We consider a class of interpolation algorithms, including the least-squares optimal Yen interpolator, and we derive a closed-form expression for the interpolation error for interpolators of this type. The error depends on the eigenvalue distribution of a matrix which is specified for each set of sampling points. The error expression can be used to prove that the Yen interpolator is optimal. The implementation of the Yen algorithm suffers from numerical ill-conditioning, forcing the use of a regularized, approximate solution. We suggest a new, approximate solution, consisting of a sinc-kernel interpolator with specially chosen weighting coefficients. The newly designed sinc-kernel interpolator is compared with the usual sinc interpolator using Jacobian (area) weighting, through numerical simulations. We show that the sinc interpolator with Jacobian weighting works well only when the sampling is nearly uniform. The newly designed sinc-kernel interpolator is shown to perform better than ...

Abstract. In many applications one seeks to recover an entire function of exponential type from its non-uniformly spaced samples. Whereas the mathematical theory usually addresses the question of when such a function in L 2 (R) can be recovered, numerical methods operate with a finite-dimensional model. The numerical reconstruction or approximation of the original function amounts to the solution of a large linear system. We show that the solutions of a particularly efficient discrete model in which the data are fit by trigonometric polynomials converge to the solution of the original infinite-dimensional reconstruction problem. This legitimatizes the numerical computations and explains why the algorithms employed produce reasonable results. The main mathematical result is a new type of approximation theorem for entire functions of exponential type from a finite number of values. From another point of view our approach provides a new method for proving sampling theorems. A standard problem in many applications requires one to find a reconstruction of a function f from a collection of samples f(xn). In most applications the assumption that f is band-limited, or equivalently that f is an entire function of exponential type, is well justified, and frequently the sampling points are non-uniformly spaced or distributed quite randomly. Then the mathematical problem is to find conditions under which f can be reconstructed completely from its samples f(xn). This problem is almost completely understood thanks to the work of Duffin-Schaeffer,

space whose dimension is measured A measure F A field V A vector space U Open sets H s Hausdor# measure Appendix B # A Gaussian probability density function # # A Gaussian distribution tangent to a manifold ## A Gaussian distribution normal to a manifold # Sample covariance matrix E, E Error cost functions x x Center of a#ne subspace x Sample mean X A shifted data matrix Appendix C No special symbols Appendix D # Set of all images I An image I(x) Pixel brightness of image I at x P() A morph between two images Z(, , ) A general morph between images # A control line # Unit vector along the control line # # Vector perpendicular to the control line # # 1 ,# 0 The destination and source endpoints of # # The perpendicular proportion of a point to a control line # The signed perpendicular distance of a point to a control line d The Euclidean distance of a point to a control line #(#, , x) The point with the same relation to the control line # as ...

FOV) (4,5), finite-support extrapolation (6 --13), spectral localization by imaging (SLIM) (14), spectral localization with optimal pointspread function (SLOOP) (15), reduced-encoded imaging by generalized-series reconstruction (RIGR) (16), two-reference RIGR (TRIGR) (17), generalized SLIM (GSLIM) (18), feature-recognizing MRI (FrMRI) (19 --22), locally-focused MRI (LfMRI) (21,23--25), singular value decomposition MRI (SVD-MRI) (26), functional image reconstruction enhancement (FIRE) (27,28), and wavelet MRI (29 --31). Despite the apparent variety, this work shows that all of the above methods (1--31) can be represented by a single common equation, which provides a unifying conceptual link. Limitations of existing methods can be identified within this unified framework and new methods can be rationally designed to suit particular needs. Based on this unified framework, a new member in this class of methods has been developed, which is termed the broad-use linear acquisition speed-u

In magnetic resonance (MR) imaging, limited data sampling in-space leads to the well-known Fourier truncation artifact, which includes ringing and blurring. This problem is particularly severe for MR spectroscopic imaging, where only 16--24 points are typically acquired along each spatial dimension. Several methods have been proposed to overcome this problem by incorporating prior information in the image reconstruction. These include the generalized series (GS) model and the finite-support extrapolation method.