A Spatially Coherent Discrete Wavelet Transform - Accessing the Localization Property for Data Compression

Wavelets intrinsically give us the capability of localized signal decomposition and analysis in space and frequency. However, the popular Fast Wavelet Transform (FWT) that is typically used as an “off the shelf” component in most wavelet based compression algorithms like the Embedded Zerotree (EZW), organizes the intermediate coefficients only by frequency, and not by space. We present the Recursive Merge Filter (RMF) discrete wavelet transform (DWT) algorithm, that organizes intermediate coefficients with respect to both space and frequency. This allows close coupling and pipelining with the encoder, fine grained coding, and “cheap growing” of larger DWTs from smaller ones in a constant number of filter operations. The RMF algorithm computes the DWT of an array of length N in a bottom-up fashion, by successively “merging” two smaller DWTs (four in 2-D), and applying the wavelet filter only on the “smooth” or DC coefficients (Figure 1).