The FFT is used widely in signal processing for efficient computation of the Fourier transform (FT) of finitelength signals over a set of uniformly-spaced frequency locations. However, in many applications, one requires nonuniform sampling in the frequency domain, i.e.,a nonuniform FT . Several papers have described fast approximations for the nonuniform FT based on interpolating an oversampled FFT. This paper presents an interpolation method for the nonuniform FT that is optimal in the min-max sense of minimizing the worst-case approximation error over all signals of unit norm. The proposed method easily generalizes to multidimensional signals. Numerical results show that the min-max approach provides substantially lower approximation errors than conventional interpolation methods. The min-max criterion is also useful for optimizing the parameters of interpolation kernels such as the Kaiser-Bessel function.

......................................................................................................... viii CHAPTERS 1 INTRODUCTION ............................................................................... 1 2 AN APPROACH TO SUPERRESOLUTION ...................................... 5 2.1 Introducing the Problem .......................................................... 5 2.2 Comments and Observations ................................................... 9 2.3 Motivation and Description of the Superresolution Architecture ............................................................................ 15 3 EXISTING APPROACHES TO RECONSTRUCTION OF OPTICAL AND SYNTHETIC APERTURE RADAR IMAGES .......................... 22 3.1 Optical Image Interpolation with a Single Kernel ..................... 23 3.1.1 Common Kernels ............................................................ 24 3.1.2 Increasing the Sample Density ........................................ 28 3.1.3 Interpolation as Projec...

Image and volume data rotation with 1- and 3pass algorithms Three different implementations of the 3-pass algorithm of image and volume data rotation are illustrated and discussed. The three protocols use interpolation in real domain, with a peculiar implementation of the Shannon reconstruction, or phase shifts in Fourier domain. Accuracy and speed of the three methods are compared with corresponding values obtained with a 1-pass method. The results indicate that for low or moderate accuracy, 1-pass is more convenient than 3-pass rotation for both accuracy and speed. Very accurate rotations can be obtained in reasonable time if all steps of 3-pass rotation are performed in the Fourier domain. The computer processing of large arrays of experimental data involves two conflicting aspects: computing time and accuracy. Rotation and projections of large images or of volume data are typical tasks in which the two aspects are to be carefully balanced. For this reason, multi-step algorithms (Catmull and Smith, 1980; Paeth, 1986) to perform rotations in a seemingly fast and accurate way have been welcomed with enthusiasm (Wolberg, 1990), and raise continuing interest (see, for example, Unser et al., 1995). In this report, we present our experience with methods for the rotation of images and volume data, with 1- and 3-pass algorithms. It is worth recalling some tasks which require two- (2D) and three-dimensional (3D) functions to be accurately rotated stepwise to obtain sets of projections; typical examples are the computation of sinograms for the angular reconstitution method (van Heel, 1987) or for checking reconstruction algorithms (Bellon and Lanzavecchia, 1995). Projections of preliminary 3D reconstructions are used to improve tomographic results (Gilbert, 1972; Harauz and Ottensmeyer, 1984) or to obtain a sampled 3D Radon transform (Radermacher, 1994). Another field requiring rotations of volume data is multimodal imaging, a frontier in clinical diagnosis. Multimodal information is obtained by merging the results of X-ray, MR and emission tomography (Meyer et al., 1995; van den Elsen Diparlimento di Chimica Slrutturale e Slereochtmica Inorganica,