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Discretization of Total Variation in Optimization with Integrality Constraints

Annika Schiemann, Paul Manns

TL;DR

This work advances the discretization of the dual formulation of the total variation term with Raviart--Thomas functions which is known from literature for certain convex problems and introduces discretizations of infinite-dimensional optimization problems with total variation regularization and integrality constraints on the optimization variables.

Abstract

We introduce discretizations of infinite-dimensional optimization problems with total variation regularization and integrality constraints on the optimization variables. We advance the discretization of the dual formulation of the total variation term with Raviart--Thomas functions which is known from literature for certain convex problems. Since we have an integrality constraint, the previous analysis from Caillaud and Chambolle [10] does not hold anymore. Even weaker $Γ$-convergence results do not hold anymore because the recovery sequences generally need to attain non-integer values to recover the total variation of the limit function. We solve this issue by introducing a discretization of the input functions on an embedded, finer mesh. A superlinear coupling of the mesh sizes implies an averaging on the coarser mesh of the Raviart--Thomas ansatz, which enables to recover the total variation of integer-valued limit functions with integer-valued discretized input functions. Moreover, we are able to estimate the discretized total variation of the recovery sequence by the total variation of its limit and an error depending on the mesh size ratio. For the discretized optimization problems, we additionally add a constraint that vanishes in the limit and enforces compactness of the sequence of minimizers, which yields their convergence to a minimizer of the original problem. This constraint contains a degree of freedom whose admissible range is determined. Its choice may have a strong impact on the solutions in practice as we demonstrate with an example from imaging.

Discretization of Total Variation in Optimization with Integrality Constraints

TL;DR

This work advances the discretization of the dual formulation of the total variation term with Raviart--Thomas functions which is known from literature for certain convex problems and introduces discretizations of infinite-dimensional optimization problems with total variation regularization and integrality constraints on the optimization variables.

Abstract

We introduce discretizations of infinite-dimensional optimization problems with total variation regularization and integrality constraints on the optimization variables. We advance the discretization of the dual formulation of the total variation term with Raviart--Thomas functions which is known from literature for certain convex problems. Since we have an integrality constraint, the previous analysis from Caillaud and Chambolle [10] does not hold anymore. Even weaker -convergence results do not hold anymore because the recovery sequences generally need to attain non-integer values to recover the total variation of the limit function. We solve this issue by introducing a discretization of the input functions on an embedded, finer mesh. A superlinear coupling of the mesh sizes implies an averaging on the coarser mesh of the Raviart--Thomas ansatz, which enables to recover the total variation of integer-valued limit functions with integer-valued discretized input functions. Moreover, we are able to estimate the discretized total variation of the recovery sequence by the total variation of its limit and an error depending on the mesh size ratio. For the discretized optimization problems, we additionally add a constraint that vanishes in the limit and enforces compactness of the sequence of minimizers, which yields their convergence to a minimizer of the original problem. This constraint contains a degree of freedom whose admissible range is determined. Its choice may have a strong impact on the solutions in practice as we demonstrate with an example from imaging.
Paper Structure (12 sections, 24 theorems, 54 equations, 4 figures, 3 tables, 1 algorithm)

This paper contains 12 sections, 24 theorems, 54 equations, 4 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Discretized total variation
  3. Discretization of the total variation
  4. Lim inf inequality for $\mathop{\mathrm{TV^h}}\limits$
  5. Lim sup inequality for $\mathop{\mathrm{TV^h}}\limits$
  6. Discretization of problem (P)
  7. Existence of minimizers and compactness
  8. Lim sup inequality
  9. Outer Approximation
  10. Numerical experiments
  11. Conclusion and outlook
  12. Proof of \ref{['thm:tvh_tv_d=2']}

Key Result

Lemma 2.1

Let $\{w^h\}_{h>0} \subset L^1_W(\Omega)$ be given such that $w^h \to w$ in $L^1(\Omega)$ as $h \searrow 0$. Then we have that $w \in L^1_W(\Omega)$ and $w^h \to w$ in $L^p(\Omega)$ for all $1 \leq p < \infty$.

Figures (4)

  • Figure 1: Example for the construction in \ref{['expl:TVh']} with $k = 1$. The level sets of the limit function $w$ are separated by the solid line and the level sets of the rounding of $\Pi_{P0^h} w$ are separated by the dotted line. Nonzero fluxes of $\phi$ are indicated next to the corresponding edges of the grid cells.
  • Figure 2: Errors $| \mathop{\mathrm{TV}}\limits(w) - {\mathop{\mathrm{TV}}\limits}^\tau(w^\tau)|$ and $|\mathop{\mathrm{TV}}\limits(w) - \mathop{\mathrm{TV^h}}\limits(w^\tau) |$ for the pairs $(h,\tau)$ listed in \ref{['fig:table_TV']} with $w^\tau = R^W_{P0^\tau}(w)$.
  • Figure 3: Original image, noisy image, and resulting images obtained by applying \ref{['alg:outer_approx']} to the discretization of \ref{['eq:P_I']} with $(h,\tau) = (\frac{1}{64},\frac{1}{512})$ and different values $c$.
  • Figure 4: Example for a function $w \in \mathop{\mathrm{BV_W}}\limits(\Omega)$ with level sets that have polygonal boundaries with the notations from the proof of \ref{['thm:TV_polygonal']}.

Theorems & Definitions (57)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Remark 2.6
  • Lemma 2.7
  • proof
  • ...and 47 more