In this paper we improve, by almost doubling, the existing lower bound for the number of limit cycles of the family of complex differential equations with three monomials, z(center dot) = Azkzl + Bzmzn + Czpzq, being k,l, m, n, p, q non-negative integers and A, B, C is an element of C. More concretely, if N = max (k + l, m + n, p + q) and H3(N) is an element of N boolean OR {infinity} denotes the maximum number of limit cycles of the above equations, we show that for N >= 4, H3(N) >= N - 3 and that for some values of N this new lower bound is N + 1. We also present examples with many limit cycles and different configurations. Finally, we show that H 3 ( 2 ) >= 2 and study in detail the quadratic case with three monomials proving in some of them non-existence, uniqueness or existence of exactly two limit cycles. (c) 2024 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons .org /licenses/by /4 .0/).

Pansharpening aims to fuse the geometry of a high-resolution panchromatic image with the color information of a low-resolution multispectral image to generate a high-resolution multispectral image. Classical variational methods are more interpretable and flexible than pure deep learning approaches, but their performance is limited by the use of rigid priors. In this paper, we efficiently combine both techniques by introducing a shallow residual network to learn the regularization term of a variational pansharpening model. The proposed energy includes the classical observation model for the multispectral data and a constraint to preserve the geometry encoded in the panchromatic. The experiments demonstrate that our method achieves state-of-the-art results.

In this supplementary material we provide more details about the hypersharpening algorithm, the implementation of the unfolded network along with information about the deep-learning settings and additional visual and objective results

The fusion of multi-source data with different spatial and spectral resolutions is a crucial task in many remote sensing and computer vision applications. Model-based fusion methods are more interpretable and. flexible than pure data-driven networks, but their performance depends greatly on the established fusion model and. the hand-crafted, prior. In this work, we propose an end-to-end trainable model-based. network for hyperspectral and panchromatic image fusion. We introduce an energy functional that takes into account classical observation models and. incorporates a high-frequency injection constraint. The resulting optimization function is solved by a forward-backward splitting algorithm and. unfolded into a deep-learning framework that uses two modules trained, in parallel to ensure both data observation fitting and constraint compliance. Extensive experiments are conducted, on the remote-sensing hyperspectral PRISMA dataset and on the CAVE dataset, proving the superiority of the proposed deep unfolding network qualitatively and quantitatively.

The goal of this work is the study of the probability of occurrence of limit cycles for a family of planar differential systems that are a natural extension of linear ones. To prove our results we first develop several results of non-existence, existence, uniqueness and non-uniqueness of limit cycles for this family. They are obtained by studying some Abelian integrals, via degenerate Andronov-Hopf bifurcations or by using the Bendixson-Dulac criterion. To the best of our knowledge, this is the first time that the probability of existence of limit cycles for a non-trivial family of planar systems is obtained analytically. In particular, we give vector fields for which the probability of having limit cycles is positive, but as small as desired.

The objective of this work is the study of the probability of occurrence of phase portraits in a family of planar quasi-homogeneous vector fields of quasi degree q, that is a natural extension of planar linear vector fields, which correspond to q=1. We obtain the exact values of the corresponding probabilities in terms of a simple one-variable definite integral that only depends on q. This integral is explicitly computable in the linear case, recovering known results, and it can be expressed in terms of either complete elliptic integrals or of generalized hypergeometric functions in the non-linear one. Moreover, it appears a remarkable phenomenon when q is even: the probability to have a center is positive, in contrast with what happens in the linear case, or also when q is odd, where this probability is zero.

The fusion of multisensor data has attracted a lot of attention in computer vision, particularly among the remote sensing community. Hyperspectral image fusion consists in merging the spectral information of a hyperspectral image with the geometry of a multispectral one in order to infer an image with high spatial and spectral resolutions. In this paper, we propose a variational fusion model with a nonlocal regularization term that encodes patch-based filtering conditioned to the geometry of the multispectral data. We further incorporate a radiometric constraint that injects the high frequencies of the scene into the fused product with a band per band modulation according to the energy levels of the multispectral and hyperspectral images. The proposed approach proved robust to noise and aliasing. The experimental results demonstrate the performance of our method with respect to the state-of-the-art techniques on data acquired by commercial hyperspectral cameras and Earth observation satellites.

The aim of this work is the study of the asymptotic dynamical behaviour, of solutions that approach parabolic fixed points in difference equations. In one dimensional difference equations, we present the asymptotic development for positive solutions tending to the fixed point. For higher dimensions, through the study of two families of difference equations in the two and three dimensional case, we take a look at the asymptotic dynamic behaviour. To show the existence of solutions we rely on the parametrization method.

In this work we give an asymptotic lower bound for the Hilbert number for real planar polynomial differential systems. This lower bound equals, up to leading order, to the best existing one, but the method we provide is new and involves slow-fast systems. The construction strongly relies on generalized Liénard systems.

We propose a new convex variational model for hyperspectral and multispectral image fusion. Our approach introduces nonlocal regularization conditioned to the geometry of the multispectral image and incorporates a constraint forcing the fusion product and the multispectral data to share modulated high frequencies. The proposed method is compared with state-of-the-art fusion techniques, showing competitive results for several quality metrics on different data.

A stereo algorithm based on the matching of line segments between two images is proposed. We extract several characteristics of the segments which permit its matching across the two images. A depth ordering computed from the line segments of the reference image allows us to attribute the match disparity to the correct pixels. This depth sketch is computed by joining close line segments and identifying T-junctions and convexity points. The disparity computed for segments is then extrapolated to the rest of the image by means of a diffusion process. The performance of the proposed algorithm is illustrated by applying the procedure to synthetic stereo pairs.

Most satellites decouple the acquisition of a panchromatic image at high spatial resolution from the acquisition of a multispectral image at lower spatial resolution. Pansharpening is a fusion technique used to increase the spatial resolution of the multispectral data while simultaneously preserving its spectral information. In this paper, we consider pansharpening as an optimization problem minimizing a cost function with a nonlocal regularization term. The energy functional which is to be minimized decouples for each band, thus permitting the application to misregistered spectral components. This requirement is achieved by dropping the, commonly used, assumption that relates the spectral and panchromatic modalities by a linear transformation. Instead, a new constraint that preserves the radiometric ratio between the panchromatic and each spectral component is introduced. An exhaustive performance comparison of the proposed fusion method with several classical and state-of-the-art pansharpening techniques illustrates its superiority in preserving spatial details, reducing color distortions, and avoiding the creation of aliasing artifacts.

We reconsider the classic problem of estimating accurately a 2D transformation from point matches between images containing outliers. RANSAC discriminates outliers by randomly generating minimalistic sampled hypotheses and verifying their consensus over the input data. Its response is based on the single hypothesis that obtained the largest inlier support. In this article we show that the resulting accuracy can be improved by aggregating all generated hypotheses. This yields RANSAAC, a framework that improves systematically over RANSAC and its state-of-the-art variants by statistically aggregating hypotheses. To this end, we introduce a simple strategy that allows to rapidly average 2D transformations, leading to an almost negligible extra computational cost. We give practical applications on projective transforms and homography+distortion models and demonstrate a significant performance gain in both cases.

The fusion of hyperspectral and multispectral images is a crucial task nowadays for it allows the extraction of relevant information from the fused image. Fusion consists of the combination of the spectral information of the hypespectral image (h) and the spatial information of the multispectral image (m). The fused image (f) has both good spatial and spectral information. In this paper we suggest a new hyperspectral and multispectral image (h-m) fusion approach based on Optimal Transport (OT) which highlights the idea of energy transfer from the starting images m and h to the resulting image f. The simulations show that the suggested method is effective and compares competitively with other state-of-the-art methods.

Common satellite imagery products consist of a panchromatic image at high spatial resolution and several misregistered spectral bands at lower resolution. Pansharpening is the fusion process by which a high-resolution multispectral image is inferred. We propose a variational model for which pan-sharpening is defined as an optimization problem minimizing a cost function with nonlocal regularization. We incorporate a new term preserving the radiometric ratio between the panchromatic and each spectral band. The resulting model is channel-decoupled, thus permitting the application to misregistered spectral data. The experimental results illustrate the superiority of the proposed method to preserve spatial details, reduce color artifacts, and avoid aliasing.

In this work we propose a method for the fusion of hyperspectral (HS) and multispectral (MS) satellite images. The aim of the fusion process is to merge the spectral quality of the HS images with the better spatial resolution of the MS images. The final result is an image having both high spectral and spatial resolution. In order to perform the fusion task, we suggest an approach based on optimal transport theory that highlights the idea of energy transfer from the starting images HS and MS to the resulting final image. In this sense, the map transport can be thought as the transfer of characteristics of one image to another.

Image restoration is the problem of recovering an original image from an observation of it in order to extract the most meaningful information. In this paper, we study this problem from a variational point of view through the minimization of energies composed of a quadratic data-fidelity term and a nonsmooth nonconvex regularization term. In the discrete setting, existence of minimizer is proved for arbitrary linear operators. For this kind of problems, fully segmented solutions can be found by minimizing objective nonconvex functionals. We propose a dual formulation of the model by introducing an auxiliary variable with a double function. On one hand, it marks the edges and it ensures their preservation from smoothing. On the other hand, it makes the criterion half-linear in the sense that the dual energy depends linearly on the gradient of the image to be recovered. This leads to design an efficient optimization algorithm with wide applicability to several image restoration tasks such as denoising and deconvolution. Finally, we present experimental results and we compare them with TV-based image restoration algorithms.

In this paper, we propose a new dual algorithm for the minimization of discrete nonconvex functionals, called half-linear regularization. Our approach alternates the calculation of a explicit weight with the minimization of a convex functional with respect to the solution. This minimization corresponds to the weighted total variation which is solved via the well-known Chambolle's algorithm. Finally, we present experimental results by applying it to some image restoration problems as denoising and deconvolution.

This paper focuses on the implementation of the pansharpened image fusion technique proposed in the companion paper [A Nonlocal Variational Model for Pansharpening Image Fusion, SIAM Journal on Imaging Sciences, 2014, to appear]. Pansharpening refers to the process of inferring a high resolution multispectral image from a high resolution panchromatic image and low resolution multispectral one. Although quite successful in terms of relative error, state-of-the-art pansharpening methods still introduce relevant color artifacts. The variational pansharpening model proposed by Duran et al. incorporates a nonlocal regularization term that takes advantage of image self-similarity, leading to significant reduction of the above-mentioned color artifacts.