Abstract

A new class of phase-calibration algorithms is presented. The originality of these algorithms, as well as their efficiency, results from certain particular structures, the analysis of which calls on algebraic graph theory. The corresponding optimization process, which is based on the principle of the trust-region methods, proves to be well suited to these structures. The main message that emerges from the study is very clear: The traditional notions of phase closure imaging can be understood and refined in a wider framework. The implications of this research therefore concern all the fields in which the notion of phase closure plays a key role: weak-phase imaging in optical interferometry, radio imaging, remote sensing by aperture synthesis, etc.

© 1998 Optical Society of America

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References

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  1. R. Merris, “A survey of graph Laplacians,” Linear Multilinear Algebra 39, 19–31 (1995). When the author began discovering the algebraic structures of phase closure imaging,2,3 he was completely unaware of the existence of algebraic graph theory. Likewise, the mathematicians working in this very particular field of research were not aware of a possible application in interferometry (see for instance the long list of references given in this review paper).
  2. 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]
  3. A. Lannes, “Phase and amplitude calibration in aperture synthesis. Algebraic structures,” Inv. Probl. 7, 261–298 (1991).
    [CrossRef]
  4. 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]
  5. 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]
  6. A. Lannes, E. Anterrieu, P. Maréchal, “Clean and wipe,” Astron. Astrophys. Suppl. Ser. 123, 183–198 (1997).
    [CrossRef]
  7. 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.

1997

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

1996

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

R. Merris, “A survey of graph Laplacians,” Linear Multilinear Algebra 39, 19–31 (1995). When the author began discovering the algebraic structures of phase closure imaging,2,3 he was completely unaware of the existence of algebraic graph theory. Likewise, the mathematicians working in this very particular field of research were not aware of a possible application in interferometry (see for instance the long list of references given in this review paper).

1994

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]

1991

A. Lannes, “Phase and amplitude calibration in aperture synthesis. Algebraic structures,” Inv. Probl. 7, 261–298 (1991).
[CrossRef]

1990

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]

Lannes, A.

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, “Phase and amplitude calibration in aperture synthesis. Algebraic structures,” Inv. Probl. 7, 261–298 (1991).
[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, “A survey of graph Laplacians,” Linear Multilinear Algebra 39, 19–31 (1995). When the author began discovering the algebraic structures of phase closure imaging,2,3 he was completely unaware of the existence of algebraic graph theory. Likewise, the mathematicians working in this very particular field of research were not aware of a possible application in interferometry (see for instance the long list of references given in this review paper).

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.

Astron. Astrophys. Suppl. Ser.

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

Inv. Probl.

A. Lannes, “Phase and amplitude calibration in aperture synthesis. Algebraic structures,” Inv. Probl. 7, 261–298 (1991).
[CrossRef]

J. Mod. Opt.

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

Linear Multilinear Algebra

R. Merris, “A survey of graph Laplacians,” Linear Multilinear Algebra 39, 19–31 (1995). When the author began discovering the algebraic structures of phase closure imaging,2,3 he was completely unaware of the existence of algebraic graph theory. Likewise, the mathematicians working in this very particular field of research were not aware of a possible application in interferometry (see for instance the long list of references given in this review paper).

Other

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.

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Figures (5)

Fig. 1
Fig. 1

A graph and its Laplacian (or connectivity) matrix.

Fig. 2
Fig. 2

Example of complementary graphs.

Fig. 3
Fig. 3

Connected components of the graphs involved in the inversion of Lτ(G). Note that G2c is connected, hence the corresponding termination point in the structural tree shown in Fig. 4.

Fig. 4
Fig. 4

Structural tree involved in the inversion of Lτ(G) in the case of the graph G shown in Fig. 3. The termination points are encircled (see properties 3, 5 and 7). The corresponding recursive programs were written in C.

Fig. 5
Fig. 5

Phase calibration of a three-element array. For each baseline (j, k), the phasors of the complex visibilities Ve(j, k), Vm(j, k), Vc(j, k), and Vo(j, k)ϕˆo[u(j, k)] are respectively defined by the position of the letters e, m, c, and o on the corresponding circles with radius ρe(j, k)=ρm(j, k). The strong baselines, here, (1, 2) and (1, 3), behave almost as independent (or incoherent) baselines. To illustrate this point, the phasors ζm(1, 2) and ζm(1, 3) were chosen as indicated, very different from ζo(1, 2) and ζo(1, 3), respectively.

Equations (116)

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ϕˆo(u)2ϕo(ξ)exp(-2iπu·ξ)dξ,
uu(j, k)=r(j)-r(k)λ.
Ve(j, k)=ϕˆo[u(j, k)]exp[iβe(j, k)]+errorterms,
βe(j, k)=αe(j)-αe(k).
Vm(j, k)ϕˆ[u(j, k)].
c(β)(j,k)Be|Ve(j, k)-Vm(j, k)exp[iβ(j, k)]|2×ϖ(j, k),
VcVe exp(-iβopt).
Ve=ρeζe,Vm=ρmζm,Vc=ρcζc.
c(α)c(Beα),(Beα)(j, k)α(j)-α(k), (j, k)Be.
exp[iβopt(j, k)]=ζ¯m(j, k)ζe(j, k).
ζc(j, k)=ζm(j, k).
(α(1) | α(2))FjAα(1)(j)α(2)(j),
(β(1) | β(2))G12(j,k)Bβ(1)(j, k)β(2)(j, k),
A: F,(Aω)(j)ω(jA);
B: FG,(Bα)(j, k)α(j)-α(k).
A*α=jAα(j),(B*β)(j)=kAkjβ(j, k).
PAA*n,QB*Bn,
P+Q=I,
Fˇker A*,
A+=A*/n,B+=B*/n.
w(j, k)1if(j, k)W0otherwise.
L(G)αB*wBα.
(α | L(G)α)F=(Bα | wBα)G=12(j,k)Bw(j, k)[α(j)-α(k)]2,
(L(G)α)(j)=α(j)kAkjw(j, k)-kAkjw(j, k)α(k).
F=r1Fr.
Ar : Fr,(Arω)(j)ω;
Br : FrGr,(Brα)(j, k)α(j)-α(k).
Jr : FrF,(Jrα)(j) α(j)ifjAr0otherwise.
Jr* : FFr,(Jr*α)(j)=α(j).
Jr*Js=δrsIr,
JrJr*=Tr,
Eλ(G)ker[L(G)-λI](λ0).
λ0,Eλ(G)=r1Eλ(Gr),
Eλ(Gr)ker[L(Gr)-λIr].
Lˇ(G) : FˇFˇ,Lˇ(G)αL(G)α.
Eˇλ(G)ker[Lˇ(G)-λIˇ](λ0),
Eλ(G)=Eˇ0(G)ker Bifλ=0Eˇλ(G)ifλ>0.
B*wB+B*wcB=B*B;
L(G)+L(Gc)=L(H).
3-1-1-1-12-10-1-13-1-10-12+0000010-100000-101=3-1-1-1-13-1-1-1-13-1-1-1-13
L(G)+L(Gc)=nQ,
L(G)=M(Gc)-nP,M(Gc)nI-L(Gc).
Lˇ(G)+Lˇ(Gc)=nIˇ.
Eˇλ(G)=Eˇn-λ(Gc).
Eλ(G)=En(Gc)kerBifλ=0En-λ(Gc)if0<λ<nE0(Gc)Fˇifλ=n.
Be: FGe,(Beα)(j, k)α(j)-α(k),(j, k)Be,
(Be*βe)(j)=(B*β)(j),β(j, k)βe(j, k)if(j, k)Be0otherwiseonB.
c: Fˇ,c(α)14(j,k)Be|Ve(j, k)-Vm(j, k)exp[i(Beα)(j, k)]|2ϖ(j, k).
ψ1|ψ2Ge(j,k)Beψ¯1(j, k)ψ2(j, k)ϖ(j, k)=(j,k)Be Re{ψ¯1(j, k)ψ2(j, k)}ϖ(j, k),
c(α)=14Ve-Vm exp(iBeα)Ge2.
q: Fˇ,
q(η)c(α)+(c(α) | η)F+12(c(α)η | η)F.
c(α)=Be*[ϖ Im{VmV¯c}],
c(α)η=Be*[ϖ Re{VmV¯c}]Beη,
VcVe exp(-iBeα)
cc(α),gc(α),Hˇc(α),
q: Fˇ,q(η)=c+(g | η)F+12(Hˇη | η)F.
qt: Fˇ,qt(η)q(η)+12t|η|F2(t0).
qt(η)=c+(g | η)F+12(Hˇtη | η)F,HˇtHˇ+tIˇ.
ηt=-Hˇt-1g,tt0max(-μ+, 0).
q(η)+12tηF2>q(ηt)+12tηtF2;
q(η)>q(ηt)+12t(ηtF2-ηF2)(tt0).
c(α)-c(α+ηt)>a[q(0)-q(ηt)](0<a<1).
q(0)-q(ηt)=-(g | ηt)F-12(Hˇηt | ηt)F=12[tηtF2-(g | ηt)F].
Hˇ: FˇFˇ,Hˇη=B*hBη,
hϖ Re{VmV¯c}onBe0otherwiseonB.
e=B*hBd+td,ω=σ/(d | e)F;
η:=η+ωd,r:=r-ωe,
σ=rF2.
γ=σ/σ,d:=r+γd.
(Beαl)(j,k)=(BeQαl(j,k)=2π(lj-lk)
η˜t-H˜t-1g(tt˜0).
c(α)-c(α+η˜t)>a2[tη˜tF2-(g | η˜t)F]
(0<a<1).
h˜=t2w+t1wc=(t2-t1)w+t1(t2>t1).
Hˇtη(t2-t1)B*wBη+nt1+tt2-t1η(ηFˇ).
H˜t=κLˇτ(G),κt2-t1,
Lˇτ(G)Lˇ(G)+τIˇ,G=(A, W),
τnt1+tt2-t1.
Lτ(G)L(G)+τI(τ).
Mτ(Gc)M(Gc)+τI=(n+τ)I-L(Gc)(τ).
Lτ(G)=Mτ(Gc)-nP(τ).
[wB]+=[Lˇ(G)]-1B*w=[M(Gc)]-1B*w.
Lτ(Gr)L(Gr)+τIr(r1).
[Lτ(G)]-1=r1Jr[Lτ(Gr)]-1Jr*.
Lτ(G)=r1[JrL(Gr)Jr*+τJrIrJr*]=r1Jr[L(Gr)+τIr]Jr*=r1JrLτ(Gr)Jr*.
r1JrLτ(Gr)Jr*s1Js[Lτ(Gs)]-1Js*=r1Tr=I.
[Mτ(Gc)]-1=-r1J·rc[L-n-τ(G·rc)]-1J·rc*.
[Lτ(G°)]-1=n°P°τ(n°+τ)-r1J°rc[L-n°-τ(G°rc)]-1J°rc*,
[Lτ(G°)]-1-P°τ=[Mτ(G°c)]-1-P°n°+τ.
a=1+ττ(2+τ),b=1τ(2+τ),
c=1τ(τ>0).
a=2+τ(1+τ)(3+τ),b=-1(1+τ)(3+τ),
c=13+τ(τ>-1).
c(α0)8.77×10-1;
h(1, 2)0.95,h(1, 3)-0.17,h(2, 3)0.03;
c(α1)1.79×10-2;
h(1, 2)0.99,h(1, 3)0.64,h(2, 3)0.03;
c(α2)8.899×10-3;
h(1,2)1.00,h(1, 3)0.64,h(2, 3)0.03;
c(α3)c(α4)8.898810×10-3.
ζc(1, 2)ζm(1, 2),ζc(1, 3)ζm(1, 3).
ζc(2, 3)ζe(2, 3)[ζe(1, 2)ζ¯m(1, 2)][ζ¯e(1, 3)ζm(1, 3)].
[δg](1)=sin(δθ)δθ,
[δg](2)=-sin(δθ)-δθ,[δg](3)=0.
δα=-Hˇ-1δg.
δθc=-Bδα=BHˇ-1δg.
HˇκLˇ(G),κ=t2-t1,
t2=1.00+0.64245,t1=0.030,
δθc(1, 2)δθc(1, 3)δθc(2, 3)1-1010-101-1×54×1310002-10-12δθ-δθ0,
δθc(1, 2)54δθ, δθc(1, 3)0, δθc(2, 3)-54δθ.
c(α+η)=14Ve-Vmz(1+iBeη-)Ge2=c(α)+12Ve-Vmz |-iVmzBeηGe+ ,
Ve-Vmz |-iVmzBeηGe=i(V¯mVez¯-|Vm|2) | BeηGe=(j,k)Be Re{-iVm(j, k)V¯e(j, k)z(j, k)}×(Beη)(j, k)ϖ(j, k)=2(ϖ Im{VmV¯ez}|Beη)Ge=2(Be*[ϖ Im{VmV¯ez}]|η)F.
c(α)=Be*[ϖ Im{VmV¯c}].
c(α+η)=Be*[ϖ Im{VmV¯c(1+iBeη-}],
c(α)η=Be*[ϖ Re{VmV¯c}]Beη.

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