Eckhard Hitzer’s research while affiliated with International Christian University and other places

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 two-dimensional planes) for natural and biomedical image analysis through the following computational workflows and outcomes: natural and biomedical image re-colorization, natural image de-colorization, natural and 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 in each instance. 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. Furthermore, 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 notable potential of feature-rich mathematical/computational frameworks for natural and biomedical images.

Complex-valued and, more generally, hypercomplex-valued neural networks (HVNNs) constitute a rapidly growing research area that has attracted continued interest for the last decade. Besides their natural ability to treat multidimensional data, hypercomplex-valued neural networks can benefit from hypercomplex numbers’ geometric and algebraic properties. For example, complex-valued neural networks are essential for adequately treating phase and the information contained in phase, including the treatment of wave- and rotation-related phenomena such as electromagnetism, light waves, quantum waves, and oscillatory phenomena. Quaternion-valued neural networks, which naturally incorporate spacial rotations and have potential applications in three- and four-dimensional data modeling, have been effectively used to process and analyze multivariate images such as color and polarimetric SAR images. Despite significant theoretical development and successful applications, many research directions in HVNNs remain in progress. These include formally generalizing the commonly used real-valued network architectures and training algorithms to the hypercomplex-valued case. There are also many exciting applications in pattern recognition and classification, nonlinear filtering, intelligent image processing, brain-computer interfaces, time-series prediction, bioinformatics, and robotics, to list a few. This special session aims to be the proper forum for a systematic and comprehensive exchange of ideas, presenting recent research results and discussing future trends in complex- and hypercomplex-valued neural networks. We hope the proposed session will attract potential speakers and researchers interested in joining the community. We also expect this session to benefit and inspire computational intelligence researchers and other specialties that need sophisticated neural network tools.

We compute and explore the full geometric product of two oriented points in conformal geometric algebra Cl(4, 1) of three-dimensional Euclidean space. We comment on the symmetry of the various components, and state for all expressions also a representation in terms of point pair center and radius vectors.

From viewpoints of crystallography and of elementary particles, we explore symmetries of multivectors in the geometric algebra Cl(3, 1) that can be used to describe space-time.

Hypercomplex signal and image processing extends upon conventional methods by using hypercomplex numbers in a unified framework for algebra and geometry. The special issue is divided into two parts and is focused on current advances and applications in computational signal and image processing in the hypercomplex domain. The first part offered well-rounded coverage of the field, with seven articles that focused on overviews of current research, color image processing, signal filtering, and machine learning.

Novel computational signal and image analysis methodologies based on feature-rich mathematical/computational frameworks continue to push the limits of the technological envelope, thus providing optimized and efficient solutions. Hypercomplex signal and image processing is a fascinating field that extends conventional methods by using hypercomplex numbers in a unified framework for algebra and geometry. Methodologies that are developed within this field can lead to more effective and powerful ways to analyze signals and images. Processing audio, video, images, and other types of data in the hypercomplex domain allows for more complex and intuitive representations with algebraic properties that can lead to new insights and optimizations. Applications in image processing, signal filtering, and deep learning (just to name a few) have shown that working in the hypercomplex domain can lead to more efficient and robust outcomes. As research in this field progresses and software tools become more widely available, we can expect to see increasingly sophisticated applications in many areas of research, e.g., computer vision, machine learning, and so on.