Multidimensional Signal Processing Research Articles

On General Factorizations for n-D Polynomial Matrices

Multivariate (n-D) polynomial matrix factorizations are basic research subjects in multidimensional (n-D) systems and signal processing. In this paper, several results on general matrix factorizations are provided for extracting a matrix factor from a given n-D polynomial matrix whose lower order minors satisfy certain conditions. These results are further generalizations of previous results in (Lin et al. in Circuits Syst. Signal Process. 20(6):601–618, 2001). As a consequence, the application range of the constructive algorithm in (Lin et al. in Circuits Syst. Signal Process. 20(6):601–618, 2001) has been greatly extended. Three examples are worked out in detail to show the practical value of the proposed method for obtaining general factorizations for a class of n-D polynomial matrices.

Realization of multidimensional signal processing systems given in natural k-D state space description

The paper addresses design, analysis, and realization of linear, discrete domain k-D signal processing systems based on a natural state space description. The latter is chosen to overcome restrictions by widely known Givone-Roesser and Fornasini-Marchesini models. It is capable of describing noncausal systems as well as shift operations with respect to more than one coordinate direction. It thus provides a system description that offers better compliance with possible hardware realizations than existing models and thus allows evaluation and consideration of many realization specific effects.

Signal Assignment to Hierarchical Memory Organizations for Embedded Multidimensional Signal Processing Systems

The storage requirements of the array-dominated and loop-organized algorithmic specifications running on embedded systems can be significant. Employing a data memory space much larger than needed has negative consequences on the energy consumption, latency, and chip area. Finding an optimized storage of the usually large arrays from these algorithmic specifications is an essential task of memory allocation. This paper proposes an efficient algorithm for mapping multidimensional arrays to the data memory. Similarly to [1], it computes bounding windows for live elements in the index space of arrays, but this algorithm is several times faster. More important, since this algorithm works not only for <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">entire</i> arrays, but also <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">parts</i> of arrays - like, for instance, array references or, more general, sets of array elements represented by lattices [2], this signal-to-memory mapping technique can be also applied in hierarchical memory architectures.

Signal Assignment Model for the Memory Management of Multidimensional Signal Processing Applications

Many signal processing systems, particularly in the multimedia and telecom domains, are synthesized to execute data-dominated applications. Their behavior is described in a high-level programming language, where the code is typically organized in sequences of loop nests and the main data structures are multidimensional arrays. Since data transfer and storage have a significant impact on both the system performance and the major cost parameters--power consumption and chip area, the designer must spend a significant effort during the system development process on the exploration of the memory subsystem in order to achieve a cost-optimized design. This paper presents a memory allocation methodology for multidimensional signal processing applications, focusing on the problem of efficiently mapping the multidimensional signals from the algorithmic specification into the physical memory. In a first phase, two previous mapping models are implemented within a common theoretical framework, which is advantageous from both the point of view of computational efficiency and the amount of allocated data storage. Different from all the previous mapping models that aim to optimize the memory sharing between the elements of a same array (creating separate windows in the physical memory for distinct arrays), this proposed mapping model exploit--in a second phase--the possibility of memory sharing between the elements of different arrays. As a consequence, this signal assignment approach yields significant savings in the amount of data storage resulted after mapping.

Notes on factor prime factorizations for n-D polynomial matrices

Multidimensional (n?D) polynomial matrix factorizations have been widely investigated during the past years due to the fundamental importance in the areas of multidimensional systems and signal processing. This note is a continuation of the papers in Wang (Circuits Syst I 54: 1398---1405, 2007 and Circuits Syst II 55: 61---64, 2008). A complete constructive criterion is given for a matrix of 2 × m full row rank to be FLP. We also present a reduction method for dealing with the factorization problems with cases of non-regular divisors.

Array-OL with delays, a domain specific specification language for multidimensional intensive signal processing

Intensive signal processing applications appear in many application domains such as video processing or detection systems. These applications handle multidimensional data structures (mainly arrays) to deal with the various dimensions of the data (space, time, frequency). A specification language allowing the direct manipulation of these different dimensions with a high level of abstraction is a key to handling the complexity of these applications and to benefit from their massive potential parallelism. The Array-OL specification language is designed to do just that. We introduce here an extension of Array-OL to deal with states or delays by the way of uniform inter-repetition dependences. We show that this specification language is able to express the main patterns of computation of the intensive signal processing domain.

Energy-Aware Memory Allocation Framework for Embedded Data-Intensive Signal Processing Applications

The storage requirements in data-intensive signal processing systems (video and image processing, artificial vision, medical imaging, real-time 3-D rendering, advanced audio and speech coding, etc.) have an important impact on both the system performance and the essential design parameters—the overall power consumption and chip area. Therefore, the optimization of the memory subsystem is an important step in the early stages of the system-level exploration. This work proposes an energy-aware memory allocation methodology for embedded data-intensive signal processing applications. Starting from the high-level behavioral specification of a given application, the framework proposed in the thesis performs the assignment of the multidimensional signals to the memory layers—the on-chip scratch-pad memories (SPM) and the off-chip main memory—the goal being the reduction of the dynamic energy consumption in the hierarchical memory subsystem. Based on the assignment results, the framework subsequently performs the mapping of signals into both memory layers such that the overall amount of data storage be reduced. The multidimensional signal processing applications are algorithmically specified in high-level programming languages, the main data structures being multidimensional arrays. The major contributions of the Ph.D. thesis are (1) a formal model (operating with multidimensional polyhedra and linearly bounded lattices) for the reduction of the dynamic energy consumption in the hierarchical memory subsystem of embedded data-intensive signal processing applications, and (2) a novel methodology (using inter-array memory sharing) for mapping the multidimensional arrays from the high-level algorithmic specifications into the physical memory. This software system yields a complete allocation solution: the exact storage amount on each memory layer, the mapping functions that determine the exact locations for any array element (scalar signal) in the specification, and an estimation of the dynamic energy consumption in the memory subsystem.

Formal Model for the Reduction of the Dynamic Energy Consumption in Multi-Layer Memory Subsystems

In real-time data-dominated communication and multimedia processing applications, a multi-layer memory hierarchy is typically used to enhance the system performance and also to reduce the energy consumption. Savings of dynamic energy can be obtained by accessing frequently used data from smaller on-chip memories rather than from large background memories. This paper focuses on the reduction of the dynamic energy consumption in the memory subsystem of multidimensional signal processing systems, starting from the high-level algorithmic specification of the application. The paper presents a formal model which identifies those parts of arrays more intensely accessed, taking also into account the relative lifetimes of the signals. Tested on a two-layer memory hierarchy, this model led to savings of dynamic energy from 40% to over 70% relative to the energy used in the case of flat memory designs.

REVERSIBLE SYSTOLIC ARRAYS: m-ARY BIJECTIVE SINGLE-INSTRUCTION MULTIPLE-DATA (SIMD) ARCHITECTURES AND THEIR QUANTUM CIRCUITS

New type of m-ary systolic arrays called reversible systolic arrays is introduced in this paper. The m-ary quantum systolic architectures' realizations and computations of the new type of systolic arrays are also introduced. A systolic array is an example of a single-instruction multiple-data (SIMD) machine in which each processing element (PE) performs a single simple operation. Systolic devices provide inexpensive but massive computation power, and are cost-effective, high-performance, and special-purpose systems that have wide range of applications such as in solving several regular and compute-bound problems containing repetitive multiple operations on large arrays of data. Similar to the classical case, information in a reversible and quantum systolic circuit flows between cells in a pipelined fashion, and communication with the outside world occurs only at the boundary cells. Since basic PEs used in the construction of arithmetic systolic arrays are the add–multiply cells, the results introduced in this paper are general and apply to a very wide range of add–multiply-based systolic arrays. Since the reduction of power consumption is a major requirement for the circuit design in future technologies, such as in quantum computing, the main features of several future technologies will include reversibility. Consequently, the new systolic circuits can play an important task in the design of future circuits that consume minimal power. It is also shown that the new systolic arrays maintain the high level of regularity while exhibiting the new fundamental bijectivity (reversibility) and quantum superposition properties. These new properties will be essential in performing super-fast arithmetic-intensive computations that are fundamental in several future applications such as in multi-dimensional quantum signal processing (QSP).

Adaptive Flattening for Multidimensional Image Restoration

Whereas most previous works treating color or hyper- spectral image restoration use hybrid filters or data splitting, some new approaches consider multidimensional or tensor signal processing techniques. Tensor processing methods are based on multilinear algebra and are more efficient than 2-D filtering. However, they rely on orthogonal tensor flattening. The aim of this letter is to show that this orthogonal flattening may not be optimal for multidimensional images. We propose a method to adapt the flattening depending on the data set. Our proposed method is based on the estimation of main directions in multidimensional data. For this purpose, we extend the straight line detection algorithm. Multidimensional filtering method HOSVD - (K1,..., K <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</sub> ) is applied along the estimated directions. We also adapt a quadtree partitioning in order to split tensors into homogeneous sub-tensors to keep local characteristics. Considering some examples of color and hyperspectral images, we present some promising results.

Remarks on $n$-D Polynomial Matrix Factorization Problems

Multidimensional (n-D) polynomial matrix factorizations are intimately linked to many problems of multidimensional systems and signal processing. This paper gives a new result for a n-D polynomial matrix to have an minor prime factorization using methods from computer algebra. This result may be regarded as a generalization of a previous criterion under a special restriction [IEEE Trans. Circuits Syst. II: Exp Briefs, vol. 52, no. 9 (2005)]. Examples are given to illustrate results using computer algebra software system Singular.

PARAMETRIZATION OF COMPACTLY SUPPORTED TRIVARIATE ORTHOGONAL WAVELET FILTER

Multivariate wavelets analysis is a powerful tool for multi-dimensional signal processing, but tensor product wavelets have a number of drawbacks. In this paper, we give an algorithm of parametric representation compactly supported trivariate orthogonal wavelet filter, which simplifies the study of trivariate orthogonal wavelet. Four examples are also given to demonstrate the method.

Power and Area Minimization for Multidimensional Signal Processing

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> Sensitivity-based methodology is applied to optimization of performance, power and area across several levels of design abstraction for a complex wireless baseband signal processing algorithm. The design framework is based on a unified, block-based graphical description of the algorithm to avoid design re-entry in various phases of chip development. The use of architectural techniques for minimization of power and area for complex signal processing algorithms is demonstrated using this framework. As a proof of concept, an ASIC realization of the MIMO baseband signal processing for a multi-antenna WLAN is described. The chip implements a 4<formula formulatype="inline"><tex>$\,\times\,$</tex> </formula>4 adaptive singular value decomposition (SVD) algorithm with combined power and area minimization achieving a power efficiency of 2.1 GOPS/mW (12-bit add equivalent) in just 3.5 <formula formulatype="inline"><tex>${\hbox {mm}}^{2}$</tex> </formula> in a standard 90 nm CMOS process. The computational throughput of 70 GOPS is implemented with 0.5 M cells at a 100 MHz clock and 385 mV supply, dissipating 34 mW of power. With optimal channel conditions the algorithm implemented can deliver up to 250 Mb/s over 16 sub-carriers. </para>

Computation of Storage Requirements for Multi-Dimensional Signal Processing Applications

Many integrated circuit systems, particularly in the multimedia and telecom domains, are inherently data dominant. For this class of systems, a large part of the power consumption is due to the data storage and data transfer. Moreover, a significant part of the chip area is occupied by memory. The computation of the memory size is an important step in the system-level exploration, in the early stage of designing an optimized (for area and/or power) memory architecture for this class of systems. This paper presents a novel nonscalar approach for computing exactly the minimum size of the data memory for high-level procedural specifications of multidimensional signal processing applications. In contrast with all the previous works which are estimation methods, this approach can perform exact memory computations even for applications with numerous and complex array references, and also with large numbers of scalars

Multidimensional Signal, Image and Video Processing and Coding [Book Reviews

This new book covers the processing and coding of multidimensional signals, in particular, still images and video. The book is accompanied by a CD-ROM and a companion Web site. The book is well designed and its material is timely. The only drawback is its lack of depth in covering color. The book would work with an introductory graduate course on still-image processing and compression or for an advanced graduate course on video processing and compression. The fact that the book takes a signal processing, rather than computer vision, viewpoint to image manipulation will make it a valuable resource for both researchers and students alike.

The role of independent component analysis in the signal processing of ECG recordings

Independent component analysis (ICA) is an emerging technique for multidimensional signal processing. In recent years, these techniques have been proposed for solving a large number of biomedical applications. This work reviews current knowledge on ICA in electrocardiographic (ECG) analysis. The benefits that ICA can bring to clinical practice are illustrated with four relevant clinical applications: foetal ECG extraction from maternal ECG recordings, analysis of atrial fibrillation, ECG denoising and removal of pacemaker artefacts.

The Fast Fourier Transform

Fast Fourier transforms (FFTs) are fast algorithms, i.e., of low complexity, for the computation of the discrete Fourier transform (DFT) on a finite abelian group. They are among the most important algorithms in applied and engineering mathematics and in computer science, in particular for one‐ and multidimensional systems theory and signal processing. We give a relatively short survey of the FFT for arbitrary finite abelian groups, cyclic or not, with complete and partially novel proofs, the main distinction being explicit induction formulas for the FFT in all cases which generalize the original FFT‐algorithm due to Cooley and Tukey and, much earlier, to Gauß. We believe that our approach has didactic advantages over the usual ones. We also present the application of the FFT to fast convolution algorithms, and the so‐called number theoretic transforms over finite coefficient rings. We do not treat those algorithms which decrease the multiplicative complexity at the expense of many more rational linear combinations, which in this context are considered costless, nor do we discuss the DFT for nonabelian finite groups.

Feature - Multidimensional systems and signal processing: Good theory for good practice

The subject of multidimensional systems, as originally conceived, is concerned with a mathematical framework for tackling a broad range of paradigms whose analysis or synthesis require the use of functions and polynomials in either several complex or real variables. Its applications, which may range from the processing of spatial and temporal signals of diverse physical origin to the design of linear continuous-discrete multidimensional control systems, are already plentiful. This paper aims to promote interaction between a broad spectrum of scientists and engineers through clear exposition and interpretation of fundamentals, albeit a daunting task, so that complex theoretical results become useful to practitioners in distinct but related disciplines. The expanding arena of theory and applications of the original aims and scopes attest to the importance and vitality of the subject-matter.

Multidimensional systems and signal processing

Multidimensional systems and signal processing

Information rates of multidimensional front ends for digital storage channels with data-dependent transition noise

This paper presents a simulation-based method for the computation of the information rates of recently proposed multidimensional front ends for digital storage channels with transition noise. First, we propose an algorithm, based on linear prediction and state reduction techniques, that extends recent work on the information rates of magnetic recording channels affected by colored Gaussian thermal noise, intersymbol interference, and signal-dependent transition noise. Following a previous study on statistical sufficiency, we apply this algorithm to compute the information rates of digital storage systems featuring a multidimensional detection front end. The resulting information rates indicate that significant gains may be achievable by multidimensional signal processing techniques in transition-noise-limited digital storage channels