The Quaternion Fourier Transform of Finite Measure and Its Properties

Bahri, Mawardi; Rahim, Amran; Nur, Muh.; Amir, Amir Kamal

doi:10.1007/978-3-030-79606-8_6

Cited by 2 publications

(1 citation statement)

References 13 publications

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“…It was only in 1990s that both these independent works begin to be referred jointly in the open literature. The linear canonical transform (LCT) is a three-free-parameter class of linear integral transforms, which encompasses a number of well-known unitary transformations as well as signal processing and optics-related mathematical operations, for example, the Fourier transform, the fractional Fourier transform, the Fresnel transform and the scaling operations [3][4][5][6][7][8][9][10]. Due to the extra degrees of freedom and simple geometrical manifestation, the LCT [30] is more flexible than other transforms and is as such suitable as well as powerful tool for investigating deep problems in science and engineering.…”

Section: Introductionmentioning

confidence: 99%

Generalized sampling expansion for the quaternion linear canonical transform

Siddiqui,

Bing-Zhao,

Samad

2024

SIViP

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The theory of quaternions has gained firm ground in recent times and is being widely explored, with the field of signal and image processing being no exception. However, many important aspects of quaternionic signals are yet to be explored, particularly the formulation of Generalized Sampling Expansions (GSE). In the present article, our aim is to formulate the GSE in the realm of a one-dimensional quaternion linear canonical transform (QLCT). To facilitate the intent, we construct a set of quaternion-valued filter functions which are used to construct a system of equations determining the synthesis functions for the process of reconstruction. Besides, as a special case, another sampling formula involving the derivatives of the quaternionic signal is also obtained in the sequel. Since derivatives contain information about the edges and curves appearing in images, therefore, by invoking the higher degrees of freedom pertaining to QLCT, the obtained sampling formula can be used to develop new image scaling algorithms with state-of-the-art flexibility. As an endorsement of the obtained results, an example with simulations demonstrating the signal reconstruction is presented at the end.

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Section: Introductionmentioning

confidence: 99%

Generalized sampling expansion for the quaternion linear canonical transform

Siddiqui,

Bing-Zhao,

Samad

2024

SIViP

View full text Add to dashboard Cite

show abstract

Generalized Sampling Expansion for the Quaternion Linear Canonical Transform

Siddiqui,

Li,

Samad

2024

Preprint

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The theory of quaternions has gained firm ground in recent times and is being widely explored, with the field ofsignal and image processing being no exception. However, many important aspects of quaternionic signals are yet tobe explored, particularly the formulation of Generalized Sampling Expansions (GSE). In the present article, our aim is to formulate the GSE in the realm of a one-dimensional quaternion linear canonical transform (QLCT). To facilitate the intent, we construct a set of quaternion-valued filter functions which are used to construct a system of equations determining the synthesis functions for the process of reconstruction. Besides, as a special case, another sampling formula involving the derivatives of the quaternionic signal is also obtained in the sequel. Since derivatives contain information about the edges and curves appearing in images, therefore, by invoking the higher degrees of freedom pertaining to QLCT, the obtained sampling formula can be used to develop new image scaling algorithms with state-of-the-art flexibility. As an endorsement of the obtained results, an example with simulations demonstrating the signal reconstruction is presented at the end.

show abstract

The Quaternion Fourier Transform of Finite Measure and Its Properties

Cited by 2 publications

References 13 publications

Generalized sampling expansion for the quaternion linear canonical transform

Generalized sampling expansion for the quaternion linear canonical transform

Generalized Sampling Expansion for the Quaternion Linear Canonical Transform

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