Content-adaptive Multiresolution Analyses

In this paper we present a technique for building adaptive wavelets by means of an extension of the lifting scheme and analyze the stability of the resulting decompositions. Our scheme comprises an adaptive update lifting and a fixed prediction lifting step. The adaptivity consists hereof that the system can choose between two different 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 sufficient conditions for the invertibility of such an adaptive system for various scenarios. Furthermore, we present some examples to illustrate our theoretical results. We also discuss the effects of quantization in such an adaptive wavelet decomposition and provide conditions for recovering the original decisions at the synthesis and for relating the reconstruction error to the quantization error. Such an analysis is essential for the application of these adaptive decompositions in image compression.