We describe the development of adapted waveform analysis (AWA) as a tool for fast processing of the various identification tasks involved in medical diagnostics and automatic target recognition. Such tasks consist of steps: representing the signal as a superposition of component functions, choosing to retain some of the components and discard the others, then reconstructing a new, approximate signal from what was kept. AWA provides tools for each of these steps, accelerating the decomposition and reconstruction computations, providing new functions for analysis and modeling, and extracting new features for recognition and classification. AWA extends Fourier analysis by providing new libraries of standard waveforms with properties akin to windowed sines and cosines, and it extends principal component analysis and eigen-function expansions by adapting the standard functions to individual operators. The cost of representing a function can be measured by how many components must be superposed to obtain a desired degree of approximation, and this cost can be minimized by a fast search through the library of representations. The analysis can be iterated to sift coherent signals from noise. We consider applications to signal and image compression, feature detection, and medical image denoising.