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We describe some improvements to a proposed algorithm which encodes the error due to motion compensation in a videoconferencing codec. The algorithm involves lattice quantization using the 8-dimensional E8 lattice. We show that by using a different but equivalent form of E8 based on the extended Hamming code, we can significantly reduce storage requirements and computational complexity at the receiver. We also show that it is easy to perform lattice quantization using the well known 24-dimensional Leech lattice, which is an even better quantizer than E8. Index Terms--Lattice quantization, Leech lattice, motion compensation, vector quantization, video compression, videoconferencing.

A study was conducted to characterize a method for proving the convergence of the CORDIC adaptive lattice filtering algorithm (CALF). The convergence of the CALF algorithm was represented with a Markov chain supporting a limiting state probability distribution. Stochastic gradient lattice algorithms were determined by reducing an instantaneous prediction error. Results showed that the CALF algorithm supports an exponential convergence rate.

A study was conducted to analyze the design of two-channel linear-phase filter banks using a novel algebraic framework. Canonical solutions were determined for the filter banks supporting odd-order symmetric and antisymmetric filters. The computational complexity of transversal and lattice implementations were also examined. Results indicated that the algebraic approach can effectively analyze categories of nontrivial linear-phase perfect reconstruction filter banks.

An alternate Givens rotation-based least-squares lattice algorithm is proposed. Its main advantage is that it reduces storage requirements by 50%, thereby ridding the algorithm of redundancy. The new algorithm propagates a minimal number of internal variables, all of magnitude bounded by 1. It satisfies the backward consistency property and thus features stable error propagation.

A modified two-dimensional lattice algorithm that does not constrain reflection coefficients due to symmetry or stationarity is proposed. The algorithm updates autocorrelations and cross-correlations in the vertical and horizontal directions. Tests prove that the algorithm is effective in predicting correlated clutter in image data. Its resulting filter has been compared with an ideal Wiener-Hopf filter.

A realization method of an ARMAX lattice filter that can do frequency-weighting model identification is proposed. The lattice uses an exponentially weighted sliding window for the same application as the extended least squares (ELS) achieves. The proposed method can be performed with fewer computational costs than the ELS algorithm. It can easily perform ARMAX model identification while other recursive algorithms for the ARMAX cannot perform such as task.

Two- and three-dimensional nonseparable wavelets, obtained from discrete-time bases by iterating filter banks, allow for the analyzation of single-input single-output periodically time-varying systems as if they were multiple-input multiple-output time-invariant systems. Three sampling lattices, quincunx, separable by two in two dimensions, and FCO, are designed using McClellan transformation or cascade structures and then analyzed using the technique. Solutions for previously impossible samplings, including linear phase and orthonormal filters, can be achieved by the multiple dimension systems.

A study conducted on the properties of the partial correlation coefficients of the adaptive lattice filter based on recursive least squares algorithms showed that the forward and backward partial coefficients (PARCOR) are complex conjugates. The PARCOR coefficients for a general lattice filter were derived and expressions for the expected values of the PARCOR coefficients in the nonoptimal cases is found. It was observed that for a given chirp rate there is a threshold adaptation constant below which the total tracking error is negligible.

We present an encoding algorithm for lattices that significantly reduces space requirements while allowing fast computations of least upper bounds and greatest lower bounds of pairs of elements. We analyze the algorithms for encoding, LUB and GLB computations, and prove their correctness. Empirical experiments reveal that our method is significantly more space efficient than the transitive closure method, and the saving becomes increasingly more important as the size of the lattice increases. Index Terms - Lattices, least upper bound, greatest lower bound, encoding algorithm, partially ordered sets

Unified time-recursive lattice structures and their inverse transforms which can be applied to discrete orthogonal transforms such as the discrete cosine, sine and Hartley transforms are proposed. Results show that the area-time complexity is efficient when the block m is small and resulting parallel structures are modular, regular and locally connected. Also, the transform size N is not constrained such that the design complexity is relatively low.

The gradient adaptive lattice and least squares lattice algorithms are evaluated for proof of global convergence via the ordinary differential equation method. The forward decoupling characteristic of the lattice structures is used in the analysis which also examined convergence speed. Analytical and simulation results are presented to show that both algorithms are not independent of data condition or the input correlation matrix.

Adaptive lattice filters are developed to solve two initialization problems associated with unwindowed parameter estimation for autoregressive and finite impulse response models using the standard least squares algorithm. A techniquewhich exploits embedding of the problem in a larger prewindowed lattice is demonstrated. The advantage of the unwindowed lattice over the prewindowed lattice is demonstrated in a numerical example.

A statistical framework for multiscale signal modeling and processing is reported. Based on the concept of wavelet transform, it makes use of dyadic homogenous trees to represent stochastic processes. Generalized lattice structures are introduced to whiten and model the autoregressive, isotropic mechanisms on the trees. An analysis of the processes as a function of the reflection coefficient sequence is also conducted.

A proposed design for a multichannel digital filter uses the lattice structure to derive the forward and backward prediction error vectors and the Gram-Schmidt orthogonalization method for decorrelating the vectors' components. The adaptive generalized sidelob canceller is derived, in turn, for the simulation of the resulting adaptive multichannel lattice-escalator (LAT-ESC) filter. Results show that the proposed design allows faster rate of convergence, better percentage of rate error. The LAT-ESC performs better than the LAT filter and equally with the ESC filter.

A study of the two-dimensional joint process lattice and its image recovery applications is presented. Adaptive lattice algorithms are developed for designing a two-dimensional joint process lattice estimator (TDJPL). An adaptive line enhancer and an adaptive noise canceller, both realizations of the TDJPL, are presented and evaluated using noisy images. Results show the efficacy of TDJPL for adaptive image restoration.

A proposed adaptive lattice filter algorithm is shown to be as effective as the steepest descent method although it converges faster. The algorithm uses only the first derivatives of (B)(j)(n) instead of the Hessian matrix and does not compute for the reflection coefficients on any iteration since these do not change in the optimization. The filter is stabilized by keeping the values of Ks less than unity.

Covariance lattice method and split version of the Burg method were developed to reduce the number of multiplications in split lattice algorithms. The split lattice algorithm, connected with the autocorrelation matrix, was superior to conventional lattice algorithms in terms of a reduction in the number of multiplications. The split algorithms were generated from the addition or subtraction of forward and backward predictors.

Subsampling offers an effective and simple way to implement a data compression technique. Its use in the video context is rather attractive, as its application to current video broadcasting schemes, such as MUSE and HD-MAC, clearly shows. A sensible exploitation of subsampling's potential requires the systematic evaluation of the image degradation effects due to the specific subsampling lattice. Such a possibility is equivalent to the determination of all the sublattices of a given index within a lattice, a problem that can be solved, as the work shows, on the basis of the Hermite normal-form theorem. The practical use of the result in the subsampling context is exemplified for a data compression case.

We conceive the problem of multiple semantic video object (SVO) extraction as an issue of designing extensive operators on a complete lattice of partitions. As a result, we propose a framework based on spatial partition generation and application of optimal operators on the generated partitions. Based on a statistical analysis of the watershed algorithm, we develop a multivalued morphological spatial segmentation method that incorporates an edge-driven marker extraction algorithm and a growing method which integrates both color and edge information. Having embedded the problem in the partition lattice framework, we propose a spatio-temporal regional maximum likelihood operator for extraction purposes. Some theoretical properties of the operator are established. Experimental results on several MPEG-4 test video sequences show that our scheme improves the precision of the extracted SVO boundaries compared to traditional watershed algorithms and provides accurate tracking of multiple SVOs in both static and moving camera scenarios. Furthermore, this scheme can be extended to deal with more general interactive video authoring systems. Index Terms--Mathematical morphology, multivalued watershed, partition lattice operators, semantic video object.

A connectionist scheme, namely, [Sigma]-Fuzzy Lattice Neurocomputing scheme or [Sigma]-FLN for short, which has been introduced in the literature lately for clustering in a lattice data domain, is employed in this work for computing clusters of directed graphs in a master-graph. New tools are presented and used here, including a convenient inclusion measure function for clustering graphs. A directed graph is treated by [Sigma]-FLN as a single datum in the mathematical lattice of subgraphs stemming from a master-graph. A series of experiments is detailed where the master-graph emanates from a Thesaurus of spoken language synonyms. The words of the Thesaurus are fed to [Sigma]-FLN in order to compute clusters of semantically related words, namely, hyperwords. The arithmetic parameters of [Sigma]-FLN can be adjusted so as to calibrate the total number of hyperwords computed in a specific application. It is demonstrated how the employment of hyperwords implies a reduction, based on the a priori knowledge of semantics contained in the Thesaurus, in the number of features to be used for document classification. In a series of comparative experiments for document classification, it appears that the proposed method favorably improves classification accuracy in problems involving longer documents, whereas performance deteriorates in problems involving short documents. Index Terms--Text classification, neural networks, clustering, graphs, framework of fuzzy lattices.

Research is presented concerning the investigation of the complex linear phase filterbank and the symmetric or antisymmetric filterbank through the use of the theory of the real-coefficient linear-phase filter-bank.

Research is presented concerning the development of an algorithm which is computationally efficient. The use of the algorithm in order to compute the 2-D Capon spectral estimator is discussed.