arXiv subjects

Julián Velasco

Publications and source records attributed to Julián Velasco.

10 recordsLinked to original sources

arXiv2025

Bifurcation in dynamic problems with seasonal succession

We investigate the bifurcation structure of equilibria in a class of non-autonomous ordinary differential equations governed by a season length parameter, $τ$, which determines the alternation between growth and decline dynamics. This structure models biological systems exhibiting seasonal variation, such as insect population dynamics or infectious disease transmission. Using the Crandall-Rabinowitz bifurcation theorem, we establish the existence of a critical threshold $τ^*$ at which a bifurcation from the extinction equilibrium occurs. We also explore the emergence of secondary bifurcations from, in general, explicitly unknown non-trivial equilibria which can only be treated numerically. Our results are illustrated with a two-species competitive Lotka-Volterra model for the growth season and a Malthusian model for the decline season for which primary and secondary bifurcations may be computed analytically, allowing the validation of numerical approximations. Our analysis shows how seasonality drives transitions between extinction of both populations, of only one, and coexistence of both populations.

math.DS↗

arXiv2024

On a cross-diffusion system arising in image denosing

We study a generalization of a cross-diffusion problem deduced from a nonlinear complex-variable diffusion model for signal and image denoising. We prove the existence of weak solutions of the time-independent problem with fidelity terms under mild conditions on the data problem. Then, we show that this translates on the well-posedness of a quasi-steady state approximation of the evolution problem, and also prove the existence of weak solutions of the latter under more restrictive hypothesis. We finally perform some numerical simulations for image denoising, comparing the performance of the cross-diffusion model and its corresponding scalar Perona-Malik equation.

math.AP↗

arXiv2024

Well-posedness of a cross-diffusion population model with nonlocal diffusion

We prove the existence and uniqueness of solution of a nonlocal cross-diffusion competitive population model for two species. The model may be considered as a version, or even an approximation, of the paradigmatic Shigesada-Kawasaki-Teramoto cross-diffusion model, in which the usual diffusion differential operator is replaced by an integral diffusion operator. The proof of existence of solutions is based on a compactness argument, while the uniqueness of solution is achieved through a duality technique.

math.AP↗

arXiv2021

Convergence of solutions of a rescaled evolution nonlocal cross-diffusion problem to its local diffusion counterpart

We prove that, under a suitable rescaling of the integrable kernel defining the nonlocal diffusion terms, the corresponding sequence of solutions of the Shigesada-Kawasaki-Teramoto nonlocal cross-diffusion problem converges to a solution of the usual problem with local diffusion. In particular, the result may be regarded as a new proof of existence of solutions for the local diffusion problem.

math.AP↗

arXiv2016

Well-posedness of a nonlinear integro-differential problem and its rearranged formulation

We study the existence and uniqueness of solutions of a nonlinear integro-differential problem which we reformulate introducing the notion of the decreasing rearrangement of the solution. A dimensional reduction of the problem is obtained and a detailed analysis of the properties of the solutions of the model is provided. Finally, a fast numerical method is devised and implemented to show the performance of the model when typical image processing tasks such as filtering and segmentation are performed.

cs.CV↗

arXiv2015

On a fast bilateral filtering formulation using functional rearrangements

We introduce an exact reformulation of a broad class of neighborhood filters, among which the bilateral filters, in terms of two functional rearrangements: the decreasing and the relative rearrangements. Independently of the image spatial dimension (one-dimensional signal, image, volume of images, etc.), we reformulate these filters as integral operators defined in a one-dimensional space corresponding to the level sets measures. We prove the equivalence between the usual pixel-based version and the rearranged version of the filter. When restricted to the discrete setting, our reformulation of bilateral filters extends previous results for the so-called fast bilateral filtering. We, in addition, prove that the solution of the discrete setting, understood as constant-wise interpolators, converges to the solution of the continuous setting. Finally, we numerically illustrate computational aspects concerning quality approximation and execution time provided by the rearranged formulation.

cs.CV↗

arXiv2014

On a new formulation of nonlocal image filters involving the relative rearrangement

Nonlocal filters are simple and powerful techniques for image denoising. In this paper we study the reformulation of a broad class of nonlocal filters in terms of two functional rearrangements: the decreasing and the relative rearrangements. Independently of the dimension of the image, we reformulate these filters as integral operators defined in a one-dimensional space corresponding to the level sets measures. We prove the equivalency between the original and the rearranged versions of the filters and propose a discretization in terms of constant-wise interpolators, which we prove to be convergent to the solution of the continuous setting. For some particular cases, this new formulation allows us to perform a detailed analysis of the filtering properties. Among others, we prove that the filtered image is a contrast change of the original image, and that the filtering procedure behaves asymptotically as a shock filter combined with a border diffusive term, responsible for the staircaising effect and the loss of contrast.

cs.CV↗

arXiv2014

Neighborhood filters and the decreasing rearrangement

Nonlocal filters are simple and powerful techniques for image denoising. In this paper, we give new insights into the analysis of one kind of them, the Neighborhood filter, by using a classical although not very common transformation: the decreasing rearrangement of a function (the image). Independently of the dimension of the image, we reformulate the Neighborhood filter and its iterative variants as an integral operator defined in a one-dimensional space. The simplicity of this formulation allows to perform a detailed analysis of its properties. Among others, we prove that the filter behaves asymptotically as a shock filter combined with a border diffusive term, responsible for the staircaising effect and the loss of contrast.

cs.CV↗

arXiv2013

Existence and nonuniqueness of segregated solutions to a class of cross-diffusion systems

We study the the Dirichlet problem for the cross-diffusion system \[ \partial_tu_i=\operatorname{div}\left(a_iu_i\nabla (u_1+u_2)\right)+f_i(u_1,u_2),\quad i=1,2,\quad a_i=const>0, \] in the cylinder $Q=Ω\times (0,T]$. The functions $f_i$ are assumed to satisfy the conditions $f_1(0,r)=0$, $f_2(s,0)=0$, $f_1(0,r)$, $f_2(s,0)$ are locally Lipschitz-continuous. It is proved that for suitable initial data $u_0$, $v_0$ the system admits segregated solutions $(u_1,u_2)$ such that $u_i\in L^{\infty}(Q)$, $u_1+u_2\in C^{0}(\overline{Q})$, $u_1+u_2>0$ and $u_1\cdot u_2=0$ everywhere in $Q$. We show that the segregated solution is not unique and derive the equation of motion of the surface $Γ$ which separates the parts of $Q$ where $u_1>0$, or $u_2>0$. The equation of motion of $Γ$ is a modification of the Darcy law in filtration theory. Results of numerical simulation are presented.

math.AP↗

arXiv2013

On a non-local spectrogram for denoising one-dimensional signals

In previous works, we investigated the use of local filters based on partial differential equations (PDE) to denoise one-dimensional signals through the image processing of time-frequency representations, such as the spectrogram. In this image denoising algorithms, the particularity of the image was hardly taken into account. We turn, in this paper, to study the performance of non-local filters, like Neighborhood or Yaroslavsky filters, in the same problem. We show that, for certain iterative schemes involving the Neighborhood filter, the computational time is drastically reduced with respect to Yaroslavsky or nonlinear PDE based filters, while the outputs of the filtering processes are similar. This is heuristically justified by the connection between the (fast) Neighborhood filter applied to a spectrogram and the corresponding Nonlocal Means filter (accurate) applied to the Wigner-Ville distribution of the signal. This correspondence holds only for time-frequency representations of one-dimensional signals, not to usual images, and in this sense the particularity of the image is exploited. We compare though a series of experiments on synthetic and biomedical signals the performance of local and non-local filters.

cs.CV↗