Optimal Restoration of Multiple Signals in Quaternion Algebra Artyom - PowerPoint PPT Presentation

Optimal Restoration of Multiple Signals in Quaternion Algebra Artyom M. Grigoryan a and Sos S. Agaian b a Department of Electrical and Computer Engineering The University of Texas at San Antonio, San Antonio, Texas, USA, and b Computer Science Department, College of Staten Island and the Graduate Center, Staten Island, NY, USA amgrigoryan@utsa.edu, sos.agaian@csi.cuny.edu 7/8/2020 1

OUTLINE • Introduction • Problem Formulation • Quaternion Numbers • 1-D Convolution in Quaternion Algebra (QA) • Inverse Problem in QA and Solution • Quaternion Convolution plus Noise: Solution • Example of Restored Quaternion Signal • Summary • References 7/8/2020 2

Abstract • This paper offers a new multiple signal restoration tool to solve the inverse problem, when signals are convoluted with a multiple impulse response and then degraded by an additive noise signal with multiple components. Inverse problems arise practically in all areas of science and engineering and refers to as methods of estimating data/parameters, in our case of multiple signals that cannot directly be observed. • The presented tool is based on the mapping multiple signals into the quaternion domain, and then solving the inverse problem. • Due to the non-commutativity of quaternion arithmetic, it is difficult to find the optimal filter in the frequency domain for degraded quaternion signals. 7/8/2020 3

Presented Work • As an alternative, we introduce an optimal filter by using special 4 × 4 matrices on the discrete Fourier transforms of signal components, at each frequency-point. The optimality of the solution is with respect to the mean-square-root error, as in the classical theory of the signal restoration by the Wiener filter. • The Illustrative example of optimal filtration of multiple degraded signals in the quaternion domain is given. The computer simulations validate the effectiveness of the proposed method. 7/8/2020 4

1. PROBLEM OF MULTIPLE SIGNAL RESTORATION In the space of quaternion signals, in the model described the signal 𝑟(𝑢) convoluted with the function ℎ(𝑢) plus a noise 𝑜(𝑢) 𝑗(𝑢) = 𝑟(𝑢) ∗ ℎ(𝑢) + 𝑜(𝑢), (1) the signal 𝑟(𝑢) is restoring from the degraded signal 𝑗(𝑢) . The classic case: the inverse problem is solving by the optimal filter 𝐼(𝜕) 𝑍 𝜕 = + 𝜚 𝑂/𝑅 (𝜕) . (2) 𝐼 𝜕 Here, 𝐼(𝜕) is the Fourier transform of ℎ(𝑢) , and 𝜚 𝑂/𝑅 (𝜕) is the noise-signal 2 > and 𝜚 𝑅 𝜕 =< 𝑅 𝜕 2 > are spatial spectral ratio, and 𝜚 𝑂 𝜕 =< 𝑂 𝜕 densities of the signal 𝑟(𝑢) and noise 𝑜(𝑢) , respectively. 7/8/2020 5

Inroduction to Quanterions The quaternion number is composed by one real part and three-component imaginary part, 𝑟 = 𝑏 + (𝑐𝑗 + 𝑑𝑘 + 𝑒𝑙) = 𝑏 + 𝑐𝑗 + 𝑑𝑘 + 𝑒𝑙, where 𝑏, 𝑐, 𝑑, and 𝑒 are real numbers. Together with unit 1, three imaginary units 𝑗 , 𝑘 , and 𝑙 are used with the multiplication laws, which are following: 𝑗 2 = 𝑘 2 = 𝑙 2 = −1, 𝑙𝑗 = −𝑗𝑙 = 𝑘, 𝑗𝑘 = −𝑘𝑗 = 𝑙, 𝑘𝑙 = −𝑙𝑘 = 𝑗. The quaternion conjugate and modulus of 𝑟 are defined as 𝑟 = 𝑏 2 + 𝑐 2 + 𝑑 2 + 𝑒 2 . 𝑟 = 𝑏 − (𝑐𝑗 + 𝑑𝑘 + 𝑒𝑙) and The multiplication of quaternions is not a commutative operation, i.e., 𝑟 1 𝑟 2 ≠ 𝑟 2 𝑟 1 for many quaternions 𝑟 2 ≠ 𝑟 1 . 7/8/2020 6

In the definition of the 𝑂 -point quaternion DFT (QDFT), the exponential kernel is used the exponential kernel is used 𝑋 𝜈 = exp(−𝜈2𝜌/𝑂) = cos(2𝜌/𝑂) − 𝜈 sin(2𝜌/𝑂), where 𝜈 is a pure unit quaternion, 𝜈 = 𝑛 1 𝑗 + 𝑛 2 𝑘 + 𝑛 3 𝑙 . For a such number, |𝜈| = 1 and 𝜈 2 = −1 . The 𝑂 -point right-side QDFT 𝑂−1 𝑜𝑞 , 𝑅 𝑞 = ෍ 𝑟 𝑜 𝑋 𝑞 = 0: 𝑂 − 1 . 𝜈 𝑜=0 The fast algorithms for the 𝑂 -points QDFT exist [1] . Because of not commutativity of multiplication in quaternion arithmetic, the main operation of the cyclic convolution is not reduced to the multiplication of the QDFTs, as for the traditional 𝑂 -point DFT. 7/8/2020 7

1-D CONVOLUTION IN QUATERNIOIN ALGEBRA In this section, we show a new technique for calculating the quaternion convolution by the Fourier transforms, which was developed by Grigoryan in 2015. The convolution will be transformed into the frequency domain and a new operation of multiplication of transforms by 4×4 matrices will be performed, instead of point-wise multiplication of the DFTs in the traditional method. The quaternion signal 𝑔 𝑜 , 𝑜 = 0: (𝑂 − 1) , of length 𝑂 𝑔 𝑜 = (𝑔 1 ) 𝑜 , (𝑔 2 ) 𝑜 , (𝑔 3 ) 𝑜 , (𝑔 4 ) 𝑜 = (𝑔 1 ) 𝑜 +𝑗(𝑔 2 ) 𝑜 +𝑘(𝑔 3 ) 𝑜 +𝑙(𝑔 4 ) 𝑜 can be considered in the following matrix representation. 7/8/2020 8

We introduce the following 4 matrices with the multiplications shown in the table 1 0 0 0 0 −1 0 0 0 0 −1 0 0 0 0 −1 0 1 0 0 1 0 0 0 0 0 0 −1 0 0 1 0 𝐹 = 𝐽 = , 𝐾 = 𝐹 = , (4) , , 0 0 1 0 0 0 0 1 1 0 0 0 0 −1 0 0 0 0 0 1 0 0 −1 0 0 1 0 0 1 0 0 0 . The quaternion signal at the point 𝑜 can be written as a matrix, (we use the same notation 𝑔 𝑜 ), 𝑔 − 𝑔 − 𝑔 − 𝑔 1 𝑜 2 𝑜 3 𝑜 4 𝑜 𝑔 𝑔 𝑔 − 𝑔 2 𝑜 1 𝑜 4 𝑜 3 𝑜 𝑔 𝑜 = 𝑔 1 𝑜 𝐹 + 𝑔 2 𝑜 𝐽 + 𝑔 3 𝑜 𝐾 + 𝑔 4 𝑜 𝐿 = , (5) 𝑔 − 𝑔 𝑔 𝑔 3 𝑜 4 𝑜 1 𝑜 2 𝑜 𝑔 𝑔 − 𝑔 𝑔 4 𝑜 3 𝑜 2 𝑜 1 𝑜 7/8/2020 9

Another quaternion sequence, which we call the impulse response characteristic of a linear system, is denoted by ℎ 𝑜 = ((ℎ 1 ) 𝑜 , (ℎ 2 ) 𝑜 , (ℎ 3 ) 𝑜 , (ℎ 4 ) 𝑜 ) and can be presented in matrix form as ℎ 𝑜 = ℎ 1 𝑜 𝐹 + ℎ 2 𝑜 𝐽 + ℎ 3 𝑜 𝐾 + ℎ 4 𝑜 𝐿. (6) We define the circular linear convolution as 𝑂−1 𝑂−1 𝑕 𝑜 = 𝑔 ∗ ℎ 𝑜 = ෍ 𝑔 𝑜−𝑛 ℎ 𝑛 ≜ ෍ 𝑔 (𝑜−𝑛)mod𝑂 ℎ 𝑛 . (7) 𝑛=0 𝑛=0 The following two sums are different: 𝑔 ∗ ℎ 𝑜 ≠ ℎ ∗ 𝑔 𝑜 . 7/8/2020 10

To simplify calculations, we separate all 𝐹, 𝐽, 𝐾 , 𝐿 -components of the signals 𝑔 𝐹 = 𝑔 1 0 , 𝑔 1 1 , 𝑔 1 2 , … , 𝑔 1 𝑂−1 , 𝑔 𝐽 = (𝑔 2 ) 0 , (𝑔 2 ) 1 , (𝑔 2 ) 2 , … , (𝑔 2 ) 𝑂−1 , 𝑔 𝐾 = (𝑔 3 ) 0 , (𝑔 3 ) 1 , (𝑔 3 ) 2 , … , (𝑔 3 ) 𝑂−1 , 𝑔 𝐿 = (𝑔 4 ) 0 , (𝑔 4 ) 1 , (𝑔 4 ) 2 , … , (𝑔 4 ) 𝑂−1 , ℎ 𝐹 = ℎ 1 0 , ℎ 1 1 , ℎ 1 2 , … , ℎ 1 𝑂−1 , ℎ 𝐽 = (ℎ 2 ) 0 , (ℎ 2 ) 1 , (ℎ 2 ) 2 , … , (ℎ 2 ) 𝑂−1 , ℎ 𝐾 = (ℎ 3 ) 0 ,(ℎ 3 ) 1 , (ℎ 3 ) 2 , … , (ℎ 3 ) 𝑂−1 , ℎ 𝐿 = (ℎ 4 ) 0 , (ℎ 4 ) 1 , (ℎ 4 ) 2 , … , (ℎ 4 ) 𝑂−1 . The 𝐹, 𝐽, 𝐾 -, and 𝐿 -components of the quaternion convolution 𝑕 𝑜 are denoted by 𝑕 𝐹 , 𝑕 𝐽 , 𝑕 𝐾 , and 𝑕 𝐿 , respectively. By using the table of multiplications for the basic four quaternion matrices, we can open the convolution 𝑕 𝑜 = 𝑔 1 𝑜 𝐹 + 𝑔 2 𝑜 𝐽 + 𝑔 3 𝑜 𝐾 + 𝑔 4 𝑜 𝐿 ∗ ℎ 1 𝑜 𝐹 + ℎ 2 𝑜 𝐽 + ℎ 3 𝑜 𝐾 + ℎ 4 𝑜 𝐿 and consider the following system of Eqs for the 𝐹, 𝐽, 𝐾 , 𝐿 -components of 𝑕 𝑜 : 𝑕 𝐹 = 𝑔 𝐹 ∗ ℎ 𝐹 − 𝑔 𝐽 ∗ ℎ 𝐽 − 𝑔 𝐾 ∗ ℎ 𝐾 − 𝑔 𝐿 ∗ ℎ 𝐿 , 𝑕 𝐽 = 𝑔 𝐹 ∗ ℎ 𝐽 + 𝑔 𝐽 ∗ ℎ 𝐹 − 𝑔 𝐾 ∗ ℎ 𝐿 + 𝑔 𝐿 ∗ ℎ 𝐾 , ቋ 𝑕 𝐾 = 𝑔 𝐹 ∗ ℎ 𝐾 + 𝑔 𝐽 ∗ ℎ 𝐿 + 𝑔 𝐾 ∗ ℎ 𝐹 − 𝑔 𝐿 ∗ ℎ 𝐽 , 𝑕 𝐿 = 𝑔 𝐹 ∗ ℎ 𝐿 − 𝑔 𝐽 ∗ ℎ 𝐾 + 𝑔 𝐾 ∗ ℎ 𝐽 + 𝑔 𝐿 ∗ ℎ 𝐹 . 7/8/2020 11

For each real component of the signal 𝑔 𝑜 and the impulse response sequence ℎ 𝑜 , we will use the corresponding capital letters for the corresponding DFTs. In the frequency domain, the system of Eq. 8 for frequency-point 𝑞 can be written as 𝐼 𝐹 −𝐼 𝐽 −𝐼 𝐾 −𝐼 𝐿 𝐻 𝐹 𝐺 𝐹 𝐼 𝐽 𝐼 𝐹 −𝐼 𝐿 𝐼 𝐾 𝐻 𝐽 𝐺 𝐽 = . 𝐻 𝐾 𝐺 𝐼 𝐾 𝐼 𝐿 𝐼 𝐹 −𝐼 𝐽 𝐾 𝐻 𝐿 𝐺 𝐼 𝐿 −𝐼 𝐾 𝐼 𝐽 𝐼 𝐹 𝐿 This is a compact form of the equation 𝐻 = 𝑰𝐺 at one frequency-point 𝑞 , which is omitted from the notation. For this point, the matrix 𝑰 is the 4 × 4 matrix 𝐼 𝐹 −𝐼 𝐽 −𝐼 𝐾 −𝐼 𝐿 𝐼 𝐽 𝐼 𝐹 −𝐼 𝐿 𝐼 𝐾 𝑰 = . 𝐼 𝐾 𝐼 𝐿 𝐼 𝐹 −𝐼 𝐽 𝐼 𝐿 −𝐼 𝐾 𝐼 𝐽 𝐼 𝐹 This matrix is orthogonal. 7/8/2020 12

The convoluted quaternion signal 𝑕 𝑜 = 𝑕 1 𝑜 𝐹 + 𝑕 2 𝑜 𝐽 + 𝑕 3 𝑜 𝐾 + 𝑕 4 𝑜 𝐿 can be calculated as 𝑕 1 = 𝑮 −1 𝐻 𝐹 , 𝑕 2 = 𝑮 −1 𝐻 𝐽 , 𝑕 3 = 𝑮 −1 𝐻 𝐾 , 𝑕 4 = 𝑮 −1 𝐻 𝐿 . Here, 𝑮 −1 is the matrix of the inverse 𝑂 -point DFT. Together with the unite matrix 𝐹, we consider the following three new quaternion matrices: 0 −1 0 0 0 0 −1 0 0 0 0 −1 1 0 0 0 0 0 0 1 0 0 −1 0 𝐽 ∗ = 𝐾 ∗ = 𝐿 ∗ = , , . 0 0 0 −1 1 0 0 0 0 1 0 0 0 0 1 0 0 −1 0 0 1 0 0 0 7/8/2020 13