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

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

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

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

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

Differential invariant signatures and flows in computer vision : a symmetry group approach

Includes bibliographical references (p. 40-44).Supported by the National Science Foundation. DMS-9204192 DMS-8811084 ECS-9122106 Supported by the Air Force Office of Scientific Research. AFOSR-90-0024 Supported by the Rothschild Foundation-Yad Hanadiv and by Image Evolutions, Ltd.Peter J. Olver, Guillermo Sapiro, Allen Tannenbaum

Virtual prototyping with surface reconstruction and freeform geometric modeling using level-set method

More and more products with complex geometries are being designed and manufactured by computer aided design (CAD) and rapid prototyping (RP) technologies. Freeform surface is a geometrical feature widely used in modern products like car bodies, airfoils and turbine blades as well as in aesthetic artifacts. How to efficiently design and generate digital prototypes with freeform surfaces is an important issue in CAD. This paper presents the development of a Virtual Sculpting system and addresses the issues of surface reconstruction from dexel data structures and freeform geometric modeling using the level-set method from distance field structure. Our virtual sculpting method is based on the metaphor of carving a solid block into a 3D freeform object using a 3D haptic input device integrated with the computer visualization. This dissertation presents the result of the study and consists primarily of four papers --Abstract, page iv

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

Pattern Recognition Society IMPLEMENTING CONTINUOUS-SCALE MORPHOLOGY VIA CURVE EVOLUTION

Abstraet--A new approach to digital implementation of continuous-scale mathematical morphology is presented. The approach is based on discretization of evolution equations associated with continuous mul-tiscale morphological operations. Those equations, and their corresponding numerical implementation, can be derived either directly from mathematical morphology definitions or from curve evolution theory. The advantages of the proposed approach over the classical discrete morphology are demonstrated. Mathematical morphology Scale-space Curve evolution Digital implementation Numerical algorithms I