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

One- and two-level filter-bank convolvers

In a recent paper, it was shown in detail that in the case of orthonormal and biorthogonal filter banks we can convolve two signals by directly convolving the subband signals and combining the results. In this paper, we further generalize the result. We also derive the statistical coding gain for the generalized subband convolver. As an application, we derive a novel low sensitivity structure for FIR filters from the convolution theorem. We define and derive a deterministic coding gain of the subband convolver over direct convolution for a fixed wordlength implementation. This gain serves as a figure of merit for the low sensitivity structure. Several numerical examples are included to demonstrate the usefulness of these ideas. By using the generalized polyphase representation, we show that the subband convolvers, linear periodically time varying systems, and digital block filtering can be viewed in a unified manner. Furthermore, the scheme called IFIR filtering is shown to be a special case of the convolver

Linear phase cosine modulated maximally decimated filter banks with perfect reconstruction

We propose a novel way to design maximally decimated FIR cosine modulated filter banks, in which each analysis and synthesis filter has a linear phase. The system can be designed to have either the approximate reconstruction property (pseudo-QMF system) or perfect reconstruction property (PR system). In the PR case, the system is a paraunitary filter bank. As in earlier work on cosine modulated systems, all the analysis filters come from an FIR prototype filter. However, unlike in any of the previous designs, all but two of the analysis filters have a total bandwidth of 2π/M rather than π/M (where 2M is the number of channels in our notation). A simple interpretation is possible in terms of the complex (hypothetical) analytic signal corresponding to each bandpass subband. The coding gain of the new system is comparable with that of a traditional M-channel system (rather than a 2M-channel system). This is primarily because there are typically two bandpass filters with the same passband support. Correspondingly, the cost of the system (in terms of complexity of implementation) is also comparable with that of an M-channel system. We also demonstrate that very good attenuation characteristics can be obtained with the new system

Coding gain in paraunitary analysis/synthesis systems

A formal proof that bit allocation results hold for the entire class of paraunitary subband coders is presented. The problem of finding an optimal paraunitary subband coder, so as to maximize the coding gain of the system, is discussed. The bit allocation problem is analyzed for the case of the paraunitary tree-structured filter banks, such as those used for generating orthonormal wavelets. The even more general case of nonuniform filter banks is also considered. In all cases it is shown that under optimal bit allocation, the variances of the errors introduced by each of the quantizers have to be equal. Expressions for coding gains for these systems are derived

Vector space framework for unification of one- and multidimensional filter bank theory

A number of results in filter bank theory can be viewed using vector space notations. This simplifies the proofs of many important results. In this paper, we first introduce the framework of vector space, and then use this framework to derive some known and some new filter bank results as well. For example, the relation among the Hermitian image property, orthonormality, and the perfect reconstruction (PR) property is well-known for the case of one-dimensional (1-D) analysis/synthesis filter banks. We can prove the same result in a more general vector space setting. This vector space framework has the advantage that even the most general filter banks, namely, multidimensional nonuniform filter banks with rational decimation matrices, become a special case. Many results in 1-D filter bank theory are hence extended to the multidimensional case, with some algebraic manipulations of integer matrices. Some examples are: the equivalence of biorthonormality and the PR property, the interchangeability of analysis and synthesis filters, the connection between analysis/synthesis filter banks and synthesis/analysis transmultiplexers, etc. Furthermore, we obtain the subband convolution scheme by starting from the generalized Parseval's relation in vector space. Several theoretical results of wavelet transform can also be derived using this framework. In particular, we derive the wavelet convolution theorem

Statistically optimum pre- and postfiltering in quantization

We consider the optimization of pre- and postfilters surrounding a quantization system. The goal is to optimize the filters such that the mean square error is minimized under the key constraint that the quantization noise variance is directly proportional to the variance of the quantization system input. Unlike some previous work, the postfilter is not restricted to be the inverse of the prefilter. With no order constraint on the filters, we present closed-form solutions for the optimum pre- and postfilters when the quantization system is a uniform quantizer. Using these optimum solutions, we obtain a coding gain expression for the system under study. The coding gain expression clearly indicates that, at high bit rates, there is no loss in generality in restricting the postfilter to be the inverse of the prefilter. We then repeat the same analysis with first-order pre- and postfilters in the form 1+αz-1 and 1/(1+γz^-1 ). In specific, we study two cases: 1) FIR prefilter, IIR postfilter and 2) IIR prefilter, FIR postfilter. For each case, we obtain a mean square error expression, optimize the coefficients α and γ and provide some examples where we compare the coding gain performance with the case of α=γ. In the last section, we assume that the quantization system is an orthonormal perfect reconstruction filter bank. To apply the optimum preand postfilters derived earlier, the output of the filter bank must be wide-sense stationary WSS which, in general, is not true. We provide two theorems, each under a different set of assumptions, that guarantee the wide sense stationarity of the filter bank output. We then propose a suboptimum procedure to increase the coding gain of the orthonormal filter bank

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

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

Fixed-analysis adaptive-synthesis filter banks

Subband/Wavelet filter analysis-synthesis filters are a major component in many compression algorithms. Such compression algorithms have been applied to images, voice, and video. These algorithms have achieved high performance. Typically, the configuration for such compression algorithms involves a bank of analysis filters whose coefficients have been designed in advance to enable high quality reconstruction. The analysis system is then followed by subband quantization and decoding on the synthesis side. Decoding is performed using a corresponding set of synthesis filters and the subbands are merged together. For many years, there has been interest in improving the analysis-synthesis filters in order to achieve better coding quality. Adaptive filter banks have been explored by a number of authors where by the analysis filters and synthesis filters coefficients are changed dynamically in response to the input. A degree of performance improvement has been reported but this approach does require that the analysis system dynamically maintain synchronization with the synthesis system in order to perform reconstruction. In this thesis, we explore a variant of the adaptive filter bank idea. We will refer to this approach as fixed-analysis adaptive-synthesis filter banks. Unlike the adaptive filter banks proposed previously, there is no analysis synthesis synchronization issue involved. This implies less coder complexity and more coder flexibility. Such an approach can be compatible with existing subband wavelet encoders. The design methodology and a performance analysis are presented.Ph.D.Committee Chair: Smith, Mark J. T.; Committee Co-Chair: Mersereau, Russell M.; Committee Member: Anderson, David; Committee Member: Lanterman, Aaron; Committee Member: Rosen, Gail; Committee Member: Wardi, Yora