Smooth Boundary Curve Research Articles

Nonparametric boundary detection

We propose a two-step nonparametric method for detecting the boundary curve of an object in an image. First we treat boundary points as change-points on lines across the image, and identify them by the one-sided kernel smoothing method. After obtaining potential boundary points, we use the principal curve method to smooth these points in order to obtain an estimate of smooth boundary curve, Computer simulations are provided to illustrate the effectiveness of the method.

Integral equations of the first kind in plane elasticity

A modified single layer potential is used to solve the interior and exterior Dirichlet problems of plane strain by means of integral equations of the first kind for all smooth boundary curves, including those on which the classical method fails to operate.

Curve segmentation under partial occlusion

2-D shape boundary segmentation is required as a fundamental and important stage in the recognition of partially occluded objects. We introduce here a new segmentation method capable of extracting a controlled number of segments along a smooth boundary curve. This new approach is invariant to similarity transformation, and partial occlusion has only marginal influence on the segmentation of the visible part. The basic concept is to transform the curve into another one which intersects itself. Points of intersection of the new curve are re-transformed to the original curve and serve as endpoints of segments. Properties of the transform are discussed, and conditions for existence of intersection points are given. Simulation results of gray level images are presented, and advantages of our method over conventional approaches relying on singular points of the curvature are discussed. >

ON NON-UNIQUE SOLUTIONS OF WEAKLY SINGULAR INTEGRAL EQUATIONS IN PLANE ELASTICITY

A unique constant matrix is constructed for each smooth boundary curve, which generalizes the concept of logarithmic capacity and indicates when the Dirichlet problem of plane elasticity cannot be solved by means of integral equations of the first kind

Boundary integral equations in time-harmonic acoustic scattering

We first review the basic existence results for exterior boundary value problems for the Helmholtz equation via boundary integral equations. Then we describe the numerical solution of these integral equations in two dimensions for a smooth boundary curve using trigonometric polynomials on an equidistant mesh. We provide a comparison of the Nyström method, the collocation method and the Galerkin method. In each case we take proper care of the logarithmic singularity of the kernel of the integral equation by choosing appropriate quadrature rules. In the case of analytic data the convergence order is exponential. The Nyström method is the most efficient since it requires the least computational effort. Finally, we consider boundary curves with corners. Here, we use a graded mesh based on the idea of transforming the nonsmooth case to a smooth periodic case via an appropriate substitution. Then, the application of Nyström method again yields rapid convergence.

An iterative method for the conformal mapping of doubly connected regions

An iterative method is described for the determination of the conformal mapping of a circular annulus R q ≔ { z: q < | z| < 1} onto a doubly connected region G with smooth boundary curves. The method is analogous to the method of [7] for simply connected regions. It requires in each step the solution of a Riemann-Hilbert problem on R q . This problem can be solved explicitly in terms of a generalized conjugation operator K. A degeneracy of this problem leads in a natural way to an adjustment of the parameter q in each step. If the boundary curves of G admit parametrizations by functions η ν with second derivatives which are Hölder continuous with exponent μ, 0 < μ ⩽ 1, then the method converges locally in the Sobolev space W of 2π-periodic absolutely continuous functions with square-integrable derivatives. The order of convergence is at least 1 + μ. The convergence is quadratic if the η ν have Lipschitz-continuous second derivatives. For the numerical implementation of the method the operator K can be approximated by a Wittich type method, which can be performed very effectively using FFT. A calculation with N = 2 m grid points on the computer requires storage of the order O( N) and computing time O( mN). The results of some test calculations are reported and compared with results of calculations with the method of Theodorsen-Garrick.

Integralgleichungen für einige Randwertprobleme für Gebiete mit Ecken

Concerning conformal mapping and problems of the theory of elasticity, the classical integral equations are modified in order to treat the cases of not smooth boundary curves. As the kernels are still bounded, no difficulties of numerical computation arise.