This thesis is concerned with the estimation of edges in magnetic resonance images (MRI), which may be seen as a first step in the automatic classification of such data. The estimation is taken as a two-stage process. A set of points lying on a single edge is first identified. Secondly, some form of closed curve is fitted to this set of points to describe the edge. The data analysed in this study are MRI of cross-sections through human thighs. Although the subject of the images exists in continuous two-dimensional space, in practice data values are only recorded at discrete, sampled points. This is due to quantisation of the underlying continuous function for storage on a computer. A major theme for this study is the recovery of the underlying continuous function from the sampled data: it is expected that this will allow edges to be estimated more accurately. Bivariate kernel regression is used in the first stage to fit a smooth function to the observed data. Edge points are identified as positions of zero-crossings of the smooth function. The accuracy with which edge points are located is influenced by the amount of smoothing, and several data-based methods are discussed for estimating an appropriate smoothing parameter. In the second stage, an edge is modelled as a simple, closed curve by fitting a Fourier series (FS) to the set of edge points. Geometric properties, such as perimeter length, can be determined from the fitted series. The accuracy of the estimation of such properties is used as a criterion to determine the number of terms to be included in the series. The choice of variable with which to label consecutive points prior to fitting the FS is also discussed.