Dyadic partition-based training schemes for TV/TGV denoising

TL;DR

This work proves existence of minimizers for fixed discontinuous parameters under mild assumptions on the data, which lead to existence of finite optimal partitions, and establishes that these assumptions are equivalent to the commonly used box constraints on the parameters.

Abstract

Due to their ability to handle discontinuous images while having a well-understood behavior, regularizations with total variation (TV) and total generalized variation (TGV) are some of the best-known methods in image denoising. However, like other variational models including a fidelity term, they crucially depend on the choice of their tuning parameters. A remedy is to choose these automatically through multilevel approaches, for example by optimizing performance on noisy/clean image pairs. In this work, we consider such methods with space-dependent parameters which are piecewise constant on dyadic grids, with the grid itself being part of the minimization. We prove existence of minimizers for fixed discontinuous parameters under mild assumptions on the data, which lead to existence of finite optimal partitions. We further establish that these assumptions are equivalent to the commonly used box constraints on the parameters. On the numerical side, we consider a simple subdivision scheme for optimal partitions built on top of any other bilevel optimization method for scalar parameters, and demonstrate its improved performance on some representative test images when compared with constant optimized parameters.

Figures (6)

• Figure 1: Example of two partitions, $\mathscr{L}^*$ and $\bar{\mathscr{L}}^*$, that yield the same solution at Level 2.
• Figure 2: Oversmoothed denoising with a sharp change of weight and schemes corresponding to Level 2 of $(\mathscr{L}\!\mathscr{S})_{{TV\!}_{\omega}}$, $(\mathscr{L}\!\mathscr{S})_{{TV\!}_{\omega_\varepsilon}}$, and $(\mathscr{L}\!\mathscr{S})_{{TV-Fid}_\omega}$, from left to right. Top row: weights $\omega(x)=\omega(x_1)$. Bottom row: results with each denoising scheme and the corresponding (not optimal) weight.
• Figure 3: Synthetic example. Top row: Clean and noisy images $u_c, u_\eta$. Middle row, left to right: TV result with global parameter, partition and spatially-dependent $\lambda$ arising from Algorithm \ref{['alg:subdivision']}, and corresponding result with weighted fidelity. Bottom row: TGV results, same order as in the middle row and with $\alpha_0=1, \alpha_1 = 10$.
• Figure 4: Lighthouse example. Top row: Clean and noisy images $u_c, u_\eta$. Middle row, left to right: TV result with global parameter, partition and spatially-dependent $\lambda$ arising from Algorithm \ref{['alg:subdivision']}, and corresponding result with weighted fidelity. Bottom row: TGV results, same order as in the middle row and with $\alpha_0=1, \alpha_1 = 2$.
• Figure 5: Cameraman example. Top row: Clean and noisy images $u_c, u_\eta$. Middle row, left to right: TV result with global parameter, partition and spatially-dependent $\lambda$ arising from Algorithm \ref{['alg:subdivision']}, and corresponding result with weighted fidelity. Bottom row: TGV results, same order as in the middle row and with $\alpha_0=1, \alpha_1 = 10$.

Theorems & Definitions (69)

• Remark 1.1
• Definition 1.2: stopping criterion for the refinement of the admissible partitions
• Remark 1.3: box constraint as a stopping criterion
• Theorem 1.4: Equivalence between box constraint and stopping criterion
• Theorem 1.5: Existence of solutions to $(\mathscr{L}\!\mathscr{S})_{{TV\!}_\omega}$
• Proposition 1.6: On the energies in $(\mathscr{L}\!\mathscr{S})_{{TV\!}_{\omega_\varepsilon}}$ as $\varepsilon\to0^+$
• Theorem 1.7: Existence of solutions to $(\mathscr{L}\!\mathscr{S})_{{TGV\!}_\omega}$
• Theorem 1.8: Existence of solutions to $(\mathscr{L}\!\mathscr{S})_{{TGV-Fid}_\omega}$
• Theorem 3.1
• proof