9 research outputs found
The Color Clifford Hardy Signal: Application to Color Edge Detection and Optical Flow
This paper introduces the idea of the color Clifford Hardy signal, which can
be used to process color images. As a complex analytic function's
high-dimensional analogue, the color Clifford Hardy signal inherits many
desirable qualities of analyticity. A crucial tool for getting the color and
structural data is the local feature representation of a color image in the
color Clifford Hardy signal. By looking at the extended Cauchy-Riemann
equations in the high-dimensional space, it is possible to see the connection
between the different parts of the color Clifford Hardy signal. Based on the
distinctive and important local amplitude and local phase generated by the
color Clifford Hardy signal, we propose five methods to identify the edges of
color images with relation to a certain color. To prove the superiority of the
offered methodologies, numerous comparative studies employing image quality
assessment criteria are used. Specifically by using the multi-scale structure
of the color Clifford Hardy signal, the proposed approaches are resistant to a
variety of noises. In addition, a color optical flow detection method with
anti-noise ability is provided as an example of application.Comment: 13 page
Convolution theorems associated with quaternion linear canonical transform and applications
Novel types of convolution operators for quaternion linear canonical
transform (QLCT) are proposed. Type one and two are defined in the spatial and
QLCT spectral domains, respectively. They are distinct in the quaternion space
and are consistent once in complex or real space. Various types of convolution
formulas are discussed. Consequently, the QLCT of the convolution of two
quaternionic functions can be implemented by the product of their QLCTs, or the
summation of the products of their QLCTs. As applications, correlation
operators and theorems of the QLCT are derived. The proposed convolution
formulas are used to solve Fredholm integral equations with special kernels.
Some systems of second-order partial differential equations, which can be
transformed into the second-order quaternion partial differential equations,
can be solved by the convolution formulas as well. As a final point, we
demonstrate that the convolution theorem facilitates the design of
multiplicative filters
Hyperbolic linear canonical transforms of quaternion signals and uncertainty
This paper is concerned with Linear Canonical Transforms (LCTs) associated with two-dimensional quaternion-valued signals defined in an open rectangle of the Euclidean plane endowed with a hyperbolic measure, which we call Quaternion Hyperbolic Linear Canonical Transforms (QHLCTs). These transforms are defined by replacing the Euclidean plane wave with a corresponding hyperbolic relativistic plane wave in one dimension multiplied by quadratic modulations in both the hyperbolic spatial and frequency domains, giving the hyperbolic counterpart of the Euclidean LCTs. We prove the fundamental properties of the partial QHLCTs and the right-sided QHLCT by employing hyperbolic geometry tools and establish main results such as the Riemann-Lebesgue Lemma, the Plancherel and Parseval Theorems, and inversion formulas. The analysis is carried out in terms of novel hyperbolic derivative and hyperbolic primitive concepts, which lead to the differentiation and integration properties of the QHLCTs. The results are applied to establish two quaternionic versions of the Heisenberg uncertainty principle for the right-sided QHLCT. These uncertainty principles prescribe a lower bound on the product of the effective widths of quaternion-valued signals in the hyperbolic spatial and frequency domains. It is shown that only hyperbolic Gaussian quaternion functions minimize the uncertainty relations.publishe
Hyperbolic linear canonical transforms of quaternion signals and uncertainty
*The final version is published in Applied Mathematics and Computation (450), 2023, Article 127971. It as available via the website https://doi.org/10.1016/j.amc.2023.127971Acknowledgements:
The first author’s work was supported by the Asociaci´on Mexicana de Cultura, A. C.. The work of M. Ferreira was supported by Portuguese funds through CIDMA-Center for Research and Development in Mathematics
and Applications, and FCT – Fundação para a Ciência e a Tecnologia, within projects UIDB/04106/2020 and UIDP/04106/202.This paper is concerned with Linear Canonical Transforms (LCTs) associated with two-dimensional quaternion-valued signals defined in an open rectangle of the Euclidean plane endowed with a hyperbolic measure, which we call Quaternion Hyperbolic Linear Canonical Transforms (QHLCTs). These transforms are defined by replacing the Euclidean plane wave with a corresponding hyperbolic relativistic plane wave in one dimension multiplied by quadratic modulations in both the hyperbolic spatial and frequency domains, giving the hyperbolic counterpart of the Euclidean LCTs. We prove the fundamental properties of the partial QHLCTs and the right-sided QHLCT by employing hyperbolic geometry tools and establish main results such as the Riemann-Lebesgue Lemma, the Plancherel and Parseval Theorems, and inversion formulas. The analysis is carried out in terms of novel hyperbolic derivative and hyperbolic primitive concepts, which lead to the differentiation and integration properties of the QHLCTs. The results are applied to establish two quaternionic versions of the Heisenberg uncertainty principle for the right-sided QHLCT. These uncertainty principles prescribe a lower bound on the product of the effective widths of quaternion-valued signals in the hyperbolic spatial and frequency domains. It is shown that only hyperbolic Gaussian quaternion functions minimize the uncertainty relations.info:eu-repo/semantics/publishedVersio