This paper aims to present a review of recent as well as classic image registration methods. Image registration is the process of overlaying images (two or more) of the same scene taken at different times, from different viewpoints, and/or by different sensors. The registration geometrically align two images (the reference and sensed images). The reviewed approaches are classified according to their nature (areabased and feature-based) and according to four basic steps of image registration procedure: feature detection, feature matching, mapping function design, and image transformation and resampling. Main contributions, advantages, and drawbacks of the methods are mentioned in the paper. Problematic issues of image registration and outlook for the future research are discussed too. The major goal of the paper is to provide a comprehensive reference source for the researchers involved in image registration, regardless of particular application areas. q 2003 Elsevier B.V. All rights reserved.

Learning an input-output mapping from a set of examples, of the type that many neural networks have been constructed to perform, can be regarded as synthesizing an approximation of a multi-dimensional function, that is solving the problem of hypersurface reconstruction. From this point of view, this form of learning is closely related to classical approximation techniques, such as generalized splines and regularization theory. This paper considers the problems of an exact representation and, in more detail, of the approximation of linear and nonlinear mappings in terms of simpler functions of fewer variables. Kolmogorov's theorem concerning the representation of functions of several variables in terms of functions of one variable turns out to be almost irrelevant in the context of networks for learning. Wedevelop a theoretical framework for approximation based on regularization techniques that leads to a class of three-layer networks that we call Generalized Radial Basis Functions (GRBF), since they are mathematically related to the well-known Radial Basis Functions, mainly used for strict interpolation tasks. GRBF networks are not only equivalent to generalized splines, but are also closely related to pattern recognition methods suchasParzen windows and potential functions and to several neural network algorithms, suchas Kanerva's associative memory,backpropagation and Kohonen's topology preserving map. They also haveaninteresting interpretation in terms of prototypes that are synthesized and optimally combined during the learning stage. The paper introduces several extensions and applications of the technique and discusses intriguing analogies with neurobiological data.

Abstract. A hierarchical scheme is presented for smoothly interpolating scattered data with radial basis functions of compact support. A nested sequence of subsets of the data is computed efficiently using successive Delaunay triangulations. The scale of the basis function at each level is determined from the current density of the points using information from the triangulation. The method is rotationally invariant and has good reproduction properties. Moreover the solution can be calculated and evaluated in acceptable computing time. During the last two decades radial basis functions have become a well established tool for multivariate interpolation of both scattered and gridded data; see [2,7,8,22,25] for some surveys. The major part

We consider elastic registration of medical image data based on thin-plate splines using a set of corresponding anatomical point landmarks. Previous work on this topic has concentrated on using interpolation schemes. Such schemes force the corresponding landmarks to exactly match each other and assume that the landmark positions are known exactly. However, in real applications the localization of landmarks is always prone to some error. Therefore, to take into account these localization errors, we have investigated the application of an approximation scheme which is based on regularization theory. This approach generally leads to a more accurate and robust registration result. In particular, outliers do not disturb the registration result as much as is the case with an interpolation scheme. Also, it is possible to individually weight the landmarks according to their localization uncertainty. In addition to this study, we report on investigations into semi-automatic extraction of anat...

This is a survey of techniques for the interpolation of scattered data in three or more independent variables. It covers schemes that can be used for any number of variables as well as schemes specifically designed for three variables. Emphasis is on breadth rather than depth, but there are explicit illustrations of different techniques used in the solution of multivariate interpolation problems. List of Contents 1. Introduction 2. Rendering of Trivariate Functions 3. Tensor Product Schemes 4. Point Schemes 4.1 Shepard's Methods 4.2 Radial Interpolants 4.2.1 Hardy Multiquadrics 4.2.2 Duchon Thin Plate Splines 5. Natural Neighbor Interpolation 6. k-dimensional Triangulations 7. Tetrahedral Schemes 7.1 Polynomial Schemes 7.2 Rational Schemes 8. Simplicial Schemes 8.1 Polynomial Schemes 8.2 Rational Schemes 8.3 A Transfinite Scheme 9. Multivariate Splines 10. Transfinite Hypercubal Methods 11. Derivative Generation 12. Interpolation on the sphere and other surfa...

A general approach to the computation of basic topographic parameters independent of the spatial distribution of given elevation data is developed. The approach is based on an interpolation function with regular first and second order derivatives and on application of basic principles of differential geometry. General equations for computation of profile, plan, and tangential curvatures are derived. A new algorithm for construction of slope curves is developed using a combined grid and vector approach. Resulting slope curves better fulfil the condition of orthogonality to contours than standard grid algorithms. Presented methods are applied to topographic analysis of a watershed in central Illinois.

This paper presents a literature survey of automatic 3D surface registration techniques emphasizing the mathematical and algorithmic underpinnings of the subject. The relevance of surface registration to medical imaging is that there is much useful anatomical information in the form of collected surface points which originate from complimentary modalities and which must be reconciled. Surface registration

. This contribution gives a partial survey over the native spaces associated to (not necessarily radial) basis functions. Starting from reproducing kernel Hilbert spaces and invariance properties, the general construction of native spaces is carried out for both the unconditionally and the conditionally positive definite case. The definitions of the latter are based on finitely supported functionals only. Fourier or other transforms are not required. The dependence of native spaces on the domain is studied, and criteria for functions and functionals to be in the native space are given. x1. Introduction and Overview For the numerical treatment of functions of many variables, radial basis functions are useful tools. They have the form OE(kx \Gamma yk 2 ) for vectors x; y 2 IR d with a univariate function OE defined on [0; 1) and the Euclidean norm k \Delta k 2 on IR d . This allows to work efficiently for large dimensions d, because the function boils the multivariate setting down...

. We discuss several approaches to the problem of interpolating or approximating data given at scattered points lying on the surface of the sphere. These include methods based on spherical harmonics, tensorproduct spaces on a rectangular map of the sphere, functions defined over spherical triangulations, spherical splines, spherical radial basis functions, and some associated multi-resolution methods. In addition, we briefly discuss sphere-like surfaces, visualization, and methods for more general surfaces. The paper includes a total of 206 references. x1. Introduction Let S be the unit sphere in IR 3 , and suppose that fv i g n i=1 is a set of scattered points lying on S. In this paper we are interested in the following problem: Problem 1. Given real numbers fr i g n i=1 , find a (smooth) function s defined on S which interpolates the data in the sense that s(v i ) = r i ; i = 1; : : : ; n; (1) or approximates it in the sense that s(v i ) ß r i ; i = 1; : : : ; n: (2) Data f...

This paper compares radial basis function interpolants on different spaces. The spaces are generated by other radial basis functions, and comparison is done via an explicit representation of the norm of the error functional. The results pose some new questions for further research. x1. Introduction We consider interpolation of real-valued functions f defined on a set \Omega ` IR d ; d 1. These functions are evaluated on a set X := fx 1 ; : : : ; xNX g of NX 1 pairwise distinct points x 1 ; : : : ; xNX in \Omega\Gamma If N 2; d 2 and\Omega ` IR d are given with\Omega containing at least an interior point, it is well known that there is no N-dimensional space of continuous functions on\Omega that contains a unique interpolant for every f and every set X = fx 1 ; : : : ; xNX g ae\Omega ` IR d consisting of N = NX data points. Thus the family of interpolants must necessarily depend on X. This can easily be achieved by using translates \Phi(x \Gamma x j ) of a single continu...