A method and apparatus are provided for interpolating a first set of multi-dimensional sample values from a second set of multi-dimensional sample values. The method includes the steps of determining a set of relationships between a second sampling space from which the second set of samples are obtained and a first sampling space into which the first set of samples are interpolated and calculating a partial Fourier transform of the second set of samples. The method further includes the steps of obtaining a partial Fourier transform of the first set of samples using the calculated partial Fourier transform of the second set of samples based upon the determined relationship between the first and second space and performing an inverse partial Fourier transform on the obtained partial Fourier transform of the first set of sample values to recover the first set of sample values.
A system and methods for converting data for an object of interest, which is characterized by a function f, between a digital form and an analog form. In one embodiment, the method includes the steps of locally selecting a plurality of data sample points in the order of N, N being an integer, wherein the data sample points are in a first form of data type, performing a transformation in a shift invariant space to the locally selected data sample points to obtain a new data set that is in a second form of data type different from the first form of data type, and reconstructing the object of interest f from the new data set. In one embodiment, the first form of data type is one of the the digital form and the analog form, and the second form of data type is one of the digital form and the analog form that is different from the first form of data type. In other words, the first form of data type can be in digital form, and the corresponding second form of data type is in analog form, and vice versa.
