International Journal of Analysis and Applications Volume 19, Number 4 (2021), 561-575 URL: https://doi.org/10.28924/2291-8639 DOI: 10.28924/2291-8639-19-2021-561 APPLICATION AND GRAPHICAL INTERPRETATION OF A NEW TWO-DIMENSIONAL QUATERNION FRACTIONAL FOURIER TRANSFORM KHINAL PARMAR∗, V. R. LAKSHMI GORTY NMIMS, MPSTME, Vile Parle - West, Mumbai - 400056, India ∗Corresponding author: khinal.parmar@nmims.edu 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. 1. Introduction In 1853, quaternions were developed by W. R. Hamilton [10]. The necessity of enlarging the operations on three-dimensional vectors to include multiplication and division led Hamilton to introduce the four- dimensional algebra of quaternions. In 1993, Ell [6] introduced quaternion Fourier transform for application to two-dimensional linear time-invariant systems of partial differential equations. In 2001 [3], authors de- fined non-commutative hypercomplex Fourier transforms of multidimensional Signals. In 2007 [9], author introduced right side quaternion Fourier transform. In 2008 [8], the concept of fractional quaternion Fourier transform was presented. In [11], the author studied the uncertainty principle for the quaternion Fourier transform. Authors in [1] developed quaternion domain Fourier transforms and its application in mathemat- ical statistics. In [4], Plancherel theorem and quaternion Fourier transform for square-integrable functions were studied. Received April 7th, 2021; accepted May 13th, 2021; published May 25th, 2021. 2010 Mathematics Subject Classification. 44A35, 44A05, 46S10. Key words and phrases. quaternion fractional Fourier transform; convolution; operational calculus; graphical representation. ©2021 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 561 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-19-2021-561 Int. J. Anal. Appl. 19 (4) (2021) 562 Quaternion Fourier transform transfers signals from the real-valued time domain to quaternion-valued fre- quency domain. But the proposed two-dimensional quaternion fractional Fourier transform will transfer the signal to unified time-frequency domains. Hence, it has a wide range of applications in the field of optics and signal processing. The organization of the paper is as follows: In section 2, some basic facts of quaternions and quaternion- valued functions are illustrated. In section 3, the two-dimensional quaternion fractional Fourier transform is defined and its inversion formula and operational properties are developed. Graphical interpretation of two-dimensional quaternion fractional Fourier transform is also illustrated. In Section 4, the application of the two-dimensional quaternion fractional Fourier transform is shown. 2. Preliminary results In quaternions, every element is a linear combination of a real scalar and three imaginary units i, j and k with real coefficients. Let q be a quaternion defined in (2.1) H = {q = x0 + ix1 + jx2 + kx3 : x0,x1,x2,x3 ∈ R} be the division ring of quaternions, where i, j, k satisfy Hamilton’s multiplication rules (see, e.g. [9]) (2.2) ij = −ji = k, jk = −kj = i, ki = −ik = j, i2 = j2 = k2 = ijk = −1. The quaternion conjugate of q is defined by (2.3) q̄ = x0 − ix1 − jx2 − kx3; x0,x1,x2,x3 ∈ R. The norm of q ∈ H is defined as (2.4) |q| = √ qq̄ = √ x20 + x 2 1 + x 2 2 + x 2 3. Alternatively, in [13] the quaternions are defined as (2.5) H = {q = q1 + jq2 : q1,q2 ∈ C} where j is the imaginary number satisfying following conditions: j2 = −1, jr = rj, ∀ r ∈ R, ji = −ij, where i is the imaginary number. From [13] f ∈ L2(R2; H), then the function is expressed as (2.6) f(u,v) = f0(u,v) + if1(u,v) + jf2(u,v) + kf3(u,v). For some applications the quaternions can be rewritten by replacing k with ij as given in [9], q = x0 + ix1 + jx2 + ix3j. Int. J. Anal. Appl. 19 (4) (2021) 563 Another way of rewritting quaternion is q = x+ + x−; x± = 1 2 (q ± iqj) . x± can also be expressed as x± = {x0 ±x3 + i(x1 ∓x2)} 1 ± k 2 = 1 ± k 2 {x0 ±x3 + j(x2 ∓x1)} . The real scalar part of the quaternion can be written as [9], (2.7) x0 = 〈q〉0 . We can also rewrite the function f ∈ L2(R2,H) as [9], f = f0 + if1 + jf2 + if3j. We can also split the function as [9], f = f+ + f−; f+ = 1 2 (f + ifj) , f− = 1 2 (f − ifj) . For f,g ∈ L2(R2,H) and u = (u,v) = ue1 + ve2 ∈ R2 with {e1,e2} as the basis of R2, the quaternion-valued inner product is defined in [9] as (2.8) (f,g) = ∫ R2 f(u)ḡ(u)d2u, with real symmetric part (2.9) 〈f,g〉 = 1 2 [(f,g) + (g,f)] = ∫ R2 〈f(u)ḡ(u)〉0 d 2u. The norm of f ∈ L2(R2,H) is defined as (2.10) ||f|| = √ (f,f) = √ 〈f,f〉 = ∫ R2 |f(u)|2d2u. 3. Main Results Definition 3.1. Let f ∈ L2(R2,H), then two-dimensional quaternion fractional Fourier transform (2D- QFrFT) of particular order α,β using [9, 12] is defined as (3.1) f̂α,β (w1,w2) = Fα,β [f (u,v) ; w1,w2] = ∞∫ −∞ ∞∫ −∞ e−iw 1 α 1 uf (u,v) e−jw 1 β 2 vdudv where 0 < α,β ≤ 1. Analogous to [5, page 112],the integral will converge for values of w1 and w2 in the strips −s1 < Im(w1) < s1 and −s2 < Im(w2) < s2 respectively; where s1 < Re(p1), s2 < Re(p2), for p1 = iw1,p2 = jw2. Int. J. Anal. Appl. 19 (4) (2021) 564 The sufficient condition for f(u,v) to have 2D-QFrFT is that ∞∫ −∞ ∞∫ −∞ |f(u,v)|dudv exists. Inversion formula: Consider the inverse formula of quaternion Fourier transform as defined in [9] f(u,v) = 1 (2π) 2 ∞∫ −∞ ∞∫ −∞ eixuf̂(x,y)ejyvdxdy. Substituting x = w 1 α 1 and y = w 1 β 2 . Then, f(u,v) = 1 (2π) 2 ∞∫ −∞ ∞∫ −∞ eiw 1 α 1 uw 1 α −1 1 f̂α,β (w1,w2) e jw 1 β 2 vw 1 β −1 2 dw1 α dw2 β f(u,v) = 1 (2π) 2 αβ ∞∫ −∞ ∞∫ −∞ eiw 1 α 1 uw 1−α α 1 f̂α,β (w1,w2) e jw 1 β 2 vw 1−β β 2 dw1dw2. Hence, the inversion formula is defined as (3.2) F−1α,β [ f̂α,β (w1,w2) ] = f(u,v) = 1 (2π) 2 αβ ∞∫ −∞ ∞∫ −∞ eiw 1 α 1 uw 1−α α 1 f̂α,β (w1,w2) e jw 1 β 2 vw 1−β β 2 dw1dw2. Property 3.1 (Left linearity). For f1,f2 ∈ L2(R2,H) and k1,k2 ∈{q|q = x0 + ix1,x0,x1 ∈ R}; (3.3) Fα,β [k1f1(u,v) + k2f2(u,v)] = k1Fα,β [f1(u,v)] + k2Fα,β [f2(u,v)] . Proof. For f1,f2 ∈ L2(R2,H); k1,k2 ∈ R and using (3.1), we get Fα,β [k1f1(u,v) + k2f2(u,v)] = ∞∫ −∞ ∞∫ −∞ e−iw 1 α 1 u [k1f1(u,v) + k2f2(u,v)] e −jw 1 β 2 vdudv = k1 ∞∫ −∞ ∞∫ −∞ e−iw 1 α 1 u [f1(u,v)] e −jw 1 β 2 vdudv + k2 ∞∫ −∞ ∞∫ −∞ e−iw 1 α 1 u [f2(u,v)] e −jw 1 β 2 vdudv = k1Fα,β [f1(u,v)] + k2Fα,β [f2(u,v)] . � Property 3.2 (Right linearity). For f1,f2 ∈ L2(R2,H) and k′1,k ′ 2 ∈{q|q = x0 + jx2,x0,x2 ∈ R}; (3.4) Fα,β [f1(u,v)k ′ 1 + f2(u,v)k ′ 2] = Fα,β [f1(u,v)] k ′ 1 + Fα,β [f2(u,v)] k ′ 2. Int. J. Anal. Appl. 19 (4) (2021) 565 The proof is similar to property 3.1. Property 3.3 (Shifting). For f ∈ L2(R2,H) and a,b ∈ R; (3.5) Fα,β [f(u−a,v − b)] = e−iw 1 α 1 aFα,β [f(u,v)] e −jw 1 β 2 b Proof. For f ∈ L2(R2,H); a,b ∈ R and using (3.1), we get Fα,β [f(u−a,v − b)] = ∞∫ −∞ ∞∫ −∞ e−iw 1 α 1 uf(u−a,v − b)e−jw 1 β 2 vdudv. Substituting u−a = s and v − b = t gives = ∞∫ −∞ ∞∫ −∞ e−iw 1 α 1 (s+a)f(s,t)e−jw 1 β 2 (t+b)dsdt = e−iw 1 α 1 aFα,β [f(s,t)] e −jw 1 β 2 b. � Property 3.4 (2D-QFrFT of derivatives). For f ∈ L2(R2,H), the two-dimensional quaternion fractional Fourier transform with derivatives of f(u,v) are as follows: (3.6) i) Fα,β [ ∂ ∂u f(u,v) ] = ( iw 1 α 1 ) Fα,β [f(u,v)] . (3.7) ii) Fα,β [ ∂ ∂v f(u,v) ] = Fα,β [f(u,v)] ( jw 1 β 2 ) . (3.8) iii) Fα,β [ ∂2 ∂u∂v f(u,v) ] = ( iw 1 α 1 ) Fα,β [f(u,v)] ( jw 1 β 2 ) . In general (3.9) iv) Fα,β [ ∂n ∂un f(u,v) ] = ( iw 1 α 1 )n Fα,β [f(u,v)] . (3.10) v) Fα,β [ ∂n ∂vn f(u,v) ] = Fα,β [f(u,v)] ( jw 1 β 2 )n . (3.11) vi) Fα,β [ ∂n ∂un ∂m ∂vm f(u,v) ] = ( iw 1 α 1 )n Fα,β [f(u,v)] ( jw 1 β 2 )m . Int. J. Anal. Appl. 19 (4) (2021) 566 Proof. i) For f ∈ L2(R2,H), the first order derivative over f(u,v) w.r.t. u is given by Fα,β [ ∂ ∂u f(u,v) ] = ∞∫ −∞ ∞∫ −∞ e−iw 1 α 1 u ∂ ∂u f(u,v)e−jw 1 β 2 vdudv = ∞∫ −∞   [ e−iw 1 α 1 uf(u,v) ] − ∞∫ −∞ −iw 1 α 1 e −iw 1 α 1 uf(u,v)du  e−jw 1 β 2 vdv = iw 1 α 1 ∞∫ −∞ ∞∫ −∞ e−iw 1 α 1 uf(u,v)e−jw 1 β 2 vdudv = ( iw 1 α 1 ) Fα,β [f(u,v)] . ii) For f ∈ L2(R2,H), the first order derivative over f(u,v) w.r.t. v is given by Fα,β [ ∂ ∂v f(u,v) ] = ∞∫ −∞ ∞∫ −∞ e−iw 1 α 1 u ∂ ∂v f(u,v)e−jw 1 β 2 vdudv = ∞∫ −∞ e−iw 1 α 1 u   [ f(u,v)e−jw 1 β 2 v ] − ∞∫ −∞ f(u,v)e−jw 1 β 2 v ( −jw 1 β 2 ) dv  du = ∞∫ −∞ ∞∫ −∞ e−iw 1 α 1 uf(u,v)e−jw 1 β 2 vdudv ( jw 1 β 2 ) = Fα,β [f(u,v)] ( jw 1 β 2 ) . iii) For f ∈ L2(R2,H), the second order derivative over f(u,v) w.r.t. u,v is given by Fα,β [ ∂2 ∂u∂v f(u,v) ] = ∞∫ −∞ ∞∫ −∞ e−iw 1 α 1 u ∂2 ∂u∂v f(u,v)e−jw 1 β 2 vdudv = ∞∫ −∞   [ e−iw 1 α 1 u ∂ ∂v f(u,v) ] − ∞∫ −∞ −iw 1 α 1 e −iw 1 α 1 u ∂ ∂v f(u,v)du  e−jw 1 β 2 vdv = ( iw 1 α 1 ) ∞∫ −∞ ∞∫ −∞ e−iw 1 α 1 u ∂ ∂v f(u,v)e−jw 1 β 2 vdudv Int. J. Anal. Appl. 19 (4) (2021) 567 = ( iw 1 α 1 ) ∞∫ −∞ e−iw 1 α 1 u   [ f(u,v)e−jw 1 β 2 v ] − ∞∫ −∞ f(u,v)e−jw 1 β 2 v ( −jw 1 β 2 ) dv  du = ( iw 1 α 1 ) ∞∫ −∞ ∞∫ −∞ e−iw 1 α 1 uf(u,v)e−jw 1 β 2 vdudv ( jw 1 β 2 ) = ( iw 1 α 1 ) Fα,β [f(u,v)] ( jw 1 β 2 ) . iv) By using mathematical induction for n = 1 by (3.6), we get Fα,β [ ∂ ∂u f(u,v) ] = iw 1 α 1 Fα,β [f(u,v)] . For n = 2, the result holds true. Fα,β [ ∂2 ∂u2 f(u,v) ] = ∞∫ −∞ ∞∫ −∞ e−iw 1 α 1 u ∂2 ∂u2 f(u,v)e−jw 1 β 2 vdudv = ∞∫ −∞   [ e−iw 1 α 1 u ∂ ∂u f(u,v) ] − ∞∫ −∞ −iw 1 α 1 e −iw 1 α 1 u ∂ ∂u f(u,v)du  e−jw 1 β 2 vdv = iw 1 α 1 ∞∫ −∞ ∞∫ −∞ e−iw 1 α 1 u ∂ ∂u f(u,v)e−jw 1 β 2 vdudv = iw 1 α 1 ∞∫ −∞   [ e−iw 1 α 1 uf(u,v) ] − ∞∫ −∞ −iw 1 α 1 e −iw 1 α 1 uf(u,v)du  e−jw 1 β 2 vdv = ( iw 1 α 1 )2 ∞∫ −∞ ∞∫ −∞ e−iw 1 α 1 uf(u,v)e−jw 1 β 2 vdudv = ( iw 1 α 1 )2 Fα,β [f(u,v)] . For n = k − 1, Fα,β [ ∂k−1 ∂uk−1 f(u,v) ] = ∞∫ −∞ ∞∫ −∞ e−iw 1 α 1 u ∂k−1 ∂uk−1 f(u,v)e−jw 1 β 2 vdudv = ∞∫ −∞   [ e−iw 1 α 1 u ∂k−2 ∂uk−2 f(u,v) ] − ∞∫ −∞ −iw 1 α 1 e −iw 1 α 1 u ∂k−2 ∂uk−2 f(u,v)du  e−jw 1 β 2 vdv = iw 1 α 1 ∞∫ −∞ ∞∫ −∞ e−iw 1 α 1 u ∂k−2 ∂uk−2 f(u,v)e−jw 1 β 2 vdudv. Int. J. Anal. Appl. 19 (4) (2021) 568 On repeating the integration by parts, we get Fα,β [ ∂k−1 ∂uk−1 f(u,v) ] = ( iw 1 α 1 )k−1 Fα,β [f(u,v)] . By method of mathematical induction, the result is true for all n = k. Fα,β [ ∂k ∂uk f(u,v) ] = ( iw 1 α 1 )k Fα,β [f(u,v)] . Thus, it is true for all n. Similarly, v) and vi) can be proved. � Property 3.5 (Power of u,v). For f ∈ L2(R2,H) (3.12) i) Fα,β [uf(u,v)] = ( i α w 1−α α 1 ) ∂ ∂w1 Fα,β [f(u,v)] . (3.13) ii) Fα,β [vf(u,v)] = ∂ ∂w2 Fα,β [f(u,v)]  j β w 1−β β 2   . Proof. For f ∈ L2(R2,H) and using (3.1), we get Fα,β [uf(u,v)] = ∞∫ −∞ ∞∫ −∞ e−iw 1 α 1 uuf (u,v) e−jw 1 β 2 vdudv = ∞∫ −∞ ∞∫ −∞ ( i α w 1−α α 1 ) ∂ ∂w1 e−iw 1 α 1 uf (u,v) e−jw 1 β 2 vdudv = ( i α w 1−α α 1 ) ∂ ∂w1 ∞∫ −∞ ∞∫ −∞ e−iw 1 α 1 uf (u,v) e−jw 1 β 2 vdudv = ( i α w 1−α α 1 ) ∂ ∂w1 Fα,β [f(u,v)] . Fα,β [vf(u,v)] = ∞∫ −∞ ∞∫ −∞ e−iw 1 α 1 uvf (u,v) e−jw 1 β 2 vdudv = ∞∫ −∞ ∞∫ −∞ e−iw 1 α 1 uf (u,v) ∂ ∂w2 e−jw 1 β 2 v  j β w 1−β β 2  dudv = ∂ ∂w2 ∞∫ −∞ ∞∫ −∞ e−iw 1 α 1 uf (u,v) e−jw 1 β 2 vdudv  j β w 1−β β 2   = ∂ ∂w2 Fα,β [f(u,v)]  j β w 1−β β 2   . Hence the proof. � Int. J. Anal. Appl. 19 (4) (2021) 569 Property 3.6 (Power of i, j). For f ∈ L2(R2,H); m,n ∈ N (3.14) Fα,β [i mf(u,v)jn] = imFα,β [f(u,v)] j n. Proof. For f ∈ L2(R2,H) and using (3.1), we get Fα,β [i mf(u,v)jn] = ∞∫ −∞ ∞∫ −∞ e−iw 1 α 1 uimf(u,v)jne−jw 1 β 2 vdudv = im ∞∫ −∞ ∞∫ −∞ e−iw 1 α 1 uf(u,v)e−jw 1 β 2 vdudvjn = imFα,β [f(u,v)] j n. Hence the proof. � Definition 3.2. The convolution for the quaternion valued functions f,g ∈ L2(R2,H) is defined [14] by (3.15) f ∗g = ∞∫ −∞ ∞∫ −∞ f(u,v)g(x−u,y −v)dudv. Theorem 3.7 (Convolution theorem). For f,g ∈ L2(R2,H); (3.16) Fα,β [f ∗g] = Fα,β [f] Fα,β [g] . Proof. For f,g ∈ L2(R2,H); X = (x1,x2), Y = (y1,y2) and Z = (z1,z2); Fα,β [f ∗g] = ∞∫ −∞ ∞∫ −∞ e−iw 1 α 1 x1 (f ∗g) (X) e−jw 1 β 2 x2dX = ∞∫ −∞ ∞∫ −∞ e−iw 1 α 1 x1   ∞∫ −∞ ∞∫ −∞ f(Y )g(X −Y )dY  e−jw 1β2 x2dX = ∞∫ −∞ ∞∫ −∞ e−iw 1 α 1 x1f(Y )   ∞∫ −∞ ∞∫ −∞ g(X −Y )e−jw 1 β 2 x2dX  dY . Substituting Z = X −Y , we get Fα,β [f ∗g] = ∞∫ −∞ ∞∫ −∞ e−iw 1 α 1 (y1+z1)f(Y )   ∞∫ −∞ ∞∫ −∞ g(Z)e−jw 1 β 2 (y2+z2)dZ  dY =   ∞∫ −∞ ∞∫ −∞ e−iw 1 α 1 y1f(Y )e−jw 1 β 2 y2dY     ∞∫ −∞ ∞∫ −∞ e−iw 1 α 1 z1g(Z)e−jw 1 β 2 z2dZ   = Fα,β [f] Fα,β [g] . � Int. J. Anal. Appl. 19 (4) (2021) 570 Theorem 3.8. The scalar product of two quaternion-valued functions f,g ∈ L2(R2,H) is given by the scalar product of the corresponding 2D-QFrFTs f̂ and ĝ: (3.17) 〈f,g〉 = 1 (2π) 2 αβ 〈Fα,β(w1,w2),Gα,β(w1,w2)〉 . Proof. For f, g ∈ L2(R2,H) and using (2.8), we get 〈f, g〉 = ∫ ∞ ∞ ∫ ∞ ∞ 〈 f(u,v)g(u,v) 〉 dudv = ∫ ∞ ∞ ∫ ∞ ∞ 〈 1 (2π) 2 αβ ∞∫ −∞ ∞∫ −∞ eiw 1 α 1 uw 1−α α 1 f̂α,β (w1,w2) × ejw 1 β 2 vw 1−β β 2 dw1dw2g(u,v) 〉 dudv = 1 (2π) 2 αβ ∫ ∞ ∞ ∫ ∞ ∞ 〈 f̂α,β(w1,w2) ∞∫ −∞ ∞∫ −∞ eiw 1 α 1 uw 1−α α 1 e jw 1 β 2 vw 1−β β 2 × g(u,v)dudv 〉 dw1dw2 = 1 (2π) 2 αβ ∫ ∞ ∞ ∫ ∞ ∞ 〈 f̂α,β(w1,w2)w 1−α α 1 w 1−β β 2 × ∞∫ −∞ ∞∫ −∞ ejw 1 β 2 vg(u,v)eiw 1 α 1 ududv 〉 dw1dw2 = 1 (2π) 2 αβ ∫ ∞ ∞ ∫ ∞ ∞ 〈 f̂α,β(w1,w2)w 1−α α 1 w 1−β β 2 × ∞∫ −∞ ∞∫ −∞ e−iw 1 α 1 vg(u,v)e−jw 1 β 2 ududv 〉 dw1dw2 = 1 (2π) 2 αβ ∫ ∞ ∞ ∫ ∞ ∞ 〈 w 1−α α 1 f̂α,β(w1,w2)w 1−β β 2 ĝα,β(w1,w2) 〉 dw1dw2 = 1 (2π) 2 αβ 〈Fα,β(w1,w2),Gα,β(w1,w2)〉 where Fα,β(w1,w2) = w 1−α α 1 f̂α,β(w1,w2); Gα,β(w1,w2) = w 1−β β 2 ĝα,β(w1,w2). Thus, the theorem holds true. � Int. J. Anal. Appl. 19 (4) (2021) 571 Figure 1. Kernel of 2D-QFrFT at α = 1 and β = 1. Figure 2. Kernel of 2D-QFrFT at α = 1/2 and β = 1/2. Figure 1 shows the kernel of 2D-QFrFT for various values of w1,w2 at order α = 1 and β = 1 which is a particular case of the study developed in this paper. Figure 2 shows the kernel of 2D-QFrFT for various values of w1,w2 at order α = 1/2 and β = 1/2. For both the figures the range of x and y is between −3 and 3. The 2D-QFrFT is superior in disparity estimation and analyzing genuine 2D texture as compared to other fractional Fourier transforms and [7]. Int. J. Anal. Appl. 19 (4) (2021) 572 Figure 3. 2D-QFrFT Kernel. Left: Top row: (1 + ij)/2 and (i−j)/2 components. Bottom row: (1 − ij)/2 and (i + j)/2 components at α = 1, β = 1 ; Right: Top row: (1 + ij)/2 and (i−j)/2 components. Bottom row: (1−ij)/2 and (i + j)/2 components at α = 1/2, β = 1/2 The components (1 + ij)/2, (i − j)/2, (1 − ij)/2 and (i + j)/2 are shown in Figure 3 at α = 1, β = 1 and α = 1/2, β = 1/2 which represents 2D-QFT extended to 2D-QFrFT. We can also observe the scale-invariant feature of 2D-QFrFT. Example 3.1. Find the quaternion fractional Fourier transform of the function: (3.18) f(x,y) =   1; |x| < 1, |y| < 1 0; otherwise. By using (3.1), we get Fα,β[f(x,y)] = ∞∫ −∞ ∞∫ −∞ e−iw 1 α 1 xf (x,y) e−jw 1 β 2 ydxdy. Fα,β[f(x,y)] = 1∫ −1 1∫ −1 e−iw 1 α 1 xe−jw 1 β 2 ydxdy Fα,β[f(x,y)] = 4 sin w 1 α 1 w 1 α 1 · sin w 1 β 2 w 1 β 2 .(3.19) The graphical representation of the quaternion fractional Fourier transform of the function (3.18) obtained using α = 1 and β = 1 in (3.19), is now a particular case of (3.1) which is represented in the following figure: Int. J. Anal. Appl. 19 (4) (2021) 573 Figure 4. Graph of Fα,β[f(x,y)] with α = 1 and β = 1. The graphical representation of the quaternion fractional Fourier transform of the function (3.18) obtained using α = 1/2 and β = 1/2 in (3.19) is represented in the following figure: Figure 5. Graph of Fα,β[f(x,y)] with α = 1/2 and β = 1/2. The graphical representation of the quaternion fractional Fourier transform of the function (3.18) obtained using α = 1/2 and β = 1 in (3.19) is represented in the following figure: Figure 6. Graph of Fα,β[f(x,y)] with α = 1/2 and β = 1. Int. J. Anal. Appl. 19 (4) (2021) 574 Figures 1-3 are plotted using online freeware version of wolframalpha. Figures 4-6 are plotted using online freeware version 3D surface plotter of academo. 4. Application Let us consider the initial value problem from [2]: (4.1) ∂h ∂t −O2h = 0, on R0, 2 × (0,∞), and (4.2) h(u,v) = f(u,v), f ∈S(R0, 2; H) at t = 0, where S(R0, 2; H) is the quaternion Schwartz space and O2 = ∂2 ∂2u + ∂2 ∂2v . Applying the definition of 2D-QFrFT to both sides of (4.1), we get Fα,β [ ∂h ∂t ] = ( iw 1 α 1 )2 Fα,β[h] + Fα,β[h] ( jw 1 β 2 )2 (4.3) ∂ ∂t Fα,β [h] = − ( w 2 α 1 + w 2 β 2 ) Fα,β[h]. The general solution of (4.3) is given by (4.4) Fα,β [h] = Ce − ( w 2 α 1 +w 2 β 2 ) t , where C is a quaternion constant. By using the initial value condition, we get (4.5) Fα,β [h] = e − ( w 2 α 1 +w 2 β 2 ) t Fα,β [f] . Analogous to [2, equation 6.6], we have (4.6) 1 4πt Fα,β [ e − ( u 2 α +v 2 β ) /4t ] = e − ( w 2 α 1 +w 2 β 2 ) t . Applying the inversion formula of 2D-QFrFT to (4.5), we get h = F−1α,β  e− ( w 2 α 1 +w 2 β 2 ) t Fα,β [f]   = F−1α,β [ 1 4πt Fα,β [ e − ( u 2 α +v 2 β ) /4t ] Fα,β [f] ] . Using convolution theorem, we have (4.7) h = Kt ∗f Int. J. Anal. Appl. 19 (4) (2021) 575 where Kt = 1 4πt e − ( u 2 α +v 2 β ) /4t . 5. Conclusion The authors developed a new two-dimensional quaternion fractional Fourier transform in this study. 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