Robustly adaptive wavelet filter bank using L1 norm

Sparse representation of signals is the key for many applications, such as denoising, compression, or compressive sensing. In this paper, we propose an original adaptive wavelet filter bank that, for a class of signals, provides better compaction of information. Previously reported 1D and 2D point-wise adaptive wavelets were based on minimization of the L₂ error norm. Now, we introduce minimum of the L₁ norm on a sliding window as the adaptation criterion. Its main advantages are robustness to outliers and sparser representation of the input data. The proposed algorithm was tested on synthetic signals. It shows significant improvement over known methods, which is paid with somewhat increased numerical complexity. Still, there is some room for improvements, by further development of the adaptive criterion and its efficient realization.