##### Title: Application and graphical interpretation of a new two-dimensional quaternion fractional Fourier transform

##### Pages: 561-575

##### Cite as:

Khinal Parmar, V. R. Lakshmi Gorty, Application and graphical interpretation of a new two-dimensional quaternion fractional Fourier transform, Int. J. Anal. Appl., 19 (4) (2021), 561-575.#### Abstract

In this paper, a new two-dimensional quaternion fractional Fourier transform is developed. The properties such as linearity, shifting and derivatives of the quaternion-valued function are studied. The convolution theorem and inversion formula are also established. An example with graphical representation is solved. An application related to two-dimensional quaternion Fourier transform is also demonstrated.

##### Full Text: PDF

#### References

- M. Bahri, A.K. Amir, C. Lande, The quaternion domain Fourier transform and its application in mathematical statistics, IAENG Int. J. Appl. Math. 48 (2018), 1-7.
- M. Bahri, R. Ashino, R. Vaillancourt, Continuous quaternion Fourier and wavelet transforms, Int. J. Wavelets Multiresolut. Inf. Process. 12 (2014), 1460003.
- T. B¨ulow, M. Felsberg, G. Sommer, Non-commutative hyper complex Fourier transforms of multidimensional signals, In: G. Sommer (eds) Geometric Computing with Clifford Algebras, Springer, Berlin, Heidelberg, 2001.
- D. Cheng, K.I. Kou, Plancherel theorem and quaternion Fourier transform for square integrable functions, Complex Var. Elliptic Equ. 64 (2019), 223–242.
- B. Davies, Integral transforms and their Applications, Springer, New York, 1978.
- T.A. Ell, Quaternion-Fourier transforms for analysis of two-dimensional linear time-invariant partial differential systems, in: Proceedings of 32nd IEEE Conference on Decision and Control, IEEE, San Antonio, TX, USA, 1993: pp. 1830–1841.
- T.A. Ell, S.J. Sangwine, Hypercomplex Fourier Transforms of Color Images, IEEE Trans. Image Process. 16 (2007), 22–35.
- X. Guanlei, W. Xiaotong, X. Xiaogang, Fractional quaternion Fourier transform, convolution and correlation, Signal Processing. 88 (2008), 2511–2517.
- E.M.S. Hitzer, Quaternion Fourier Transform on Quaternion Fields and Generalizations, Adv. Appl. Clifford Alg. 17 (2007), 497–517.
- A.C. Lewis, William Rown Hamilton, Lectures on quaternions (1853), in: Landmark Writings in Western Mathematics 1640-1940, Elsevier, 2005: pp. 460–469.
- P. Lian, Uncertainty principle for the quaternion Fourier transform, J. Math. Anal. Appl. 467 (2018), 1258–1269.
- L. Romero, R. Cerutti, L. Luque, A new Fractional Fourier Transform and convolutions products, Int. J. Pure Appl. Math. 66 (2011), 397-408.
- R. Roopkumar, Quaternionic one-dimensional fractional Fourier transform, Optik, 127 (2016), 11657-11661.
- A.H. Zemanian, Generalized integral transformations, John Wiley Sons Inc., New York, 1968.