Julián Velasco

A parabolic cross-diffusion system for granular materials

We analyze a cross-diffusion system of parabolic equations for the relative concentration and the dynamic repose angle of a mixture of two different granular materials in a long rotating drum. The main feature of the system is the ability to describe the axial segregation of the two granular components. The existence of global-in-time weak solutions is shown for arbitrary large cross-diffusion by using entropy-type inequalities and approximation arguments. The uniqueness of solutions is proved if cross-diffusion is not too large. Furthermore, we derive a sufficient condition on the parameters to have nonsegregation. Finally, numerical simulations illustrate the long-time coarsening of the segregation bands in the drum.

On a chirplet transform-based method applied to separating and counting wolf howls

We propose a parametric model based in chirp decomposition to modelize wolf chorus emissions. The problem consists on estimating the phase and amplitude of chirps corresponding to each individual, as well as the number of individuals. For the first task, we use a well known technique, the Chirplet Transform, which allows us to obtain a first order approximation of the phase, improving the zero order approximation given by the Short Time Fourier Transform (STFT). This gain in accuracy allows to use criteria for a better chirp tracking, which is specially important at crossing points and in the determination of harmonics of a fundamental tone. We explore the efficiency of the method applying it to synthetic signals and to wolves choruses recordings (original motivation of this work). The results show good performance for chirps tracking even under strong noise corruption.

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 commonly used transformation: the decreasing rearrangement of a function. 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 filtered image is a contrast change of the original image, an 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.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 commonly used transformation: the decreasing rearrangement of a function. 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 filtered image is a contrast change of the original image, an 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.

Implementation of a diffusive differential reassignment method for signal enhancement : An application to wolf population counting

We investigate the combination of two PDE-based image processing techniques applied to the image produced by time-frequency representations of one-dimensional signals, such as the spectrogram. Specifically, we consider the energy transport equation associated to the lagrangian coordinates corresponding to the spectrogram differential reassignment proposed by Chassandre-Mottin, Daubechies, Auger and Flandrin as a spectrogram readability improving method, together with the image restoration model proposed by Álvarez, Lions and Morel for noise reduction and edge enhancement. Our aim is to produce a transformation of the spectrogram in which the instantaneous frequency lines are easier to track, for using it as an input for a (wolves howls) counting algorithm. After presenting the model derivation, we show some analytical properties of it, such as the existence of a unique solution and a comparison principle, and perform later a discretization to numerically investigate its performance for the cases of synthetic signals and field recorded wolves choruses. We finally compare our results with those obtained from well established techniques.

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.

A Dynamic Boundary Value Problem Arising in the Ecology of Mangroves

We consider an evolution model describing the vertical movement of water and salt in a domain split in two parts: a water reservoir and a saturated porous medium below it, in which a continuous extraction of fresh water takes place (by the roots of mangroves). The problem is formulated in terms of a coupled system of partial differential equations for the salt concentration and the water flow in the porous medium, with a dynamic boundary condition which connects both subdomains.

Some Nonlocal Filters Formulation Using Functional Rearrangements

We present an exact reformulation of a broad class of nonlocal filters, among which the bilateral filters, in terms of two functional rearrangements: the decreasing and the relative rearrangements.We present an exact reformulation of a broad class of nonlocal filters, among which the bilateral filters, in terms of two functional rearrangements: the decreasing and the relative rearrangements. Independently of the image spatial dimension, these filters are expressed as integral operators defined in a one-dimensional space, corresponding to the level sets measures. We provide some insight into the properties of this new formulation and show some numerical demonstrations to illustrate them.

Finite element approximation of a population spatial adaptation model

In [18], Sighesada, Kawasaki and Teramoto presented a system of partial differential equations for modeling spatial segregation of interacting species. Apart from competitive Lotka-Volterra (reaction) and population pressure (cross-diffusion) terms, a convective term modeling the populations attraction to more favorable environmental regions was included. In this article, we study numerically a modification of their convective term to take account for the notion of spatial adaptation of populations. After describing the model, in which a time non-local drift term is considered, we propose a numerical discretization in terms of a mass-preserving time semi-implicit finite element method. Finally, we provied the results of some biologically inspired numerical experiments showing qualitative differences between the original model of [18] and the model proposed in this article.

Evolution Nonlinear Diffusion‐Convection PDE Models for Spectrogram Enhancement

In previous works we studied the application of PDE‐based image processing techniques applied to the spectrogram of audio signals in order to improve the readability of the signal. In particular we considered the implementation of the nonlinear diffusive model proposed by Alvarez, Lions and Morel [1](ALM) combined with a convective term inspired by the differential reassignment proposed by Chassandre‐Mottin, Daubechies, Auger and Flandrin [2]–[3]. In this work we consider the possibility of replacing the diffusive model of ALM by diffusive terms in divergence form. In particular we implement finite element approximations of nonlinear diffusive terms studied by Chen, Levine, Rao [4] and Antontsev, Shmarev [5]–[8] with a convective term.

Detection of Cardiac Arrhythmias Through Singular Spectrum Analysis of a Time-Distorted EGM Signal.

A new method for detecting cardiac arrhythmias is proposed. The differences between the instantaneous frequencies of signals recorded in atrium and ventricle are computed by means of a non-linear spectral transform. This transform dilates or contracts the time scale until the ventricle signal has a flat frequency spectrum in time. Singular Spectrum Analysis is used to isolate its oscillatory components. The same temporal dilations and contractions are applied to the atrium signal, that is subsequently projected onto the oscillatory components found in the ventricle signal. It is shown that the frequency spectrum of the processed atrium signal becomes uneven only at arrhythmia episodes.