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Segmentation of monotone data by Kobayashi-Warren-Carter type total variation energies

Yoshikazu Giga, Ayato Kubo, Hirotoshi Kuroda, Koya Sakakibara

Abstract

We consider a Kobayashi-Warren-Carter (KWC) type total variation energy with a fidelity term. Since the energy is non-convex, the profiles of minimizers are quite different from those of the original Rudin-Osher-Fatemi energy. In one-dimensional setting, we prove that KWC type energy (and its generalization) with fidelity must have a piecewise constant minimizer if the data in fidelity is bounded not necessarily in $BV$. Moreover, we give quantitative estimates of the energy for a monotone data in fidelity. This estimate shows that any minimizer must be piecewise constant with an improved estimate of the number of jumps for a monotone data. We also show the non-uniqueness of minimizers. Since this energy is useful from the point of segmentation or clustering, we compare with results of segmentation by the original Rudin-Osher-Fatemi energy and Mumford-Shah energy.

Segmentation of monotone data by Kobayashi-Warren-Carter type total variation energies

Abstract

We consider a Kobayashi-Warren-Carter (KWC) type total variation energy with a fidelity term. Since the energy is non-convex, the profiles of minimizers are quite different from those of the original Rudin-Osher-Fatemi energy. In one-dimensional setting, we prove that KWC type energy (and its generalization) with fidelity must have a piecewise constant minimizer if the data in fidelity is bounded not necessarily in . Moreover, we give quantitative estimates of the energy for a monotone data in fidelity. This estimate shows that any minimizer must be piecewise constant with an improved estimate of the number of jumps for a monotone data. We also show the non-uniqueness of minimizers. Since this energy is useful from the point of segmentation or clustering, we compare with results of segmentation by the original Rudin-Osher-Fatemi energy and Mumford-Shah energy.

Paper Structure

This paper contains 9 sections, 24 theorems, 158 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Approximation
  3. Basic estimates
  4. Quantitative estimates
  5. Numerical Experiments
  6. Verification of Theorem \ref{['TSJ']}: Linear Data
  7. Verification of Corollary \ref{['CSJ']}: Non-uniqueness of Minimizers
  8. Segmentation of Non-monotone Oscillating Data
  9. Robust Denoising of Piecewise Constant Signals

Key Result

Theorem 1.1

Assume that $K$ satisfies (K1), (K2) and (K3). Assume that $g\in C[a,b]$. Let $M>0$ be a number such that Let $U\in BV(a,b)$ be a minimizer of $TV_{Kg}$. Then $U$ must be a piecewise constant function with finitely many jumps satisfying $\min g\le U\le\max g$ on $[a,b]$. Let $m$ be the number of jumps of $U$. Then with $A_M:=\min\left\{ c_M / M, C_M \right\}$, where $C_M$ is in (K2) while $c_M$

Figures (9)

  • Figure 1: the graph of $u^{-1}$ and $u_m^{-1}$
  • Figure 2: $U_0$ and $U_\delta$
  • Figure 3: the graphs of $g$ and $U$
  • Figure 4: the graph of $u^z$
  • Figure 5: the graph of $-Q'_1$
  • ...and 4 more figures

Theorems & Definitions (45)

  • Theorem 1.1: GKKOS
  • Lemma 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Theorem 1.6
  • Corollary 1.7
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • ...and 35 more