Optimal construction of reduced pyramids for lossless and progressive image coding

Reduced pyramids, including in particular pyramids without analysis filters, are known to produce excellent results when used for lossless signal and image compression. The present paper presents a methodology for the optimal construction of such pyramids by selecting the interpolation synthesis post-filters so as to minimize the error variance at each level of the pyramid. This establishes optimally efficient interpolative pyramidal lossless compression. It also has the added advantage of producing lossy replicas of the original which, at lower resolutions, retain as much similarity to the original as possible. This is highly useful for the progressive coding of signals or images needed for many applications such as fast browsing through image databases or hybrid lossless/lossy medical image coding. The general optimization methodology is developed first, for a general family of reduced pyramids. Subsequently, this is applied to the optimization of pyramids in this family formed using separable, two-dimensional (2-D) quincunx and three-dimensional (3-D) face-centered orthorhombic lattice sampling matrices. It is shown that this family includes in particular the well known 2-D and 3-D "hierarchical interpolation" (HINT) techniques which have been particularly popular for the lossless compression of medical records. Optimal versions of these techniques are determined for 2-D and 3-D images characterized by separable or isotropic correlation functions. The advantages of the developed methods are demonstrated by experimental evaluation. It is shown that the method outperforms the HINT method for the lossless compression of 3-D images. It is also shown to outperform all other known interpolative coders and to produce results comparable to the best predictive lossless coder of 2-D images.