Abstract

New methods for redundant spacing calibration (RSC) are proposed. These are based on recent studies concerning phase calibration and the related phase unwrapping problem. In the corresponding theoretical framework, two subspaces of the baseline phase space, the variational spectral phase space K and the aberration baseline phase space L, play an important role. An interferometric device for which KL is reduced to {0} is said to be of full phase. For any imaging device of this type, including those for which the traditional recursive approach fails, the phase restoration problem can be solved in the least-squares sense. When the closure phases are strongly blurred, global instabilities may occur. Their analysis appeals to elementary concepts of algebraic number theory: ℤ lattice, reduced basis, and closest node. In all cases, the separation angle between K and L must be as large as possible. The imaging devices based on the RSC principle should be designed accordingly. All these points are illustrated by considering the phase restoration problems encountered in passive remote sensing by aperture synthesis. The difficulties related to possible correlator failures are examined in this context.

© 1999 Optical Society of America

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References

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  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, “Weak phase imaging in optical interferometry,” J. Opt. Soc. Am. A 15, 811–824 (1998).
    [CrossRef]
  3. A. Lannes, E. Anterrieu, “Image reconstruction methods for remote sensing by aperture synthesis,” in Proceedings of the International Geoscience and Remote Sensing Symposium, 1994 (Institute of Electrical and Electronics Engineers, New York, 1994), pp. 2892–2903.
  4. A. H. Greenaway, D. P. Cheese, J. D. Bregman, J. E. Noordam, “TOAST: a terrestrial optical aperture synthesis technique,” in Proceedings of Interferometry Imaging in Astronomy (European Space Agency/National Optical Astronomy Observatories, Garching, Germany, 1987), pp. 153–156.
  5. A. H. Greenaway, “Self-calibrating dilute-aperture optics,” in Digital Image Synthesis and Inverse Optics, A. F. Gmitro, P. S. Idell, I. J. LaHaie, eds., Proc. SPIE1351, 738–748 (1990).
    [CrossRef]
  6. A. Lannes, “Spectral reduction of phase closure data and stabilized image reconstruction via a new hybrid procedure,” in High-Resolution Imaging by Interferometry II, J. M. Beckers, F. Merkle, eds., Vol. 39 of ESO Conference and Workshop Proceedings (European Southern Observatory, Garching, Germany, 1991), pp. 881–887.
  7. A. Lannes, “Phase calibration on interferometric graphs,” J. Opt. Soc. Am. A 16, 443–454 (1999).
    [CrossRef]
  8. A. Lannes, “Abstract holography,” J. Math. Anal. Appl. 74, 530–559 (1980).
    [CrossRef]
  9. P. Roman, Some Modern Mathematics for Physicists and Other Outsiders, Vol. 1 of Algebra, Topology, and Measure Theory (Pergamon, New York, 1975).
  10. 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.
  11. 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]

1999

1998

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]

1980

A. Lannes, “Abstract holography,” J. Math. Anal. Appl. 74, 530–559 (1980).
[CrossRef]

Anterrieu, E.

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, “Image reconstruction methods for remote sensing by aperture synthesis,” in Proceedings of the International Geoscience and Remote Sensing Symposium, 1994 (Institute of Electrical and Electronics Engineers, New York, 1994), pp. 2892–2903.

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]

Bregman, J. D.

A. H. Greenaway, D. P. Cheese, J. D. Bregman, J. E. Noordam, “TOAST: a terrestrial optical aperture synthesis technique,” in Proceedings of Interferometry Imaging in Astronomy (European Space Agency/National Optical Astronomy Observatories, Garching, Germany, 1987), pp. 153–156.

Cheese, D. P.

A. H. Greenaway, D. P. Cheese, J. D. Bregman, J. E. Noordam, “TOAST: a terrestrial optical aperture synthesis technique,” in Proceedings of Interferometry Imaging in Astronomy (European Space Agency/National Optical Astronomy Observatories, Garching, Germany, 1987), pp. 153–156.

Greenaway, A. H.

A. H. Greenaway, “Self-calibrating dilute-aperture optics,” in Digital Image Synthesis and Inverse Optics, A. F. Gmitro, P. S. Idell, I. J. LaHaie, eds., Proc. SPIE1351, 738–748 (1990).
[CrossRef]

A. H. Greenaway, D. P. Cheese, J. D. Bregman, J. E. Noordam, “TOAST: a terrestrial optical aperture synthesis technique,” in Proceedings of Interferometry Imaging in Astronomy (European Space Agency/National Optical Astronomy Observatories, Garching, Germany, 1987), pp. 153–156.

Lannes, A.

A. Lannes, “Phase calibration on interferometric graphs,” J. Opt. Soc. Am. A 16, 443–454 (1999).
[CrossRef]

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, “Weak phase imaging in optical interferometry,” J. Opt. Soc. Am. A 15, 811–824 (1998).
[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, “Abstract holography,” J. Math. Anal. Appl. 74, 530–559 (1980).
[CrossRef]

A. Lannes, E. Anterrieu, “Image reconstruction methods for remote sensing by aperture synthesis,” in Proceedings of the International Geoscience and Remote Sensing Symposium, 1994 (Institute of Electrical and Electronics Engineers, New York, 1994), pp. 2892–2903.

A. Lannes, “Spectral reduction of phase closure data and stabilized image reconstruction via a new hybrid procedure,” in High-Resolution Imaging by Interferometry II, J. M. Beckers, F. Merkle, eds., Vol. 39 of ESO Conference and Workshop Proceedings (European Southern Observatory, Garching, Germany, 1991), pp. 881–887.

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.

Noordam, J. E.

A. H. Greenaway, D. P. Cheese, J. D. Bregman, J. E. Noordam, “TOAST: a terrestrial optical aperture synthesis technique,” in Proceedings of Interferometry Imaging in Astronomy (European Space Agency/National Optical Astronomy Observatories, Garching, Germany, 1987), pp. 153–156.

Roman, P.

P. Roman, Some Modern Mathematics for Physicists and Other Outsiders, Vol. 1 of Algebra, Topology, and Measure Theory (Pergamon, New York, 1975).

J. Math. Anal. Appl.

A. Lannes, “Abstract holography,” J. Math. Anal. Appl. 74, 530–559 (1980).
[CrossRef]

J. Mod. Opt.

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

Other

A. Lannes, E. Anterrieu, “Image reconstruction methods for remote sensing by aperture synthesis,” in Proceedings of the International Geoscience and Remote Sensing Symposium, 1994 (Institute of Electrical and Electronics Engineers, New York, 1994), pp. 2892–2903.

A. H. Greenaway, D. P. Cheese, J. D. Bregman, J. E. Noordam, “TOAST: a terrestrial optical aperture synthesis technique,” in Proceedings of Interferometry Imaging in Astronomy (European Space Agency/National Optical Astronomy Observatories, Garching, Germany, 1987), pp. 153–156.

A. H. Greenaway, “Self-calibrating dilute-aperture optics,” in Digital Image Synthesis and Inverse Optics, A. F. Gmitro, P. S. Idell, I. J. LaHaie, eds., Proc. SPIE1351, 738–748 (1990).
[CrossRef]

A. Lannes, “Spectral reduction of phase closure data and stabilized image reconstruction via a new hybrid procedure,” in High-Resolution Imaging by Interferometry II, J. M. Beckers, F. Merkle, eds., Vol. 39 of ESO Conference and Workshop Proceedings (European Southern Observatory, Garching, Germany, 1991), pp. 881–887.

P. Roman, Some Modern Mathematics for Physicists and Other Outsiders, Vol. 1 of Algebra, Topology, and Measure Theory (Pergamon, New York, 1975).

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 (6)

Fig. 1
Fig. 1

Top, redundant array A; bottom, corresponding complete graph (A, B). By definition, B is the set of all the baselines generated by A.

Fig. 2
Fig. 2

Graph (A, Be) involved in the special case studied in Section 5. Baselines (2, 3), (2, 4), and (4, 5) are lacking, so that dim Ge=12. The thick lines correspond to the selected spanning tree. Here such a tree includes five baselines; the remaining baselines define as many cycles: p=7 (see text).

Fig. 3
Fig. 3

Projection onto K+L. In this geometrical representation, σ and τ denote the projections of β onto K+L and (K+L), respectively. When the separation angle ψ between K and L is strictly positive (see property 5 later in the paper), the decomposition σ=g+h with gK and hL is unique.

Fig. 4
Fig. 4

Projection of lattice Lη(Z) onto (K+L). In this illustration, where L is of dimension 2, U[Lη(Z)] is a lattice of rank 1 with basis vector U(η2-η1); i.e., τ2-τ1. Note that 2τ2-3τ1=0. When KL={0}, the intersection of Lη(Z) with SK is the projection of lattice Kϱ(Z) onto L (see Fig. 3 and the proof of property 4). The closest node relative to the RSC phase restoration process is not necessarily the same as that relative to the related phase calibration operation (for further details see Fig. 5 and Subsection 3.A).

Fig. 5
Fig. 5

Example of RSC instability. For clarity, we have chosen a situation where the projection of η2 onto (K+L) coincides with that of η1:τ1=τ2. (The general case is illustrated in Fig. 4.) When the baseline phase i=1pγ[ε](i)ηi moves from zone (a) to zone (b) in a continuous manner, g jumps from its (a)-value g(a) to its (b)-value g(b).

Fig. 6
Fig. 6

Geometrical illustration of the main parameters involved in the phase restoration process; ψ is the separation angle between K and L. Note that δ0, δo, and δm are norms of calibration phase shifts. By definition, ε0e=arc(βe-ϑ0) [see Eq. (38)]. Here S(ε0e-2πκ) and U(ε0e-2πκ) are greater than δ0 and δm, respectively.

Equations (134)

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βe(j, k)=βo(j, k)+[α(j)-α(k)]+ei(j, k).
γe(j, k, l)  βe(j, k)+βe(k, l)+βe(l, j)
=βo(j, k)+βo(k, l)+βo(l, j)
 γo(j, k, l).
βo(1, 3)=2βo(1, 2)-γe(1, 2, 3)(modulo2π).
βo(1, 4)=βo(1, 2)+βo(1, 3)-γe(1, 2, 4)
(modulo2π).
(α1|α2)F  jAα1(j)α2(j),
(β1|β2)Ge  12 (j, k)Beϖ(j, k)β1(j, k)β2(j, k)
 (j, k)Beϖ(j, k)β1(j, k)β2(j, k),
βo=βr+fo,
er  βr-ϑr.
Be : FGe,(Beα)(j, k)  α(j)-α(k).
βi(j, k)1ifj=jiandk=ki-1ifj=kiandk=ji0otherwise(i=1,, p).
Sβ=i=1pγ[β](i)ηi,ηi  Sβi.
(Mβ)(j, k)
 β(j, k)ifj=jiandk=kiβ(j, k)ifj=kiandk=ji0otherwise(i=1,, p).
arc(θ)=θ-2πq{θ/(2π)},
|arc(θ)| =infkZ|θ-2πk|.
ζ(j, k)exp{iβ(j, k)}(βGe).
φ(ζ1, ζ2)  arc(β1-β2).
φ2(ζ1, ζ2)=(j, k)Beϖ(j, k)arc2{β1(j, k)-β2(j, k)}.
c:(K+L)R,c(σ)  φ{ζe, ξr exp(iσ)}.
ξm  ξr exp(igr)
ζc  ζe exp(-ihr)
KL={0}
ϖ(j, k)  ρe2(j, k)(j, k)Beρe2(j, k),
φ(ξm, ζc)=infξKm(C)ζLc(C) φ(ξ, ζ).
ϑo  ϑr+fo.
βo-ϑm=(βr-ϑr)+(fo-qr)=(βr-ϑr)+(ϑo-ϑm).
arc(βo-ϑm)2=arc(er)2+arc(ϑm-ϑo)2.
φ2(ξm, ζo)=φ2(ξr, ζr)+φ2(ξo, ξm).
βe=ϑo+Beα+e,e  ei+er.
ε  arc(β-ϑ).
β-ϑ=(β-ϑ)-(g+h)modulo2π,
ε  arc(β-ϑ)=arc{(β-ϑ)-Tε}=arc(ε-Tε)=arc(Uε).
Ge=SKL(K+L).
g=A+β,A+ SK-1V.
PSKg=PSβ.
σ=g+h(gK, hL).
(g|PSKg)=(g|SKg)=(SKg|SKg)=SKg2,
dim Kdim L,i.e.,mp,
dim(K+L)=p-m.
ε(κ)  ε-2πκ,κGe(Z).
εmc=arc(β-ϑ)-σ-2πκ
T[ε(κ)]-Tσ=0;
|arc(β-ϑ)|  |(β-ϑ)-(g+h)-2πκ|;
S[ε(κ)]=i=1p(γ[ε](i)-2πk(i))ηi,ηi  Sβi,
U[ε(κ)]=i=1p(γ[ε](i)-2πk(i))τi,τi  Uηi.
U[Lη(Z)](K+L)τ(Z).
SK-1VS[Kϱ(Z)]=SK-1S[Kϱ(Z)]=Kϱ(Z),
g=i=1pγ[ε](i)bi,bi  SK-1Vβi=A+βi.
cos ψ  supgK,g=1hL,h=1(g|h),ψ[0, π/2].
SK-1=1sin ψ
υ  infg=1SKg
SK-1  sups0 gs=1infg0 sg.
(g|h)=(Rg|h)Rgh  Rg;
cos ψ=supg=1RKg=RK,
υ2=1-supg=1RKg2=1-cos2 ψ=sin2 ψ.
PSK=P(IK-RK)=IK-PRK.
VE=cos μ,μ  (E, SK).
ϑm=ϑ+A+i=1pγ[ε](i)βi.
ε0=S[ε0e(κ)],ε0e  arc(βe-ϑ0),
ε0=infκGe(Z)S[ε0e(κ)],
S[ε0e(κ)]=i=1p(γ[ε0e](i)-2πk(i))ηi.
ε0=i=1parc(γ[ε0e](i))ηi=Si=1parc(γ[ε0e](i))βi.
ϑm=ϑ0+g0m,
g0m=A+i=1parc(γ[ε0e](i))βi.
f(σ)=(j, k)Beϖ(j, k)|ζe(j, k)-ξr(j, k)exp[iσ(j, k)]|2.
ϑm=ϑ˜m+A+[arc(β˜c-ϑ˜m)].
f(σ)=-2T[sin(β-ϑ)],
[f(σ)]ς=2T[ς cos(β-ϑ)],
ϑ  ϑr+g,β  βe-h.
ϑm=ϑo+A+i=1parc(γ[εoe](i))βi,
ϑm-ϑo=A+i=1parc(γ[e](i))βi,
ϑm-ϑo νΩe,ν  A+M,
Ωe2  i=1pϖ(ji, ki)arc2(γ[e](i)).
A+M=(PSK)-1PSM.
(A+M)*(A+M)=MS(PSK)-2PSM.
ϑm-ϑo cos μsin ψ Ωe.
ϖ(1, 2)=0.20,ϖ(1, 3)=0.06,
ϖ(1, 4)=0.05,ϖ(1, 5)=0.03,
ϖ(1, 6)=0.01.
βo(1, 2)=0°,βo(1, 3)=172°,
βo(1, 4)=40°,βo(1, 5)=-10°,
βo(1, 6)=15°.
βe(1, 2)=0.76°,βe(1, 3)49.4°,
βe(1, 4)34.8°,βe(1, 5)-133.7°,
βe(1, 6)-64.3°.
ϑm(1, 2)=0°,ϑm(1, 3)-178.9°,
ϑm(1, 4)58.6°,ϑm(1, 5)17.4°,
ϑm(1, 6)36.3°,
βc(1, 2)=-0.05°,βc(1, 3)179.4°,
βc(1, 4)60.7°,βc(1, 5)17.8°,
βc(1, 6)36.3°.
δm  φ{ξm, Lc(C)}=φ(ξm, ζc),
δ0  φ{ξ0,Lc(C)}=ε0
ϑm(1, 2)=0°,ϑm(1, 3)-151.4°,
ϑm(1, 4)114.2°,ϑm(1, 5)99.7°,
ϑm(1, 6)100.2°,
u[k(1) ,, k(p)]  i=1p(γ[ε](i)-2πk(i))τi.
ϑm=ϑ+A+i=1pγ[ε]*(i)βi.
γ[ε]*(i)=arc(γ[ε](i)).
γ[ε]*(i)=γ[ε](i).
εmc=Uεmc.
γ[εmc]*(i)=γ[εmc](i).
γ[ε˜mc]*(i)=γ[ε˜mc](i).
ϑm=ϑ˜m+A+i=1pγ[ε˜mc](i)βi.
ϑm-ϑo=A+i=1parc(γ[e](i))βi.
Ωe2  i=1pϖ(ji, ki)arc2(γ[e](i)),
ϑm-ϑo cos μsin ψ Ωe,
Ψ1|Ψ2Ge  (j, k)Beϖ(j, k)Ψ¯1(j, k)Ψ2(j, k)
=(j, k)Beϖ(j, k)Re{Ψ¯1(j, k)Ψ2(j, k)},
f(σ)=12zre-zGe2.
f(σ+ς)=12zre-z(1+iς-)Ge2=f(σ)+zre-z|-izςGe+ ,
zre-z|-izςGe=i(zrez¯-1)|ςGe=(j, k)Beϖ(j, k)×Re{izre(j, k)z¯(j, k)}ς(j, k)=2(-Im{zrez¯}|ς)Ge=2(-T Im{zrez¯}|ς)Ge.
f(σ)=-2T Im{zrez¯}.
f(σ+ς)=-2T[Im{zrez¯(1-iς+}],
[f(σ)]ς=2T[ς Re{zrez¯}].
zrez¯=exp[i(βe-ϑr-g-h)].
f(σ)=-2T sin(β-ϑ),
[f(σ)]ς=2T[ς cos(β-ϑ)].
q:[K+L]R,
q(ς)  f(σ)+(f(σ)|ς)+12[f(σ)ς|ς].
f  f(σ),g  f(σ),h  f(σ),
q(ς)=f+(g|ς)+12(hς|ς).
qt(ς)  q(ς)+12tς2(t0).
qt(ς)=f+(g|ς)+12(htς|ς),
ht  h+tI,
ςt=-ht-1g,tt0  max(-χ+, 0).
q(ς)+12tς2 >q(ςt)+12tςt2(tt0);
q(ς)>q(ςt)+12t(ςt2-ς2)(tt0).
f(σ)-f(σ+ςt)>a[q(0)-q(ςt)](0<a<1).
q(0)-q(ςt)=-(g|ςt)-12(hςt|ςt)=12[tςt2-(g|ςt)].

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