial for practical applications.
Purpose of the Study:
- To establish rigorous error bounds for the L2 distance between the ROF functional's continuous minimizer and its discrete counterpart on rectilinear grids.
- To investigate the impact of data regularity (Lq space) and dimensionality on these convergence rates.
Main Methods:
- Analysis of the L2 distance between continuous and discrete ROF minimizers.
- Derivation of Lq estimates for the ROF minimizer.
- Exploitation of a universal minimality property of the ROF minimizer.
Main Results:
- Proved convergence rates of O(h^(1/2 - q'/2q)) in dimensions d >= 1 for Lq data (q >= 2).
- Achieved improved rates of O(h^(1/2 - 1/2q)) in the 1D case.
- Demonstrated a universal minimality property of the ROF minimizer in finite and infinite dimensions.
Conclusions:
- The study provides theoretical guarantees for the accuracy of discrete ROF approximations in image processing.
- The established error bounds are essential for selecting appropriate grid sizes and understanding the fidelity of denoised images.
- The universal minimality property offers a broader perspective on the ROF minimizer's behavior and its relation to other convex functionals.
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