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.