Building nonredundant adaptive wavelets by update lifting

Adaptive wavelet decompositions appear useful in various applications in image and video processing, such as image analysis, compression, feature extraction, denoising and deconvolution, or optic flow estimation. For such tasks it may be important that the multiresolution representations take into account the characteristics of the underlying signal and do leave intact important signal characteristics such as sharp transitions, edges, singularities or other regions of interest. In this paper, we propose a technique for building adaptive wavelets by means of an extension of the lifting scheme. The classical lifting scheme provides a simple yet flexible method for building new, possibly nonlinear, wavelets from existing ones. It comprises a given wavelet transform, followed by a prediction and an update step. The update step in such a scheme computes a modification of the approximation signal, using information in the detail band. It is obvious that such an operation can be inverted, and therefore the perfect reconstruction property is guaranteed. In this paper we propose a lifting scheme including 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. In this paper we derive necessary and sufficient conditions for the invertibility of such an adaptive system for various scenarios.