Abstract

The first imaging devices of optical interferometry are likely to be of weak phase, typically: a set of three-element arrays independently observing the same object. The study of their imaging capabilities refers to appropriate optimization methods, which essentially address the self-calibration process and its stability. A general survey of these techniques is given, and it is shown, in particular, how the related algorithms can be used for examining the imaging capabilities of weak-phase interferometric devices. The phase-calibration algorithm involved in the self-calibration cycles is based on the principle underlying the trust-region methods. It benefits from certain remarkable properties, the analysis of which appeals to algebraic graph theory. The Fourier synthesis operation, which is also involved in these cycles, is performed by means of Wipe, a methodology recently introduced in radio imaging and optical interferometry. (Wipe is reminiscent of Clean, a widely used technique in astronomy). In the related theoretical framework the stability of the image-reconstruction process is controlled by considering certain elements of the singular-value decomposition of the derivative of the self-calibration operator. For example, the largest singular value of this derivative, which depends on the interferometric configuration and on the object thus imaged, provides a key indication of the observational limits of these experimental devices.

© 1998 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. A. Lannes, “Phase-closure imaging in algebraic graph theory: a new class of phase-calibration algorithms,” J. Opt. Soc. Am. A 15, 419–429 (1998).
    [CrossRef]
  2. A. Lannes, E. Anterrieu, K. Bouyoucef, “Fourier interpolation and reconstruction via Shannon-type techniques: Part I: Regularization principle,” J. Mod. Opt. 41, 1537–1574 (1994).
    [CrossRef]
  3. A. Lannes, E. Anterrieu, K. Bouyoucef, “Fourier interpolation and reconstruction via Shannon-type techniques: Part II: Technical developments and applications,” J. Mod. Opt. 43, 105–138 (1996).
    [CrossRef]
  4. A. Lannes, E. Anterrieu, P. Maréchal, “Clean and Wipe,” Astron. Astrophys. Suppl. Ser. 123, 183–198 (1997).
    [CrossRef]
  5. T. J. Cornwell, P. N. Wilkinson, “New method for making maps with unstable interferometers,” Mon. Not. R. Astron. Soc. 196, 1067–1086 (1981).
  6. A. Lannes, “Remarkable algebraic structures of phase-closure imaging and their algorithmic implications in aperture synthesis,” J. Opt. Soc. Am. A 7, 500–512 (1990).
    [CrossRef]
  7. R. Merris, “Survey of graph Laplacians,” Linear Multilinear Algebra 39, 19–31 (1995).
  8. J. Moré, “Recent developments in algorithms and software for trust region methods,” in Mathematical Programming: the State of the Art, A. Bachem, M. Grötschel, B. Korte, eds. (Springer-Verlag, Berlin, 1983), pp. 258–287.

1998 (1)

1997 (1)

A. Lannes, E. Anterrieu, P. Maréchal, “Clean and Wipe,” Astron. Astrophys. Suppl. Ser. 123, 183–198 (1997).
[CrossRef]

1996 (1)

A. Lannes, E. Anterrieu, K. Bouyoucef, “Fourier interpolation and reconstruction via Shannon-type techniques: Part II: Technical developments and applications,” J. Mod. Opt. 43, 105–138 (1996).
[CrossRef]

1995 (1)

R. Merris, “Survey of graph Laplacians,” Linear Multilinear Algebra 39, 19–31 (1995).

1994 (1)

A. Lannes, E. Anterrieu, K. Bouyoucef, “Fourier interpolation and reconstruction via Shannon-type techniques: Part I: Regularization principle,” J. Mod. Opt. 41, 1537–1574 (1994).
[CrossRef]

1990 (1)

1981 (1)

T. J. Cornwell, P. N. Wilkinson, “New method for making maps with unstable interferometers,” Mon. Not. R. Astron. Soc. 196, 1067–1086 (1981).

Anterrieu, E.

A. Lannes, E. Anterrieu, P. Maréchal, “Clean and Wipe,” Astron. Astrophys. Suppl. Ser. 123, 183–198 (1997).
[CrossRef]

A. Lannes, E. Anterrieu, K. Bouyoucef, “Fourier interpolation and reconstruction via Shannon-type techniques: Part II: Technical developments and applications,” J. Mod. Opt. 43, 105–138 (1996).
[CrossRef]

A. Lannes, E. Anterrieu, K. Bouyoucef, “Fourier interpolation and reconstruction via Shannon-type techniques: Part I: Regularization principle,” J. Mod. Opt. 41, 1537–1574 (1994).
[CrossRef]

Bouyoucef, K.

A. Lannes, E. Anterrieu, K. Bouyoucef, “Fourier interpolation and reconstruction via Shannon-type techniques: Part II: Technical developments and applications,” J. Mod. Opt. 43, 105–138 (1996).
[CrossRef]

A. Lannes, E. Anterrieu, K. Bouyoucef, “Fourier interpolation and reconstruction via Shannon-type techniques: Part I: Regularization principle,” J. Mod. Opt. 41, 1537–1574 (1994).
[CrossRef]

Cornwell, T. J.

T. J. Cornwell, P. N. Wilkinson, “New method for making maps with unstable interferometers,” Mon. Not. R. Astron. Soc. 196, 1067–1086 (1981).

Lannes, A.

A. Lannes, “Phase-closure imaging in algebraic graph theory: a new class of phase-calibration algorithms,” J. Opt. Soc. Am. A 15, 419–429 (1998).
[CrossRef]

A. Lannes, E. Anterrieu, P. Maréchal, “Clean and Wipe,” Astron. Astrophys. Suppl. Ser. 123, 183–198 (1997).
[CrossRef]

A. Lannes, E. Anterrieu, K. Bouyoucef, “Fourier interpolation and reconstruction via Shannon-type techniques: Part II: Technical developments and applications,” J. Mod. Opt. 43, 105–138 (1996).
[CrossRef]

A. Lannes, E. Anterrieu, K. Bouyoucef, “Fourier interpolation and reconstruction via Shannon-type techniques: Part I: Regularization principle,” J. Mod. Opt. 41, 1537–1574 (1994).
[CrossRef]

A. Lannes, “Remarkable algebraic structures of phase-closure imaging and their algorithmic implications in aperture synthesis,” J. Opt. Soc. Am. A 7, 500–512 (1990).
[CrossRef]

Maréchal, P.

A. Lannes, E. Anterrieu, P. Maréchal, “Clean and Wipe,” Astron. Astrophys. Suppl. Ser. 123, 183–198 (1997).
[CrossRef]

Merris, R.

R. Merris, “Survey of graph Laplacians,” Linear Multilinear Algebra 39, 19–31 (1995).

Moré, J.

J. Moré, “Recent developments in algorithms and software for trust region methods,” in Mathematical Programming: the State of the Art, A. Bachem, M. Grötschel, B. Korte, eds. (Springer-Verlag, Berlin, 1983), pp. 258–287.

Wilkinson, P. N.

T. J. Cornwell, P. N. Wilkinson, “New method for making maps with unstable interferometers,” Mon. Not. R. Astron. Soc. 196, 1067–1086 (1981).

Astron. Astrophys. Suppl. Ser. (1)

A. Lannes, E. Anterrieu, P. Maréchal, “Clean and Wipe,” Astron. Astrophys. Suppl. Ser. 123, 183–198 (1997).
[CrossRef]

J. Mod. Opt. (2)

A. Lannes, E. Anterrieu, K. Bouyoucef, “Fourier interpolation and reconstruction via Shannon-type techniques: Part I: Regularization principle,” J. Mod. Opt. 41, 1537–1574 (1994).
[CrossRef]

A. Lannes, E. Anterrieu, K. Bouyoucef, “Fourier interpolation and reconstruction via Shannon-type techniques: Part II: Technical developments and applications,” J. Mod. Opt. 43, 105–138 (1996).
[CrossRef]

J. Opt. Soc. Am. A (2)

Linear Multilinear Algebra (1)

R. Merris, “Survey of graph Laplacians,” Linear Multilinear Algebra 39, 19–31 (1995).

Mon. Not. R. Astron. Soc. (1)

T. J. Cornwell, P. N. Wilkinson, “New method for making maps with unstable interferometers,” Mon. Not. R. Astron. Soc. 196, 1067–1086 (1981).

Other (1)

J. Moré, “Recent developments in algorithms and software for trust region methods,” in Mathematical Programming: the State of the Art, A. Bachem, M. Grötschel, B. Korte, eds. (Springer-Verlag, Berlin, 1983), pp. 258–287.

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 (4)

Fig. 1
Fig. 1

Basic configurations involved in the definition of the interferometric devices D1 and D2: (a) four arrays of three telescopes, (b) three arrays of four telescopes.

Fig. 2
Fig. 2

Frequency coverages of the interferometric devices D1 and D2. These devices are generated by rotations of the basic configurations shown in Fig. 1: rotations of (a) π/9 for D1 and (b) π/6 for D2. The corresponding frequency lists Le have the same number of frequency points. The circle S fixes (for our study) the limits of the frequency coverage to be synthesized.

Fig. 3
Fig. 3

(a) Neat beam and (b) image to be reconstructed, at the corresponding level of resolution. The neat beam represented here corresponds to the frequency coverage to be synthesized shown in Fig. 2. Note that the portion of the field shown in (b) is twice as large as that defined in (a).

Fig. 4
Fig. 4

Image reconstruction via Self-calibrated Wipe: (a) for D1, (b) for D2. These images are to be compared with the image to be reconstructed shown in Fig. 3(b). The conditions of this simulation are specified in this section.

Tables (2)

Tables Icon

Table 1 Largest Singular Values of the Derivative of the Self-Calibration Operator for D1 and D2 [for the Image Shown in Fig. 3(b)]

Tables Icon

Table 2 Largest Singular Values of the Derivative of the Reconstruction Operator in the Ideal Case Where D1 and D2 Would Be Full-Phase Imaging Devices

Equations (138)

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

ϕˆo(u)2ϕo(ξ)exp(-2iπu·ξ)dξ.
ui(j, k)=ri(j)-ri(k)λ,
Vi,e(j, k)=ϕˆo(ui(j, k))exp[iβi,e(j, k)]+errorterms,
βi,e(j, k)=αi,e(j)-αi,e(k).
dφ=i=1is[ni(ni-1)/2]-(ni-1)i=1isni(ni-1)/2=i=1is(ni-1)(ni-2)i=1isni(ni-1).
dφ=1-2n.
GL×L,Lp:-N2pN2-1.
e0(ξ)sincξ1δξsincξ2δξ,
ϕ=pGxpep(xp).
(ϕ(1)|ϕ(2))=(δξ)2pGxp(1)xp(2).
ϕˆ=pGxpeˆp,
eˆp(u)=eˆ0(u)exp-2iπp·uΔu,
eˆ0(u)=1(Δu)2rectu1Δurectu2Δu.
ϕssϕo,
1s2S|sˆ(u)|2du=χ2.
2ϕs(ξ)dξ=2ϕo(ξ)dξ.
ϕ=pDxpep(xp).
ϕpGxpepEo,(Pϕϕ)p=xpifpD0otherwise·
Bi{(j, k)Ai2:jk};
ψ0,esˆ(0)V0,e=V0,e,
ψi,e(j, k)sˆ(ui(j, k))Vi,e(j, k),
(j, k)Bi,e(i=1, , is).
fe(ϕ, α)12|ψ0,e-ϕˆ(0)|2ϖ0,e
+12i=1is(j, k)Bi,e|ψi,e(j, k)-ϕˆ[ui(j, k)]×exp{i[αi(j)-αi(k)]}|2ϖi,e(j, k).
Le{{0};L1,e, , Lis,e}.
we(u)1/σe2(u)uLe 1σe2(u)sinc2u1-u1δusinc2u2-u2δu×(δu)2.
ψi(1)|ψi(2)Gi,e(j, k)Bi,eψ¯i(1)(j, k)ψi(2)(j, k)ϖi,e(j, k)=(j, k)Bi,e Re{ψ¯i(1)(j, k)ψi(2)(j, k)}×ϖi,e(j, k),
ψ0(1)|ψ0(2)G0,eψ0(1)ψ0(2)ϖ0,e.
Gei=0isGi,e
ψ(1)|ψ(2)Gei=0isψi(1)|ψi(2)Gi,e.
Aeϕ(A0,eϕ, A1,eϕ, , Ai,eϕ, , Ais,eϕ)
Ai,e:EoGi,e,
(Ai,eϕ)(j, k)ϕˆ(0)ifi=0ϕˆ(ui(j, k))otherwise·
Aeϕ | ψGe=i=0isAi,eϕ|ψiGi,e,
Ae*ψ=i=0isAi,e*ψi.
(αi(1)|αi(2))FijAiαi(1)(j)αi(2)(j),
Fi=1isFi
(α(1)|α(2))Fi=1is(αi(1)|αi(2))Fi·
α˘i(j)1nik=1niαi(k)(foralljAi).
FˇiαiFi:j=1niαi(j)=0.
Fˇi=1isFˇi.
Eα=Fˇ.
(βi(1)|βi(2))Gi12(j, k)Biβi(1)(j, k)βi(2)(j, k),
Gei=1isGi,e,
(βi(1)|βi(2))Gi,e12(j, k)Bi,eβi(1)(j, k)βi(2)(j, k).
Bi:FiGi,Biαiαi(j)-αi(k),
Bi,e:FiGi,e,Bi,eαiαi(j)-αi(k).
ziexp(iBi,eαi)(i=1, , is).
(Bi*βi)(j)=kAikjβi(j, k).
(Bi,e*βi)(j)=(Bi*βi(0))(j),
βi(0)(j, k)βi(j, k)if(j, k)Bi,e0otherwise.
[Z(α)]ψ(ψ0, z1ψ1, , zisψis).
fe(ϕ, α)12ψe-[Z(α)]AeϕGe2.
Gr{q:qδuS}.
(ψ(1)|ψ(2))GrqGrψ¯(1)(q)ψ(2)(q)ϖr
=qGrRe{ψ¯(1)(q)ψ(2)(q)}ϖr
fr(ϕ)12ψr-ArϕGr2,
ψr0,Ar:EoGr,(Arϕ)(q)ϕˆ(qδu).
fr(ϕ)=12qGr|ϕˆ(qδu)|2(δu)2.
f:EϕEα,f(ϕ, α)fe(ϕ, α)+fr(ϕ).
fe(ϕ_, α_)+fr(ϕ_)fe(ϕ, α)+fr(ϕ)fe(ϕ_, α_)+fr(ϕ),
j=1niαi(1)(j)=0j=1niαi(2)(j)=0.
α(ξ)ξ1α(1)+ξ2α(2)
[Z(α+α(ξ))]Aeϕ
=[A0,eϕ,(A1,eϕ)z1(ξ)z1,,(Ais,eϕ)zis(ξ)zis],
zi(ξ)(j, k)exp[2iπui(j, k)·ξ](i=1, , is).
c(α)12i=1is(j, k)Bi,e|ψi,e(j, k)-ψi,m(j, k)×exp{i(Bi,eαi)(j, k)}|2ϖi,e(j, k),
ψi,m(j, k)ϕˆ[ui(j, k)](i=1, , is).
ci(αi)12(j, k)Bi,e|ψi,e(j, k)-ψi,m(j, k)×exp[i(Bi,eαi)(j, k)]|2ϖi,e(j, k).
ψc[Z(-α)]ψe,
c(ϕ)ce(ϕ)+cr(ϕ),
ce(ϕ)12ψc-AeϕGe2,cr(ϕ)fr(ϕ)12ArϕGr2.
c(ϕ)=-Pϕ(Ae*ψc-A*Aϕ),
A*AAe*Ae+Ar*Ar;
Ae*Ae=i=0isAi,e*Ai,e.
Pϕ(Ae*ψc-A*Aϕ)=0.
r(l)Ae*ψc-A*Aϕ(l).
rp(l)(ep|r(l))(pG),
Ceψc-Aeϕ(l)Ge/σeGe
D(l+1)=D(l)D.
f:Xu×Yd,f(x, y)12y-d(x)2,
XuEϕEα,YdGeGr.
d:XuYd,d(ϕ, α)([Z(α)]Aeϕ; Arϕ).
[d(ϕ, α)]·(υ, η)=[dϕ(ϕ, α)]υ+[dα(ϕ, α)]η,
[dϕ(ϕ, α)]υ=([Z(α)]Aeυ; Arυ)=(A0,eυ, (A1,eυ)z1, , (Ais,eυ)zis; Arυ),
[dα(ϕ, α)]η=(0, i(A1,eϕ)z1B1,eη1, , i(Ais,eϕ)zisBis,eηis; 0).
[d(ϕ, α)]·(υ, η)=[A0,eυ,(A1,eυ)z1+i(A1,eϕ)z1B1,eη1, , (Ais,eυ)zis+i(Ais,eϕ)zisBis,eηis; Arυ],
[d(ϕ, α)]·(υ, η)2=|υˆ(0)|2ϖ0,e+i=1is(j, k)Bi,e|υˆ[ui(j, k))+iϕˆ(ui(j, k)](Bi,eηi)(j, k)|2ϖi,e(j, k)+qGr|υˆ(qδu)|2ϖr.
i=1is(j, k)Bi,e|ϕˆ(ui(j, k))|2|(Bi,eηi)(j, k)|2ϖi,e(j, k)=0,
ϕˆ(ui(j, k))0onBi,e(i=1, , is).
fx(xu+δxu, yd+δyd)=0.
fxx(xu, yd)δxu+fxy(xu, yd)δyd=0.
fx(x, y)=-[d(x)]* [y-d(x)],
fxx(xu, yd)=[d(xu)]* [d(xu)],
fxy(xu, yd)=-[d(xu)]*.
du*duδxu=du*δyd,dud(xu).
δxu=du+δyd,
du+=[du*du]-1du*.
δx=P[du+]δy=P[du+]Qδy.
D=P[du+]Q.
δyY,δx=Dδy,
δx=k(ϕk | Dδy)ϕk=kD*ϕk | δyϕk,
δyY,δx=kγkψk|δyϕk.
δyY,δxγδy,
γmaxk{γk}=D.
DD*=P[du+]QQ*[du+]*P*=P[du+]Q[du+]*P=P[du*du]-1du*Qdu[du*du]-1P,
DD*=P[du*du]-1[Qdu]*[Qdu][du*du]-1P.
f˜:Xu×Yd,f˜(x, y)12Q[y-d(x)]2.
DD*=PH-1H˜H-1P,
Hfxx(xu, yd),H˜f˜xx(xu, yd).
H=HϕϕHϕαHαϕHαα.
Hϕϕυ=Pϕ(Ae*Ae+Ar*Ar)υ,
Hϕαiηi=PϕAi,e*[i(Ai,eϕ)]Bi,eηi,
Hαiϕυ=2Bi,e*[ϖi,eIm{(Ai,eυ)Ai,eϕ¯}],
Hαiαiηi=2Bi,e*[ϖi,e|Ai,eϕ|2]Bi,eηi
H˜ϕϕυ=PϕAe˜*Ae˜υ,
H˜ϕαiηi=PϕAi,e˜*[i(Ai,e˜ϕ)]Bi,e˜ηi
H˜αiϕυ=2Bi,e˜*[ϖi,e˜Im{(Ai,e˜υ)Ai,e˜ϕ¯}],
H˜αiαiηi=2Bi,e˜*[ϖi,e˜|Ai,e˜ϕ|2]Bi,e˜ηi.
fx(x, y)=-[d(x)]*[y-d(x)].
d(ϕ, α)(A0,eϕ, (A1,eϕ)z1, , (Ais,eϕ)zis; Arϕ).
[d(ϕ, α)]·(υ, η)=[dϕ(ϕ, α)]υ+[dα(ϕ, α)]η,
[dϕ(ϕ, α)]υ=(A0,eυ, (A1,eυ)z1, , (Ais,eυ)zis; Arυ),
[dα(ϕ, α)]η=(0, i(A1,eϕ)z1B1,eη1, , i(Ais,eϕ)zisBis,eηis; 0).
[dϕ(ϕ, α)]*·(Ψe, Ψr)=PϕA0,e*Ψ0,e+i=1isAi,e*(z¯iΨi,e)+Ar*Ψr,
{[dα(ϕ, α)]*·(Ψe, Ψr)}i=-2Bi,e*{ϖi,eIm[(Ai,eϕ)ziΨ¯i,e]}(i=1, , is).
[dϕ(ϕ, α)]υ | (Ψe, Ψr)=A0,eυ | Ψ0,eG0,e+i=1is(Ai,eυ)zi | Ψi,e)Gi,e+Arυ | ΨrGr=(υ | PϕA0,e*Ψ0,e)+i=1is(υ | PϕAi,e*z¯iΨi,e)+(υ | PϕAr*Ψr),
[dα(ϕ, α)]η | (Ψe, Ψr)=i=1isi(Ai,eϕ)ziBi,eηi | Ψi,eGi,e=i=1is(j, k)Bi,eRe[i(Ai,eϕ)(j, k)zi(j, k)Ψ¯i,e(j, k)]×(Bi,eηi)(j, k)ϖi,e(j, k)=-2i=1is{Bi,eηi | ϖi,e Im[(Ai,eϕ)ziΨ¯i,e]}Gi,e=-2i=1is{ηi | Bi,e*[ϖi,e Im[(Ai,eϕ)ziΨ¯i,e]}Fi.
fϕ(ϕ, α; ψe, ψr)=-[dϕ(ϕ, α)]*·[(ψe, ψr)-d(ϕ, α)]=-PϕA0,e*(ψ0,e-A0,eϕ)+i=1isAi,e*z¯i(ψi,e-ziAi,eϕ)+Ar*(ψr-Arϕ);
fϕ(ϕ, α;ψe, ψr)=-PϕA0,e*(ψ0,e-A0,eϕ)+i=1isAi,e*(z¯iψi,e-Ai,eϕ)+Ar*(ψr-Arϕ).
fαi(ϕ, α;ψe, ψr)=-{[dα(ϕ, α)]*·[(ψe, ψr)-d(ϕ, α)]}i=2Bi,e*{ϖi,e Im[(Ai,eϕ)zi(ψ¯i,e-z¯iAi,eϕ¯)]};
fαi(ϕ, α;ψe, ψr)=2Bi,e*[ϖi,e Im{(Ai,eϕ)ziψ¯i,e}]
(i=1, , is).
Ae*Ae=i=0isAi,e*Ai,e,
fϕϕυ=Pϕ(Ae*Ae+Ar*Ar)υ,
fϕαiηi=PϕAi,e*[iz¯iψi,e]Bi,eηi,
fαiϕυ=2Bi,e*[ϖi,eIm{(Ai,eυ)ziψ¯i,e}],
fαiαiηi=2Bi,e*[ϖi,eRe{(Ai,eϕ)ziψ¯i,e}]Bi,eηi.

Metrics