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

Adaptive continuous-scale morphology for matrix fields

In this article we consider adaptive, PDE-driven morphological operations for 3D matrix fields arising e.g. in diffusion tensor magnetic resonance imaging (DT-MRI). The anisotropic evolution is steered by a matrix constructed from a structure tensor for matrix valued data. An important novelty is an intrinsically one-dimensional directional variant of the matrix-valued upwind schemes such as the Rouy-Tourin scheme. It enables our method to complete or enhance anisotropic structures effectively. A special advantage of our approach is that upwind schemes are utilised only in their basic one-dimensional version. No higher dimensional variants of the schemes themselves are required. Experiments with synthetic and real-world data substantiate the gap-closing and line-completing properties of the proposed method

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

Morphology for matrix data : ordering versus PDE-based approach

Matrix fields are becoming increasingly important in digital imaging. In order to perform shape analysis, enhancement or segmentation of such matrix fields, appropriate image processing tools must be developed. This paper extends fundamental morphological operations to the setting of matrices, in the literature sometimes referred to as tensors despite the fact that matrices are only rank two tensors. The goal of this paper is to introduce and explore two approaches to mathematical morphology for matrix-valued data: One is based on a partial ordering, the other utilises nonlinear partial differential equations (PDEs). We start by presenting definitions for the maximum and minimum of a set of symmetric matrices since these notions are the cornerstones of the morphological operations. Our first approach is based on the Loewner ordering for symmetric matrices, and is in contrast to the unsatisfactory component-wise techniques. The notions of maximum and minimum deduced from the Loewner ordering satisfy desirable properties such as rotation invariance, preservation of positive semidefiniteness, and continuous dependence on the input data. These properties are also shared by the dilation and erosion processes governed by a novel nonlinear system of PDEs we are proposing for our second approach to morphology on matrix data. These PDEs are a suitable counterpart of the nonlinear equations known from scalar continuous-scale morphology. Both approaches incorporate information simultaneously from all matrix channels rather than treating them independently. In experiments on artificial and real medical positive semidefinite matrix-valued images we contrast the resulting notions of erosion, dilation, opening, closing, top hats, morphological derivatives, and shock filters stemming from these two alternatives. Using a ball shaped structuring element we illustrate the properties and performance of our ordering- or PDE-driven morphological operators for matrix-valued data

Shock filters based on implicit cluster separation

One of the classic problems in low level vision is image restoration. An important contribution toward this effort has been the development of shock filters by Osher and Rudin (1990). It performs image deblurring using hyperbolic partial differential equations. In this paper we relate the notion of cluster separation from the field of pattern recognition to the shock filter formulation. A kind of shock filter is proposed based on the idea of gradient based separation of clusters. The proposed formulation is general enough as it can allow various models of density functions in the cluster separation process. The efficacy of the method is demonstrated through various examples

Nonlocal smoothing and adaptive morphology for scalar- and matrix-valued images

In this work we deal with two classic degradation processes in image analysis, namely noise contamination and incomplete data. Standard greyscale and colour photographs as well as matrix-valued images, e.g. diffusion-tensor magnetic resonance imaging, may be corrupted by Gaussian or impulse noise, and may suffer from missing data. In this thesis we develop novel reconstruction approaches to image smoothing and image completion that are applicable to both scalar- and matrix-valued images. For the image smoothing problem, we propose discrete variational methods consisting of nonlocal data and smoothness constraints that penalise general dissimilarity measures. We obtain edge-preserving filters by the joint use of such measures rich in texture content together with robust non-convex penalisers. For the image completion problem, we introduce adaptive, anisotropic morphological partial differential equations modelling the dilation and erosion processes. They adjust themselves to the local geometry to adaptively fill in missing data, complete broken directional structures and even enhance flow-like patterns in an anisotropic manner. The excellent reconstruction capabilities of the proposed techniques are tested on various synthetic and real-world data sets.In dieser Arbeit beschäftigen wir uns mit zwei klassischen Störungsquellen in der Bildanalyse, nämlich mit Rauschen und unvollständigen Daten. Klassische Grauwert- und Farb-Fotografien wie auch matrixwertige Bilder, zum Beispiel Diffusionstensor-Magnetresonanz-Aufnahmen, können durch Gauß- oder Impulsrauschen gestört werden, oder können durch fehlende Daten gestört sein. In dieser Arbeit entwickeln wir neue Rekonstruktionsverfahren zum zur Bildglättung und zur Bildvervollständigung, die sowohl auf skalar- als auch auf matrixwertige Bilddaten anwendbar sind. Zur Lösung des Bildglättungsproblems schlagen wir diskrete Variationsverfahren vor, die aus nichtlokalen Daten- und Glattheitstermen bestehen und allgemeine auf Bildausschnitten definierte Unähnlichkeitsmaße bestrafen. Kantenerhaltende Filter werden durch die gemeinsame Verwendung solcher Maße in stark texturierten Regionen zusammen mit robusten nichtkonvexen Straffunktionen möglich. Für das Problem der Datenvervollständigung führen wir adaptive anisotrope morphologische partielle Differentialgleichungen ein, die Dilatations- und Erosionsprozesse modellieren. Diese passen sich der lokalen Geometrie an, um adaptiv fehlende Daten aufzufüllen, unterbrochene gerichtet Strukturen zu schließen und sogar flussartige Strukturen anisotrop zu verstärken. Die ausgezeichneten Rekonstruktionseigenschaften der vorgestellten Techniken werden anhand verschiedener synthetischer und realer Datensätze demonstriert

Efficient Simulation Tools (EST) for sediment transport in geomorphological shallow flows

Entre los fenómenos superficiales geofísicos y medioambientales, los flujos rápidos de mezclas de agua y sedimentos son probablemente los más exigentes y desconocidos de los procesos movidos por gravedad. El transporte de sedimentos es ubicuo en los cuerpos de agua naturales, como ríos, crecidas, costas o estuarios, además de ser el principal proceso en deslizamientos, flujos de detritos y coladas barro. En este tipo de flujos, el material fluidificado en movimiento consiste en una mezcla de agua y múltiples fases sólidas, que pueden ser de distinta naturaleza como diferentes clases de sedimento, materiales orgánicos, solutos químicos o metales pesados en lodos mineros. El modelado del transporte de sedimentos involucra una alta complejidad debido a las propiedades variables de la mezcla agua-sólidos, el acoplamiento de procesos físicos y la presencia de fenómenos multicapa. Los modelos matemáticos bidimensionales promediados en la vertical ('shallow-type') se construyen en el contexto de flujos superficiales y son aplicables a un amplio rango de estos procesos geofísicos que involucran transporte de sedimentos. Su resolución numérica en el marco de los métodos de Volúmenes Finitos (VF) está controlada por el conjunto de ecuaciones escogido, las propiedades dinámicas del sistema, el acoplamiento entre las variables del flujo y la malla computacional seleccionada. Además, la estimación de los términos fuente de masa y momento puede también afectar la robustez y precisión de la solución. La complejidad de la resolución numérica y el coste computacional de simulación crecen considerablemente con el número de ecuaciones involucradas. Además, la mayor parte de estos flujos son altamente transitorios y ocurren en terrenos irregulares con altas pendientes, requiriendo el uso de una discretización espacial no-estructurada refinada para capturar la complejidad del terreno e incrementando exponencialmente el tiempo computacional. Por tanto, el esfuerzo computacional es uno de los grandes retos para la aplicación de modelos promediados 2D en flujos realistas con grandes escalas espaciales y largas duraciones de evento. En esta tesis, modelos matemáticos superficiales 2D apropiados, algoritmos numéricos de VF robustos y precisos, y códigos eficientes de computación de alto rendimiento son combinados para desarrollar Herramientas Eficientes de Simulación (HES) para procesos medioambientales superficiales involucrando transporte de sedimentos con escalas temporales y espaciales realistas. Nuevas HES capaces de trabajar en mallas estructuradas y no-estructuradas son propuestas para el flujos de lodo/detritos con densidad variable, transporte pasivo en suspensión y transporte de fondo generalizado. Una atención especial es puesta en el acoplamiento entre las variables del sistema y en la integración de los términos fuente de masa y momento. Las propiedades de cada HES han sido cuidadosamente analizadas y sus capacidades demostradas usando tests de validación analíticos y experimentales, así como mediciones en eventos reales.Among the geophysical and environmental surface phenomena, rapid flows of water and sediment mixtures are probably the most challenging and unknown gravity-driven processes. Sediment transport is ubiquitous in environmental water bodies such as rivers, floods, coasts and estuaries, but also is the main process in wet landslides, debris flows and muddy slurries. In this kind of flows, the fluidized material in motion consists of a mixture of water and multiple solid phases which might be of different nature, such as different sediment size-classes, organic materials, chemical solutes or heavy metals in mine tailings. Modeling sediment transport involves an increasing complexity due to the variable bulk properties in the sediment-water mixture, the coupling of physical processes and the presence of multiple layers phenomena. Two-dimensional shallow-type mathematical models are built in the context of free surface flows and are applicable to a large number of these geophysical surface processes involving sediment transport. Their numerical solution in the Finite Volume (FV) framework is governed by the particular set of equations chosen, by the dynamical properties of the system, by the coupling between flow variables and by the computational grid choice. Moreover, the estimation of the mass and momentum source terms can also affect the robustness and accuracy of the solution. The complexity of the numerical resolution and the computational cost of simulation tools increase considerably with the number of equations involved. Furthermore, most of these highly unsteady flows usually occur along very steep and irregular terrains which require to use a refined non-structured spatial discretization in order to capture the terrain complexity, increasing exponentially the computational times. So that, the computational effort required is one of the biggest challenges for the application of depth-averaged 2D models to realistic large-scale long-term flows. Throughout this thesis, proper 2D shallow-type mathematical models, robust and accurate FV numerical algorithms and efficient high-performance computational codes are combined to develop Efficient Simulation Tools (EST's) for environmental surface processes involving sediment transport with realistic temporal and spatial scales. New EST's able to deal with structured and unstructured meshes are proposed for variable-density mud/debris flows, passive suspended transport and generalized bedload transport. Special attention is paid to the coupling between system variables and to the integration of mass and momentum source terms. The features of each EST have been carefully analyzed and their capabilities have been demonstrated using analytical and experimental benchmark tests, as well as observations in real events.<br /

Mesh generation using a correspondence distance field

The central tool of this work is a correspondence distance field to discrete surface points embedded within a quadtree data structure. The theory, development, and implementation of the distance field tool are described, and two main applications to two-dimensional mesh generation are presented with extension to three-dimensional capabilities in mind. First is a method for surface-oriented mesh generation from a sufficiently dense set of discrete surface points without connectivity information. Contour levels of distance from the body are specified and correspondences oriented normally to the contours are created. Regions of merging fronts inside and between objects are detected in the correspondence distance field and incorporated automatically. Second, the boundaries in a Voronoi diagram between specified coordinates are detected adaptively and used to make Delaunay tessellation. Tessellation of regions with holes is performed using ghost nodes. Images of meshed for each method are given for a sample set of test cases. Possible extensions, future work, and CFD applications are also discussed