JEOS RP ISSN03 | Page 329

322 J. Eur. Opt. Society-Rapid Publ. 22, 31( 2026)
Appendix B: Forward and inverse transformation matrices, up to order 6
Table B1. The forward transformation matrix T, up to order 6. n 0 1 2 3 4 5 6 i = m 0 −1 1 −2 0 2 −3 −1 1 3 −4 −2 0 2 4 −5 −3 −1 1 3 5 −6 −4 −2 0 2 4 6
0
0
1
0
0
0
−1
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
−1
0
0
0
1
0
0
0
1
0
0
0
0
0
−2
0
0
0
0
0
0
0
0
0
3
0
0
0
0
0
0
0
0
0
1
0
1
0
0
0
0
0
−2
0
0
0
0
0
0
0
0
0
3
0
0
0
0
0
0
0
0
0
0
2
0
0
0
0
0
2
1
0
0
0
0
0
0
−6
−3
0
0
0
0
0
0
0
0
0
0
12
6
0
0
1
0
0
0
2
0
0
0
0
0
0
0
−6
0
0
0
0
0
0
0
0
0
0
0
12
0
0
0
0
2
0
0
0
0
2
−1
0
0
0
0
0
0
−6
3
0
0
0
0
0
0
0
0
0
0
12
−6
0
0
3
0
0
0
0
0
0
0
0
0
3
1
0
0
0
0
0
0
0
0
−12
−4
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
3
3
0
0
0
0
0
0
0
0
−12
−12
0
0
0
0
0
0
0
0
0
0
2
0
0
0
0
0
0
0
0
3
−3
0
0
0
0
0
0
0
0
−12
12
0
0
0
0
0
0
0
0
3
0
0
0
0
0
0
−1
3
0
0
0
0
0
0
0
0
4
−12
0
0
0
0
0
0
0
0
0
0
4
0
0
0
0
0
0
0
0
0
0
0
0
0
6
4
1
0
0
0
0
0
0
0
0
0
−30
−20
−5
0
1
0
0
0
0
0
0
0
0
0
0
4
8
0
0
0
0
0
0
0
0
0
0
−20
−40
0
0
0
0
2
0
0
0
0
0
0
0
0
0
0
0
0
12
0
−6
0
0
0
0
0
0
0
0
0
−60
0
30
0
3
0
0
0
0
0
0
0
0
0
0
−4
8
0
0
0
0
0
0
0
0
0
0
20
−40
0
0
0
0
4
0
0
0
0
0
0
0
0
0
0
0
0
6
−4
1
0
0
0
0
0
0
0
0
0
−30
20
−5
0
5
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
10
5
1
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
5
15
10
0
0
0
0
0
0
0
0
0
0
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
20
−10
−10
0
0
0
0
0
0
0
3
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
−10
10
20
0
0
0
0
0
0
0
0
0
0
4
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
10
−15
5
0
0
0
0
0
0
0
5
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
−5
10
0
0
0
0
0
0
0
0
0
0
6
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
20
15
6
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
6
24
30
0
0
0
0
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
60
15
−30
−15
3
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
−20
0
60
0
0
0
0
4
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
60
−15
−30
15
5
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
6
−24
30
0
0
0
0
6
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
20
−15
6
−1