Optimal hierarchical coding is sought, for progressive or scalable multidimensional signal transmission, by minimizing the variance of the error difference between the original image and its lower resolution renditions. The optimal, according to the above criterion, pyramidal coders are determined for images quantized using the optimal vector Lloyd-Max quantizers. A rigorous general statistical model of a vector Lloyd-Max quantizer is used, consisting of a linear time-invariant filter followed by additive noise uncorrelated with the input. Given arbitrary analysis filters, the optimal synthesis filters are found. The optimal analysis filters are subsequently determined, leading to formulas for globally optimal structures for pyramidal multidimensional signal decompositions. These structures produce replicas of the original image, which at lower resolutions retain as much similarity to the original as possible. This is highly useful for the progressive coding of two- or three-dimensional (2-D or 3-D) images needed in applications such as fast browsing through image databases. Furthermore, the minimization of the variance of the error image leads to minimization of the variance of the quantization noise for this image and, hence, to its optimally efficient compression. Experimental results illustrate the implementation and performance of the optimal pyramids in application for the coding of still 2-D images.
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