arXiv subjects

Eckhard Hitzer

Publications and source records attributed to Eckhard Hitzer.

At least 19 recordsLinked to original sources

arXiv2025

Novel computational workflows for natural and biomedical image processing based on hypercomplex algebras

Hypercomplex image processing extends conventional techniques in a unified paradigm encompassing algebraic and geometric principles. This work leverages quaternions and the two-dimensional orthogonal planes split framework (splitting of a quaternion - representing a pixel - into pairs of orthogonal 2D planes) for natural/biomedical image analysis through the following computational workflows and outcomes: natural/biomedical image re-colorization, natural image de-colorization, natural/biomedical image contrast enhancement, computational re-staining and stain separation in histological images, and performance gains in machine/deep learning pipelines for histological images. The workflows are analyzed separately for natural and biomedical images to showcase the effectiveness of the proposed approaches. The proposed workflows can regulate color appearance (e.g. with alternative renditions and grayscale conversion) and image contrast, be part of automated image processing pipelines (e.g. isolating stain components, boosting learning models), and assist in digital pathology applications (e.g. enhancing biomarker visibility, enabling colorblind-friendly renditions). Employing only basic arithmetic and matrix operations, this work offers a computationally accessible methodology - in the hypercomplex domain - that showcases versatility and consistency across image processing tasks and a range of computer vision and biomedical applications. The proposed non-data-driven methods achieve comparable or better results (particularly in cases involving well-known methods) to those reported in the literature, showcasing the potential of robust theoretical frameworks with practical effectiveness. Results, methods, and limitations are detailed alongside discussion of promising extensions, emphasizing the potential of feature-rich mathematical/computational frameworks for natural and biomedical images.

cs.CV↗

arXiv2019

Generalized Uncertainty Principles associated with the Quaternionic Offset Linear Canonical Transform

The quaternionic offset linear canonical transform (QOLCT) can be thought as a generalization of the quaternionic linear canonical transform (QLCT). In this paper we define the QOLCT, we derive the relationship between the QOLCT and the quaternion Fourier transform (QFT). Based on this fact we prove the Plancherel formula, and some properties related to the QOLCT, then we generalize some different uncertainty principles (UPs), including Heisenberg-Weyls UP, Hardys UP, Beurlings UP, and logarithmic UP to the QOLCT domain in a broader sense

math.CA↗

arXiv2013

Uncertainty Principle for the Clifford Geometric Algebra Cl(3,0) based on Clifford Fourier Transform

In the field of applied mathematics the Fourier transform has developed into an important tool. It is a powerful method for solving partial differential equations. The Fourier transform provides also a technique for signal analysis where the signal from the original domain is transformed to the spectral or frequency domain. In the frequency domain many characteristics of the signal are revealed. With these facts in mind, we extend the Fourier transform in geometric algebra. We explicitly show detailed properties of the real Clifford geometric algebra Fourier transform (CFT), which we subsequently use to define and prove the uncertainty principle for Cl(3,0) multivector functions.

math.RA↗

arXiv2013

Two-sided Clifford Fourier transform with two square roots of -1 in Cl(p,q)

We generalize quaternion and Clifford Fourier transforms to general two-sided Clifford Fourier transforms (CFT), and study their properties (from linearity to convolution). Two general \textit{multivector square roots} $\in \cl{p,q}$ \textit{of} -1 are used to split multivector signals, and to construct the left and right CFT kernel factors. Keywords: Clifford Fourier transform, Clifford algebra, signal processing, square roots of -1 .

math.RA↗

arXiv2013

The Orthogonal 2D Planes Split of Quaternions and Steerable Quaternion Fourier Transformations

The two-sided quaternionic Fourier transformation (QFT) was introduced in \cite{Ell:1993} for the analysis of 2D linear time-invariant partial-differential systems. In further theoretical investigations \cite{10.1007/s00006-007-0037-8, EH:DirUP_QFT} a special split of quaternions was introduced, then called $\pm$split. In the current \change{chapter} we analyze this split further, interpret it geometrically as \change{an} \emph{orthogonal 2D planes split} (OPS), and generalize it to a freely steerable split of $\H$ into two orthogonal 2D analysis planes. The new general form of the OPS split allows us to find new geometric interpretations for the action of the QFT on the signal. The second major result of this work is a variety of \emph{new steerable forms} of the QFT, their geometric interpretation, and for each form\change{,} OPS split theorems, which allow fast and efficient numerical implementation with standard FFT software.

math.RA↗

arXiv2013

A General Geometric Fourier Transform

The increasing demand for Fourier transforms on geometric algebras has resulted in a large variety. Here we introduce one single straight forward definition of a general geometric Fourier transform covering most versions in the literature. We show which constraints are additionally necessary to obtain certain features like linearity or a shift theorem. As a result, we provide guidelines for the target-oriented design of yet unconsidered transforms that fulfill requirements in a specific application context. Furthermore, the standard theorems do not need to be shown in a slightly different form every time a new geometric Fourier transform is developed since they are proved here once and for all.

math.AG↗

arXiv2013

Detection of Outer Rotations on 3D-Vector Fields with Iterative Geometric Correlation

Correlation is a common technique for the detection of shifts. Its generalization to the multidimensional geometric correlation in Clifford algebras has proven a useful tool for color image processing, because it additionally contains information about rotational misalignment. In this paper we prove that applying the geometric correlation iteratively can detect the outer rotational misalignment for arbitrary three-dimensional vector fields. Thus, it develops a foundation applicable for image registration and pattern matching. Based on the theoretical work we have developed a new algorithm and tested it on some principle examples.

math.AG↗

arXiv2013

Detection of Outer Rotations on 3D-Vector Fields with Iterative Geometric Correlation and its Efficiency

Correlation is a common technique for the detection of shifts. Its generalization to the multidimensional geometric correlation in Clifford algebras has been proven a useful tool for color image processing, because it additionally contains information about a rotational misalignment. But so far the exact correction of a three-dimensional outer rotation could only be achieved in certain special cases. In this paper we prove that applying the geometric correlation iteratively has the potential to detect the outer rotational misalignment for arbitrary three-dimensional vector fields. We further present the explicit iterative algorithm, analyze its efficiency detecting the rotational misalignment in the color space of a color image. The experiments suggest a method for the acceleration of the algorithm, which is practically tested with great success.

cs.CV↗

arXiv2013

New views of crystal symmetry guided by profound admiration of the extraordinary works of Grassmann and Clifford

This paper shows how beginning with Justus Grassmann's work, Hermann Grassmann was influenced in his mathematical thinking by crystallography. H. Grassmann's Ausdehnungslehre in turn had a decisive influence on W.K. Clifford in the genesis of geometric algebras. Geometric algebras have been expanded to conformal geometric algebras, which provide an ideal framework for modern computer graphics. Within this framework a new visualization of three-dimensional crystallographic space groups has been created. The complex beauty of this new visualization is shown by a range of images of a diamond cell. Mathematical details are given in an appendix.

cond-mat.mtrl-sci↗

arXiv2013

A General Geometric Fourier Transform Convolution Theorem

The large variety of Fourier transforms in geometric algebras inspired the straight forward definition of ``A General Geometric Fourier Transform`` in Bujack et al., Proc. of ICCA9, covering most versions in the literature. We showed which constraints are additionally necessary to obtain certain features like linearity, a scaling, or a shift theorem. In this paper we extend the former results by a convolution theorem.

math.AG↗

arXiv2013

Representation of Crystallographic Subperiodic Groups in Clifford's Geometric Algebra

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 .

cond-mat.mtrl-sci↗

arXiv2013

Detection of Total Rotations on 2D-Vector Fields with Geometric Correlation

Correlation is a common technique for the detection of shifts. Its generalization to the multidimensional geometric correlation in Clifford algebras additionally contains information with respect to rotational misalignment. It has been proven a useful tool for the registration of vector fields that differ by an outer rotation. In this paper we proof that applying the geometric correlation iteratively has the potential to detect the total rotational misalignment for linear two-dimensional vector fields. We further analyze its effect on general analytic vector fields and show how the rotation can be calculated from their power series expansions.

math.AG↗

arXiv2013

New Views of Crystal Symmetry

Already Hermann Grassmann's father Justus (1829, 1830) published two works on the geometrical description of crystals, influenced by the earlier works of C.S. Weiss (1780-1856) on three main crystal forces governing crystal formation. In his 1840 essay on the derivation of crystal shapes from the general law of crystal formation Hermann established the notion of a three-dimensional vectorial system of forces with rational coefficients, that represent the interior crystal structure, regulate its formation, its shape and physical behavior. In the Ausdehnungslehre 1844 (Paragraph 171) he finally writes: I shall conclude this presentation by one of the most beautiful applications which can be made of the science treated, i.e. the application to crystal figures (Scholz, 1996). The geometry of crystals thus certainly influenced the Ausdehnungslehre. In this paper we see how Grassmann's work influenced Clifford's creation of geometric algebras, which in turn leads to a new geometric description of crystal symmetry suitable for modern computer algebra graphics.

cond-mat.mtrl-sci↗

arXiv2013

Clifford (Geometric) Algebra Wavelet Transform

While the Clifford (geometric) algebra Fourier Transform (CFT) is global, we introduce here the local Clifford (geometric) algebra (GA) wavelet concept. We show how for $n=2,3 (\mod 4)$ continuous $Cl_n$-valued admissible wavelets can be constructed using the similitude group $SIM(n)$. We strictly aim for real geometric interpretation, and replace the imaginary unit $i \in \C$ therefore with a GA blade squaring to $-1$. Consequences due to non-commutativity arise. We express the admissibility condition in terms of a $Cl_{n}$ CFT and then derive a set of important properties such as dilation, translation and rotation covariance, a reproducing kernel, and show how to invert the Clifford wavelet transform. As an explicit example, we introduce Clifford Gabor wavelets. We further invent a generalized Clifford wavelet uncertainty principle. Extensions of CFTs and Clifford wavelets to $Cl_{0,n'}, n' = 1,2 (\mod 4)$ appear straight forward. Keywords: Clifford geometric algebra, Clifford wavelet transform, multidimensional wavelets, continuous wavelets, similitude group.

math.RA↗

arXiv2013

Angles between subspaces

We first review the definition of the angle between subspaces and how it is computed using matrix algebra. Then we introduce the Grassmann and Clifford algebra description of subspaces. The geometric product of two subspaces yields the full relative angular information in an explicit manner. We explain and interpret the result of the geometric product of subspaces gaining thus full access to the relative orientation information.

math.MG↗

arXiv2013

Non-constant bounded holomorphic functions of hyperbolic numbers - Candidates for hyperbolic activation functions

The Liouville theorem states that bounded holomorphic complex functions are necessarily constant. Holomorphic functions fulfill the socalled Cauchy-Riemann (CR) conditions. The CR conditions mean that a complex $z$-derivative is independent of the direction. Holomorphic functions are ideal for activation functions of complex neural networks, but the Liouville theorem makes them useless. Yet recently the use of hyperbolic numbers, lead to the construction of hyperbolic number neural networks. We will describe the Cauchy-Riemann conditions for hyperbolic numbers and show that there exists a new interesting type of bounded holomorphic functions of hyperbolic numbers, which are not constant. We give examples of such functions. They therefore substantially expand the available candidates for holomorphic activation functions for hyperbolic number neural networks. Keywords: Hyperbolic numbers, Liouville theorem, Cauchy-Riemann conditions, bounded holomorphic functions

cs.NE↗

arXiv2013

Introduction to Clifford's Geometric Algebra

Geometric algebra was initiated by W.K. Clifford over 130 years ago. It unifies all branches of physics, and has found rich applications in robotics, signal processing, ray tracing, virtual reality, computer vision, vector field processing, tracking, geographic information systems and neural computing. This tutorial explains the basics of geometric algebra, with concrete examples of the plane, of 3D space, of spacetime, and the popular conformal model. Geometric algebras are ideal to represent geometric transformations in the general framework of Clifford groups (also called versor or Lipschitz groups). Geometric (algebra based) calculus allows, e.g., to optimize learning algorithms of Clifford neurons, etc. Keywords: Hypercomplex algebra, hypercomplex analysis, geometry, science, engineering.

math.RA↗

arXiv2013

Quaternionic Fourier-Mellin Transform

In this contribution we generalize the classical Fourier Mellin transform [S. Dorrode and F. Ghorbel, Robust and efficient Fourier-Mellin transform approximations for gray-level image reconstruction and complete invariant description, Computer Vision and Image Understanding, 83(1) (2001), 57-78, DOI 10.1006/cviu.2001.0922.], which transforms functions $f$ representing, e.g., a gray level image defined over a compact set of $\mathbb{R}^2$. The quaternionic Fourier Mellin transform (QFMT) applies to functions $f: \mathbb{R}^2 \rightarrow \mathbb{H}$, for which $|f|$ is summable over $\mathbb{R}_+^* \times \mathbb{S}^1$ under the measure $dθ\frac{dr}{r}$. $\mathbb{R}_+^*$ is the multiplicative group of positive and non-zero real numbers. We investigate the properties of the QFMT similar to the investigation of the quaternionic Fourier Transform (QFT) in [E. Hitzer, Quaternion Fourier Transform on Quaternion Fields and Generalizations, Advances in Applied Clifford Algebras, 17(3) (2007), 497-517.; E. Hitzer, Directional Uncertainty Principle for Quaternion Fourier Transforms, Advances in Applied Clifford Algebras, 20(2) (2010), 271-284, online since 08 July 2009.].

math.RA↗