Centered Pyramids

Quadtree-like pyramids have the advantage of resulting in a multiresolution representation where each pyramid node has four unambiguous parents. Such a centered topology guarantees a clearly defined up-projection of labels. This concept has been successfully and extensively used in applications of contour detection, object recognition and segmentation. Unfortunately, the quadtree-like type of pyramid has poor approximation powers because of the employed piecewise-constant image model. This paper deals with the construction of improved centered image pyramids in terms of general approximation functions. The advantages of the centered topology such a symmetry, consistent boundary conditions and accurate up-projection of labels are combined with a more faithful image representation at coarser pyramid levels. We start by introducing a general framework for the design of least squares pyramids using the standard filtering and decimation tools. We give the most general explicit formulas for the computation of the filter coefficients by any (well behaving) approximation function in both the continuous $(L _{ \infty } )$ and the discrete $(l _{ \infty } )$ norm. We then define centered pyramids and provide the filter coefficients for odd spline approximation functions. Finally, we compare the centered pyramid to the ordinary one and highlight some applications.