Clifford‐Valued Bendlet Transforms: Theory and Localization Operators

Computer graphics applications require efficient tools to model geometric objects. The traditional approach based on compute-intensive matrix calculations is error-prone due to a lack of integration between geometric reasoning and matrix-based algorithms. Clifford algebra offers a solution to these issues since it permits specification of geometry at a coordinate-free level. The best way to exploit the symbolic computing power of geometric (Clifford) algebra is supporting its data types and operators directly in hardware. This paper outlines the architecture of S-CliffoSor (Sliced Clifford coprocessor), a parallelizable embedded coprocessor that executes native Clifford algebra operations. S-CliffoSor is a sliced coprocessor that can be replicated for parallel execution of concurrent Clifford operations. A single slice has been designed, implemented and tested on the Celoxica Inc. RC1000 board. The experimental results show the potential to achieve a 3times speedup for Clifford sums and 4times speedup for Clifford products compared to against the analogous operations in the software library generator GAIGEN.

Many concepts that are used in multi-dimensional signal processing are derived from one-dimensional signal processing. As a consequence, they are only suited to multi-dimensional signals which are intrinsically one-dimensional. We claim that this restriction is due to the restricted algebraic frame used in signal processing, especially to the use of the complex numbers in the frequency domain. We propose a generalization of the two-dimensional Fourier transform which yields a quaternionic signal representation. We call this transform quaternionic Fourier transform (QFT). Based on the QFT, we generalize the conceptions of the analytic signal, Gabor filters, instantaneous and local phase to two dimensions in a novel way which is intrinsically two-dimensional. Experimental results are presented.

: This is an annual report on research conducted under the auspices of the Joint Services Electronics Program. Specific topics covered are: multidimensional digital signal processing, signal restoration and detection, morphological systems for multidimensional signal processing, multidimensional processing for sensory arrays, multiprocessor systems and tools for digital signal processing, linear and nonlinear image processing, two-dimensional optical storage and processing, semiconductor quantum wave devices, electromagnetic measurements in the time and frequency domains, and automated radiation measurements for near- and far-field transformations.

The intense requirements of high-speed implementations of MultiDimensional (MD) Digital Signal Processing (DSP) systems justify the Application Specific Integrated Circuits (ASIC) designs and/or multiprocessor implementations. MD retiming has been recently proposed to improve the circuitry performance in high-level synthesis of single-rate MD DSP systems. This paper has conducted new theoretical analysis of MD multirate DSP systems modeled in data-flow graphs, and proposes intercalation of MD multirate systems so that unified MD retiming operations can be applied on multidimensional multirate DSP systems. By retiming and intercalation, full intra-iteration parallelism is achieved and functional elements can be executed simultaneously on circuits for the generic class of MD multirate DSP systems.KeywordsMultirate Signal ProcessingMD Data Flow GraphMultidimenional Retiming

Neural networks are a soft computing technique with wide application in signal processing as well as system and process modeling. In the present study, multilayer perceptron (MLP) neural networks were employed to process multidimensional signals generated in metal machining operations (including three-dimensional cutting force signals and three-dimensional cutting vibration signals) and to establish a model for predicting the machined surface roughness. This paper describes in detail our methods of multidimensional signal processing and modeling with MLP neural networks. The MLP neural network model developed in the present study fills an important research gap by taking into account the critical effect of tool-edge radius in machining. As compared to regression models, the MLP neural network model developed in the present study has significantly higher accuracy in predicting the machined surface roughness.

In this article, a new multidimensional signal processing scheme for a heated-dot magnetic recording (HDMR) system using double-layered bit-patterned media (BPM) is proposed. This proposed signal processing scheme is capable of simultaneously detecting data sequences recorded on two tracks in each layer sequentially. The effective transmission rate of the HDMR system with two-track simultaneous log-likelihood ratio (LLR) detection and double-layered recording can be increased about four times that of the HDMR system with single-track LLR detection and single-layered recording if the data sequence recorded on each track has the same rate. It is shown that the error rate performance of the HDMR system with two-track simultaneous LLR detection and double-layered recording shows a significant improvement over the HDMR system with single-track LLR detection and single-layered recording by computer simulation.

This work introduces a geometrical interpretation of the Chinese remainder theorem (CRT) which is novel and which can be profitably used in the field of multidimensional signal processing. The practical application of the proposed result in some multidimensional signal processing procedures is demonstrated.

In real-time data-intensive multimedia processing applications, data transfer and storage significantly influence, if not dominate, all the major cost parameters of the design space—namely energy consumption, performance, and chip area. This paper presents an electronic design automation (EDA) methodology for the high-level design of hierarchical memory architectures in embedded data-intensive applications, mainly in the area of multidimensional signal processing. Different from the previous works, the problems of data assignment to the memory layers, of mapping the signals into the physical memories, and of banking the on-chip memory are addressed in a consistent way, based on the same formal model. This memory management framework employs techniques specific to the integral polyhedra based dependence analysis. The main design target is the reduction of the static and dynamic energy consumption in the hierarchical memory subsystem.

The Composition Algebra-based Methodology (CAM) [B. Wolk, Pap. Phys. 9, 090002 (2017); Phys. Scr. 94, 025301 (2019); Adv. Appl. Clifford Algebras 27, 3225 (2017); J. Appl. Math. Phys. 6, 1537 (2018); Phys. Scr. 94, 105301 (2019), Adv. Appl. Clifford Algebras 30, 4 (2020)], which provides a new model for generating the interactions of the Standard Model, is geometrically modeled for the electromagnetic and weak interactions on the parallelizable sphere operator fiber bundle [Formula: see text] consisting of base space, the tangent bundle [Formula: see text] of space–time [Formula: see text], projection operator [Formula: see text], the parallelizable spheres [Formula: see text] conceived as operator fibers [Formula: see text] attaching to and operating on [Formula: see text] [Formula: see text] as [Formula: see text] varies over [Formula: see text], and as structure group, the norm-preserving symmetry group [Formula: see text] for each of the division algebras which is simultaneously the isometry group of the associated unit sphere. The massless electroweak [Formula: see text] Lagrangian is shown to arise from [Formula: see text]’s generation of a local coupling operation on sections of Dirac spinor and Clifford algebra bundles over [Formula: see text]. Importantly, CAM is shown to be a new genre of gauge theory which subsumes Yang–Mills Standard Model gauge theory. Local gauge symmetry is shown to be at its core a geometric phenomenon inherent to CAM gauge theory. Lastly, the higher-dimensional, topological architecture which generates CAM from within a unified eleven [Formula: see text]-dimensional geometro-topological structure is introduced.

Special Section on Multidimensional Signal Processing and Its Application

This paper nonce a volitional path to the Reciprocity Attributes of the Multidimensional Complex Fourier Transform (IFT) or the Infiite Fourier Transform (IFT) finis the Multidimensional Infinite Hartley Transform (IHT). The provenances for the reciprocity attributes of multidimensional IFT are conferred out of the reciprocity attributes of the multidimensional IHT, and this is doable mainly due to the mathematical liaison betwixt the IFT and the IHT. Apres the IHT is a real transform, it is viable to deduce the reciprocity attributes for IFT throughout the whilom in multidimensions. It is quotidian run-through in signal processing and communication engineering to chance upon signals having the same fettle in the temporal- and frequency-bailiwicks, such as the Gaussian function. In such paradigms, the reciprocity attributes can be blisteringly wont for working out the onward and rearward transforms of the signals under query. In multidimensional signal processing where the outgo of perpertration is of superemeinent momentousness and the IFT is the exemplar, and where the reciprocity attributes of the IFT are to be wont, it is propounded to execute these reciprocity attributes of the IFT by using the reciprocity attributes of the IHT which have been philosophized in this paper. This props up in the penny-pinching of the utraexpensive reckoning time and disbursement by moiety of one-c as it is a well-established fact that real transforms take less time for execution than those the complex transforms. Thus, the reciprocity attributes unfolded in this paper can be used in digital video processing and in the solutions of differential equations and integral equations.

The Canonical Polyadic (CP) decomposition of R-way arrays is a powerful tool in multi-dimensional signal processing. There exists many methods to compute the CP decomposition. In particular, the Semi-Algebraic framework for the approximate Canonical Polyadic (CP) decomposition via SImultaneaous matrix diagonalization (SECSI) is an efficient and flexible framework for the computation of the CP decomposition. In this work, we perform a first-order performance analysis of the SECSI framework for the computation of the approximate CP decomposition of a noise corrupted low-rank 3-way tensor. We provide closed-form expressions of the relative Mean Square Factor Error (rMSFE) for each of the estimated factor matrices. The derived expressions are formulated in terms of the second-order moments of the noise, such that apart from a zero mean, no assumptions on the noise statistics are required. The numerical results depict the excellent match between the closed-form expressions and the empirical results.

Multidimensional Signal Processing: Methods and Applications

In this paper a method for mechanical theorem proving in geometries is proposed. We first discuss how to describe geometric objects and geometric relations in 2D and/or 3D Euclidean space with Clifford algebraic expression. Then we present some rules to simplify Clifford algebraic polynomials to the so-called final Clifford algebraic polynomials. The key step for proving the theorems is to check if a Clifford algebraic expression can be simplified to zero. With the help of introducing coordinates, we can prove mechanically most of the geometric theorems about lines, conics, planes and so on in plane and/or solid geometry. The proofs produced by machine with our method are readable and geometrically interpretable. Finally, some interesting examples are given.

The application of methods from multidimensional systems theory and digital signal processing opens new ways for the efficient simulation of time and space dependent problems like wave propagation or heat and mass transfer. The basic idea, is the application of suitably chosen functional transformations for the time and space coordinates. Initial and boundary conditions are considered in a systematic way. This leads to the structure of a discrete system which closely models the behaviour of the given continuous system and which is suitable for computer implementation. Numerical results show considerable savings in computer time over conventional finite difference methods.