Shift-invariance in the Discrete Wavelet Transform

In this paper we review a number of approaches to reducing, or removing, the problem of shift variance in the discrete wavelet transform (DWT). We describe a generalization of the critically sampled DWT and the fully sampled algorithme à trous that provides approximate shift-invariance with an acceptable level of redundancy. The proposed over complete DWT (OCDWT) is critically sub-sampled to a given level of the decomposition, below which it is then fully sampled. The efficacy of the proposed algorithm is illustrated in an edge detection context and directly compared to a number of other shift-invariant transforms in terms of complexity and redundancy.