Similarity and consimilarity of elements in the real quaternion, octonion, and sedenion algebras, as well as in the general real Cayley-Dickson algebras are considered by solving the two fundamental equations ax = xb and ax = xb in these algebras. Some consequences are also presented. AMS mathematics subject classifications: 17A05, 17A35.

This paper concerns the estimation problem of attitude, position, and linear velocity a rigid-body autonomously navigating with six degrees freedom (6 DoF). The navigation dynamics are highly nonlinear modeled on matrix Lie group extended Special Euclidean Group SE2(3). A computationally cheap geometric stochastic filter is proposed SE2(3) guaranteed transient steady-state performance. operates...

Color in an image is resolved to 3 or 4 color components and 2-Dimages of these components are stored in separate channels. Most of the color image enhancement algorithms are applied channel-by-channel on each image. But such a system of color image processing is not processing the original color. When a color image is represented as a quaternion image, processing is done in original colors. Th...

We propose a general paradigm for generating optimal coordinate frame fields that may be exploited to annotate and display curves and surfaces. Parallel-transport framings, which work well for open curves, generally fail to have desirable properties for cyclic curves and for surfaces. We suggest that minimal quaternion measure provides an appropriate generalization of parallel transport. Our fu...

The quaternion group Q8 is one of the two non-abelian groups of size 8 (up to isomorphism). The other one, D4, can be constructed as a semi-direct product: D4 ∼= Aff(Z/(4)) ∼= Z/(4) o (Z/(4))× ∼= Z/(4) o Z/(2), where the elements of Z/(2) act on Z/(4) as the identity and negation. While Q8 is not a semi-direct product, it can be constructed as the quotient group of a semi-direct product. We wil...

The additive identity is (0, 0), the multiplicative identity is (1, 0), and from addition and scalar multiplication of real vectors we have (a, b) = (a, 0) + (0, b) = a(1, 0) + b(0, 1), which looks like a+ bi if we define i to be (0, 1). Real numbers occur as the pairs (a, 0). Hamilton asked himself if it was possible to multiply triples (a, b, c) in a nice way that extends multiplication of co...