Abstract

Using a previous technique to rotate two-dimensional images on an N×N square pixellated screen unitarily, we can rotate three-dimensional pixellated cubes of side N, and also generally D-dimensional Cartesian data arrays, also unitarily. Although the number of operations inevitably grows as N2D (because each rotated pixel depends on all others), and Gibbs-like oscillations are inevitable, the result is a strictly unitary and real transformation (thus orthogonal) that is invertible (thus no loss of information) and could be used as a standard.

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References

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  1. See, for example, the Image Processing tutorial in WOLFRAM MATHEMATICA, www.wolfram.com/mathematica .
  2. S.-C. Pei and C.-L. Liu, “Discrete spherical harmonic oscillator transforms on the Cartesian grids using transformation coefficients,” IEEE Trans. Signal Process. 61, 1149–1164 (2013).
    [CrossRef]
  3. M. Moshinsky, The Harmonic Oscillator in Modern Physics: From Atoms to Quarks (Gordon & Breach, 1969).
  4. V. D. Efros, “Some properties of the Moshinsky coefficients,” Nucl. Phys. A 202, 180–190 (1973).
    [CrossRef]
  5. N. M. Atakishiyev and K. B. Wolf, “Fractional Fourier–Kravchuk transform,” J. Opt. Soc. Am. A 14, 1467–1477 (1997).
    [CrossRef]
  6. L. Barker, Ç. Çandan, T. Hakioğlu, A. Kutay, and H. M. Ozaktas, “The discrete harmonic oscillator, Harper’s equation, and the discrete fractional Fourier transform,” J. Phys. A 33, 2209–2222 (2000).
    [CrossRef]
  7. N. M. Atakishiyev, G. S. Pogosyan, L. E. Vicent, and K. B. Wolf, “Finite two-dimensional oscillator. I: the Cartesian model,” J. Phys. A 34, 9381–9398 (2001).
    [CrossRef]
  8. L. E. Vicent and K. B. Wolf, “Analysis of digital images into energy-angular momentum modes,” J. Opt. Soc. Am. A 28, 808–814 (2011).
    [CrossRef]
  9. R. Simon and K. B. Wolf, “Fractional Fourier transforms in two dimensions,” J. Opt. Soc. Am. A 17, 2368–2381 (2000).
    [CrossRef]
  10. K. B. Wolf and T. Alieva, “Rotation and gyration of finite two-dimensional modes,” J. Opt. Soc. Am. A 25, 365–370 (2008).
    [CrossRef]
  11. L. C. Biedenharn and J. D. Louck, Angular Momentum in Quantum Physics, Theory and Application, G.-C. Rota, ed., Vol. 8 of Encyclopedia of Mathematics and Its Applications (Addison-Wesley, 1981).
  12. K. B. Wolf, “A recursive method for the calculation of the SOn, SOn,1, and ISOn representation matrices,” J. Math. Phys. 12, 197–206 (1971).
    [CrossRef]
  13. K. B. Wolf, “Linear transformations and aberrations in continuous and in finite systems,” J. Phys. A 41, 304026 (2008).
    [CrossRef]

2013

S.-C. Pei and C.-L. Liu, “Discrete spherical harmonic oscillator transforms on the Cartesian grids using transformation coefficients,” IEEE Trans. Signal Process. 61, 1149–1164 (2013).
[CrossRef]

2011

2008

K. B. Wolf and T. Alieva, “Rotation and gyration of finite two-dimensional modes,” J. Opt. Soc. Am. A 25, 365–370 (2008).
[CrossRef]

K. B. Wolf, “Linear transformations and aberrations in continuous and in finite systems,” J. Phys. A 41, 304026 (2008).
[CrossRef]

2001

N. M. Atakishiyev, G. S. Pogosyan, L. E. Vicent, and K. B. Wolf, “Finite two-dimensional oscillator. I: the Cartesian model,” J. Phys. A 34, 9381–9398 (2001).
[CrossRef]

2000

L. Barker, Ç. Çandan, T. Hakioğlu, A. Kutay, and H. M. Ozaktas, “The discrete harmonic oscillator, Harper’s equation, and the discrete fractional Fourier transform,” J. Phys. A 33, 2209–2222 (2000).
[CrossRef]

R. Simon and K. B. Wolf, “Fractional Fourier transforms in two dimensions,” J. Opt. Soc. Am. A 17, 2368–2381 (2000).
[CrossRef]

1997

1973

V. D. Efros, “Some properties of the Moshinsky coefficients,” Nucl. Phys. A 202, 180–190 (1973).
[CrossRef]

1971

K. B. Wolf, “A recursive method for the calculation of the SOn, SOn,1, and ISOn representation matrices,” J. Math. Phys. 12, 197–206 (1971).
[CrossRef]

Alieva, T.

Atakishiyev, N. M.

N. M. Atakishiyev, G. S. Pogosyan, L. E. Vicent, and K. B. Wolf, “Finite two-dimensional oscillator. I: the Cartesian model,” J. Phys. A 34, 9381–9398 (2001).
[CrossRef]

N. M. Atakishiyev and K. B. Wolf, “Fractional Fourier–Kravchuk transform,” J. Opt. Soc. Am. A 14, 1467–1477 (1997).
[CrossRef]

Barker, L.

L. Barker, Ç. Çandan, T. Hakioğlu, A. Kutay, and H. M. Ozaktas, “The discrete harmonic oscillator, Harper’s equation, and the discrete fractional Fourier transform,” J. Phys. A 33, 2209–2222 (2000).
[CrossRef]

Biedenharn, L. C.

L. C. Biedenharn and J. D. Louck, Angular Momentum in Quantum Physics, Theory and Application, G.-C. Rota, ed., Vol. 8 of Encyclopedia of Mathematics and Its Applications (Addison-Wesley, 1981).

Çandan, Ç.

L. Barker, Ç. Çandan, T. Hakioğlu, A. Kutay, and H. M. Ozaktas, “The discrete harmonic oscillator, Harper’s equation, and the discrete fractional Fourier transform,” J. Phys. A 33, 2209–2222 (2000).
[CrossRef]

Efros, V. D.

V. D. Efros, “Some properties of the Moshinsky coefficients,” Nucl. Phys. A 202, 180–190 (1973).
[CrossRef]

Hakioglu, T.

L. Barker, Ç. Çandan, T. Hakioğlu, A. Kutay, and H. M. Ozaktas, “The discrete harmonic oscillator, Harper’s equation, and the discrete fractional Fourier transform,” J. Phys. A 33, 2209–2222 (2000).
[CrossRef]

Kutay, A.

L. Barker, Ç. Çandan, T. Hakioğlu, A. Kutay, and H. M. Ozaktas, “The discrete harmonic oscillator, Harper’s equation, and the discrete fractional Fourier transform,” J. Phys. A 33, 2209–2222 (2000).
[CrossRef]

Liu, C.-L.

S.-C. Pei and C.-L. Liu, “Discrete spherical harmonic oscillator transforms on the Cartesian grids using transformation coefficients,” IEEE Trans. Signal Process. 61, 1149–1164 (2013).
[CrossRef]

Louck, J. D.

L. C. Biedenharn and J. D. Louck, Angular Momentum in Quantum Physics, Theory and Application, G.-C. Rota, ed., Vol. 8 of Encyclopedia of Mathematics and Its Applications (Addison-Wesley, 1981).

Moshinsky, M.

M. Moshinsky, The Harmonic Oscillator in Modern Physics: From Atoms to Quarks (Gordon & Breach, 1969).

Ozaktas, H. M.

L. Barker, Ç. Çandan, T. Hakioğlu, A. Kutay, and H. M. Ozaktas, “The discrete harmonic oscillator, Harper’s equation, and the discrete fractional Fourier transform,” J. Phys. A 33, 2209–2222 (2000).
[CrossRef]

Pei, S.-C.

S.-C. Pei and C.-L. Liu, “Discrete spherical harmonic oscillator transforms on the Cartesian grids using transformation coefficients,” IEEE Trans. Signal Process. 61, 1149–1164 (2013).
[CrossRef]

Pogosyan, G. S.

N. M. Atakishiyev, G. S. Pogosyan, L. E. Vicent, and K. B. Wolf, “Finite two-dimensional oscillator. I: the Cartesian model,” J. Phys. A 34, 9381–9398 (2001).
[CrossRef]

Simon, R.

Vicent, L. E.

L. E. Vicent and K. B. Wolf, “Analysis of digital images into energy-angular momentum modes,” J. Opt. Soc. Am. A 28, 808–814 (2011).
[CrossRef]

N. M. Atakishiyev, G. S. Pogosyan, L. E. Vicent, and K. B. Wolf, “Finite two-dimensional oscillator. I: the Cartesian model,” J. Phys. A 34, 9381–9398 (2001).
[CrossRef]

Wolf, K. B.

L. E. Vicent and K. B. Wolf, “Analysis of digital images into energy-angular momentum modes,” J. Opt. Soc. Am. A 28, 808–814 (2011).
[CrossRef]

K. B. Wolf and T. Alieva, “Rotation and gyration of finite two-dimensional modes,” J. Opt. Soc. Am. A 25, 365–370 (2008).
[CrossRef]

K. B. Wolf, “Linear transformations and aberrations in continuous and in finite systems,” J. Phys. A 41, 304026 (2008).
[CrossRef]

N. M. Atakishiyev, G. S. Pogosyan, L. E. Vicent, and K. B. Wolf, “Finite two-dimensional oscillator. I: the Cartesian model,” J. Phys. A 34, 9381–9398 (2001).
[CrossRef]

R. Simon and K. B. Wolf, “Fractional Fourier transforms in two dimensions,” J. Opt. Soc. Am. A 17, 2368–2381 (2000).
[CrossRef]

N. M. Atakishiyev and K. B. Wolf, “Fractional Fourier–Kravchuk transform,” J. Opt. Soc. Am. A 14, 1467–1477 (1997).
[CrossRef]

K. B. Wolf, “A recursive method for the calculation of the SOn, SOn,1, and ISOn representation matrices,” J. Math. Phys. 12, 197–206 (1971).
[CrossRef]

IEEE Trans. Signal Process.

S.-C. Pei and C.-L. Liu, “Discrete spherical harmonic oscillator transforms on the Cartesian grids using transformation coefficients,” IEEE Trans. Signal Process. 61, 1149–1164 (2013).
[CrossRef]

J. Math. Phys.

K. B. Wolf, “A recursive method for the calculation of the SOn, SOn,1, and ISOn representation matrices,” J. Math. Phys. 12, 197–206 (1971).
[CrossRef]

J. Opt. Soc. Am. A

J. Phys. A

K. B. Wolf, “Linear transformations and aberrations in continuous and in finite systems,” J. Phys. A 41, 304026 (2008).
[CrossRef]

L. Barker, Ç. Çandan, T. Hakioğlu, A. Kutay, and H. M. Ozaktas, “The discrete harmonic oscillator, Harper’s equation, and the discrete fractional Fourier transform,” J. Phys. A 33, 2209–2222 (2000).
[CrossRef]

N. M. Atakishiyev, G. S. Pogosyan, L. E. Vicent, and K. B. Wolf, “Finite two-dimensional oscillator. I: the Cartesian model,” J. Phys. A 34, 9381–9398 (2001).
[CrossRef]

Nucl. Phys. A

V. D. Efros, “Some properties of the Moshinsky coefficients,” Nucl. Phys. A 202, 180–190 (1973).
[CrossRef]

Other

See, for example, the Image Processing tutorial in WOLFRAM MATHEMATICA, www.wolfram.com/mathematica .

M. Moshinsky, The Harmonic Oscillator in Modern Physics: From Atoms to Quarks (Gordon & Breach, 1969).

L. C. Biedenharn and J. D. Louck, Angular Momentum in Quantum Physics, Theory and Application, G.-C. Rota, ed., Vol. 8 of Encyclopedia of Mathematics and Its Applications (Addison-Wesley, 1981).

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Figures (3)

Fig. 1.
Fig. 1.

Left: Images of the 2D Cartesian Kravchuk functions Ψnx,ny(mx,my) in Eq. (8), of points mx, my|jj for j=8 on 17×17 pixellated screens, arranged by modes (nx,ny). Right: The Laguerre–Kravchuk functions Λn,μj(mx,my) in Eq. (11) on the same pixellated screens. At its center (μ=0) the functions are real; to the right (μ>0) are the real parts of Λn,μj, and to the left (μ<0) are the imaginary parts of Λn,μj, which can be seen to be off by a phase of (1/4)π.

Fig. 2.
Fig. 2.

Rotation of a pixellated letter “R” in steps of 6°, from 0° to 90°, for j=8; i.e., N=17. The N2×N2 matrix kernel Rmx,my;mx,my(j)(α) was computed once for α=6° and applied successively to each image to obtain the next one. At right is the gray-tone range for the pixels in the figure.

Fig. 3.
Fig. 3.

Rotation of a pixellated 1-and-0 three-dimensional image in steps of 8° from 0° to 120°, around the axis n^=(1,1,1)/3, for j=8; i.e., N=17. The color bar at right indicates the density range of the partially transparent pixels in the figure. The “TL” figure is not centered in the cube.

Equations (28)

Equations on this page are rendered with MathJax. Learn more.

positionm:J1|j,m1=m|j,m1,m|jj,moden:J3|j,n3=(nj)|j,n3,n|02j,
[J3,J1]=iJ2,[J3,J2]=iJ1,[J1,J2]=iJ3,
Ψn(m)j,m|j,n31=j,j+m|e+i12πJ2|j,n33=dnj,mj(12π)
=(1)nΨn(m)=(1)mΨ2jn(m)
=(1)n2j(2jn)(2jj+m)Kn(j+m;12;2j),
exp(iβJ2)|j,n3=n=02jdnj,njj(β)|j,n3,
dμ,μj(β)=j,j+μ|eiβJ2|j,j+μ33=dμ,μj(β)=(j+μ)!(jμ)!(j+μ)!(jμ)!×k(cos12β)2j2k+μμ(sin12β)2kμ+μk!(j+μk)!(jμk)!(μμ+k)!,
Ψnx,ny(mx,my):=j;mx,my|j;nx,ny13=Ψnx(mx)Ψny(my),
{|j;0,n3,|j;1,n13,,|j;n,03},n|02j,
{|j;n2j,2j3,|j;n2j+1,2j13,,|j;2j,n2j3},n|2j4j,
Λn,μj(mx,my):=j;mx,my|j;n,μ)1=nx+ny=nei(nxny)/4dμ/2,(nxny)/2n/2(12π)×Ψnx,ny(mx,my),
μ|ννforν={n,when0n2j,4jn,when2jn4j.
Λn,μj(mx,my)=Λn,μj(mx,my)*=(1)nΛn,μj(mx,my)=(1)mx+myΛ4jn,μj(mx,my),
R(α):Λn,μj(mx,my):=j;mx,my|R(α)|j;n,μ)1=eiμαΛn,μj(mx,my).
R(α):fmx,my=mx,my=jjfmx,myRmx,my;mx,my(j)(α),
Rmx,my;mx,my(j)(α):=n,μΛn,μj(mx,my)eiμαΛn,μj(mx,my)*.
mx,myRmx,my;mx,my(j)(α1)Rmx,my;mx,my(j)(α2)=Rmx,my;mx,my(j)(α1+α2).
R3(α,β,γ)=Rx,y(α)Rz,x(β)Rx,y(γ),
R3(α,β,γ):fmx,my,my=mx,my,mz=jjfmx,my,mz×Rmx,my,mz;mx,my,mz(j)(α,β,γ),
Rmx,my,mz;mx,my,mz(j)(α,β,γ):=mx,mx,my=jjRmx,my;mx,my(j)(α)×Rmz,mx;mz,mx(j)(β)Rmx,my;mx,my(j)(γ),
cos212β=cos212ψ+nz2sin212ψ,
cos12(γ+α)=cos12ψ/cos12β,
tan12(γα)=ny/nx.
SO(D)=SO(D1)SD1,
{θ·,·(d)}:={θ1,2(d),θ2,3(d),,θd,d+1(d)},0θ1,2(d)<2π,0θi,i+1(d)π,2id.
{θ·,··}(d):={θ1,2(1),{θ·,·(2)},,{θ·,·(d)}}.
Rd+1({θ·,··}(d))=Rd({θ·,··}(d1))Sd({θ·,·(d)}),
Sd({θ·,·(d)}):=R(θd,d+1(d))R(θd1,d(d))R(θ2,3(d))R(θ1,2(d)).

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