Non-Euclidean Pyramids

We propose to design the reduction operator of an image pyramid so as to minimize the approximation error in the $l _{ p }$ sense (not restricted to the usual p = 2), where p can take non-integer values. The underlying image model is specified using arbitrary shift-invariant basis functions such as splines. The solution is determined by an iterative optimization algorithm, based on digital filtering. Its convergence is accelerated by the use of first and second derivatives. For p = 1, our modified pyramid is robust to outliers; edges are preserved better than in the standard case where p = 2. For 1 < p < 2, the pyramid decomposition combines the qualities of $l _{ 1 }$ and $l _{ 2 }$ approximations. The method is applied to edge detection and its improved performance over the standard formulation is determined.