The strictly linear quaternion valued affine projection algorithm (QAPA) and its widely linear counterpart (WLQAPA) are introduced, in order to provide fast converging stochastic gradient learning in the quaternion domain, for the processing of both second order circular (proper) and second order noncircular (improper) signals. This is achieved based on the recent advances in augmented quaterni...

This paper presents a digital color image watermarking scheme using the hypercomplex numbers representation and the Quaternion Fourier Transform (QFT). Previous color image watermarking methods are first presented and the quaternion representation is then described. In this framework RGB pixel values are associated with an unique quaternion number having three imaginary parts. The QFT is presen...

In the paper a new structure of Multi-Layer Perceptron, able to deal with quaternion-valued signal, is proposed. A learning algorithm for the proposed Quaternion MLP (QMLP) is also derived. Such a neural network allows to interpolate functions of quaternion variable with a smaller number of connections with respect to the corresponding real valued MLP. INTRODUCTION In the last few years, neural...

The roots of -1 in the set of biquaternions (quaternions with complex components, or complex numbers with quaternion real and imaginary parts) are studied and it is shown that there is an infinite number of non-trivial complexified quaternion roots (and two degenerate solutions which are the complex imaginary operator and the set of unit pure real quaternions). The non-trivial roots are shown t...

A revised version of the quaternion approach for numerical integration of the equations of motion for rigid polyatomic molecules is proposed. The modified approach is based on a formulation of the quaternion dynamics with constraints. This allows to resolve the rigidity problem rigorously using constraint forces. It is shown that the procedure for preservation of molecular rigidity can be reali...

A neural network for extracting color information from gloomy images is presented in this paper. We adopt a feedforward quaternion neural network, where neuronal parameters are represented as 4-dimensional vectors. The network is trained by imposing a gloomy image as input and its finer one as target by quaternion version of back propagation algorithm, and then relations between them are establ...

We prove that the Scott module whose vertex is isomorphic to a direct product of generalized quaternion $2$-group and cyclic Brauer indecomposable. This result generalizes similar results which are obtained for abelian, dihedral, quaternion, semidihedral wreathed vertices.

In recent years, several authors have reported that spectral saliency detection methods provide state-of-the-art performance in predicting human gaze in images (see, e.g., [1 3]). We systematically integrate and evaluate quaternion DCTand FFT-based spectral saliency detection [3,4], weighted quaternion color space components [5], and the use of multiple resolutions [1]. Furthermore, we propose ...