Over the past century, a correlation has been an essential mathematical
technique utilized in engineering sciences, including practically every
signal/image processing field. This paper describes an effective method of
calculating the correlation function of signals and color images in quaternion
algebra. We propose using the quaternions with a commutative multiplication
operation and defining the corresponding correlation function in this
arithmetic. The correlation between quaternion signals and images can be
calculated by multiplying two quaternion DFTs of signals and images. The
complexity of the correlation of color images is three times higher than in
complex algebra.
An Efficient Calculation of Quaternion Correlation
四元相関の効率的な計算
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of Signals and Color Images Artyom M. Grigoryana, Sos S. Agaianb
信号とカラー画像の Artyom M. Grigoryana, Sos S. Agaianb
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aThe University of Texas at San Antonio, bCity University of New York / CSI
aThe University of Texas at San Antonio, bCity University of New York / CSI
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amgrigoryan@utsa.edu , sos.agaian@csi.cuny. edu
amgrigoryan@utsa.edu , sos.agaian@csi.cuny. edu
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ABSTRACT Over the past century, a correlation has been an essential mathematical technique utilized in engineering sciences, including practically every signal/image processing field.
1. INTRODUCTION Correlation is one of the most fundamental concepts used in almost any engineering science.
1. 導入 相関は、ほとんどすべての工学科学で使われている最も基本的な概念の1つである。
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Correlation measures the degree to which two variables are related and expresses in quantitative terms the strength and direction of the relationship between these variables.
相関は2つの変数が関係する度合いを測り、これらの変数間の関係の強さと方向を定量的に表す。
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Practically, correlation, autocorrelation, and phase-correlation become basic tools in many applications, including computer vision, robotics, sport, biomedicine, acoustic, human movement and rehabilitation research, quantum computation, geophysical applications, and image and signal processing applications, to name only a few [1]-[4].
For example, cross-correlation is a key statistical method for getting the degree of relationship/similar ity among two templates/images [1] or analyses in the human movement and rehabilitation sciences [2].
Quaternions, an extension of complex numbers, were launched in the nineteenth century by Gauss.
複素数の拡張である四元数は19世紀にガウスによって開始された。
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However, quaternion information processing has only recently become popular in engineering.
しかし、四元数情報処理は工学で最近普及したばかりである。
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Recently, quaternion algebra has become a very effective tool for color image application and for describing rotations and orientations in 3D spaces [5]-[9].
Unlike commonly used image processing applications, where color images deal with all color channels separately, quaternion algebra allows them to be one unit [10,12].
Also, many signal processing procedures have been expanded to the quaternion domain, including quaternion Fourier transforms, wavelets, Kalman filtering, color image restoration and enhancement, regression, quaternion neural networks, quaternion moments, and least mean square adaptive filtering [10]-[19].
It is natural to extend the correlation procedure to the quaternion domain.
相関手順を四元数領域に拡張することは自然である。
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The challenge here is that traditional quaternion algebra, which is noncommutative, does not provide effective methods of calculating convolution and correlation.
Because of the noncommutativity of the multiplication, the correlation of images has two forms; when multiplying images in the double sum from left and right; the results are different.
To overcome these limitations, we consider the 4-D model of signals and color images in the commutative quaternion algebra, called the (2,2)-model of quaternions, and describe an effective method of calculating the correlation function for signals and images in this paper.
3. We estimate the complexity of the correlation of color images and show that
3. 我々はカラー画像の相関の複雑さを推定し、そのことを示す。
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a. The complexity of calculating the correlation function based on quaternion algebra is two times higher
aだ 四元数代数に基づく相関関数の計算の複雑さは2倍である
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than in the case of complex algebra.
複雑な代数の場合よりも
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b. Calculating the correlation function in the frequency domain using the commutative (2,2)-model can be
bだ 可換(2,2)-モデルを用いた周波数領域の相関関数の計算
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accomplished without quaternion DFTs.
四元数DFT無しで達成。
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c. The quaternion correlation needs only six complex 2-D DFTs plus 4𝑁𝑀 complex multiplications or
cだ 四元相関は6つの複素2次元DFTと4NM複素乗算を必要とする。
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16𝑁𝑀 real multiplications. The rest of the paper is organized in the following way.
16nm実乗算。 残りの論文は以下の方法で整理されている。
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In Section 2, a brief introduction to quaternion numbers is given in the (1,3)- and (2,2)-models.
第2節では、(1,3)-および(2,2)-モデルで四元数について簡単に紹介する。
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In Section 3, the quaternion correlation of signals is described with an example.
第3節では、信号の四元相関を例に示す。
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The quaternion correlation implementation block-diagram is illustrated.
四元相関実装ブロックダイアグラムを例示する。
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Section 4 presents the correlation of color images in the (2,2)-model.
第4節は(2,2)-モデルにおけるカラーイメージの相関を示す。
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The complexity of the quaternion correlation is compared with the known methods in the traditional commutative (1,3)-model of quaternions.
四元数相関の複雑さは、四元数の伝統的な可換(1,3)-モデルにおける既知の方法と比較される。
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Finally, Section 5 presents the conclusion and future work.
最後に、第5節は、結論と今後の作業を示す。
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2. QUATERNION NUMBERS
2. クォータニオン数
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This section briefly introduces quaternion numbers in the noncommutative quaternion algebra and the (2,2)-model [23].
この節では、非可換四元環と (2,2)-モデル[23] における四元数を簡単に紹介する。
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This material and subsequent sections will allow us to work with the concept of the correlation function of quaternion images.
この材料とその後のセクションは、四元数画像の相関関数の概念を扱うことができる。
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To simplify the discussion of this topic, we will first work on the one-dimensional case and then generalize the results to the two-dimensional case.
この話題の議論を単純化するために、まず1次元の場合に取り組み、2次元の場合に結果を一般化する。
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As we know, fast correlation methods are based on the fast Fourier transform since the correlation can be represented as a linear convolution and then reduced the latter to a cyclic convolution.
It should be recalled that the linear convolution and DFT do not have unique definitions in traditional quaternion arithmetic because it is not commutative.
All these obstacles could be removed if we had commutative multiplication.
これらすべての障害は、可換乗算によって取り除くことができる。
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Therefore, we first describe these concepts in commutative quantum arithmetic, which we call the (2,2)-mode, and then introduce the solution to the problem of calculating the quaternion correlation.
It should be noted, that these two matrices are orthogonal.
この2つの行列は直交であることに注意する必要がある。
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2.2. The (1,3)-model of quaternions
2.2. 四元数の(1,3)-モデル
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In this section, we consider the commutative (2,2)-model of quaternions and compare briefly it with the noncommutative (1,3)-model, for which the multiplication rules are defined as in Eq 5.
The multiplications of these quaternions are given in Table 1, which we call 𝑇(1, 𝑒2, 𝑒3, 𝑒4) table.
これらの四元数の乗法はテーブル 1 で与えられ、T(1, e2, e3, e4) テーブルと呼ぶ。
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𝑒2 𝑒3 𝑒4
𝑒2 𝑒3 𝑒4
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𝑒3 𝑒2 𝑒4 −1 𝑒4 −1 −𝑒3 −𝑒2
𝑒3 𝑒2 𝑒4 −1 𝑒4 −1 −𝑒3 −𝑒2
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𝑒4 −𝑒3 −𝑒2 1
𝑒4 −𝑒3 −𝑒2 1
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Table 1. 𝑇(1, 𝑒2, 𝑒3, 𝑒4).
表1。 𝑇(1, 𝑒2, 𝑒3, 𝑒4).
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In matrix form, the product of two quaternions 𝑞 = 𝑞1𝑞2 = [𝑎1, 𝑎2][𝑏1, 𝑏2] can be written as
行列形式では、2つの四元数 q = q1q2 = [a1, a2][b1, b2] の積は書ける。
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𝑞 = [ 𝑎 𝑏 𝑐 𝑑
𝑞 = [ 𝑎 𝑏 𝑐 𝑑
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] = 𝑴𝑞1𝑞2 = [
] = 𝑴𝑞1𝑞2 = [
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𝑎1,1 −𝑎1,2 −𝑎2,1 𝑎1,2 𝑎2,1 −𝑎2,2 𝑎2,2 𝑎2,1
𝑎1,1 −𝑎1,2 −𝑎2,1 𝑎1,2 𝑎2,1 −𝑎2,2 𝑎2,2 𝑎2,1
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𝑎2,2 𝑎1,1 −𝑎2,2 −𝑎2,1 𝑎1,1 −𝑎1,2 𝑎1,2 𝑎1,1
𝑎2,2 𝑎1,1 −𝑎2,2 −𝑎2,1 𝑎1,1 −𝑎1,2 𝑎1,2 𝑎1,1
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] 𝑏1,1 𝑏1,2 𝑏2,1 𝑏2,2]
] 𝑏1,1 𝑏1,2 𝑏2,1 𝑏2,2]
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[ .
[ .
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(9) One can notice the quaternion number 𝑞1 in the first column of this matrix.
(9) この行列の最初の列で四元数 q1 に気づくことができる。
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The matrix 𝑴𝑞1 is not orthogonal.
行列 Mq1 は直交ではない。
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Example 1. Consider two quaternion numbers 𝑞1 = (𝑎1, 𝑎2) = [(1,4), (−1,2)]
例1。 考察 2つの四元数 q1 = (a1, a2) = [(1,4), (−1,2)]
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and 𝑞2 = (𝑏1, 𝑏2) =
and 𝑞2 = (𝑏1, 𝑏2) =
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[(2,5), (3, −1)].
[(2,5), (3, −1)].
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The product 𝑞 = 𝑞1𝑞2 is calculated by the matrix 𝑴 of multiplication
積 q = q1q2 は乗算の行列 M によって計算される
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The inverse matrix 𝑴−1 as 𝑴 = 𝑴𝑞1 = [
逆行列 M−1 は 𝑴 = 𝑴𝑞1 = [
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−1 −2 2 −1
−1 −2 2 −1
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1 −4 4 1 1 −2
1 −4 4 1 1 −2
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2 1 1 −4 4 1
2 1 1 −4 4 1
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], det 𝑴 = 340.
], det m = 340。
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𝑴−1 = 1 170
𝑴−1 = 1 170
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[ −1 −38 −13 −16
[ −1 −38 −13 −16
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16 13 38 16 −13 −38 −1
16 13 38 16 −13 −38 −1
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13 38 −1 −16 −1
13 38 −1 −16 −1
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]. The 𝑞 = 𝑞1𝑞2 is calculated by
]. q = q1q2 が計算される
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𝒒 = [ 1 −4 4
𝒒 = [ 1 −4 4
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1 1 −2 −1 −2 2 −1
1 1 −2 −1 −2 2 −1
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2 1 1 −4 4 1
2 1 1 −4 4 1
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2 ] [ 5 3 −1
2 ] [ 5 3 −1
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] = [ −17 6 −5 10
] = [ −17 6 −5 10
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]. Therefore, 𝑞 = 𝑞1𝑞2 = [(−17,6), (−5,10)].
]. Therefore, 𝑞 = 𝑞1𝑞2 = [(−17,6), (−5,10)].
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We note for comparison that in the (1,3)-model, the product of these two quaternions is equal to [−13,8,5,20] which is the quaternion −13 + (8𝑖 + 5𝑗 + 20𝑘).
(17) The cross-correlation functions of components of both signals are calculated, and then the sum and difference of the mixed correlations are calculated.
(17) 両信号の成分の相互相関関数を算出し、混合相関の和と差を算出する。
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Thus, to calculate the quaternion correlation, four aperiodic convolutions in complex arithmetic can be used, as shown in Fig 1
1 𝐾1 . Note that the maximum correlation of two real parts of these signals, i.e., the 80th and 90th columns of the ‘jetplane’ image, is 0.9688 at point 𝑛 = 0.
1 𝐾1 . これらの信号の2つの実部、すなわち「ジェットプレーン」画像の80番目と90番目のカラムの最大相関は、点 n = 0 において 0.9688 である。
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For columns number 81 and 91 such correlation number is 0.9672, and so on.
柱番号81、91については、相関数は0.9672等である。
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The maximum 0.9696 of quaternion correlation is close to these numbers.
四元数相関の最大 0.9696 はこれらの数に近い。
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This fact shows that the normalization of the quaternion correlation function given in Eq 19 can be considered successful.
この事実は、eq 19で与えられる四元相関関数の正規化が成功することを示している。
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Perhaps there are other better normalization coefficients for signal correlation.
信号相関の正規化係数は他にもあるかもしれない。
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The normalization of the correlation function in Eq 20 results in almost the same result, since the constant 𝐸[𝑣]𝐸[𝑞] = 5.7577 × 107.
(21) Here, the capital letters are used for the 𝑁-point DFTs of the corresponding complex signals.
(21) ここで、大文字は対応する複素信号のN点DFTに使用される。
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In the (2,2)-model, this expression equals to the product [𝑉1,𝑁−𝑝, 𝑉2,𝑁−𝑝][𝐹𝑝, 𝐺𝑝], which is the quaternion DFT, namely the 𝑒2-QDFT (see for detail [23]).
Here, the 𝑁-point 𝑒2-QDFT of a time-reversal signal 𝑣̂𝑛 = 𝑣−𝑛 is calculated by
ここでは、時間反転信号 v = n = v−n の N 点 e2-QDFT を算出する。
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𝑁−1 𝑁−1 𝑁−1
𝑁−1 𝑁−1 𝑁−1
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𝑁−1 ℱ[𝑣̂]𝑝 = ∑ 𝑣−𝑛𝑊𝜇
𝑁−1 ℱ[𝑣̂]𝑝 = ∑ 𝑣−𝑛𝑊𝜇
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𝑛𝑝 = ∑ 𝑣𝑁−𝑛𝑊𝜇
𝑛𝑝 = ∑ 𝑣𝑁−𝑛𝑊𝜇
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−(𝑁−𝑛)𝑝 = ∑ 𝑣𝑛𝑊𝜇
−(𝑁−𝑛)𝑝 = ∑ 𝑣𝑛𝑊𝜇
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−𝑛𝑝 = ∑ 𝑣𝑛𝑊𝜇
−𝑛𝑝 = ∑ 𝑣𝑛𝑊𝜇
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𝑛(𝑁−𝑝) = 𝑉𝑁−𝑝, (23)
𝑛(𝑁−𝑝) = 𝑉𝑁−𝑝, (23)
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where 𝑝 = 0: (𝑁 − 1).
where 𝑝 = 0: (𝑁 − 1).
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The exponential function is calculated for 𝜇 = −𝑒2 by
指数関数は μ = −e2 で計算される
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𝑛=0 𝑛=0 𝑛=0
𝑛=0 𝑛=0 𝑛=0
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𝑛=0 𝑊𝜇 = exp (
𝑛=0 wμ = exp (
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𝜇2𝜋 𝑁 ) = [(cos (
𝜇2𝜋 𝑁 ) = [(cos)
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2𝜋 𝑁 ) , −sin (
2𝜋 𝑁 )、-sin()
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2𝜋 𝑁 )) , (0,0)] = [𝑒−𝑖(
2𝜋 𝑁 )) , (0,0)] = [𝑒−𝑖(
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2𝜋 𝑁 ), 0].
2𝜋 𝑁 ), 0].
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(24) Therefore, the 𝑁-point QDFT of the correlation can be calculated by
(24) したがって、相関のN点QDFTを計算できる。
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𝑁−1 ℱ[𝑟]𝑝 = ∑ 𝑟𝑛𝑊𝜇
𝑁−1 ℱ[𝑟]𝑝 = ∑ 𝑟𝑛𝑊𝜇
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𝑛𝑝 = 𝑉𝑁−𝑝𝑄𝑝,
𝑛𝑝 = 𝑉𝑁−𝑝𝑄𝑝,
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𝑝 = 0: (𝑁 − 1).
𝑝 = 0: (𝑁 − 1).
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(25) 𝑛=0 Here, 𝑉𝑝 and 𝑄𝑝 are coefficients of the 𝑁-point QDFT of signals 𝑣𝑛 and 𝑞𝑛, respectively.
(25) 𝑛=0 ここで、VpとQpはそれぞれ信号vnとqnのN点QDFTの係数である。
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A similar equation holds for the traditional correlation function.
同様の方程式は伝統的な相関関数に当てはまる。
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3.1 Comparison Now, we compare the computation of the quaternion correlation in the noncommutative algebra, or (1,3)-model, which is considered in the form
Here, 𝑁 = 𝐿 + 𝑁 − 1 and the 𝑁-point QDFT in the (1,3)-model is defined by the equation
ここで、(1,3)-モデルの N = L + N − 1 と N-点 QDFT は方程式によって定義される。
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𝑁−1 ℱ𝑗[𝑟]𝑝 = ∑ 𝑟𝑛𝑊𝜇
𝑁−1 ℱ𝑗[𝑟]𝑝 = ∑ 𝑟𝑛𝑊𝜇
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𝑛𝑝 , 𝑝 = 0: (𝑁 − 1)
𝑛𝑝 , 𝑝 = 0: (𝑁 − 1)
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𝑛=0 𝑛𝑝 The exponential coefficients 𝑊𝜇 are defined for the quaternion basis unit 𝜇 = −𝑗.
𝑛=0 𝑛𝑝 指数係数 wμ は四元基底単位 μ = −j に対して定義される。
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As follows from Eq 27, the correlation requires the 𝑁-point QDFT of 𝑞𝑛, two DFTs of complex signals 𝑧1 = (𝑣1,1 + 𝑗𝑣2,1) and 𝑧2 = (𝑣1,2 + 𝑗𝑣2,2), and 2𝑁 operations of quaternion multiplication.
Note that the 𝑁-point QDFT can be calculated by two 𝑁-point DFTs [12].
N点QDFTを2つのN点DFT[12]で計算できることに注意。
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The quaternion transforms ℱ𝑗[𝑧1] and ℱ𝑗[𝑧2] can be considered as the complex 𝑁point DFT each.
四元数変換 Fj[z1] と Fj[z2] はそれぞれ複素 N 点 DFT とみなすことができる。
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Multiplying of quaternions by complex numbers requires 8 real multiplications.
複素数による四元数の乗算には8つの実乗法が必要である。
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Therefore, the minimum number of quaternion multiplications can be estimated as 𝑚𝑄𝐶 = 6𝑚𝐷𝐹𝑇 + 16𝑁.
したがって、四元数最小の乗算は mQC = 6mDFT + 16N と推定できる。
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Here, 𝑚𝐷𝐹𝑇 is the number of real multiplications for the complex DFT.
ここで、mDFT は複素 DFT に対する実乗法の数である。
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The diagram of the calculation of the QDFT of the correlation by Eq 27 is shown in Fig 4.
Eq 27による相関のQDFTの計算図を図4に示す。
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The transform 𝑣𝑛 → (𝑧1,𝑛, 𝑧2,𝑛) is denoted by 𝑇.
変換 vn → (z1,n,z2,n) は T で表される。
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Signal 𝑞𝑛 𝑇 𝑧1,𝑛
信号qn 𝑇 𝑧1,𝑛
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1D QDFT 1D QDFT
1次元QDFT 1次元QDFT
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Signal 𝑣𝑛 = [𝑣1,𝑛, 𝑣2,𝑛]
信号 𝑣𝑛 = [𝑣1,𝑛, 𝑣2,𝑛]
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𝑧2,𝑛 1D QDFT
𝑧2,𝑛 1次元QDFT
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𝑄𝑝 𝑍1,𝑝
𝑄𝑝 𝑍1,𝑝
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𝑍2,𝑝 𝑋𝑁−𝑝
𝑍2,𝑝 𝑋𝑁−𝑝
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𝑋𝑝 𝑅𝑝 1D QDFT of quaternion correlation
𝑋𝑝 𝑅𝑝 四元相関の1次元qdft
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Fig. 4 The operation of the calculation of the 1-D QDFT of the quaternion correlation 𝑅𝑝 by Eq 27.
第4図 四元相関Rpの1次元QDFTのEq27による演算
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In the commutative (2,2)-model, the calculation of the correlation function in the frequency domain is the multiplication of the QDFTs, i.e., 𝑅𝑝 = 𝑉𝑁−𝑝𝑄𝑝.
可換(2,2)-モデルでは、周波数領域における相関関数の計算はQDFTの乗算、すなわち Rp = VN−pQp である。
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Thus, the quaternion correlation requires three QDFTs and number of quaternion
したがって、四元相関は3つのqdftと四元数を必要とする。
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multiplications is estimated as 𝑚𝑄𝐶 = 3𝑚𝑄𝐷𝐹𝑇.
乗算は mQC = 3mQDFT と推定される。
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The inverse 1-D QDFT requires two complex DFTs [12].
逆1次元QDFTは2つの複素DFTを必要とする[12]。
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To estimate the number of complex multiplications, we can use Eq 21.
複素乗算の数を推定するために、Eq 21 を用いることができる。
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The calculations at each frequency-point are performed by the butterfly-type operation shown in Fig 5.
図5に示す蝶型演算により、各周波数点での演算を行う。
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Thus, the calculation of the correlation function in the frequency domain does not require quaternion DFTs, only six complex DFTs plus 4𝑁 complex multiplications, or 16𝑁 real multiplications.
Thus, the estimation of complex multiplications also equals 𝑚𝑄𝐶 = 6𝑚𝐷𝐹𝑇 + 16𝑁.
したがって、複素乗法の推定は mQC = 6mDFT + 16N と等しい。
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Quaternion multiplication operations are not required.
四元数演算は不要である。
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Signal 𝑞𝑛 = [𝑓𝑛, 𝑔𝑛]
信号 𝑞𝑛 = [𝑓𝑛, 𝑔𝑛]
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Signal 𝑣𝑛 = [𝑣1,𝑛, 𝑣2,𝑛]
信号 𝑣𝑛 = [𝑣1,𝑛, 𝑣2,𝑛]
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𝑓𝑛 𝑔𝑛 𝑣1,𝑛
𝑓𝑛 𝑔𝑛 𝑣1,𝑛
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𝑣2,𝑛 1D DFT
𝑣2,𝑛 1次元DFT
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1D DFT 1D DFT
1次元DFT 1次元DFT
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1D DFT 𝐹𝑝 𝐺𝑝
1次元DFT 𝐹𝑝 𝐺𝑝
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𝑉1,𝑝 𝑉2,𝑝
𝑉1,𝑝 𝑉2,𝑝
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𝑋𝑝 𝑅1,𝑝 −1
𝑋𝑝 𝑅1,𝑝 −1
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𝑋𝑁−𝑝 𝑅𝑝 𝑅2,𝑝
𝑋𝑁−𝑝 𝑅𝑝 𝑅2,𝑝
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1D QDFT of quaternion correlation
四元相関の1次元qdft
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Fig. 5 The block-diagram of calculation of the 1-D QDFT of the quaternion correlation 𝑅𝑝(𝑣, 𝑞).
第5図 四元相関 rp(v, q) の 1 次元 qdft の計算のブロックダイアグラム
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The same operation of a butterfly is shown in Fig 6 in a more compact form.
蝶の操作は、よりコンパクトな形で図6に示される。
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1st quaternion signal QDFT
第1四元信号QDFT
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𝑄𝑝 𝑉1,𝑁−𝑝
𝑄𝑝 𝑉1,𝑁−𝑝
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𝐹𝑝 𝐺𝑝 𝑅1,𝑝
𝐹𝑝 𝐺𝑝 𝑅1,𝑝
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𝑅2,𝑝 ∓𝑉2,𝑁−𝑝
𝑅2,𝑝 ∓𝑉2,𝑁−𝑝
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𝑉1,𝑁−𝑝 nd
𝑉1,𝑁−𝑝 nd
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2 quaternion signal QDFT
2 四元数信号QDFT
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𝑅𝑝 QDFT of the quaternion correlation
𝑅𝑝 四元数相関のQDFT
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Fig. 6 The operation of the 2×2 butterfly for the quaternion correlation 𝑅𝑝(𝑣, 𝑞).
第6図 四元相関 rp(v, q) に対する 2×2 蝶の操作
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The main difference between these two methods is the following:
これら2つの方法の主な違いは次のとおりである。
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• In the (2,2)-model, the quaternion correlation s 𝑟𝑛 of two signals 𝑣𝑛 and 𝑞𝑛 is defined by four traditional cross correlation functions of their components.
• 2,2)-モデルでは、2つの信号 vn と qn の四元相関 s rn はそれらの成分の4つの伝統的な交叉相関関数によって定義される。
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No quaternion operations are required. • Also, in the frequency domain, the correlation of two signals is the operation of multiplication of their QDFTs,
Thus, two autocorrelations of 𝑓 and 𝑔 are required, plus the cross-correlation of these components.
したがって、f と g の2つの自己相関とこれらの成分の相互相関が必要である。
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The block-diagram similar to one shown in Fig 5 can be used to calculate this 2-D quaternion correlation function.
図5に示すブロックダイアグラムは、この2次元四元数相関関数を計算するのに使うことができる。
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It follows from Eq 30, that the correlation 𝑟𝑛,𝑚 can be calculated by four complex 𝑁 × 𝑀-point 2-D DFTs of components of the quaternion images, 𝑓𝑛,𝑚, 𝑔𝑛,𝑚, 𝑣1;𝑛,𝑚, and 𝑣2;𝑛,𝑚.
eq30から、相関 rn,m は四元画像の成分 fn,m,gn,m,v1;n,m,v2;n,m の4つの複素 n × m 点 2-d dft によって計算できる。 訳抜け防止モード: eq 30 から、相関 rn,m は四元画像の成分の4つの複素 n × m-点 2-d dft によって計算できる。 𝑓𝑛,𝑚 , 𝑔𝑛,𝑚 , 𝑣1;𝑛,𝑚 , and 𝑣2;𝑛,𝑚.
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Also, two inverse complex 𝑁 × 𝑀-point 2-D DFTs.
また、2つの逆複素 N × M-点 2-D DFT も成り立つ。
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Therefore, the total number of complex operations of real multiplication for quaternion convolution in the (2,2)-model can be estimated as 𝑚2𝐷𝑄𝐶 = 6𝑚2𝐷𝐷𝐹𝑇 + 4(4𝑀𝑁).
The authors suggest calculating this operation by using the type-3 QDFT.
著者らは,タイプ3QDFTを用いて,この操作を計算することを提案する。
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Type-3 QDFT is the right-side transform
Type-3 QDFTは右辺変換である
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𝑀−1 𝑁−1 𝑄𝑝,𝑠 = ∑ ∑ 𝑞𝑛,𝑚(𝑣, 𝑞)𝑒−𝜇2𝜋[𝑠𝑚/𝑀+𝑛𝑝/𝑁]
𝑀−1 𝑁−1 𝑄𝑝,𝑠 = ∑ ∑ 𝑞𝑛,𝑚(𝑣, 𝑞)𝑒−𝜇2𝜋[𝑠𝑚/𝑀+𝑛𝑝/𝑁]
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, 𝑝 = 0: (𝑁 − 1), 𝑠 = 0: (𝑀 − 1).
, 𝑝 = 0: (𝑁 − 1), 𝑠 = 0: (𝑀 − 1).
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𝑚=0 𝑛=0 Here, 𝜇 is a pure quaternion unit.
𝑚=0 𝑛=0 ここで μ は純粋四元数単位である。
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This transform can be calculated by two complex 2-D DFT, when using the method of symplectic decomposition [26] of the imaginary part 𝑞𝑛,𝑚 of the quaternion image 𝑞𝑛,𝑚.
For that, the image is presented in the new basis {1, 𝜇1, 𝜇2, 𝜇3}, where quaternion units 𝜇1 and 𝜇2 are orthogonal to each other, and 𝜇3 = 𝜇1𝜇2 (see detail in [12]).
Table 1. Comparison of two models of quantum algebra.
表1。 量子代数の2つのモデルの比較
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5. CONCLUSION An effective calculation method of the correlation function in the commutative (2,2)-model of quaternions is described for signals and color images.
Conflicts of Interest: The author(s) declare that there are no conflicts of interest regarding the publication of this paper.
conflicts of interest: the author(s)は、本論文の出版に関して利害関係の衝突はないと宣言する。
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