The Fourier transforms play a critical role in a broad range of image processing applications, including enhancement, analysis, restoration, and compression. Until recently, it was common to use the conventional methods to deal with colored images. These methods are based on RGB decomposition of the colored image by separating it into three separate scalar images and computing the Fourier trans...

Recently the general orthogonal planes split with respect to any two pure unit quaternions f, g ∈ H, f = g = −1, including the case f = g, has proved extremely useful for the construction and geometric interpretation of general classes of double-kernel quaternion Fourier transformations (QFT) [7]. Applications include color image processing, where the orthogonal planes split with f = g = the gr...

Variations in illumination is a well-known affecting factor on face recognition system performance. Features extraction is one of the principal steps on a face recognition framework, where it is possible to alleviate the illumination effects on face images. The aim of this work is to study the illumination invariant properties of an hypercomplex image representation. A quaternion description fr...

‎Let $n$ and $k$ be two positive integers‎, ‎$kleq n$ and $A$ be an $n$-square quaternion matrix‎. ‎In this paper‎, ‎some results on the $k-$numerical range of $A$ are investigated‎. ‎Moreover‎, ‎the notions of $k$-numerical radius‎, ‎right $k$-spectral radius and $k$-norm of $A$ are introduced‎, ‎and some of their algebraic properties are studied‎.

This paper presents a compact diierential formula for the rst derivative of a unit quaternion curve deened on SO(3) or S 3. The formula provides a convenient way to compute the angular velocity of a rotating 3D solid. We demonstrate the eeectiveness of this formula by deriving the diierential properties of various unit quaternion curves 4, 5, 6, 7, 8] at the curve end points.

This paper derives an approach to combine estimates and covariances for decentralized attitude estimation using a quaternion parameterization. The approach is based on the covariance intersection method, which is modified to maintain quaternion normalization in the combination process. A simple simulation result is provided where local extended Kalman filters are used on two star trackers, each...

In 1988, J.R. Faulkner has given a procedure to construct an octonion algebra on a finite dimensional unitary alternative algebra of degree three over a field K. Here we use a similar procedure to get a quaternion algebra. Then we obtain some conditions for these octonion and quaternion algebras to be split or division algebras. Then we consider the implications of the found conditions to the u...

Many attitude estimation methods are based on an optimization problem posed in 1965 by Grace Wahba. All these methods will yield the same optimal estimate, except for inevitable computer roundoff errors. This note shows that Shuster’s Quaternion Estimator (QUEST) and Mortari’s Estimator of the Optimal Quaternion (ESOQ) are essentially identical even in the presence of computer roundoff errors. ...

We discuss the relationship between quaternion algebras and quadratic forms with a focus on computational aspects. Our basic motivating problem is to determine if a given algebra of rank 4 over a commutative ring R embeds in the 2 × 2-matrix ring M2(R) and, if so, to compute such an embedding. We discuss many variants of this problem, including algorithmic recognition of quaternion algebras amo...

This paper derives an approach to combine estimates and covariances for decentralized attitude estimation using a quaternion parameterization. The approach is based on the covariance intersection method, which is modified to maintain quaternion normalization in the combination process. A practical simulation result is provided where local extended Kalman filters are used on two star trackers, e...