Some Results on the Combination of Linear Transforms with Order-Statistic Filters

Since the most efficient waveform coding methods use linear transforms before quantization and entropy coding, the methods designed to allow random access to compressed data normally have to deal with the result of these transforms. However, in some specialized technical applications the information needed is the result of non-linear operations. For instance, it is useful to have fast access to the minimum and maximum values in the compression of elevation maps. In this document we show that some linear transforms have the property of preserving order if certain conditions are satisfied. We provide a proof that this property can be used not only for maximum and minimum, but also for the very general class of non-linear order-statistic filters (which includes median filters). We show that this result is valid for a set of commonly used transforms, including the discrete cosine, Walsh-Hadamard, and dyadic Haar transforms, and also valid for any type of order-statistic filter output.