Snakes, or active contours, are used extensively in computer vision and image processing applications, particularly to locate object boundaries. Problems associated with initialization and poor convergence to boundary concavities, however, have limited their utility. This paper presents a new external force for active contours, largely solving both problems. This external force, which we call gradient vector flow (GVF), is computed as a diffusion of the gradient vectors of a gray-level or binary edge map derived from the image. It differs fundamentally from traditional snake external forces in that it cannot be written as the negative gradient of a potential function, and the corresponding snake is formulated directly from a force balance condition rather than a variational formulation. Using several two-dimensional (2-D) examples and one three-dimensional (3-D) example, we show that GVF has a large capture range and is able to move snakes into boundary concavities.

/pdf/snakes-shapes-and-gradient-vector-flow-29x5p2ndjo.pdf

Snakes, shapes, and gradient vector flow

Large linear systems of saddle point type arise in a wide variety of applications throughout computational science and engineering. Due to their indefiniteness and often poor spectral properties, such linear systems represent a significant challenge for solver developers. In recent years there has been a surge of interest in saddle point problems, and numerous solution techniques have been proposed for this type of system. The aim of this paper is to present and discuss a large selection of solution methods for linear systems in saddle point form, with an emphasis on iterative methods for large and sparse problems.

/pdf/numerical-solution-of-saddle-point-problems-l583iob5c2.pdf

Numerical solution of saddle point problems

Basic error estimates for elliptic problems

Snakes, or active contours, are used extensively in computer vision and image processing applications, particularly to locate object boundaries. Problems associated with initialization and poor convergence to concave boundaries, however, have limited their utility. This paper develops a new external force for active contours, largely solving both problems. This external force, which we call gradient vector flow (GVF) is computed as a diffusion of the gradient vectors of a gray-level or binary edge map derived from the image. The resultant field has a large capture range and forces active contours into concave regions. Examples on simulated images and one real image are presented.

/pdf/gradient-vector-flow-a-new-external-force-for-snakes-1bovn07lhh.pdf

Gradient vector flow: a new external force for snakes

The aim of the present chapter is to provide an in-depth survey of arbitrary Lagrangian–Eulerian (ALE) methods, including both conceptual aspects of the mixed kinematical description and numerical implementation details. Applications are discussed in fluid dynamics, nonlinear solid mechanics and coupled problems describing fluid–structure interaction. The need for an adequate mesh-update strategy is underlined, and various automatic mesh-displacement prescription algorithms are reviewed. This includes mesh-regularization methods essentially based on geometrical concepts, as well as mesh-adaptation techniques aimed at optimizing the computational mesh according to some error indicator. Emphasis is then placed on particular issues related to the modeling of compressible and incompressible flow and nonlinear solid mechanics problems. This includes the treatment of convective terms in the conservation equations for mass, momentum, and energy, as well as a discussion of stress-update procedures for materials with history-dependent constitutive behavior.


Keywords:

ALE description;
convective transport;
finite elements;
stabilization techniques;
mesh regularization and adaptation;
fluid dynamics;
nonlinear solid mechanics;
stress-update procedures;
fluid–structure interaction

https://onlinelibrary.wiley.com/doi/pdf/10.1002/0470091355.ecm009

Arbitrary Lagrangian–Eulerian Methods

Computer-oriented mesh generators, which serve as pre-processors to finite element programs, have recently been developed by several investigators to alleviate the frustration and to reduce the amount of time involved in the tedious manual subdividing of a complex structure into finite elements. Our purpose here is to describe how the techniques of bivariate ‘blending-function’ interpolation, which were originally developed for, and applied to, geometric problems of computer-aided design and numerically controlled machining of free-form surfaces such as automobile exterior panels, can be adapted and applied to the problems of mesh generation for finite element analyses. We concentrate attention on the problem of curvilinearly co-ordinating simply connected planar domains ℛ by constructing invertible maps of the unit square [0, 1] × [0, 1] onto ℛ. Extensions of the methods described herein to shells in 3-space is straightforward and is illustrated by a practical example taken from the automobile industry. Analogous mesh generators for three-dimensional solids can be developed on the basis of the trivariate ‘blending-function’ formulae found at the end of the second section.

Construction of curvilinear co-ordinate systems and applications to mesh generation

In order to better conform to curved boundaries and material interfaces, curved finite elements have been widely applied in recent years by practicing engineering analysts. The most well known of such elements are the "isoparametric elements". As Zienkiewicz points out in [18, p. 132] there has been a certain parallel between the development of "element types" as used in finite element analyses and the independent development of methods for the mathematical description of general free-form surfaces. One of the purposes of this paper is to show that the relationship between these two areas of recent mathematical activity is indeed quite intimate. In order to establish this relationship, we introduce the notion of a "transfinite element" which, in brief, is an invertible mapping $$\vec T$$ from a square parameter domainJ onto a closed, bounded and simply connected region? in thexy-plane together with a "transfinite" blending-function type interpolant to the dependent variablef defined over?. The "subparametric", "isoparametric" and "superparametric" element types discussed by Zienkiewicz in [18, pp. 137---138] can all be shown to be special cases obtainable by various discretizations of transfinite elements Actual error bounds are derived for a wide class of semi-discretized transfinite elements (with the nature of the mapping $$\vec T$$ :J?? remaining unspecified) as applied to two types of boundary value problems. These bounds for semi-discretized elements are then specialized to obtain bounds for the familiar isoparametric elements. While we consider only two dimensional elements, extensions to higher dimensions is straightforward.

Charles A. Hall

Papers

Construction of curvilinear co-ordinate systems and applications to mesh generation

Transfinite element methods: Blending-function interpolation over arbitrary curved element domains

Optimal error bounds for cubic spline interpolation

Numerical Analysis of Partial Differential Equations

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