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Michał Łasica

Publications and source records attributed to Michał Łasica.

16 recordsLinked to original sources
arXiv2026

Adaptive double-phase Rudin--Osher--Fatemi denoising model

Even though more than 30 years have passed since the seminal Rudin--Osher--Fatemi (ROF) paper on total variation (TV) denoising, it remains relevant due to its simplicity, robustness and interpretability. However, it is known to suffer from artifacts such as the staircasing effect. Many variants of the model have been proposed with the aim of countering this. Recently, against the backdrop of immense research output on double-phase problems in the mathematical analysis community, a double-phase type integral functional, comprising of TV and a weighted term of quadratic growth, was suggested as a regularizer for image restoration. Here, we propose an adaptive variant of the ROF denoising model based on that regularizer. Variable growth of the double-phase functional allows for qualitatively different behavior at image contours, which are captured by an initial ROF reconstruction step. The model is designed to reduce staircasing with respect to the classical ROF model, while preserving the edges of the image in a similar fashion. We derive a closed-form resolvent formula and adapt the primal-dual Chambolle--Pock scheme for the numerical solution of the model. We also propose a practical noise-dependent parameter prescription and evaluate its performance on synthetic and natural images over a range of noise levels. Compared to established models with similar interpretability, we observe an improved or similar performance in terms of similarity metrics SSIM, PSNR, and LPIPS, while the staircasing effect is visibly reduced.

eess.IV↗
arXiv2025

Mosco convergence framework for singular limits of gradient flows on Hilbert spaces with applications

We consider the question of convergence of a sequence of gradient flows defined on different Hilbert spaces. In order to give meaning to this idea, we introduce a notion of connecting operators. This permits us to generalize the concept of Mosco convergence of functionals to our present setting, and state a desired convergence result for gradient flows, which we then prove. We present a variety of examples, including thin domains, dynamic boundary conditions, and discrete-to-continuum limits.

math.AP↗
arXiv2025

Total variation flow of curves in Riemannian manifolds

We consider the functional of total variation of maps from an interval into a Riemannian submanifold of $\mathbb R^N$. We define a notion of strong solution to the system of equations corresponding to the $L^2$-gradient flow of this functional. We prove global existence of strong solutions for initial data of bounded variation. We show that the solutions satisfy a variational equality, and deduce uniqueness in the case of non-positive sectional curvature. We prove convergence of strong solutions to a constant map in finite time.

math.AP↗
arXiv2025

Euler-Lagrange equations for variable-growth total variation

We consider a class of integral functionals with Musielak-Orlicz type variable growth, possibly linear in some regions of the domain. This includes $p(x)$ power-type integrands with $p(x)\ge 1$ as well as double-phase $p\!-\!q$ integrands with $p=1$. The main goal of this paper is to identify the $L^2$-subdifferential of the functional, including a local characterisation in terms of a variant of the Anzellotti product defined through the Young's inequality. As an application, we obtain the Euler-Lagrange equation for the variant of the Rudin-Osher-Fatemi image denoising problem with variable growth regularising term.

math.AP↗
arXiv2024

Inclusion and estimates for the jumps of minimizers in variational denoising

We study stability and inclusion of the jump set of minimizers of convex denoising functionals, such as the celebrated "Rudin-Osher-Fatemi" functional, for scalar or vectorial signals. We show that under mild regularity assumptions on the data fidelity term and the regularizer, the jump set of the minimizer is essentially a subset of the original jump set. Moreover, we give an estimate on the magnitude of jumps in terms of the data. This extends old results, in particular of the first author (with V. Caselles and M. Novaga) and of T. Valkonen, to much more general cases. We also consider the case where the original datum has unbounded variation, and define a notion of its jump set which, again, must contain the jump set of the solution.

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

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↗
arXiv2022

Local estimates for vectorial Rudin-Osher-Fatemi type problems in one dimension

We consider the Rudin-Osher-Fatemi variational denoising model with general regularizing term in one-dimensional, vector-valued setting. We obtain local estimates on the singular part of the variation measure of the minimizer in terms of the singular part of the variation measure of the datum. In the case of homogeneous regularizer, we prove local estimates on the whole variation measure of the minimizer and deduce an analogous result for the gradient flow of the regularizer.

math.AP↗
arXiv2021

Existence of $W^{1,1}$ solutions to a class of variational problems with linear growth on convex domains

We consider a class of convex integral functionals composed of a term of linear growth in the gradient of the argument, and a fidelity term involving $L^2$ distance from a datum. Such functionals are known to attain their infima in the $BV$ space. Under the assumption that the domain of integration is convex, we prove that if the datum is in $W^{1,1}$, then the functional has a minimizer in $W^{1,1}$. In fact, the minimizer inherits $W^{1,p}$ regularity from the datum for any $p \in [1, +\infty]$. We also obtain a quantitative bound on the singular part of the gradient of the minimizer in the case that the datum is in $BV$. We infer analogous results for the gradient flow of the underlying functional of linear growth. We admit any convex integrand of linear growth.

math.AP↗
arXiv2018

A local estimate for vectorial total variation minimization in one dimension

Let $\boldsymbol u$ be the minimizer of vectorial total variation ($VTV$) with $L^2$ data-fidelity term on an interval $I$. We show that the total variation of $\boldsymbol u$ over any subinterval of $I$ is bounded by that of the datum over the same subinterval. We deduce analogous statement for the vectorial total variation flow on $I$.

math.AP↗
arXiv2017

Regular 1-harmonic flow

We consider the 1-harmonic flow of maps from a bounded domain into a submanifold of a Euclidean space, i.e. the gradient flow of the total variation functional restricted to maps taking values in the manifold. We restrict ourselves to Lipschitz initial data. We prove uniqueness and, in the case of a convex domain, local existence of solutions to the flow equations. If the target manifold has non-positive sectional curvature or in the case that the datum is small, solutions are shown to exist globally and to become constant in finite time. We also consider the case where the domain is a compact Riemannian manifold without boundary, solving the homotopy problem for 1-harmonic maps under some assumptions.

math.AP↗
arXiv2017

Total variation denoising in $l^1$ anisotropy

We aim at constructing solutions to the minimizing problem for the variant of Rudin-Osher-Fatemi denoising model with rectilinear anisotropy and to the gradient flow of its underlying anisotropic total variation functional. We consider a naturally defined class of functions piecewise constant on rectangles (PCR). This class forms a strictly dense subset of the space of functions of bounded variation with an anisotropic norm. The main result shows that if the given noisy image is a PCR function, then solutions to both considered problems also have this property. For PCR data the problem of finding the solution is reduced to a finite algorithm. We discuss some implications of this result, for instance we use it to prove that continuity is preserved by both considered problems.

math.AP↗
arXiv2015

A conditional regularity result for p-harmonic flows

We prove an $\varepsilon$-regularity result for a wide class of parabolic systems $$ u_t-\text{div}\big(|\nabla u|^{p-2}\nabla u) = B(u, \nabla u) $$ with the right hand side $B$ growing like $|\nabla u|^p$. It is assumed that the solution $u(t,\cdot)$ is uniformly small in the space of functions of bounded mean oscillation. The crucial tool is provided by a sharp nonlinear version of the Gagliardo-Nirenberg inequality which has been used earlier in an elliptic context by T. Rivière and the last named author.

math.AP↗
arXiv2015

Analysis of solutions to a model parabolic equation with very singular diffusion

We consider a singular parabolic equation of form \[ u_t = u_{xx} + \fracα{2}(\mathrm{sgn}\,u_x)_x \] with periodic boundary conditions. Solutions to this kind of equations exhibit competition between smoothing due to one-dimensional Laplace operator and tendency to create flat facets due to strongly nonlinear operator $(\mathrm{sgn}\,u_x)_x$ coming from the total variation flow. We present results concerning analysis of qualitative behaviour and regularity of the solutions. Our main result states that locally (between moments when facets merge), the evolution is described by a system of free boundary problems for $u$ in intervals between facets coupled with equations of evolution of facets. In particular, we provide a proper law governing evolution of endpoints of facets. This leads to local smoothness of the motion of endpoints and the unfaceted part of the solution.

math.AP↗
arXiv2013

On steady solutions to vacuumless Newtonian models of compressible flow

We prove the existence of weak solutions to the steady compressible Navier-Stokes system in the barotropic case for a class of pressure laws singular at vacuum. We consider the problem in a bounded domain in R^2 with slip boundary conditions. Due to appropriate construction of approximate solutions used in proof, obtained density is bounded away from 0 (and infinity). Owing to a classical result by P.-L. Lions, this implies that density and gradient of velocity are at least Hölder continuous, which does not generally hold for the classical isentropic model in the presence of vacuum.

math.AP↗