The decomposition of deformations by principal warps is demonstrated. The method is extended to deal with curving edges between landmarks. This formulation is related to other applications of splines current in computer vision. How they might aid in the extraction of features for analysis, comparison, and diagnosis of biological and medical images in indicated. >

Principal warps: thin-plate splines and the decomposition of deformations

The problem of the approximation of nonlinear mapping, (especially continuous mappings) is considered. Regularization theory and a theoretical framework for approximation (based on regularization techniques) that leads to a class of three-layer networks called regularization networks are discussed. Regularization networks are mathematically related to the radial basis functions, mainly used for strict interpolation tasks. Learning as approximation and learning as hypersurface reconstruction are discussed. Two extensions of the regularization approach are presented, along with the approach's corrections to splines, regularization, Bayes formulation, and clustering. The theory of regularization networks is generalized to a formulation that includes task-dependent clustering and dimensionality reduction. Applications of regularization networks are discussed. >

Networks for approximation and learning

A modified conjugate gradient algorithm for geometry optimization is outlined for use with ab initioMO methods. Since the computation time for analytical energy gradients is approximately the same as for the energy, the optimization algorithm evaluates and utilizes the gradients each time the energy is computed. The second derivative matrix, rather than its inverse, is updated employing the gradients. At each step, a one-dimensional minimization using a quartic polynomial is carried out, followed by an n-dimensional search using the second derivative matrix. By suitably controlling the number of negative eigenvalues of the second derivative matrix, the algorithm can also be used to locate transition structures. Representative timing data for optimizations of equilibrium geometries and transition structures are reported for ab initioSCF–MO calculations.

Optimization of equilibrium geometries and transition structures

We construct a new class of positive definite and compactly supported radial functions which consist of a univariate polynomial within their support. For given smoothness and space dimension it is proved that they are of minimal degree and unique up to a constant factor. Finally, we establish connections between already known functions of this kind.

Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree

A new finite element method is presented that features the ability to include in the finite element space knowledge about the partial differential equation being solved This new method can therefore be more efficient than the usual finite element methods An additional feature of the partition-of-unity method is that finite element spaces of any desired regularity can be constructed very easily This paper includes a convergence proof of this method and illustrates its efficiency by an application to the Helmholtz equation for high wave numbers The basic estimates for a posteriori error estimation for this new method are also proved © 1997 by John Wiley & Sons, Ltd

The Partition of Unity Method

Absract. This paper is concerned with the evaluation of methods for scattered data interpolation and some of the results of the tests when applied to a number of methods. The process involves evaluation of the methods in terms of timing, storage, accuracy, visual pleasantness of the surface, and ease of implementation. To indicate the flavor of the type of results obtained, we give a summary table and representative perspective plots of several surfaces.

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Scattered data interpolation: tests of some methods

A. H. Stroud has recently shown the existence of cubature formulas for planar regions which use m2 points and have polynomial precision 2m - 1. In this paper, the author gives sufficient conditions for the existence of formulas using fewer than m2 points, and having polynomial precision 2m - 1. An algorithm is given for computing such for- mulas, and is shown to be useful in a more general setting than given in the theorem. Numer- ical examples are given, both in terms of previously known and new cubature formulas. 1. Introduction. A number of authors have attempted to use the common zeros of orthogonal polynomials in two variables as evaluation points in cubature formulas for regions in the plane. A theorem given recently by Stroud (10) shows how to con- struct formulas using orthogonal polynomials of degree m if they have exactly m2 distinct common zeros. These formulas approximate the integrals of polynomials of degree ? 2m - 1 exactly. The principal result of this paper is to give sufficient con- ditions for the existence of similar formulas which use fewer than m2 points. The following notation and definitions will be used in this paper.

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Richard Franke

Papers

Scattered data interpolation: tests of some methods

Obtaining cubatures for rectangles and other planar regions by using orthogonal polynomials