The Lorentz Group Using Conformal Geometric Algebra and Split Quaternions for Color Image Processing: Theory and Practice

Abstract

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 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 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.

Solution to Several Split Quaternion Matrix Equations

“…Hence, using the inverse of (•) σ , we have AXB + (AXB) η = E; that is, X in ( 5) is the solution to (1). Therefore, that (3) is solvable implies that (1) is solvable, too.…”

“…It is well known that H s is an associative, non-commutative, and four-dimensional Clifford algebra that includes zero divisors, nilpotent elements, and nontrivial idempotents. H s has various applications in computer graphics [1], robotics [2], physics [3], and so on [4][5][6][7].…”