Stability properties of the ENO method

We review the currently available stability properties of the ENO reconstruction procedure, such as its monotonicity and non-oscillatory properties, the sign property, upper bounds on cell interface jumps and a total variation-type bound. We also outline how these properties can be applied to derive stability and convergence of high-order accurate schemes for conservation laws.Comment: To appear in Handbook of Numerical Methods for Hyperbolic Problem

Highly accurate schemes for PDE-based morphology with general structuring elements

The two fundamental operations in morphological image processing are dilation and erosion. These processes are defined via structuring elements. It is of practical interest to consider a variety of structuring element shapes. The realisation of dilation/erosion for convex structuring elements by use of partial differential equations (PDEs) allows for digital scalability and subpixel accuracy. However, numerical schemes suffer from blur by dissipative artifacts. In our paper we present a family of so-called flux-corrected transport (FCT) schemes that addresses this problem for arbitrary convex structuring elements. The main characteristics of the FCT-schemes are: (i) They keep edges very sharp during the morphological evolution process, and (ii) they feature a high degree of rotational invariance. We validate the FCT-scheme theoretically by proving consistency and stability. Numerical experiments with diamonds and ellipses as structuring elements show that FCT-schemes are superior to standard schemes in the field of PDE-based morphology

A shock-capturing algorithm for the differential equations of dilation and erosion

Dilation and erosion are the fundamental operations in morphological image processing. Algorithms that exploit the formulation of these processes in terms of partial differential equations offer advantages for non-digitally scalable structuring elements and allow sub-pixel accuracy. However, the widely-used schemes from the literature suffer from significant blurring at discontinuities. We address this problem by developing a novel, flux corrected transport (FCT) type algorithm for morphological dilation / erosion with a flat disc. It uses the viscosity form of an upwind scheme in order to quantify the undesired diffusive effects. In a subsequent corrector step we compensate for these artifacts by means of a stabilised inverse diffusion process that requires a specific nonlinear multidimensional formulation. We prove a discrete maximum-minimum principle in this multidimensional framework. Our experiments show that the method gives a very sharp resolution of moving fronts, and it approximates rotation invariance very well

PDE-based morphology for matrix fields : numerical solution schemes

Tensor fields are important in digital imaging and computer vision. Hence there is a demand for morphological operations to perform e.g. shape analysis, segmentation or enhancement procedures. Recently, fundamental morphological concepts have been transferred to the setting of fields of symmetric positive definite matrices, which are symmetric rank two tensors. This has been achieved by a matrix-valued extension of the nonlinear morphological partial differential equations (PDEs) for dilation and erosion known for grey scale images. Having these two basic operations at our disposal, more advanced morphological operators such as top hats or morphological derivatives for matrix fields with symmetric, positive semidefinite matrices can be constructed. The approach realises a proper coupling of the matrix channels rather than treating them independently. However, from the algorithmic side the usual scalar morphological PDEs are transport equations that require special upwind-schemes or novel high-accuracy predictor-corrector approaches for their adequate numerical treatment. In this chapter we propose the non-trivial extension of these schemes to the matrix-valued setting by exploiting the special algebraic structure available for symmetric matrices. Furthermore we compare the performance and juxtapose the results of these novel matrix-valued high-resolution-type (HRT) numerical schemes by considering top hats and morphological derivatives applied to artificial and real world data sets

Étude d'une nouvelle classe de contours actifs pour la détection de routes dans des images de télédétection

Nous proposons dans cet article d'étudier une nouvelle classe de contours actifs d'ordre supérieur. Nous utilisons une énergie quadratique sur l'espace des 1-chaînes afin d'introduire des contraintes globales sur la forme du contour. Cette énergie est proposée dans le cadre de l'extraction automatique de routes sur des images de télédétection

Evolution-Operator-Based Single-Step Method for Image Processing

This work proposes an evolution-operator-based single-time-step method for image and signal processing. The key component of the proposed method is a local spectral evolution kernel (LSEK) that analytically integrates a class of evolution partial differential equations (PDEs). From the point of view PDEs, the LSEK provides the analytical solution in a single time step, and is of spectral accuracy, free of instability constraint. From the point of image/signal processing, the LSEK gives rise to a family of lowpass filters. These filters contain controllable time delay and amplitude scaling. The new evolution operator-based method is constructed by pointwise adaptation of anisotropy to the coefficients of the LSEK. The Perona-Malik-type of anisotropic diffusion schemes is incorporated in the LSEK for image denoising. A forward-backward diffusion process is adopted to the LSEK for image deblurring or sharpening. A coupled PDE system is modified for image edge detection. The resulting image edge is utilized for image enhancement. Extensive computer experiments are carried out to demonstrate the performance of the proposed method. The major advantages of the proposed method are its single-step solution and readiness for multidimensional data analysis

Nonlinear anisotropic diffusion filters for the numerical approximation of conservation laws

This work integrates nonlinear discrete dissipation models -- namely anisotropic diffusion filters -- into the field of the numerical approximation for conservation laws is accomplished. Second-order accurate methods, namely the Lax-Wendroff scheme, are stabilised using the anisotropic dissipation filters combined with the shock detector. A new class of numerical schemes with nonlinear anisotropic dissipation models are derived. These algorithms nearly reduce all oscillations while the sharp shock resolution of the high-order scheme is maintained. Furthermore, emanating from the positivity requirement an alternative scheme is developed which is founded on the wave propagation algorithms developed by LeVeque. The corresponding dissipation model can be considered as a member of the class of anisotropic diffusion filters. The developed positive diffusion filter, based on an entropy steered coefficient correction, is nearly oscillation-free and comparable to the application of limiter functions. The class of anisotropic diffusion filters is extended to systems of conservation laws, namely the Euler equations. This is done by a characteristic splitting where the filter algorithm is applied to each characteristic field separately. Thus, it is shown that the extension of the class of anisotropic dissipation filter to systems of equations is possible in principle.In der vorliegenden Arbeit werden nichtlineare anisotrope Diffusionsfilter aus der Bildverarbeitung in Algorithmen zur numerischen Approximation von skalaren wie auch von Systemen von Erhaltungsgleichungen integriert. Dazu werden diese Filterverfahren mit einem numerischen Verfahren für Erhaltungsgleichungen von zweiter Ordnung Genauigkeit gekoppelt, namentlich dem Schema von Lax und Wendroff. Die resultierenden Schemata zur Diskretisierung von Erhaltungsgleichungen zeichnen sich durch ein neues, nichtlineares, anisotropes Diffusionsmodell aus. Dies führt zu einem oszillationsreduzierenden Verfahren von zweiter Approximationsordnung mit scharfer Auflösung der Stoßunstetigkeiten. Weiterhin werden die anisotropen Diffusionsfilter für Systeme von Erhaltungsgleichungen, namentlich den Euler-Gleichungen der Gasdynamik, angepasst. Dazu wird die charakteristische Formulierung der Euler-Gleichungen verwendet und die Filteralgorithmen auf die jeweiligen charakteristischen Felder separat angewendet. Für alle in dieser Arbeit entwickelten numerischen Verfahren zur Approximation von Erhaltungsgleichungen werden entsprechende Testfälle berechnet

A New Adaptive Image Interpolation Method to Define the Shoreline at Sub-Pixel Level

[EN] This paper presents a new methodological process for detecting the instantaneous land-water border at sub-pixel level from mid-resolution satellite images (30 m/pixel) that are freely available worldwide. The new method is based on using an iterative procedure to compute Laplacian roots of a polynomial surface that represents the radiometric response of a set of pixels. The method uses a first approximation of the shoreline at pixel level (initial pixels) and selects a set of neighbouring pixels to be part of the analysis window. This adaptive window collects those stencils in which the maximum radiometric variations are found by using the information given by divided differences. Therefore, the land-water surface is computed by a piecewise interpolating polynomial that models the strong radiometric changes between both interfaces. The assessment is tested on two coastal areas to analyse how their inherent differences may affect the method. A total of 17 Landsat 7 and 8 images (L7 and L8) were used to extract the shorelines and compare them against other highly accurate lines that act as references. Accurate quantitative coastal data from the satellite images is obtained with a mean horizontal error of 4.38 +/- 5.66 m and 1.79 +/- 2.78 m, respectively, for L7 and L8. Prior methodologies to reach the sub-pixel shoreline are analysed and the results verify the solvency of the one proposed.This study is part of the PhD dissertation of E. Sanchez-Garcia, which was supported by a grant from the Spanish Ministry of Education, Culture and Sports (I + D + i 2013-2016). The authors also appreciate the financial support provided by the Spanish Ministry of Economy and Competitiveness (CGL2015-69906-R)Sánchez-García, E.; Balaguer-Beser, Á.; Almonacid-Caballer, J.; Pardo Pascual, JE. (2019). A New Adaptive Image Interpolation Method to Define the Shoreline at Sub-Pixel Level. 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