We present two applications of wavelet and related techniques to problems arising in medical imaging. Both make considerable use of the edge detection and classification properties of wavelet-type representations. First we describe simple and effective techniques for image denoising and contrast enhancement based on the multiscale edge representation of images. These techniques are sufficiently flexible to successfully address the varying requirements posed by several different medical imaging modalities in common use today. Experimental results are presented to illustrate the application of these techniques to various types of medical images. Next we describe adapted waveform encoding, a technique for magnetic resonance imaging. One advantage of this technique is that it can be used to efficiently encode edge features of the object being imaged. This has a particular diagnostic application in tracking heart wall thickness during the cardiac cycle, which we present along with some experimental results along this line. We also present an analysis of the signal-to-noise ratios of images formed with this technique, as this is a factor of paramount importance in MRI. The fact that wavelet schemes tend to concentrate energy near edge features makes the result rather different than that found in standard Fourier based approaches. We indicate an exciting potential application of our technique: reducing spectral leakage in phosphorus spectroscopy.