Applications of wavelet-based compression to multidimensional Earth science data

A data compression algorithm involving vector quantization (VQ) and the discrete wavelet transform (DWT) is applied to two different types of multidimensional digital earth-science data. The algorithms (WVQ) is optimized for each particular application through an optimization procedure that assigns VQ parameters to the wavelet transform subbands subject to constraints on compression ratio and encoding complexity. Preliminary results of compressing global ocean model data generated on a Thinking Machines CM-200 supercomputer are presented. The WVQ scheme is used in both a predictive and nonpredictive mode. Parameters generated by the optimization algorithm are reported, as are signal-to-noise (SNR) measurements of actual quantized data. The problem of extrapolating hydrodynamic variables across the continental landmasses in order to compute the DWT on a rectangular grid is discussed. Results are also presented for compressing Landsat TM 7-band data using the WVQ scheme. The formulation of the optimization problem is presented along with SNR measurements of actual quantized data. Postprocessing applications are considered in which the seven spectral bands are clustered into 256 clusters using a k-means algorithm and analyzed using the Los Alamos multispectral data analysis program, SPECTRUM, both before and after being compressed using the WVQ program

Optimum low cost two channel IIR orthonormal filter bank

In this paper, we statistically optimize a well known class of IIR two channel orthonormal filter banks parameterized by a single coefficient when subband quantizers are present. The optimization procedure is extremely simple and very fast compared for example to the linear programming method used in the FIR case to achieve similar compaction (coding) gains. The special form of the filters assure the existence of a zero at π which can be important for some wavelet applications and eliminate some of the major concerns that arise in the FIR design case. Finally, the compaction gain obtained is high and numerically very close to two (ideal case) for low pass spectra, high pass spectra and certain cases of multiband spectra. For these cases, the use of higher order IIR filters does not increase the compaction (coding) gain

Orthonormal and biorthonormal filter banks as convolvers, and convolutional coding gain

Convolution theorems for filter bank transformers are introduced. Both uniform and nonuniform decimation ratios are considered, and orthonormal as well as biorthonormal cases are addressed. All the theorems are such that the original convolution reduces to a sum of shorter, decoupled convolutions in the subbands. That is, there is no need to have cross convolution between subbands. For the orthonormal case, expressions for optimal bit allocation and the optimized coding gain are derived. The contribution to coding gain comes partly from the nonuniformity of the signal spectrum and partly from nonuniformity of the filter spectrum. With one of the convolved sequences taken to be the unit pulse function,,e coding gain expressions reduce to those for traditional subband and transform coding. The filter-bank convolver has about the same computational complexity as a traditional convolver, if the analysis bank has small complexity compared to the convolution itself

A new class of two-channel biorthogonal filter banks and wavelet bases

We propose a novel framework for a new class of two-channel biorthogonal filter banks. The framework covers two useful subclasses: i) causal stable IIR filter banks. ii) linear phase FIR filter banks. There exists a very efficient structurally perfect reconstruction implementation for such a class. Filter banks of high frequency selectivity can be achieved by using the proposed framework with low complexity. The properties of such a class are discussed in detail. The design of the analysis/synthesis systems reduces to the design of a single transfer function. Very simple design methods are given both for FIR and IIR cases. Zeros of arbitrary multiplicity at aliasing frequency can be easily imposed, for the purpose of generating wavelets with regularity property. In the IIR case, two new classes of IIR maximally flat filters different from Butterworth filters are introduced. The filter coefficients are given in closed form. The wavelet bases corresponding to the biorthogonal systems are generated. the authors also provide a novel mapping of the proposed 1-D framework into 2-D. The mapping preserves the following: i) perfect reconstruction; ii) stability in the IIR case; iii) linear phase in the FIR case; iv) zeros at aliasing frequency; v) frequency characteristic of the filters

On the study of four-parallelogram filter banks

The most commonly used 2-D filter banks are separable filter banks, which can be obtained by cascading two 1-D filter banks in the form of a tree. The supports of the analysis and synthesis filters in the separable systems are unions of four rectangles. The natural nonseparable generalization of such supports are those that are unions of four parallelograms. We study four parallelogram filter banks, which is the class of 2-D filter banks in which the supports of the analysis and synthesis filters consist of four parallelograms. For a given a decimation matrix, there could be more than one possible configuration (the collection of passbands of the analysis filters). Various types of configuration are constructed for four-parallelogram filter banks. Conditions on the configurations are derived such that good design of analysis and synthesis filters are possible. We see that there is only one category of these filter banks. The configurations of four-parallelogram filter banks in this category can always be achieved by designing filter banks of low design cost

Flat zones filtering, connected operators, and filters by reconstruction

This correspondence deals with the notion of connected operators. Starting from the definition for operator acting on sets, it is shown how to extend it to operators acting on function. Typically, a connected operator acting on a function is a transformation that enlarges the partition of the space created by the flat zones of the functions. It is shown that from any connected operator acting on sets, one can construct a connected operator for functions (however, it is not the unique way of generating connected operators for functions). Moreover, the concept of pyramid is introduced in a formal way. It is shown that, if a pyramid is based on connected operators, the flat zones of the functions increase with the level of the pyramid. In other words, the flat zones are nested. Filters by reconstruction are defined and their main properties are presented. Finally, some examples of application of connected operators and use of flat zones are described.Peer ReviewedPostprint (published version