Reduced pyramids, including in particular pyramids withoutnanalysis filters, are known to produce excellent results when used fornlossless signal and image compression. The present paper presents anmethodology for the optimal construction of such pyramids by selectingnthe interpolation synthesis post-filters so as to minimize the errornvariance at each level of the pyramid. This establishes optimallynefficient interpolative pyramidal lossless compression. It also has thenadded advantage of producing lossy replicas of the original which, atnlower resolutions, retain as much similarity to the original asnpossible. This is highly useful for the progressive coding of signals ornimages needed for many applications such as fast browsing through imagendatabases or hybrid lossless/lossy medical image coding. The generalnoptimization methodology is developed first, for a general family ofnreduced pyramids. Subsequently, this is applied to the optimization ofnpyramids in this family formed using separable, two-dimensional (2-D)nquincunx and three-dimensional (3-D) face-centered orthorhombic latticensampling matrices. It is shown that this family includes in particularnthe well known 2-D and 3-D “hierarchical interpolation”n(HINT) techniques which have been particularly popular for the losslessncompression of medical records. Optimal versions of these techniques arendetermined for 2-D and 3-D images characterized by separable ornisotropic correlation functions. The advantages of the developed methodsnare demonstrated by experimental evaluation. It is shown that the methodnoutperforms the HINT method for the lossless compression of 3-D images.nIt is also shown to outperform all other known interpolative coders andnto produce results comparable to the best predictive lossless coder ofn2-D images
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