arXiv subjects

Hirotoshi Kuroda

Publications and source records attributed to Hirotoshi Kuroda.

6 recordsLinked to original sources

arXiv2026

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

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.

math.AP↗

arXiv2026

On segmentation by total variation type energies of Kobayashi-Warren-Carter type with fidelity

We consider a total variation type energy which measures the jump discontinuities different from usual total variation energy. Such a type of energy is obtained as a singular limit of the Kobayashi-Warren-Carter energy with minimization with respect to the order parameter. We consider the Rudin-Osher-Fatemi type energy by replacing relaxation term by this type of total variation energy. We show that all minimizers are piecewise constant if the data function in the fidelity term is continuous in one-dimensional setting. Moreover, the number of jumps is bounded by an explicit constant involving a constant related to the fidelity. This is quite different from conventional Rudin-Osher-Fatemi energy where a minimizer has no jumps if the data has no jumps. Our results give an upper bound of the number of segments in a segmentation problem. The existence of a minimizer is guaranteed in multi-dimensional setting when the data is bounded.

math.AP↗

arXiv2024

A few topics on total variation flows

Total variation gradient flows are important in several applied fields, including image analysis and materials science. In this paper, we review a few basic topics including definition of a solution, explicit examples and the notion of calibrability, finite time extinction, and some regularity properties of solutions. We focus on the second-order flow (possibly with weights) and the fourth-order flow. We also discuss the fractional cases.

math.AP↗

arXiv2023

Fractional time differential equations as a singular limit of the Kobayashi-Warren-Carter system

This paper is concerned with a singular limit of the Kobayashi-Warren-Carter system, a phase field system modelling the evolutions of structures of grains. Under a suitable scaling, the limit system is formally derived when the interface thickness parameter tends to zero. Different from many other problems, it turns out that the limit system is a system involving fractional time derivatives, although the original system is a simple gradient flow. A rigorous derivation is given when the problem is reduced to a gradient flow of a single-well Modica-Mortola functional in a one-dimensional setting.

math.AP↗

arXiv2023

The fourth-order total variation flow in $\mathbb{R}^n$

We define rigorously a solution to the fourth-order total variation flow equation in $\mathbb{R}^n$. If $n\geq3$, it can be understood as a gradient flow of the total variation energy in $D^{-1}$, the dual space of $D^1_0$, which is the completion of the space of compactly supported smooth functions in the Dirichlet norm. However, in the low dimensional case $n\leq2$, the space $D^{-1}$ does not contain characteristic functions of sets of positive measure, so we extend the notion of solution to a larger space. We characterize the solution in terms of what is called the Cahn-Hoffman vector field, based on a duality argument. This argument relies on an approximation lemma which itself is interesting. We introduce a notion of calibrability of a set in our fourth-order setting. This notion is related to whether a characteristic function preserves its form throughout the evolution. It turns out that all balls are calibrable. However, unlike in the second-order total variation flow, the outside of a ball is calibrable if and only if $n\neq2$. If $n\neq2$, all annuli are calibrable, while in the case $n=2$, if an annulus is too thick, it is not calibrable. We compute explicitly the solution emanating from the characteristic function of a ball. We also provide a description of the solution emanating from any piecewise constant, radially symmetric datum in terms of a system of ODEs.

math.AP↗

arXiv2011

The Mosco convergence of Dirichlet forms approximating the Laplace operators with the delta potential on thin domains

We consider the convergent problems of Dirichlet forms associated with the Laplace operators on thin domains. This problem appears in the field of quantum waveguides. We study that a sequence of Dirichlet forms approximating the Laplace operators with the delta potential on thin domains Mosco converges to the form associated with the Laplace operator with the delta potential on the graph in the sense of Gromov-Hausdorff topology. From this results we can make use of many results established by Kuwae and Shioya about the convergence of the semigroups and resolvents generated by the infinitesimal generators associated with the Dirichlet forms.

math.AP↗