Image resampling is the process of geometrically transforming digital images. This report considers several aspects of the process. We begin by decomposing the resampling process into three simpler sub-processes: reconstruction of a continuous intensity surface from a discrete image, transformation of that continuous surface, and sampling of the transformed surface to produce a new discrete image. We then consider the sampling process, and the subsidiary problem of intensity quantisation. Both these are well understood, and we present a summary of existing work, laying a foundation for the central body of the report where the sub-process of reconstruction is studied. The work on reconstruction divides into four parts, two general and two specific: 1. Piecewise local polynomials: the most studied group of reconstructors. We examine these, and the criteria used in their design. One new derivation is of two piecewise local quadratic reconstructors. 2. Infinite extent reconstructors: we consider these and their local approximations, the problem of finite image size, the resulting edge effects, and the solutions to these problems. Amongst the reconstructors discussed are the interpolating cubic B-spline and the interpolating Bezier cubic. We derive the filter kernels for both of these, and prove that they are the same. Given this kernel we demonstrate how the interpolating cubic B-spline can be extended from a one-dimensional to a two-dimensional reconstructor, providing a considerable speed improvement over the existing method of extension. 3. Fast Fourier transform reconstruction: it has long been known that the fast Fourier transform (FFT) can be used to generate an approximation to perfect scaling of a sample set. Donald Fraser (in 1987) took this result and generated a hybrid FFT reconstructor which can be used for general transformations, not just scaling. We modify Fraser’s method to tackle two major problems: its large time and storage requirements, and the edge effects it causes in the reconstructed intensity surface. 4. A priori knowledge reconstruction: first considering what can be done if we know how the original image was sampled, and then considering what can be done with one particular class of image coupled with one particular type of sampling. In this latter case we find that exact reconstruction of the image is possible. This is a surprising result as this class of images cannot be exactly reconstructed using classical sampling theory. The final section of the report draws all of the strands together to discuss transformations and the resampling process as a whole. Of particular note here is work on how the quality of different reconstruction and resampling methods can be assessed.