Abstract

Phase calibration is the key operation of phase closure imaging. In the case of nonredundant arrays, the related problem amounts to finding the node of a ℤ lattice closest to the end of a vector, the components of which are the differences between the closure phases of the data and those of the model. The aim of the paper is to show that this integer ambiguity problem can be solved in a very efficient manner. Its potential instabilities can also be well identified. The corresponding approach, which can be extended to redundant arrays, revisits and completes the analysis presented in a recent paper entitled “Phase calibration on interferometric graphs” [J. Opt. Soc. Am. A 16, 443 (1999)]. The procedures of self-calibration must be modified accordingly. The spinoffs of this approach also concern the integer ambiguity problems encountered in the global positioning system.

© 2001 Optical Society of America

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References

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  1. A. Lannes, “Phase calibration on interferometric graphs,” J. Opt. Soc. Am. A 16, 443–454 (1999).
    [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, “Redundant spacing calibration: phase restoration methods,” J. Opt. Soc. Am. A 16, 2866–2879 (1999).
    [CrossRef]
  4. P. Teunissen, “The least-squares ambiguity decorrelation adjustment: a method for fast GPS integer ambiguity estimation,” J. Geodesy 70, 65–82 (1995).
    [CrossRef]
  5. P. J. de Jonge, “A processing strategy for the application of the GPS in networks,” in Publication on Geodesy (Netherlands Geodetic Commission, Delft, The Netherlands, 1998), Vol. 46.
  6. L. Delage, F. Reynaud, A. Lannes, “Laboratory imaging stellar interferometer with fiber links,” Appl. Opt. 39, 6406–6420 (2000).
    [CrossRef]

2000

1999

1998

1995

P. Teunissen, “The least-squares ambiguity decorrelation adjustment: a method for fast GPS integer ambiguity estimation,” J. Geodesy 70, 65–82 (1995).
[CrossRef]

Appl. Opt.

J. Geodesy

P. Teunissen, “The least-squares ambiguity decorrelation adjustment: a method for fast GPS integer ambiguity estimation,” J. Geodesy 70, 65–82 (1995).
[CrossRef]

J. Opt. Soc. Am. A

Other

P. J. de Jonge, “A processing strategy for the application of the GPS in networks,” in Publication on Geodesy (Netherlands Geodetic Commission, Delft, The Netherlands, 1998), Vol. 46.

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Figures (5)

Fig. 1
Fig. 1

Phase calibration principle. The phase functions β represented in this figure take their values on the baselines (j, k) of the interferometric graph to be considered. Here, notation such as βe therefore stands for βe(j, k). According to the terminology adopted in this paper, βe is the experimental baseline phase, βm is the model spectral phase, ϕ is the optimal bias phase, βe is the calibrated phase, and β is the optimal model phase shift (see text).

Fig. 2
Fig. 2

Example of interferometric graph (n=6). Baselines (2, 3), (2, 4), and (4, 5) are lacking so that dimG=12. The thick lines correspond to the selected spanning tree. Here, such a tree includes five baselines; the remaining baselines define as many loops: p=7 (see text).

Fig. 3
Fig. 3

Main decompositions of the baseline phase space G. The range of B is the bias phase space L. Its orthogonal complement L is the bias-free phase space; R and S are the corresponding orthogonal projections. The spanning-tree phase space E is the space induced by the selected spanning tree. Its orthogonal complement E is the loop-entry phase space; P and Q are the corresponding orthogonal projections. The phase closure operator C is the oblique projection of G onto E along L.

Fig. 4
Fig. 4

Canonical decomposition of the integer ambiguity lattice G(). The intersection of G() with the bias phase space L is a lattice of rank n-1, the bias integer ambiguity lattice L(). The intersection of G() with the loop-entry phase space E is a lattice of rank p, the loop-entry integer ambiguity lattice E(). For a given choice of spanning tree, any μG() can be decomposed in the nonorthogonal form μ=μ0+μ1 with μ0L() and μ1CμE(). The integer ambiguity lattice G() can therefore be regarded as the direct sum of L() and E().

Fig. 5
Fig. 5

Self-calibration on interferometric graphs. For the nine-element array shown here, two levels of resolution may be basically involved in the self-calibration procedure. The low-resolution interferometric graph (dashed lines) corresponds to the first inner circle of the spatial frequency coverage (bottom right). Note that the high-resolution interferometric graph (solid and dashed lines) is incomplete (18 baselines out of 36).

Equations (104)

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βe(j, k)=βm(j, k)+ϕ(j, k)+e(j, k).
ϕ(j, k)=α(j)-α(k)
arc(θ)θ-2πθ2π
ϖ(j, k)ϖ0(j, k)ae2(j, k)(j, k)Bϖ0(j, k)ae2(j, k),
β2(j, k)Bϖ(j, k)β2(j, k).
ψβe-βm,
g(ϕ)arc(ψ-ϕ)2=(j, k)Bϖ(j, k)arc2[ψ(j, k)-ϕ(j, k)].
βeβe-ϕ.
βarc(βe-βm)=arc(ψ-ϕ)
arc2[ψ(j, k)-ϕ(j, k)]=infμ(j, k)[ψ(j, k)-ϕ(j, k)-2πμ(j, k)]2,
g(μ, ϕ)ψ-ϕ-2πμ2.
(α1|α2)FjAα1(j)α2(j),
dim G=q.
(β1|β2)G12 (j, k)Bβ1(j, k)β2(j, k)ϖ(j, k),
(j, k)Bβ1(j, k)β2(j, k)ϖ(j, k),
B:FG,(Bα) (j, k)α(j)-α(k).
p=q-(n-1).
ξi(j, k)1ifj=ji andk=ki-1ifj=ki andk=ji0otherwise(i=1, , p)
γβ(1)β(2, 5)+β(5, 1)+β(1, 2).
C: GE,Cβi=1pγβ(i)ξi.
C2=C,
dim(ker C)=dim G-dim E=q-p,
ker C=L,
Sβ=i=1pγβ(i)ηi,
ηiSξi.
ξiCηi.
μ=(j, k)Bμj,kbj,kwithμj,kμ(j, k).
bj,k(j, k)1ifj=jandk=k-1ifj=kandk=j0otherwise.
L()G()L,E()G()E
am(j)=1ifj=m0otherwise(m=1, , n).
μ1Cμ=i=1pγμ(i)ξi.
μ=μ0+μ1withμ0L(),μ1E().
g: G()×L,g(μ, ϕ)(ψ-ϕ)-2πμ2=(ψ-2πμ)-ϕ2.
μ=ψ-ϕ2π,ϕ=R(ψ-2πµ).
ψ-ϕ-2πµ=arc(ψ-ϕ)=(ψ-2πµ)-R(ψ-2πµ)=S(ψ-2πµ),
β=S(ψ-2πµ).
G()=L()E(),
Cψ=i=1pγψ(i)ξi.
μ1=i=1pn(i)ξi[n(i)].
xˆγψ2π,
c:p×L,
c(n, ϕ)2πi=1p[xˆ(i)-n(i)] ξi-ϕ2.
ϕn=2πi=1p[xˆ(i)-n(i)]φi,φiRξi.
c1:p,c1(n)4π2i=1p[xˆ(i)-n(i)]ηi2.
c1(n)=β(n)2,
β(n)2πi=1p[xˆ(i)-n(i)]ηi=i=1p[γψ(i)-2πn(i)]ηi=S(ψ-2πµ1).
c(n, ϕ)Pϕ2+2πi=1p[xˆ(i)-n(i)]ξi-Qϕ2.
Qϕ=2πi=1pxϕ(i)ξi,xϕ(i)12πϕ(ji, ki).
c(n, ϕ)=ϕ2+4π2i=1p[xˆ(i)-xϕ(i)-n(i)]2ϖ(i)
nϕ=xˆ-xϕ,
nϕ(i)=xˆ(i)-xϕ(i)(i=1, , p).
c2:L,
c2(ϕ)Pϕ2+4π2i=1p[xˆ(i)-xϕ(i)-nϕ(i)]2ϖ(i).
V:pL,Vx2πi=1px(i)ηi.
c1(n)=β(n)2=V(xˆ-n)2=Ω(xˆ-n),
Ω(x)(x|Wx)p,WV*V.
W=U*U.
Ω(x)=Uxp2.
c1(n)=i=1pri2(n),
ri(n)j=ipuij[n(j)-xˆ(j)].
E(n){x:Ω(xˆ-x)β(n)2}.
β=i=1pγ(i)ηi,γ(i)γψ(i)-2πn(i).
β=S(ψ-2πµ)=(ψ-2πµ)-ϕ,
ϕ=ψ-β-2πi=1pn(i)ξi.
α=Bref(-1)Pϕ,
n0=xˆ.
Ω[xˆ-(n0+m(i)ei)]=Ω(xˆ-n0)-2(xˆ-n0|Wei)pm(i)+Ω(ei)m(i)2.
aiΩ(ei)=(ei|Wei)p=wi,i
bi-2(xˆ-n0|Wei)p
=-2j=iptjuji,tjk=jpujk[xˆ(k)-n0(k)].
c:p×L,
c(x, ϕ)2πi=1p[xˆ(i)-x(i)]ξi-ϕ2.
ϕx2πl=1p[xˆ(l)-x(l)]φl,φlRξl.
nϕx=xˆ-xϕx,
xϕx(i)=l=1p(xˆ(l)-x(l))φl(ji, ki)(i=1, , p).
x1, 2xˆ-x1; x2, 2xˆ-x2;
υ=i=1pri2,
rij=ipuijζj,ζjn(j)-xˆ(j).
ri2+l=i+1prl2ε.
yiri2,ziε-l=i+1prl2ifi<pεifi=p,
yizi.
zi=ε-l=i+2prl2-ri+12,
zi=zi+1-yi+1.
yp=(uppζp)2,zp=ε.
-zirizi.
ri=uiiζi+r˜i
r˜ij=i+1puijζjifi<p0ifi=p.
-1uii(zi+r˜i)ζi1uii(zi-r˜i),
xˆ(i)-1uii(zi+r˜i)n(i)xˆ(i)+1uii(zi-r˜i).
xˆ(p)-1upp εn(p)xˆ(p)+1upp ε.
xˆ(1)-1u11(z1+r˜1)n(1)xˆ(1)+1u11(z1-r˜1).
y1=r12=(u11ζ1+r˜1)2[ζ1n(1)-xˆ(1)].
υ=ε-(z1-y1).
ϖ(1,2)=0.55,ϖ(1,3)=0.03,ϖ(1,4)=0.02,
ϖ(2,3)=0.06,ϖ(2,4)=0.11,ϖ(3,4)=0.23.
γ(1)=55°,γ(2)=40°,γ(3)=-30°.
γ(1)=-72°,γ(2)=54°,γ(3)=-162°,
n0(1)=0, n0(2)=0, n0(3)=-1,(180/π)β(n0)25°,
γ(1)=-6°γ(2)=-176°γ(3)=13°,
n(1)=0, n(2)=0, n(3)=-1,(180/π)β38°.
γ(1)=-8°γ(2)=-165°γ(3)=22°,
β=i=1p arc[γβe(i)]ηi.
ϕ=βe-β-2πi=1pγβe(i)2πξi.
βel(j, k)=βel(j, k)-[αl(j)-αl(k)].

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