In this paper, we propose a technique for building adaptive wavelets by means of an extension of the lifting scheme. Our scheme comprises an adaptive update lifting step and a fixed prediction lifting step. The adaptivity consists hereof that the system can choose between two di#erent update filters, and that this choice is triggered by the local gradient of the original signal. If the gradient is large (in some seminorm sense) it chooses one filter, if it is small the other. We derive necessary and su#cient conditions for the invertibility of such an adaptive system for various scenarios. Furthermore, we present some examples to illustrate our theoretical results.

The use of wavelets in the broad areas of source coding, communications, and networks is surveyed. Specifically, the impact of wavelets and wavelet theory in image coding, video coding, image interpolation, image-adaptive lifting transforms, multipledescription coding, and joint source-channel coding is overviewed. Recent contributions in these areas arising in subsequent papers of the present special issue are described.