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The processing of color images is of great interest, because the human perception of color is a very complex process, still not well understood. In this article, firstly the authors present an analysis of the well-known mathematical methods used to model color as a phenomenon and to process color images. We propose an adequate metric to process color images using the Minkowski or space-time metric. We propose that the processing of HSV images can be done using the HSV cone in the quaternion split algebra or the conformal geometric algebra frameworks. We show that the processing of RGB images using the Euclidean metric in the $\mathbb {R}^{3}$ doesn’t yield good results. The colors of images change by daylight, we understand this phenomenon as the color change follows the action of the Lorentz Lie group. Consequently, we formulate a new model for representing and processing color using split quaternions and space-time conformal geometric algebra. We propose new algorithms for practical color image processing: we formulate the novel Split Quaternion Fourier Transform for color image processing and we interpolate color images using Split Motors which belong to the space-time conformal geometric algebra. The experimental part proves that real color images have to be processed in the HSV cone. We show successful applications of the Quaternion Split Fourier Transform and the interpolation processing. We compare these computations using the RGB metric in the $\mathbb {R}^{3}$ space and using the Minkowski metric in the HSV cone. This comparison shows clearly that the color image processing using the Minkowski metric in the HSV cone performs better.

Geometric Algebra (GA) has proven to be an advanced language for mathematics, physics, computer science, and engineering. This review presents a comprehensive study of works on Quaternion Algebra and GA applications in computer science and engineering from 1995 to 2020. After a brief introduction of GA, the applications of GA are reviewed across many fields. We discuss the characteristics of the applications of GA to various problems of computer science and engineering. In addition, the challenges and prospects of various applications proposed by many researchers are analyzed. We analyze the developments using GA in image processing, computer vision, neurocomputing, quantum computing, robot modeling, control, and tracking, as well as improvement of computer hardware performance. We believe that up to now GA has proven to be a powerful geometric language for a variety of applications. Furthermore, there is evidence that this is the appropriate geometric language to tackle a variety of existing problems and that consequently, step-by-step GA-based algorithms should continue to be further developed. We also believe that this extensive review will guide and encourage researchers to continue the advancement of geometric computing for intelligent machines.

In this paper we present the design of a device to guide the visually impaired person who normally uses a cane. We propose a non-invasive device that will help blind and visually impaired people to navigate. The system uses stereoscopic vision, a RGB-D sensor and an IMU to process images and to compute the distances from obstacles relative to cameras and to search for free walking paths in the scene. This computing is done using stereo vision, vanishing points, and fuzzy rules. Vanishing points are used to obtain a main orientation in structured spaces. Since the guidance system is related to a spatial reference system, the vanishing point is used like a virtual compass that helps the blind to orient him- or herself towards a goal. Reinforced with fuzzy decision rules, the system supports the blind in avoiding obstacles, thus the blind person is able to cross structured spaces and avoid obstacles without the need for a cane.

This paper uses geometric algebra to formulate, in a single framework, the kinematics of a three finger robotic hand, a binocular robotic head, and the interactions between 3D objects, all of which are seen in stereo images. The main objective is the formulation of a kinematic control law to close the loop between perception and actions, which allows to perform a smooth visually guided object manipulation.

We have developed a method to render brain tumours from endoneurosonography. We propose to track an ultrasound probe in successive endoscopic images without relying on an external optic or magnetic tracking system. The probe is tracked using two different methods: one of them based on a generalised Hough transform and the other one based on particle filters. By estimating the pose of the ultrasound probe in several contiguous images, we use conformal geometric algebra to compute the geometric transformations that yield the 3D position of the tumour, which was segmented in the ultrasound image using morphological operators. We use images from brain phantoms to evaluate the performance of the proposed methods, and our results show that they are robust.

People tracking is an interesting topic in computer vision. It has applications in industrial areas such as surveillance or human-machine interaction. Particle Filters is a common algorithm for people tracking; challenging situations occur when the target's motion is poorly modelled or with unexpected motions. In this paper, an alternative to address people tracking is presented. The proposed algorithm is based in particle filters, but instead of using a dynamical model, it uses background subtraction to predict future locations of particles. The algorithm is able to track people in omnidirectional sequences with a low frame rate (one or two frames per second). Our approach can tackle unexpected discontinuities and changes in the direction of the motion. The main goal of the paper is to track people from laboratories, but it has applications in surveillance, mainly in controlled environments.

In this paper we show how to improve the performance of two self-organizing neural networks used to approximate the shape of a 2D or 3D object by incorporating gradient information in the adaptation stage. The methods are based on the growing versions of the Kohonen's map and the neural gas network. Also, we show that in the adaptation stage the network utilizes efficient transformations, expressed as versors in the conformal geometric algebra framework, which build the shape of the object independent of its position in space (coordinate free). Our algorithms were tested with several images, including medical images (CT and MR images). We include also some examples for the case of 3D surface estimation.