The Marr Wavelet Pyramid

We introduce a new semi-orthogonal complex wavelet basis of $L _{ 2 }$ $(\mathbb{R} ^{ 2 }$ ). The basis functions are associated to the complex gradient-Laplace operator, which plays a central role in image processing. We define analytically a single-generator wavelet that is shifted on the coset positions of the subsampling matrix. Next, we propose the "wavelet Marr pyramid" for an extension of the new basis that achieves near shift-invariance and steerability (using a Gaussian-like smoothing kernel), for a mild redundancy factor only. This new wavelet pyramid decomposition closely mimicks the basic operations of Marr's framework for early vision. The pyramid is implemented by a fast filterbank algorithm using the FFT.