Abstract

The two-dimensional nonseparable linear canonical transform (2D NSLCT), which is a generalization of the fractional Fourier transform and the linear canonical transform, is useful for analyzing optical systems. However, since the 2D NSLCT has 16 parameters and is very complicated, it is a great challenge to implement it in an efficient way. In this paper, we improved the previous work and propose an efficient way to implement the 2D NSLCT. The proposed algorithm can minimize the numerical error arising from interpolation operations and requires fewer chirp multiplications. The simulation results show that, compared with the existing algorithm, the proposed algorithms can implement the 2D NSLCT more accurately and the required computation time is also less.

© 2012 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. H. M. Ozaktas, M. A. Kutay, and Z. Zalevsky, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2000).
  2. H. M. Ozaktas and D. Mendlovic, “Fractional Fourier optics,” J. Opt. Soc. Am. A 12, 743–751 (1995).
    [CrossRef]
  3. K. B. Wolf, “Canonical Transforms,” in Integral Transforms in Science and Engineering (Plenum, 1979), pp. 381–416.
  4. L. M. Bernardo, “ABCD matrix formalism of fractional Fourier optics,” Opt. Eng. 35, 732–740 (1996).
    [CrossRef]
  5. Z. Zalevsky, D. Mendlovic, and A. W. Lohmann, “The ABCD—Bessel transformation,” Opt. Commun. 147, 39–41 (1998).
    [CrossRef]
  6. S. Abe and J. T. Sheridan, “Optical operations on wave functions as the Abelian subgroups of the special affine Fourier transformation,” Opt. Lett. 19, 1801–1803 (1994).
    [CrossRef]
  7. G. B. Folland, Harmonic Analysis in Phase Space (Princeton University, 1989).
  8. M. J. Bastiaans and T. Alieva, “Classification of lossless first order optical systems and the linear canonical transformation,” J. Opt. Soc. Am. A 24, 1053–1062 (2007).
    [CrossRef]
  9. A. Koç, H. M. Ozaktas, and L. Hesselink, “Fast and accurate computation of two-dimensional non-separable quadratic-phase integrals,” J. Opt. Soc. Am. A 27, 1288–1302 (2010).
    [CrossRef]
  10. J. J. Ding and S. C. Pei, “Eigenfunctions and self-imaging phenomena of the two dimensional nonseparable linear canonical transform,” J. Opt. Soc. Am. A 28, 82–95 (2011).
    [CrossRef]
  11. H. M. Ozaktas and O. Arikan, “Digital computation of the fractional Fourier transform,” IEEE Trans. Signal Process. 44, 2141–2150, (1996).
    [CrossRef]
  12. H. M. Ozaktas, A. Koç, I. Sari, and M. A. Kutay, “Efficient computation of quadratic-phase integrals in optics,” Opt. Lett. 31, 35–37 (2006).
    [CrossRef]
  13. A. Koc, H. M. Ozaktas, C. Candan, and M. A. Kutay, “Digital computation of linear canonical transforms,” IEEE Trans. Signal Process. 56, 2383–2394 (2008).
    [CrossRef]
  14. D. P. Kelly, B. M. Hennelly, W. T. Rhodes, and J. T. Sheridan, “Analytical and numerical analysis of linear optical systems,” Opt. Eng. 45, 088201 (2006).
    [CrossRef]
  15. J. J. Healy and J. T. Sheridan, “Reevaluation of the direct method of calculating Fresnel and other linear canonical transforms,” Opt. Lett. 35, 947–949 (2010).
    [CrossRef]
  16. J. J. Healy and J. T. Sheridan, “Fast linear canonical transform,” J. Opt. Soc. Am. A 27, 21–30 (2010).
    [CrossRef]
  17. J. J. Healy and J. T. Sheridan, “Space-bandwidth ratio as a means of choosing between Fresnel and other linear canonical transform algorithms,” J. Opt. Soc. Am. A 28, 786–790 (2011).
    [CrossRef]
  18. X. Deng, B. Bihari, J. Gan, F. Zhou, and R. T. Chen, “Fast algorithm for chirp transforms with zooming–in ability and its applications,” J. Opt. Soc. Am. A 17, 762–771 (2000).
    [CrossRef]
  19. A. Koç, H. M. Ozaktas, and L. Hesselink, “Fast and accurate computation of two-dimensional non-separable quadratic-phase integrals,” J. Opt. Soc. Am. A 27, 1288–1302 (2010).
    [CrossRef]
  20. T. Alieva and M. J. Bastiaans, “Alternative representation of the linear canonical integral transform,” Opt. Lett. 30, 3302–3304 (2005).
    [CrossRef]
  21. K. B. Wolf, Geometric Optics on Phase Space (Springer, 2004).
  22. M. R. Spiegel, Mathematical Handbook of Formulas and Tables (McGraw-Hill, 1990).

2011 (2)

2010 (4)

2008 (1)

A. Koc, H. M. Ozaktas, C. Candan, and M. A. Kutay, “Digital computation of linear canonical transforms,” IEEE Trans. Signal Process. 56, 2383–2394 (2008).
[CrossRef]

2007 (1)

2006 (2)

D. P. Kelly, B. M. Hennelly, W. T. Rhodes, and J. T. Sheridan, “Analytical and numerical analysis of linear optical systems,” Opt. Eng. 45, 088201 (2006).
[CrossRef]

H. M. Ozaktas, A. Koç, I. Sari, and M. A. Kutay, “Efficient computation of quadratic-phase integrals in optics,” Opt. Lett. 31, 35–37 (2006).
[CrossRef]

2005 (1)

2000 (1)

1998 (1)

Z. Zalevsky, D. Mendlovic, and A. W. Lohmann, “The ABCD—Bessel transformation,” Opt. Commun. 147, 39–41 (1998).
[CrossRef]

1996 (2)

L. M. Bernardo, “ABCD matrix formalism of fractional Fourier optics,” Opt. Eng. 35, 732–740 (1996).
[CrossRef]

H. M. Ozaktas and O. Arikan, “Digital computation of the fractional Fourier transform,” IEEE Trans. Signal Process. 44, 2141–2150, (1996).
[CrossRef]

1995 (1)

1994 (1)

Abe, S.

Alieva, T.

Arikan, O.

H. M. Ozaktas and O. Arikan, “Digital computation of the fractional Fourier transform,” IEEE Trans. Signal Process. 44, 2141–2150, (1996).
[CrossRef]

Bastiaans, M. J.

Bernardo, L. M.

L. M. Bernardo, “ABCD matrix formalism of fractional Fourier optics,” Opt. Eng. 35, 732–740 (1996).
[CrossRef]

Bihari, B.

Candan, C.

A. Koc, H. M. Ozaktas, C. Candan, and M. A. Kutay, “Digital computation of linear canonical transforms,” IEEE Trans. Signal Process. 56, 2383–2394 (2008).
[CrossRef]

Chen, R. T.

Deng, X.

Ding, J. J.

Folland, G. B.

G. B. Folland, Harmonic Analysis in Phase Space (Princeton University, 1989).

Gan, J.

Healy, J. J.

Hennelly, B. M.

D. P. Kelly, B. M. Hennelly, W. T. Rhodes, and J. T. Sheridan, “Analytical and numerical analysis of linear optical systems,” Opt. Eng. 45, 088201 (2006).
[CrossRef]

Hesselink, L.

Kelly, D. P.

D. P. Kelly, B. M. Hennelly, W. T. Rhodes, and J. T. Sheridan, “Analytical and numerical analysis of linear optical systems,” Opt. Eng. 45, 088201 (2006).
[CrossRef]

Koc, A.

A. Koc, H. M. Ozaktas, C. Candan, and M. A. Kutay, “Digital computation of linear canonical transforms,” IEEE Trans. Signal Process. 56, 2383–2394 (2008).
[CrossRef]

Koç, A.

Kutay, M. A.

A. Koc, H. M. Ozaktas, C. Candan, and M. A. Kutay, “Digital computation of linear canonical transforms,” IEEE Trans. Signal Process. 56, 2383–2394 (2008).
[CrossRef]

H. M. Ozaktas, A. Koç, I. Sari, and M. A. Kutay, “Efficient computation of quadratic-phase integrals in optics,” Opt. Lett. 31, 35–37 (2006).
[CrossRef]

H. M. Ozaktas, M. A. Kutay, and Z. Zalevsky, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2000).

Lohmann, A. W.

Z. Zalevsky, D. Mendlovic, and A. W. Lohmann, “The ABCD—Bessel transformation,” Opt. Commun. 147, 39–41 (1998).
[CrossRef]

Mendlovic, D.

Z. Zalevsky, D. Mendlovic, and A. W. Lohmann, “The ABCD—Bessel transformation,” Opt. Commun. 147, 39–41 (1998).
[CrossRef]

H. M. Ozaktas and D. Mendlovic, “Fractional Fourier optics,” J. Opt. Soc. Am. A 12, 743–751 (1995).
[CrossRef]

Ozaktas, H. M.

A. Koç, H. M. Ozaktas, and L. Hesselink, “Fast and accurate computation of two-dimensional non-separable quadratic-phase integrals,” J. Opt. Soc. Am. A 27, 1288–1302 (2010).
[CrossRef]

A. Koç, H. M. Ozaktas, and L. Hesselink, “Fast and accurate computation of two-dimensional non-separable quadratic-phase integrals,” J. Opt. Soc. Am. A 27, 1288–1302 (2010).
[CrossRef]

A. Koc, H. M. Ozaktas, C. Candan, and M. A. Kutay, “Digital computation of linear canonical transforms,” IEEE Trans. Signal Process. 56, 2383–2394 (2008).
[CrossRef]

H. M. Ozaktas, A. Koç, I. Sari, and M. A. Kutay, “Efficient computation of quadratic-phase integrals in optics,” Opt. Lett. 31, 35–37 (2006).
[CrossRef]

H. M. Ozaktas and O. Arikan, “Digital computation of the fractional Fourier transform,” IEEE Trans. Signal Process. 44, 2141–2150, (1996).
[CrossRef]

H. M. Ozaktas and D. Mendlovic, “Fractional Fourier optics,” J. Opt. Soc. Am. A 12, 743–751 (1995).
[CrossRef]

H. M. Ozaktas, M. A. Kutay, and Z. Zalevsky, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2000).

Pei, S. C.

Rhodes, W. T.

D. P. Kelly, B. M. Hennelly, W. T. Rhodes, and J. T. Sheridan, “Analytical and numerical analysis of linear optical systems,” Opt. Eng. 45, 088201 (2006).
[CrossRef]

Sari, I.

Sheridan, J. T.

Spiegel, M. R.

M. R. Spiegel, Mathematical Handbook of Formulas and Tables (McGraw-Hill, 1990).

Wolf, K. B.

K. B. Wolf, Geometric Optics on Phase Space (Springer, 2004).

K. B. Wolf, “Canonical Transforms,” in Integral Transforms in Science and Engineering (Plenum, 1979), pp. 381–416.

Zalevsky, Z.

Z. Zalevsky, D. Mendlovic, and A. W. Lohmann, “The ABCD—Bessel transformation,” Opt. Commun. 147, 39–41 (1998).
[CrossRef]

H. M. Ozaktas, M. A. Kutay, and Z. Zalevsky, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2000).

Zhou, F.

IEEE Trans. Signal Process. (2)

H. M. Ozaktas and O. Arikan, “Digital computation of the fractional Fourier transform,” IEEE Trans. Signal Process. 44, 2141–2150, (1996).
[CrossRef]

A. Koc, H. M. Ozaktas, C. Candan, and M. A. Kutay, “Digital computation of linear canonical transforms,” IEEE Trans. Signal Process. 56, 2383–2394 (2008).
[CrossRef]

J. Opt. Soc. Am. A (8)

Opt. Commun. (1)

Z. Zalevsky, D. Mendlovic, and A. W. Lohmann, “The ABCD—Bessel transformation,” Opt. Commun. 147, 39–41 (1998).
[CrossRef]

Opt. Eng. (2)

L. M. Bernardo, “ABCD matrix formalism of fractional Fourier optics,” Opt. Eng. 35, 732–740 (1996).
[CrossRef]

D. P. Kelly, B. M. Hennelly, W. T. Rhodes, and J. T. Sheridan, “Analytical and numerical analysis of linear optical systems,” Opt. Eng. 45, 088201 (2006).
[CrossRef]

Opt. Lett. (4)

Other (5)

G. B. Folland, Harmonic Analysis in Phase Space (Princeton University, 1989).

H. M. Ozaktas, M. A. Kutay, and Z. Zalevsky, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2000).

K. B. Wolf, “Canonical Transforms,” in Integral Transforms in Science and Engineering (Plenum, 1979), pp. 381–416.

K. B. Wolf, Geometric Optics on Phase Space (Springer, 2004).

M. R. Spiegel, Mathematical Handbook of Formulas and Tables (McGraw-Hill, 1990).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (7)

Fig. 1.
Fig. 1.

Simulation results for digitally implementing the 2D NSLCT with parameters defined in Eq. (23) for the second order Hermite–Gaussian function. (a) The input, (b) the result of direct implementation, (c) the result of Koc’s method, and (d) the result of the proposed method.

Fig. 2.
Fig. 2.

Plot of computation time versus log10(NMSE) when using the proposed method and Koc’s method for digitally implementing the 2D NSLCT with parameters defined in Eq. (23) for the second order Hermite–Gaussian function. NMSE means the normalized mean square error.

Fig. 3.
Fig. 3.

Plot of computation time versus log10(NMSE) when using the proposed method and Koc’s method for digitally implementing the 2D NSLCT with parameters defined in Eq. (27) for the second order Hermite–Gaussian function.

Fig. 4.
Fig. 4.

Plot of computation time versus log10(NMSE) when using the proposed method and Koc’s method for digitally implementing the 2D NSLCT with parameters defined in Eq. (23) for the fourth order Hermite–Gaussian function.

Fig. 5.
Fig. 5.

Simulation for the additivity property. The plot shows computation time versus NMSE when using the proposed method and Koc’s method for digitally implementing two 2D NSLCTs consequently.

Fig. 6.
Fig. 6.

Simulation for reconstruction. (a) The original Lena image, (b) the reconstruction result of the direct implementation method, (c) the reconstruction result of Koc’s method, and (d) the reconstruction result of the proposed method.

Fig. 7.
Fig. 7.

Simulation for the reconstruction property. The plot shows computation time versus NMSE when using the proposed method and Koc’s method for digitally implementing the forward and the inverse 2D NSLCTs. The NMSE for the direct implementation method is 0.00581. When the proposed method is applied, the NMSE is very close to that of the direct implementation method but the computation time is much less.

Tables (2)

Tables Icon

Table 1. Comparing the Proposed Method with the Method in [19] for Implementing the 2D NSLCT

Tables Icon

Table 2. Errors (Measured by the NMSE) and the Computation Times When Using Koc’s Method and the Proposed Method to Implement the 2D NSLCT with Parameters Defined in Eq. (23)a

Equations (43)

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

G(A,B,C,D)(u,v)=ONSLCT(A,B,C,D)[g(x,y)]=12π[det(B)]1/2exp[jk1u2+k2uv+k3v22det(B)]exp{jdet(B)[(b22u+b12v)x+(b21ub11v)y+12(p1x2+p2xy+p3y2)]}g(x,y)dxdy,
k1=d11b22d12b21,k2=2(d11b12+d12b11),k3=d21b12+d22b11,p1=a11b22a21b12,p2=2(a12b22a22b12),p3=a12b21+a22b11,
A=[a11a12a21a22],B=[b11b12b21b22],C=[c11c12c21c22],D=[d11d12d21d22].
[ABCD],whereA,B,C,andDare defined in(3).
[ABCD]T[0II0][ABCD]=[0II0],
ONSLCT(A1,B1,C1,D1){ONSLCT(A,B,C,D)[g(x,y)]}=ONSLCT(A2,B2,C2,D2)[g(x,y)],where[A2B2C2D2]=[A1B1C1D1][ABCD].
ONSLCT(A,B,C,D)[g(x,y)]=[det(D)]1/2exp{j2[(c11d11+c12d12)x2+2(c11d21+c12d22)xy+(c21d21+c22d22)y2]}g(d11x+d21y,d12x+d22y),whereB=0.
ONSLCT(A,B,C,D)[g(x,y)]=ONSLCT(A1,B1,C1,D1)[1j2πexp(jxw)g(x,y)dx],where[A1B1C1D1]=[ABCD][0010010010000001].
ONSLCT(A,B,C,D)[g(x,y)]=ONSLCT(A2,B2,C2,D2)[1j2πexp(jyv)g(x,y)dy],where[A2B2C2D2]=[ABCD][1000000100100100].
x=mΔt,y=nΔt,u=pΔu,v=qΔu.
G(A,B,C,D)(pΔu,qΔu)=12π[det(B)]1/2exp(jk1p2+k2pq+k3q22det(B)Δu2)mng(mΔt,nΔt)Δt2×exp(jdet(B){[(b22p+b12q)m+(b21pb11q)n]ΔuΔt+12(p1m2+p2mn+p3n2)Δt2}).
[ABCD]=[I0GI][S00S1][XYYX],whereS=(AAT+BBT)1/2,G=(CAT+DBT)S2,X=S1A,Y=S1B.
[XYYX]=Rr2FaxayRr1,whereRr1=[cosr1sinr100sinr1cosr10000cosr1sinr100sinr1cosr1],Rr2=[cosr2sinr200sinr2cosr20000cosr2sinr200sinr2cosr2],Faxay=[cosθx0sinθx00cosθy0sinθysinθx0cosθx00sinθy0cosθy].
exp[j(θx+θy)]=det(X+jY),cos(θxθy)=detX+detY,
exp[j(r1+r2+θx+θy2)]=X11+X22Y12+Y21+j(X12X21+Y11+Y22)2cos[(θxθy)/2],exp[j(r1+r2+θx+θy2)]=X11+X22+Y12+Y21+j(X12+X21+Y11Y22)2sin[(θxθy)/2].
[ABCD]=[I0DB1I][B00(BT)1][0II0][I0B1AI].
g1(mΔt,nΔt)=exp[j2(τ1m2+2τ2mn+τ3n2)Δt2]g(mΔt,nΔt)Δt2where[τ1τ2τ2τ3]=B1A.
g2(rΔτ,sΔτ)=mnexp[j2πN(rm+sn)]g1(mΔt,nΔt),
g3(pΔu,qΔu)=(1k)(1h)g2(p2Δτ,q2Δτ)+(1k)hg2(p2Δτ,(q2+1)Δτ)+k(1h)g2((p2+1)Δτ,q2Δτ)+khg2((p2+1)Δτ,(q2+1)Δτ),
p1=b22pb12qdet(B)ΔuΔτ,q1=b11qb21pdet(B)ΔuΔτ,p2=p1,q2=q1,means rounding toward,k=p1p2,andh=q1q2.
G(A,B,C,D)(pΔu,qΔu)=12πdet(B)exp[j2(τ4m2+2τ5mn+τ6n2)Δt2]g3(pΔu,qΔu),where[τ4τ5τ5τ6]=DB1.
ϕm,n(x,y)=exp[12(x2+y2)]Hm(x)Hn(y),
[ABCD]=[1.07361.52900.16181.51231.36511.56691.54770.60756.593710.9024010.73991.53352.94951.78751.8244].
direct implementation:30.3366s,Koc's method:0.2795s,proposed method:0.1866s.
NMSE=mn|G(A,B,C,D),1(pΔu,qΔu)G(A,B,C,D),2(pΔu,qΔu)|2pq|G(A,B,C,D),2(pΔu,qΔu)|2,
Koc's method:NMSE=1.8289×106,proposed method:NMSE=7.5147×108.
[ABCD]=[1.85301.78581.2906001.00141.52151.57872.89221.18430.96102.52732.96643.31561.79750.2787].
[ABCD]=[0.709700.01420.556100.68890.53980.89139.13980.22881.41297.45760.23570.66530.51660.4061],[A1B1C1D1]=[01.57241.85391.64430.68761.00510.65171.86170.62910.374600.28600.25511.065700.8136],[A2B2C2D2]=[A1B1C1D1][ABCD]=[17.33182.60140.921114.55946.88272.08020.57385.38190.37910.44830.35890.13210.01071.27550.99920.4775],
NMSE=mn|G1(pΔu,qΔu)G2(pΔu,qΔu)|2pq|G2(pΔu,qΔu)|2
65.615s.
[A1B1C1D1]=[DTBTCTAT],
[A1B1C1D1][ABCD]=[I00I],i.e.,ONSLCT(DT,BT,CT,AT){ONSLCT(A,B,C,D)[g(x,y)]}=g(x,y).
[ABCD]=[1.76911.51430.41451.75111.11730.47411.35701.82783.88573.482202.30271.99981.09590.09220.8276].
direct implementation:676.4327s,Koc's method:2.9200s,proposed method:1.7816s.
direct implemenataion:NMSE=0.00581Koc's method:NMSE=0.0477,proposed method:NMSE=0.00600.
[ABCD]=[A1B1C1D1][A2B2C2D2]=[I0D1B11I][B100(B1T)1][0II0][I0B11A1I][A2B2C2D2],
[A1B1C1D1]=[ABCD][A2B2C2D2]1=[ABCD][D2TB2TC2TA2T]
[A2B2C2D2]=[0010010010000001]
[A2B2C2D2]=[1000000100100100].
[A1B1C1D1]=[ABCD][0010010010000001].
[A1B1C1D1]=[ABCD][1000000100100100].
M1=[1000a21/a1110000100001][ABCD][1a12/a11b11/a11b12/a11010000100001],
det(M1)=det([ABCD]).

Metrics