An important aspect of biomedical data and image processing is to find signal representations that are adapted to the application. The Fourier transform, for example, is well adapted to analyzing the frequency components of signals. Shannon's sampling theory is well adapted to analog/digital conversions, and is therefore well suited to the processing and storage of signals and images by digital computers. However, Shannon's sampling theory is restricted to bandlimited functions. We have developed a general sampling procedure for non-bandlimited signals. In particular, by considering sampling as a problem of approximation in translation-invariant function spaces, we have shown that the least squares approximation of a signal consists of a sampling procedure generalizing the classic one. It consists of an optimal prefiltering, a pure jitter-stable sampling, and a postfiltering for the reconstruction. We have also developed a general theory of representation that uses the new concepts of multiresolutions and wavelets, which are well adapted to multiscale signal processing, edge-detection tasks, signal analyses, coding, and compression.
