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An Adaptive Finite Difference Method for Total Variation Minimization

Thomas Jacumin, Andreas Langer

TL;DR

It turns out that a finer discretization may lead to a higher value of the discrete total variation for a given function.

Abstract

In this paper, we propose an adaptive finite difference scheme in order to numerically solve total variation type problems for image processing tasks. The automatic generation of the grid relies on indicators derived from a local estimation of the primal-dual gap error. This process leads in general to a non-uniform grid for which we introduce an adjusted finite difference method. Further we quantify the impact of the grid refinement on the respective discrete total variation. In particular, it turns out that a finer discretization may lead to a higher value of the discrete total variation for a given function. To compute a numerical solution on non-uniform grids we derive a semi-smooth Newton algorithm in 2D for scalar and vector-valued total variation minimization. We present numerical experiments for image denoising and the estimation of motion in image sequences to demonstrate the applicability of our adaptive scheme.

An Adaptive Finite Difference Method for Total Variation Minimization

TL;DR

It turns out that a finer discretization may lead to a higher value of the discrete total variation for a given function.

Abstract

In this paper, we propose an adaptive finite difference scheme in order to numerically solve total variation type problems for image processing tasks. The automatic generation of the grid relies on indicators derived from a local estimation of the primal-dual gap error. This process leads in general to a non-uniform grid for which we introduce an adjusted finite difference method. Further we quantify the impact of the grid refinement on the respective discrete total variation. In particular, it turns out that a finer discretization may lead to a higher value of the discrete total variation for a given function. To compute a numerical solution on non-uniform grids we derive a semi-smooth Newton algorithm in 2D for scalar and vector-valued total variation minimization. We present numerical experiments for image denoising and the estimation of motion in image sequences to demonstrate the applicability of our adaptive scheme.

Paper Structure

This paper contains 15 sections, 2 theorems, 93 equations, 11 figures, 2 tables, 4 algorithms.

Table of Contents

  1. Mathematics Subject Classification
  2. Introduction
  3. Formulation of the Model
  4. Problem Discretization on Non-Uniform Grid
  5. Finite Difference Method on Non-uniform Grid
  6. Impact of Refinement on the Total Variation
  7. Primal-Dual Gap Error Estimator
  8. Semi-Smooth Newton Method
  9. Numerical Experiments
  10. Application to Image Reconstruction
  11. Noiseless Case
  12. Gaussian Denoising
  13. Application to Optical Flow Computation
  14. Conclusion
  15. Proof of the Primal-Dual Gap Error Estimation

Key Result

Proposition 6.1

Let $\mathbf{u}\in H^1(\Omega)^m$ be a solution of eq:primal and $\mathbf{u_h}\in\mathbb{R}^{mN}$ be a solution of eq:primal_h. Then, for all $\mathbf{v_h}\in\mathbb{R}^{mN}$ and $(q_{h,1},\mathbf{q_{h,2}})\in\mathbb{R}^{N}\times\mathbb{R}^{2mN}$, we have where $c\geq 0$ and

Figures (11)

  • Figure 1: Example of an adaptive grid.
  • Figure 2: Illustration of a regular node with $u:=u(\mathbf{x})$, $u_\text{E}:=u(\mathbf{x_\text{E}})$ and $u_\text{S}:=u(\mathbf{x_\text{S}})$.
  • Figure 3: Illustration of a dangling node 1 with $u:=u(\mathbf{x})$. For (a): $u_\text{NE}:=u(\mathbf{x_\text{NE}})$, $u_\text{SE}:=u(\mathbf{x_\text{SE}})$ and $u':=\frac{1}{2}(u_\text{NE}+u_\text{SE})$. For (b): $u_\text{SW}:=u(\mathbf{x_\text{SW}})$, $u_\text{SE}:=u(\mathbf{x_\text{SE}})$ and $u':=\frac{1}{2}(u_\text{SW}+u_\text{SE})$.
  • Figure 4: Illustration of a dangling node 2 (a)-(b) and dangling node 3 (c)-(d) with $u:=u(\mathbf{x})$. For (a): $u_\text{E}:=u(\mathbf{x_\text{E}})$, $u_\text{N}:=u(\mathbf{x_\text{N}})$ and $u':=2u_\text{E}-u_\text{N}$. For (b): $u_\text{S}:=u(\mathbf{x_\text{S}})$, $u_\text{E}:=u(\mathbf{x_\text{E}})$ and $u':=2u_\text{S}-u_\text{E}$. For (c): $u_\text{E}:=u(\mathbf{x_\text{E}})$, $u_\text{S}:=u(\mathbf{x_\text{S}})$ and $u':=2u_\text{E}-u_\text{S}$. For (d): $u_\text{S}:=u(\mathbf{x_\text{S}})$, $u_\text{W}:=u(\mathbf{x_\text{W}})$ and $u':=2u_\text{S}-u_\text{W}$.
  • Figure 5: One refinement example with $\Omega_h$ on the left and a one-refinement $\widetilde{\Omega_h}$ on the right.
  • ...and 6 more figures

Theorems & Definitions (5)

  • Example
  • Proposition 6.1
  • Remark
  • Lemma A.1
  • proof : Proof of Proposition \ref{['prop:pd-gap-error']}