Abstract : Approximation techniques are used in problems in which it's required to find
unknown functions from a set of known data. This problem appears in physics (solution of hyperbolic PDEs), medicine (medical imaging treatment) or topography (Digital Elevation Models) etc.
Reconstruction based on linear schemes, like splines, have proved to be useful in different application (DEM). However they become ineffective for approximating piecewise smooth functions (shocks in solution of hyperbolic PDEs) or edge-dominated images. For such families, nonlinear schemes may improve the approximation performance.
This MS brings together researchers from the two different communities, with the aim to generate scientific dialogue.
[01862] Stable nonlinear inversion : a general framework for interface reconstruction from cell-average
Format : Talk at Waseda University
Author(s) :
Albert Cohen (Sorbonne Universite)
Abstract : In this lecture, we present a general framework for solving inverse problem using nonlinear
approximation spaces. The main principles build up on the so called Parametrized Background
Data Weak method (PBDW) which can be thought as a linear counterpart. As a main and motivating
application we discuss the reconstruction of sharp interfaces from cell average at coarse resolutions
for which linear methods are known to be uneffective. We discuss the convergence rates of these
reconstructions and their optimality.
[02134] Adaptive Multi-Quadric Interpolation: Applications in Image Compression.
Format : Talk at Waseda University
Author(s) :
Rosa Donat (Universitat de Valencia)
Francesc Aràndiga (Universitat de Valènciacia)
Daniela Schenone (Leonardo Sistemi Integrati S.L.R.)
Abstract : Multi-Quadric interpolation techniques depend on a shape parameter that has a direct influence on its accuracy. The computation of the shape parameter can be performed using 'linear' (e.g. data-independent) estimates or by incorporating adaptive techniques, similar to those used in ENO/WENO schemes to maximize the region of accuracy when using piecewise polynomial interpolatory techniques.
We design 2D Prediction operators within Harten's Multiresolution Framework based on non-separable multi-quadric approximation that incorporates a WENO-type selection of the local shape parameter. We explore the compression properties of the resulting MR transformation, and also the combination of these techniques with the construction of edge maps of the image.
[01772] Reconstructing a Digital Elevation Model from C2 quasi-interpolation
Format : Talk at Waseda University
Author(s) :
Domingo Barrera (University of Granada)
Salah Eddargani (University of Rome Tor Vergata)
Juan Francisco Reinoso (University of Granada)
Abstract : Quasi-interpolation spline in the Bernstein basis is a low-cost computational method for approximating functions or data in one or several variables. Although linked to the degree, regularity and exactness are parameters available to the user. In this contribution we propose to define a one-dimensional quasi-interpolation operator that achieves the optimal approximation order and provides C2 continuous approximants. It will be used to reconstruct a Digital Elevation Model using a tensor product type scheme.
[01766] Low-degree quasi-interpolation in the Bernstein basis
Format : Talk at Waseda University
Author(s) :
María José Ibáñez (University of Granada)
Salah Eddargani (University of Rome Tor Vergata)
Sara Remogna (University of Torino)
Abstract : Spline quasi-interpolation is an effective tool for approximating functions or data. Usually, the quasi-interpolant is defined as a linear combination of the B-splines of a basis of the linear space in which the approximant is to be constructed. In this contribution we present a procedure for constructing quasi-interpolants by directly defining the coefficients of the expression in the Bernstein basis of its restriction to each subinterval induced by a uniform partition of the real line.
[01326] A Nonlinear B-spline quasi-interpolation method,
Format : Talk at Waseda University
Author(s) :
Francesc Aràndiga (Universitat de València)
Abstract : Quasi-interpolation based on B-spline approximation methods are used in numerous applications.
However, we observe that the Gibbs phenomenon appears when approximating near
discontinuities. We present nonlinear modifications, based on weighted essentially non-oscillatory (WENO) techniques, of well-known quasi-interpolant methods to avoid this phenomena near discontinuities and, at the same time, maintain the high-order accuracy in smooth regions.
[01853] Edge adaptive schemes and machine learning for image super-resolution
Format : Talk at Waseda University
Author(s) :
Agustin Somacal (Sorbonne University)
Abstract : In image processing Edge-adapted methods are used to reconstruct high-resolution images from coarser cell averages. When images are piece-wise smooth functions, interfaces can be approximated by a pre-specified functional class through optimization LVIRA or specific preprocessing ENO-EA. We extend the ENO-EA approach to polynomials, show two methods to treat vertices and compare with learning-based methods in which an artificial neural network is used to attain the same goal.
[01411] Univariate subdivision schemes based on local polynomial regression
Format : Talk at Waseda University
Author(s) :
Dionisio F. Yanez (UV)
Abstract : The generation of curves and surfaces from given data is a well-known problem in Computer-Aided Design that can be solved by means of subdivision schemes. They are a powerful tool that allows obtaining new data from the initial one using simple calculations. In some real applications, the initial data are given with noise and interpolatory schemes are not adequate to process them. In this talk, we present some new families of binary univariate linear subdivision schemes using weighted local polynomial regression. We study their properties, such as convergence, monotonicity and polynomial reproduction and show some examples.
[01410] Linear and nonlinear approximation rules arising from optimal denoising
Format : Talk at Waseda University
Author(s) :
Sergio López-Ureña (Universitat de València)
Abstract : We explore the design of new linear filter-like methods based on the minimization of the noise variance. But linear methods, when applied to data with large gradients, may lead to some kind of Gibbs phenomenon. To overcome this problem, we combine some of these linear methods in a WENO style to obtain a nonlinear denoising method which handles properly large gradients in the data. Some examples are performed to validate the theoretical results.
[01560] A totally C^2 quartic splines defined on mixed macro-structures
Format : Online Talk on Zoom
Author(s) :
Salah Eddargani (University of Rome "Tor Vergata")
Domingo Barrera (University of Granada)
María José Ibáñez (University of Granada)
Abstract : This work deals with the construction of normalized B-splines of degree four and C^2 smooth
everywhere on triangulations endowed with mixed splits. The main splits involved herein are
Powell-Sabin (6-), and Modified Morgan-Scott (10-) splits. With the help of Marsden identity, a
family of C^2 quartic quasi-interpolation splines of optimal orders has been provided.
[01502] Construction of quadratic and cubic orthogonal wB-spline wavelets
Format : Online Talk on Zoom
Author(s) :
Mohamed Ajeddar (MISI Laboratory, Faculty of Sciences and Technology, University Hassan First, Settat, Morocco.)
Abstract : The definition and basic properties of $\omega$B-splines, frequently used as primal scaling functions, are introduced, as well as their refinement equation. Then, a method for constructing orthogonal wavelets using $\mathcal{C}^1$ and $\mathcal{C}^2$ $\omega$B-splines is presented.
[01492] Algebraic Hyperbolic spline interpolation by means of integral values.
Format : Online Talk on Zoom
Author(s) :
Mohammed Oraiche (Department of Mathematics, University Hassan First, Settat, Morocco. )
Abstract : In this paper, a cubic Hermite spline interpolating scheme reproducing both linear polynomials and hyperbolic functions is considered. The interpolating scheme is mainly defined by means of integral values over the subintervals of a partition of the function to be approximated, rather than the function and its first derivative values. The scheme provided is $C^2$ everywhere and yields optimal order. We provide some numerical tests to illustrate the good performance of the novel approximation scheme.
