15th European Signal Processing Conference (EUSIPCO 2007), Poznan, Poland, September 3-7, 2007, copyright by EURASIP QUATERNIONS AND BIQUATERNIONS FOR SYMMETRIC MARKOV-CHAIN SYSTEM ANALYSIS Soo-Chang Pei, Jian-Jiun Ding Department of Electrical Engineering, National Taiwan University, No. 1, Sec. 4, Roosevelt Rd., 10617, Taipei, Taiwan, R.O.C TEL: 886-2-23635251-321, Fax: 886-2-23671909, Email: [email protected], [email protected] ABSTRACT In this condition, j2 = 1, k2 = −1, ij = ji, ik = ki, and jk = kj. In this conference, we use quaternions, biquaternions, and The Clifford biquaternion and the complex Tessarine have their related algebras similar to them to model symmetric idempotent elements e1 and e2: Markov systems. With these algebras, the original N×N eI1 = (1+ ) / 2 , eI2 = (1− ) / 2 , (6) Markov system can be reduced into an (N/2)×(N/2) or 2 2 e1 = e1, e2 = e2, e1e2 = 0. (N/4)×(N/4) system. It makes the system easier for imple- Idempotent elements are helpful for fast computation. If q menting and analysis. In addition to Markov chains, the is defined in (3) and p is defined as proposed idea is also helpful for simplifying the complexi- pp=+⋅+⋅+⋅ pipjpk ties of other symmetric systems whose interactions between 0 ij k , (7) two objects are determined by their distance. + pIiIjIkIIpiIpjIpkI+⋅+⋅+⋅ then p and q can be expressed as: 1. INTRODUCTION p = pe+ pe , qqeqe= + , (8) 11 2 2 11 2 2 The quaternion [1][2] is a generalization of the complex where algebra. Its application in signal and image processing were p110= pe=+⋅+⋅+⋅( p pij i p j p k k noticed in recent years [2][3][4]. The quaternion has been , p22= pe . (9) ++ppipjpk ⋅+⋅+⋅)/2 applied for filter design, feature extraction, pattern recogni- IiIjIkI tion, rotation analysis, differential equation analysis, and Similarly, SVD decomposition, etc. In this paper, we propose another qqe11= and qqe22= . (10) application of the quaternion and its related algebras, i.e., Then, the product of p and q can be expressed as: the Markov chain analysis. pqpepeqeqepqepqe= ()()++=+. (11) 11 22 11 22 111 2 22 2. QUATERNIONS AND RELATED ALGEBRAS In addition to the quaternion and the Clifford biquater- nion, there are also other types of 4-D and 8-D algebras. Quaternions [1][2] are generalizations of complex numbers. The algebras used in this paper are: A quaternion has 4 components: real, i-, j-, and k-parts: [2-D algebras] • Complex algebra. q = qr + qi ⋅i + q j ⋅ j + qk ⋅k , (1) • Tessarines (also called as split-complex numbers, and and i, j, k obey the rules as below: double complex numbers) [5][6]: 2 2 2 i = j = k = −1, ij⋅= k, jk⋅= i, ki⋅= j, 2 qa= rI+ Ia, where I = 1. (12) ji⋅=− k, kj⋅=− i, ik⋅=− j. (2) [4-D algebras] Quaternions can be used for the color image analysis, filter • Quaternions (defined in (1) and (2)) design, and 3-D object analysis. • Complex Tessarines (defined in (5)). The definition of the Clifford biquaternion [6][10] is • Reduced biquaternion: See [3][7]. It is also called the similar to that of the quaternion, but there are 8 elements: bicomplex number (C×C). Its definition is the same as the 2 qq=+⋅+⋅+⋅0 qiqjqkij k reduced biquaternion in (5) but I = −1. , (3) ++⋅+⋅+⋅qI q iI q jI q kI • 4-D purely hyperbolic complex numbers: IiIjIkI qq= ++ qIqJq +⋅ IJ, I2 = J2 = 1, IJ = JI. (13) where i, j, and k satisfy (2) and rI J IJ [8-D algebras] I2 = 1, iI⋅=⋅ Ii, jI⋅=⋅ I j, kI⋅=⋅ Ik. (4) • Octonions (standard definition [1][9]). It has 7 imaginary The Tessarine with complex coefficients [5] (we call it the parts and the multiplicative rule is shown in [6]. complex Tessarine) is defined in the similar form. • Clifford biquaternions (see (3)). qq=+⋅++⋅ri qiqIqiI I iI. (5) • 8-D Tessarines (4-D hyperbolic complex numbers with complex coefficients) ©2007 EURASIP 1337 15th European Signal Processing Conference (EUSIPCO 2007), Poznan, Poland, September 3-7, 2007, copyright by EURASIP JJG JJG q=+ ari ia + Ja J + iJa iJI + Ia + iIa iIJIiJI + JIa + iJIa , (b) If pqrs= − , then the interactions between p and q 2 2 2 where i = −1, I = J = 1, iI = Ii, iJ = Ji, and IJ = JI. (14) (denoted by hp,q) and r and s (hr,s)) has the relation of • Hamilton's biquaternions [1][3]: Its definition is the same hp,q = −hr,s as the biquaternion in (3) but I2 = −1. then • Tri-complex numbers x112[naxnbxn+=1] [ ][] − , (23) q=+ ari ia + Ia I + iIa iIJ + Ja + iJa iJK + Ka + iKa iK, x212[nbxnaxn+=1] [ ][] + , (24) where I2 = J2 = K2 = −1, Ii = iI, Ji = iJ, then, we can define a function of Ki = iK, IJ = JI, IK = KI, and JK = KJ. (15) x[nxnIxn] =+12[ ] [ ] (25) 3. 2×2 AND 4×4 SYMMETRIC SYSTEM where I2 = −1, iI = I i. (26) In this case, we can also prove that For many systems, the distance between two points will x nhxn+=⋅1 , where ha=+ι b. (27) determine the mutual effect. For example, in physics, the [ ] [ ] 2 gravitation between two objects is proportional to 1/R If x1[n] and x2[n] are complex functions, then the system is where R is the distance between two objects. Intensities of modeled by the following algebra: the electric and magnetic fields also depend on R. In social qq= 01+⋅++⋅ qiq 4ι qi 5ι . (28) science, the interaction between two cities can also be It is just the same as the reduced biquaternion algebra. modeled by a function of the distance between two points. Here, we suppose that there are N points in a Markov 1 3 system and the field in each point is denoted by xk[n], where k = 1, 2, …, N. Suppose that the interaction between two dots is determined by their distance: N xmkmk[1]nxna+=∑ [], , where afRmk,,= ( mk) , (16) 2 4 k =1 Rm,k is the distance between m and. Fig. 2 A doubly symmetric 4 × 4 rectangular system. We find that in the system is symmetric, we can model The idea can be extended to any symmetric Markov sys- the system by the complex Tessarine. tem. For the system in Fig. 2, suppose that the values at the For example, for the system of Fig. 1 points 1, 2, 3, and 4 are 1 2 x1[n], x2[n], x3[n], and x4[n] (29) and the interactions between the points m, n are denoted by hm,n. Therefore, Fig. 1 A symmetric 2 × 2 system. ⎡ x11[1]nxn+ ⎤⎡⎤⎡⎤hhhh1,1 1,2 1,3 1,4 [] ⎢⎥ If the interaction is determined by the instance between two ⎢x [1]nxn+ ⎥⎢⎥hhhh [] dots, it can be modeled by the following Markov chain: ⎢ 22⎥⎢⎥= ⎢⎥2,1 2,2 2,3 2,4 . (30) ⎢x [1]nxn+ ⎥⎢⎥⎢⎥hhhh [] x naxnbxn+=1 + , (17) 333,1 3,2 3,3 1,4 112[ ] [] [] ⎢ ⎥⎢⎥⎢⎥ hhhh ⎣x44[1]nxn+ ⎦⎣⎦⎣⎦⎢⎥4,1 4,2 4,3 4,4 [] x212[nbxnaxn+=1 ] [] + []. (18) First, note that Then, we can define a function as hm,n = hn,m. (31) x[]nxnIxn=+12 [] [], (19) Moreover, since interactions are determined by distances, 2 where I = 1. If we define h1,1 = h2,2 = h3,3 = h4,4 = a h1,2 = h2,1 = h3,4 = h4,3 = b, haIb=+ , (20) h1,3 = h2,4 = h3,1 = h4,2 = c, h1,4 = h2,3 = h3,2 = h4,1 = d. (32) then Then (30) can be rewritten as: x [1]nxn+ abcd [] h⋅= x[] n ax12 [] n + bx [] n + I() ax 21 [] n + bx [] n , (21) ⎡ 11⎤⎡⎤⎡⎤ ⎢ ⎥⎢⎥⎢⎥ i.e., . (22) x22[1]nxn+ badc [] x[]nhxn+=⋅1 [] ⎢ ⎥⎢⎥= ⎢⎥ (33) If x1[n] and x2[n] are complex functions, then (22) is a ⎢x33[1]nxn+ ⎥⎢⎥⎢⎥cdab [] complex Tessarine operation. That is, the system 1 can be ⎢ ⎥⎢⎥⎢⎥ ⎣x44[1]nxn+ ⎦⎣⎦⎣⎦dcba [] modeled by a complex Tessarine operation. If x [n] and 1 If we set that x2[n] are quaternion functions, Fig. 1 can be modeled by a Clifford biquaternion operation. yn11[ ] =+ xn[ ] Ixn 2[ ] , yn23[]=+ xn [] Ixn 4[ ] , (34) If the effect of the system is determined by the only the haIb= + , hcId= + . distance but also the direction, that is, 1 2 Then (33) can be rewritten as (a) The interaction between JJGthe pointsJJG p and q and that be- ⎡ yn1121[1]+ ⎤⎡ h h ⎤⎡ yn [] ⎤ tween r and s are the same pq and rs have the same dis- ⎢ ⎥⎢= ⎥⎢ ⎥. (35) tance and direction. ⎣ yn2212[1]+ ⎦⎣ h h ⎦⎣ yn [] ⎦ ©2007 EURASIP 1338 15th European Signal Processing Conference (EUSIPCO 2007), Poznan, Poland, September 3-7, 2007, copyright by EURASIP That is, after converting into the complex Tessarine form, Table 1 2×2 and 4×4 symmetric system simplification by quater- the 4 × 4 system become a 2×2 system. nion and octonion. Moreover, notice that (35) is also a symmetric system. original system algebra reduced system We can further reduce it into the 1 × 1 form. We can apply 2×2 symmetric complex 1-to-1 system the 8-D Tessarines algebra. We can set that system Tessarine 2×2 directional reduced 1-to-1 system yn[ ] =+ x123[ n] Jxn[ ] + Ixn[ ] + IJxn 4[ ] , (36) symmetric system biquaternion Then 2×2 symmetric Tessarine 1-to-1 system yn+=1 hyn where h=+ a Jb + Ic + IJd . (37) [ ] [ ] system, real input That is, after applying the complex Tessarines, the 4 × 4 2×2 symmetric sys- Clifford 1-to-1 system symmetric rectangular system becomes a 2 × 2 symmetric tem, quaternion input biquaternion system.