In this article, we propose a new planar shape representation, the medial branch graph representation (MBGR) which allows to easily generate, vary and edit all-new sorts of parametric shapes. Each MBGR shape is described by a collection of connected control circles organized as a graph. Each pair of connected circles generates a branch of the shape. The boundaries of the branches are designed using cubic Bézier curves which enable shape representation in a compact SVG format. Each circle is associated with a regularity parameter locally controlling the smoothness of the branch connections. In this way, we can manage shapes having both sharp corners and smooth boundaries. To illustrate the potential of the MBGR representation, we have created an online facility ( https://mbgrg.github.io/mbgrg/ ) where the user can create/edit MBGR shapes automatically or interactively.

The COVID-19 pandemy has created a radically new situation where most countries provide raw measurements of their daily incidence and disclose them in real time. This enables new machine learning forecast strategies where the prediction might no longer be based just on the past values of the current incidence curve, but could take advantage of observations in many countries. We present such a simple global machine learning procedure using all past daily incidence trend curves. Each of the 27,418 COVID-19 incidence trend curves in our database contains the values of 56 consecutive days extracted from observed incidence curves across 61 world regions and countries. Given a current incidence trend curve observed over the past four weeks, its forecast in the next four weeks is computed by matching it with the first four weeks of all samples, and ranking them by their similarity to the query curve. Then the 28 days forecast is obtained by a statistical estimation combining the values of the 28 last observed days in those similar samples. Using comparison performed by the European Covid-19 Forecast Hub with the current state of the art forecast methods, we verify that the proposed global learning method, EpiLearn, compares favorably to methods forecasting from a single past curve.

The most common reported epidemic time series in epidemiological surveillance are the daily or weekly incidence of new cases, the hospital admission count, the ICU admission count, and the death toll, which played such a prominent role in the struggle to monitor the Covid-19 pandemic. We show that pairs of such curves are related to each other by a generalized renewal equation depending on a smooth time varying delay and a smooth ratio generalizing the reproduction number. Such a functional relation is also explored for pairs of simultaneous curves measuring the same indicator in two neighboring countries. Given two such simultaneous time series, we develop, based on a signal processing approach, an efficient numerical method for computing their time varying delay and ratio curves, and we verify that its results are consistent. Indeed, they experimentally verify symmetry and transitivity requirements and we also show, using realistic simulated data, that the method faithfully recovers time delays and ratios. We discuss several real examples where the method seems to display interpretable time delays and ratios. The proposed method generalizes and unifies many recent related attempts to take advantage of the plurality of these health data across regions or countries and time, providing a better understanding of the relationship between them. An implementation of the method is publicly available at the EpiInvert CRAN package.

In this letter, we propose a new despeckling filter for fully polarimetric synthetic aperture radar (PolSAR) images defined by 3×3 complex Wishart distributions. We first generalize the well-known structure tensor to deal with PolSAR data which allows to efficiently measure the dominant direction and contrast of edges. The generalization includes stochastic distances defined in the space of the Wishart matrices. Then, we embed the formulation into an anisotropic diffusion-like schema to build a filter able to reduce speckle and preserve edges. We evaluate its performance through an innovative experimental setup that also includes Monte Carlo analysis. We compare the results with a state-of-the-art polarimetric filter.

In this letter, we propose a supervised method for the classification of fully polarimetric synthetic aperture radar (PolSAR) images based on active contour models. We use an “a priori” estimation, obtained from training data, of the complex Wishart distributions of the different types of regions in the image (for instance, water, crops, grass, forest or urban). The information of the Wishart distributions is included in the active contour models to guide the level set evolution. We study the case of two classes and the case of three or more classes separately. We present some experimental results on the synthetic data and real PolSAR images to show the performance of the proposed model. The results are compared with other well-known supervised classification methods, and, for actual PolSAR data, our method shows an overall precision of 94.31% and a $\kappa $ coefficient of 0.937.

In this paper we study the local behavior of Fully PolSAR (Polarimetric Synthetic Aperture Radar) images defined by 3×3 complex Wishart distributions. We propose a generalization of the well-known structure tensor to Fully PolSAR images, as well as a measure of the smoothness of such predominant direction. This measure provides local information as the predominant direction of maximum variation of the complex Wishart distribution in a neighborhood of an image domain point ( x, y). We use stochastic distances defined in the space of Wishart matrices to generalize de structure tensor to PolSAR images. The study of the local behavior of PolSAR images address a number of processing and analysis problems. In particular, in this paper we apply this new approach to edge estimation using the magnitude of the PolSAR image variation provided by the generalized structure tensor. We also show promising results for simulated and actual Pol-SAR data.

In this paper we study a geometric partial differential equation including a forcing term. This equation defines a hypersurface evolution that we use for level set regularization. We study the shape of the radial solutions of the equation and some qualitative properties about the level set propagations. We show that under a suitable choice of the forcing term, the geometric equation has nontrivial asymptotic states and it represents a model for level set regularization. We show that by using a forcing term which is merely a bounded Hölder continuous function, we can obtain finite time stabilization of the solutions. We introduce an explicit finite difference scheme to compute numerically the solution of the equation and we present some applications of the model to nonlinear two-dimensional image filtering and three-dimensional segmentation in the context of medical imaging.

In this paper, we study lens distortion for still images considering two well-known distortion models: the two-parameter polynomial model and the two-parameter division model. We study the invertibility of these models, and we mathematically characterize the conditions for the distortion parameters under which the distortion model defines a one-to-one transformation. This ensures that the inverse transformation is well defined and the distortion-free image can be properly computed, which provides robustness to the distortion models. A new automatic method to correct the radial distortion is proposed, and a comparative analysis for this method is extensively performed using the polynomial and the division models. With the aim of obtaining an accurate estimation of the model, we propose an optimization scheme which iteratively improves the parameters to achieve a better matching between the distorted lines and the edge points. The proposed method estimates two-parameter radial distortion models by detecting the longest distorted lines within the image. This is done by applying the Hough transform extended with a radial distortion parameter. Next, a two-parameter model is estimated using an iterative nonlinear optimization scheme. This scheme aims at minimizing the distance from the edge points to their associated lines by adjusting the two distortion parameters as well as the coordinates of the center of distortion. We present some experiments on real images with significant distortion to show the ability of the proposed approach to correct the radial distortion. A visual and quantitative comparison between both automatic two-parameter model estimations indicates that the division model is more efficient for those images showing strong distortion.

In sport scenarios like football or basketball, we often deal with central views where only the central circle and some additional primitives like the central line and the central point or a touch line are visible. In this paper we first characterize, from a mathematical point of view, the set of homographies that project a given ellipse into the unit circle, next, using some extra minimal additional information like the knowledge of the position in the image of the central line and central point or a touch line we show a method to fully determine the plane homography. We present some experiments in sport scenarios to show the ability of the proposed method to properly recover the plane homography.

We introduce new results connecting differential and morphological operators that provide a formal and theoretically grounded approach for stable and fast contour evolution. Contour evolution algorithms have been extensively used for boundary detection and tracking in computer vision. The standard solution based on partial differential equations and level-sets requires the use of numerical methods of integration that are costly computationally and may have stability issues. We present a morphological approach to contour evolution based on a new curvature morphological operator valid for surfaces of any dimension. We approximate the numerical solution of the curve evolution PDE by the successive application of a set of morphological operators defined on a binary level-set and with equivalent infinitesimal behavior. These operators are very fast, do not suffer numerical stability issues, and do not degrade the level set function, so there is no need to reinitialize it. Moreover, their implementation is much easier since they do not require the use of sophisticated numerical algorithms. We validate the approach providing a morphological implementation of the geodesic active contours, the active contours without borders, and turbopixels. In the experiments conducted, the morphological implementations converge to solutions equivalent to those achieved by traditional numerical solutions, but with significant gains in simplicity, speed, and stability.

We introduce a morphological approach to curve evolution. The differential operators used in the standard PDE snake models can be approached using morphological operations on a binary level set. By combining the morphological operators associated to the PDE components we achieve a new snakes evolution algorithm. This new solution is based on numerical methods which are very simple, fast and stable. Moreover, since the level set is just a binary piecewise constant function, this approach does not require to estimate a contour distance function. To illustrate the results obtained we present some numerical experiments on real images.

Recursive filtering structures reduce drastically the computational effort required for different tasks in image processing. These operations are done with a fixed number of operations per output point independently of the size of the neighbourhood considered. In this paper we show that implicit numerical implementations of some partial differential equations (PDEs) provide algorithms that can be interpreted in terms of recursive filters. We show, in particular, that the classical second order recursive filter introduced by Deriche (1987, 1990) is in fact a particular implementation of the heat equation. Using the well-known Neumann boundary condition for the heat equation, we propose some new implementation of the filter. We extend this linear filter to a nonlinear recursive smoothing filter, following the general idea of anisotropic diffusion. We present some comparison results with the classical Perona-Malik model.