A wavelet transform specifically designed for Fourier analysis at multiple scales is described and shown capable of providing a \em local representation which is particularly well suited to segmentation problems. It is shown that by an appropriate choice of analysis window and sampling intervals, it is possible to obtain a Fourier representation which can be computed efficiently and overcomes the limitations of using a fixed scale of window, yet by virtue of its symmetry properties allows simple estimation of such fundamental signal parameters as instantaneous frequency and onset time/position. The transform is applied to the segmentation of both image and audio signals, demonstrating its power to deal with signal events which are localised in either time/space or frequency. Feature extraction and segmentation are tackled through the introduction of a class of multiresolution Markov models, whose parameters represent the signal events underlying the segmentation. In the case of images, this provides a unified and computationally efficient approach to boundary curve segmentation; in audio analysis, it provides an effective way of note segmentation, giving accurate estimates of onset time and pitch in polyphonic musical signals.