Widely Linear Adaptive Filtering Based on Clifford Geometric Algebra: A unified framework

This paper explains how, following the representation of 3D crystallographic space groups in Clifford's geometric algebra, it is further possible to similarly represent the 162 so called subperiodic groups of crystallography in Clifford's geometric algebra. A new compact geometric algebra group representation symbol is constructed, which allows to read off the complete set of geometric algebra generators. For clarity moreover the chosen generators are stated explicitly. The group symbols are based on the representation of point groups in geometric algebra by versors (Clifford monomials, Lipschitz elements). Keywords: Subperiodic groups, Clifford's geometric algebra, versor representation, frieze groups, rod groups, layer groups .

Three-dimensional image modeling is essential in many scientific disciplines, including computer vision and precision agriculture. So far, various methods of creating three-dimensional (3D) models have been considered. However, the processing of transformation matrices of each input image data is not controlled. Site-specific crop mapping is essential because it helps farmers determine yield, biodiversity, energy, crop coverage, etc. Clifford Geometric Algebraic understanding of signaling and image processing has become increasingly important in recent years. Geometric Algebraic treats multi-dimensional signals in a holistic way to maintain relationship between side sizes and prevent loss of information. This article has used agricultural images acquired by unmanned aerial vehicles (UAVs) to construct three-dimensional models using Clifford geometric algebra. The qualitative and quantitative performance evaluation results show that Clifford geometric algebra can generate a three-dimensional geometric statistical model directly from drones' RGB images. Through peak signal-to-noise ratio (PSNR), structural similarity index measure (SSIM), and visual comparison, the proposed algorithm's performance is compared with latest algorithms. Experimental results show that proposed algorithm is better than other leading 3D modeling algorithms.

The framework for playing quantum games in an Einstein-Podolsky-Rosen (EPR) type setting is investigated using the mathematical formalism of geometric algebra (GA). The main advantage of this framework is that the players' strategy sets remain identical to the ones in the classical mixed-strategy version of the game, and hence the quantum game becomes a proper extension of the classical game, avoiding a criticism of other quantum game frameworks. We produce a general solution for two-player games, and as examples, we analyze the games of Prisoners' Dilemma and Stag Hunt in the EPR setting. The use of GA allows a quantum-mechanical analysis without the use of complex numbers or the Dirac Bra-ket notation, and hence is more accessible to the non-physicist.

This paper introduces the Clifford Support Vector Machines (CSVM) as a generalization of the real- and complex-valued Support Vector Machines using the Clifford geometric algebra. In this framework we handle the design of kernels involving the Clifford or geometric product for linear and nonlinear classification and regression. The major advantage of our approach is that we redefine the optimization variables as multivectors. This allows us to have a multivector as output therefore we can represent multiple classes according to the dimension of the geometric algebra in which we work. We conduct comparisons between CSVM and the most used approaches to solve multi-class classification to show that our approach is more suitable for practical use on certain type of multi-class classification problems.

Geometric Clifford algebra <i>G</i>_3,0 is used to describe geometric objects of 3D Euclidean space in an 8D basis with only real numbers. Basis elements are identified by their inner structure. To describe it with only +1 and -1 coefficients, a unit cube representation is used rather, then matrix ones. Two layers of information are distinguished. The first one is contained in a set of real coefficients for basis Clifford numbers. It describes the outer properties of an object. The second binary one is contained in coefficients for elements of their inner structure to implement their geometric properties in algebraic relations. It is shown, that any change of information inside the second (inner) layer preserves information inside the first layer but changes its image. Consequences for information transmission and processing are discussed.

An influence network of events is a view of the universe based on events that may be related to one another via influence. The network of events forms a partially ordered set which, when quantified consistently via a technique called chain projection, results in the emergence of spacetime and the Minkowski metric as well as the Lorentz transformation through changing an observer from one frame to another. Interestingly, using this approach, the motion of a free electron as well as the Dirac equation can be described. Indeed, the same approach can be employed to show how a discrete version of some of the features of Euclidean geometry including directions, dimensions, subspaces, Pythagorean theorem, and geometric shapes can emerge. In this paper, after reviewing the essentials of the influence network formalism, we build on some of our previous works to further develop aspects of Euclidean geometry. Specifically, we present the emergence of geometric shapes, a discrete version of the parallel postulate, the dot product, and the outer (wedge product) in 2+1 dimensions. Finally, we show that the scalar quantification of two concatenated orthogonal intervals exhibits features that are similar to those of the well-known concept of a geometric product in geometric Clifford algebras.

An approach to coherent parallel computing on finite spatial rotations is considered. It is shown that such rotations are elements of 4-dimensional (4D) space and to design any algorithms with them we need to change Boole logic to more complicated rotational one. The latter is provided by formalism of Cayley-Klein parameters, which is compatible with geometric Clifford algebra and all its elements of pure and mixed grades: vectors, scalars, paravectors, quaternions, spinors, etc.

A Clifford support vector machine (CSVM) learns the decision surface from multi distinct classes of the multiple input points using the Clifford geometric algebra. In many applications, each multiple input point may not be fully assigned to one of these multi-classes. In this paper, we apply a fuzzy membership to each multiple input point and reformulate the CSVM for multiclass classification to make different input points have their own different contributions to the learning of decision surface. We call the proposed method Clifford fuzzy SVM.

This paper introduces the Clifford support vector machines as a generalization of the real- and complex- valued support vector machines using the Clifford geometric algebra. In this framework we handle the design of kernels involving the Clifford or geometric product for linear and nonlinear classification. We present an interesting application to classify mechanical tools using the concept of Clifford moments which is invariant under affine transformations.

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.

Relevance. The execution of time-sensitive tasks in Industrial Internet of Things (IIoT) systems requires decentralized and fault-tolerant computing models that can operate under dynamic workloads, unstable node availability, and limited resources. Object of research: task management and offloading processes in edge–fog IIoT environments. Purpose of the article. Development of a method for decentralized, latency-aware task processing using adaptive filtering and dynamic computation offloading, which ensures high availability, efficient resource usage, and responsiveness in distributed edge systems. Research results. The paper presents the AFDCO method, which integrates local data filtering, node availability evaluation, latency-based offloading, and lightweight redundancy into a unified framework. This approach enables autonomous and fault-resilient task execution without relying on centralized controllers. Simulation results confirm that AFDCO reduces response time, improves task deadline compliance, and minimizes network and energy overhead. Conclusions. Compared to static or centralized task allocation models, the proposed method demonstrates better adaptability and robustness under variable conditions by dynamically adjusting execution and replication strategies based on system feedback. Scope of application of the obtained results: distributed IIoT systems, edge–fog computing platforms, and lowlatency industrial automation scenarios requiring decentralized execution and fault recovery.

Constitutive relations in optics in terms of geometric algebra

New adaptive filters for color image processing are introduced and analyzed. The proposed adaptive methodology constitutes a unifying and powerful framework for multichannel signal processing. Using the proposed methodology, color image filtering problems are treated from a global viewpoint that readily yields and unifies previous, seemingly unrelated, results. The new filters utilize Bayesian techniques and nonparametric methodologies to adapt to local data in the color image. The principles behind the new filters are explained in detail. Simulation studies indicate that the new filters are computationally attractive and have excellent performance.

Given a quadratic form on a vector space, the geometric algebra of the corresponding pseudo-euclidean space is defined in terms of a simple set of rules which characterizes the geometric product of vectors. We develop geometric algebra in such a way that it augments, but remains fully compatible with, the more traditional tools of matrix algebra. Indeed, matrix multiplication arises naturally from the geometric multiplication of vectors by introducing a spectral basis of mutually annihiliating idempotents in the geometric algebra. With the help of a few more algebraic identities, and given the proper geometric interpretation, the geometric algebra can be applied to the study of affine, projective, conformal and other geometries. The advantage of geometric algebra is that it provides a single algebraic framework with a comprehensive, but flexible, geometric interpretation. For example, the affine plane of rays is obtained from the euclidean plane of points by adding a single anti-commuting vector to the underlying vector space. The key to the study of noneuclidean geometries is the definition of the operations of meet and join, in terms of which incidence relationships are expressed. The horosphere provides a homogeneous model of euclidean space, and is obtained by adding a second anti-commuting vector to the underlying vector space of the affine plane. Linear orthogonal transformations on the higher dimensional vector space correspond to conformal or Mobius transformations on the horosphere. The horosphere was first constructed by F.A. Wachter (1792–1817), but has only recently attracted attention byo ering a host of new computational tools in projective and hyperbolic geometries when formulated in terms of geometric algebra.

ABSTRACTExplicit formulas to calculate MV functions in a basis‐free representation are presented for an arbitrary Clifford geometric algebra . The formulas are based on analysis of the roots of minimal MV polynomial and covers defective MVs, that is, the MVs that have non‐diagonalizable matrix representations. The method may be generalized straightforwardly to matrix functions and to finite dimensional linear operators. The results can find wide application in Clifford algebra analysis.