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Adaptive double-phase Rudin--Osher--Fatemi denoising model

Wojciech Górny, Michał Łasica, Alexandros Matsoukas

TL;DR

The work tackles staircasing in ROF denoising by introducing an adaptive double-phase regularizer that blends 1- and 2-growth through a locally weighted term. A two-step procedure computes $u_{ROF}$, constructs a weight $w(x)=W(|\nabla\tilde{u}_{ROF}|)$ from a mollified ROF solution, and solves a double-phase variational problem $\min_u \int |\nabla u| \,dx + \int w(x)|\nabla u|^2 \,dx + \tfrac{1}{2\lambda}\int |u-g|^2 \,dx$, yielding a unique $u_{dpROF}$. The method is implemented with an accelerated Chambolle-Pock algorithm and explicit proximal updates, enabling robust 1D and 2D denoising tests. Results show reduced staircasing and preserved edges, with competitive SSIM/PSNR compared to Huber-ROF, across synthetic and natural images and varying noise levels, highlighting practical improvements for adaptive regularization in image denoising.

Abstract

We propose a new image denoising model based on a variable-growth total variation regularization of double-phase type with adaptive weight. It is designed to reduce staircasing with respect to the classical Rudin--Osher--Fatemi model, while preserving the edges of the image in a similar fashion. We implement the model and test its performance on synthetic and natural images in 1D and 2D over a range of noise levels.

Adaptive double-phase Rudin--Osher--Fatemi denoising model

TL;DR

The work tackles staircasing in ROF denoising by introducing an adaptive double-phase regularizer that blends 1- and 2-growth through a locally weighted term. A two-step procedure computes , constructs a weight from a mollified ROF solution, and solves a double-phase variational problem , yielding a unique . The method is implemented with an accelerated Chambolle-Pock algorithm and explicit proximal updates, enabling robust 1D and 2D denoising tests. Results show reduced staircasing and preserved edges, with competitive SSIM/PSNR compared to Huber-ROF, across synthetic and natural images and varying noise levels, highlighting practical improvements for adaptive regularization in image denoising.

Abstract

We propose a new image denoising model based on a variable-growth total variation regularization of double-phase type with adaptive weight. It is designed to reduce staircasing with respect to the classical Rudin--Osher--Fatemi model, while preserving the edges of the image in a similar fashion. We implement the model and test its performance on synthetic and natural images in 1D and 2D over a range of noise levels.

Paper Structure

This paper contains 11 sections, 2 theorems, 40 equations, 18 figures, 2 tables, 4 algorithms.

Table of Contents

  1. Keywords
  2. Introduction
  3. Adaptive double-phase ROF model
  4. Numerical preliminaries
  5. Implementation of the double-phase model
  6. Results of one-dimensional experiments
  7. Heuristics and visual presentation
  8. Performance across different noise levels
  9. Comparison with the Huber-ROF model
  10. Results of two-dimensional experiments
  11. Conclusions

Key Result

Theorem 3.1

Denote $L = \| K \|$. Let $\theta = 1$. If the saddle-point problem eq:chambollepockprimaldual has at least one solution and $\tau \sigma L^2 < 1$, then the sequence $\{ (x^n,y^n) \}$ is bounded and there exists a saddle-point $(x^*,y^*)$ such that $x^n \rightarrow x^*$ and $y^n \rightarrow y^*$.

Figures (18)

  • Figure 1: Reconstruction of a one-dimensional synthetic image with classical ROF, Huber-ROF and adaptive double-phase ROF models
  • Figure 2: Reconstruction of a one-dimensional natural image with classical ROF, Huber-ROF and adaptive double-phase ROF models
  • Figure 3: Performance of the classical ROF model, the adaptive double phase ROF model, and a double-phase ROF model with weight calculated directly from the noisy image for various noise levels (for the 'saw' image: $d_{\rm TV}^{\rm image}$ and $d_{L^2}^{\rm image}$; for the 'cut1' image: $d_{\rm TV}^{\rm image}$)
  • Figure 4: Performance of the classical ROF model, the adaptive double phase ROF model, and the Huber-ROF model with $\alpha = 0.01$ (for the 'saw' image: $d_{\rm TV}^{\rm image}$; for the 'cut1' image: $d_{\rm TV}^{\rm image}$ and ${\rm PSNR}$) for the noise level $\sigma = 0.05$.
  • Figure 5: Original 'double gradient' synthetic image; image with added gaussian noise ($\sigma=0.01$); denoising results corresponding to the maximum SSIM values, respectively: classical ROF ($\lambda = 0.24$, ${\rm SSIM} = 0.926$, ${\rm PSNR} = 32.81$), double-phase ROF ($\lambda = 0.12$, ${\rm SSIM} = 0.962$, ${\rm PSNR} = 34.38$), and Huber-ROF with $\alpha = 0.01$ ($\lambda = 0.26$, ${\rm SSIM} = 0.963$, ${\rm PSNR} = 34.02$).
  • ...and 13 more figures

Theorems & Definitions (6)

  • Remark 2.1
  • Remark 2.2
  • Theorem 3.1
  • Example 1
  • Proposition 1
  • proof