Abstract

A new (to the author’s knowledge) approach to phase calibration is proposed. The corresponding theoretical framework calls on elementary concepts of algebraic graph theory (spanning trees, cycles) and computational algebraic number theory (reduced basis, closest node). The notion of phase closure is revisited in this context. One main result of this analysis is to provide all the elements for understanding the phase calibration instabilities that may occur. The phase unwrapping problem in question can be solved, directly, by minimizing the size of certain arcs. The corresponding direct methods are often quite efficient. The problem can also be solved in an indirect manner. The bulk of the work is then to minimize the length of certain chords. The pros and cons of the related methods are considered, and a link is made with the techniques developed so far. The problem of phase averaging, which is evidently much simpler, can be dealt with in a similar manner. As the corresponding results are intuitively obvious, one then has access to a more global understanding of the matter. In practice, the implications of this approach therefore concern all the methodologies in which the notions of phase closure and phase averaging prove to play an important role: self-calibration techniques in radio imaging and in optical interferometry, redundant spacing calibration in microwave imaging by aperture synthesis, etc.

© 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. T. J. Cornwell, P. N. Wilkinson, “A new method for making maps with unstable interferometers,” Mon. Not. R. Astron. Soc. 196, 1067–1086 (1981).
  4. H. Cohen, A Course in Computational Algebraic Number Theory (Springer-Verlag, Berlin, 1996).
  5. B. Vallée, “Gauss’ algorithm revisited,” J. Algorithms 12, 556–572 (1991).
    [CrossRef]
  6. B. Vallée, “An affine algorithm for minima finding in integer lattices of three dimensions,” Cah. GREYC (Université de Caen) 3, 1–22 (1995).

1998

1995

B. Vallée, “An affine algorithm for minima finding in integer lattices of three dimensions,” Cah. GREYC (Université de Caen) 3, 1–22 (1995).

1991

B. Vallée, “Gauss’ algorithm revisited,” J. Algorithms 12, 556–572 (1991).
[CrossRef]

1981

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

Cohen, H.

H. Cohen, A Course in Computational Algebraic Number Theory (Springer-Verlag, Berlin, 1996).

Cornwell, T. J.

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

Lannes, A.

Vallée, B.

B. Vallée, “An affine algorithm for minima finding in integer lattices of three dimensions,” Cah. GREYC (Université de Caen) 3, 1–22 (1995).

B. Vallée, “Gauss’ algorithm revisited,” J. Algorithms 12, 556–572 (1991).
[CrossRef]

Wilkinson, P. N.

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

Cah. GREYC (Université de Caen)

B. Vallée, “An affine algorithm for minima finding in integer lattices of three dimensions,” Cah. GREYC (Université de Caen) 3, 1–22 (1995).

J. Algorithms

B. Vallée, “Gauss’ algorithm revisited,” J. Algorithms 12, 556–572 (1991).
[CrossRef]

J. Opt. Soc. Am. A

Mon. Not. R. Astron. Soc.

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

Other

H. Cohen, A Course in Computational Algebraic Number Theory (Springer-Verlag, Berlin, 1996).

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Figures (3)

Fig. 1
Fig. 1

Example of an interferometric graph (A, e). Such a graph is connected but not necessarily homogeneous (or complete). Here e is a subset of ℬ, including only 9 baselines (out of 15).

Fig. 2
Fig. 2

Spanning tree and cycles of the interferometric graph (A, e) shown in Fig. 1 (n=6, q=9). The boldface lines correspond to the selected spanning tree. Here such a tree includes five baselines; the remaining four baselines define as many cycles (see text).

Fig. 3
Fig. 3

Notion of reduced basis. In a certain context, the ℤ lattice presented here may be defined through the initial basis {η1,η2}. In the special case under consideration the basis {η1η1, η23η1+η2} is a reduced basis: Its elements are as short as possible. As a result, such a basis is not too far from being orthogonal.

Equations (114)

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Ve(j, k)=Vo(j, k)exp{iβ(j, k)},(j, k)e.
β(j, k)=α(j)-α(k),
Vc Ve exp(-iβ)
arc(θ)=θ-2πq{θ/(2π)},
ζ(j, k)exp{iβ(j, k)}(βGe).
φ:Ge(C)×Ge(C),
φ2(ζ1, ζ2)(j, k)eϖ(j, k)arc2{β1(j, k)-β2(j, k)}.
B : FG,(Bα)(j, k)α(j)-α(k),
Be : FGe,(Beα)(j, k)α(j)-α(k).
c : BeF,c(β)φ{ζe, ζm exp(iβ)}.
ϖ(j, k)ϖe(j, k)ρe2(j, k)(j, k)eϖe(j, k)ρe2(j, k),
c : BeF,c(β)φ{ζe exp(-iβ), ζm}
c : BeF,c(β)φ{ζem, exp(iβ)},
ζemζeζ¯m.
ζcζe exp(-iβ)
φ2(ζ1, ζ2)=(j, k)eϖ(j, k)infκ(j, k)[β1(j, k)-β2(j, k)-2πκ(j, k)]2.
infκGe() (j, k)eϖ(j, k)[β1(j, k)-β2(j, k)-2πκ(j, k)]2=(j, k)eϖ(j, k)infκ(j, k)[β1(j, k)-β2(j, k)-2πκ(j, k)]2
φ(ζ1, ζ2)=infκGe()β1-β2-2πκ.
β1 | β2(β1 | ϖβ2)(j, k)eϖ(j, k)β1(j, k)β2(j, k)=12(j, k)Beϖ(j, k)β1(j, k)β2(j, k).
c(β)=infκGe()βem-β-2πκ,
c(β)=infβBeFc(β)=infβBeF infκGe()(βem-2πκ)-β=infκGe() infβBeF(βem-2πκ)-β.
β(κ)R(βem-2πκ),
(βem-2πκ)-β(κ)=S(βem-2πκ),
c(β)=infκGe()S(βem-2πκ).
c(β)=S(βem-2πκ).
S(βem-2πκ)=(βem-2πκ)-β(κ),
S(βem-2πκ)=βc-βm(modulo2π).
βsS(βem-2πκ),
βc=βm+βs(modulo2π).
φ(ζc, ζm)=βs.
ζc=ζm exp(iβs).
β=arc(β).
(j, k)eϖ(j, k)[arc2{βs(j, k)}-βs2(j, k)]=0.
β1 | β2(β1 | ϖβ2)ϖ(1, 2)β1(1, 2)β2(1, 2)+ϖ(2, 3)β1(2, 3)β2(2, 3)+ϖ(3, 1)β1(3, 1)β2(3, 1).
b(1, 2)1ϖ(1, 2),b(2, 3)1ϖ(2, 3), b(3, 1)1ϖ(3, 1).
b | Bα=ϖ(1, 2)ϖ(1, 2)[α(1)-α(2)]+ϖ(2, 3)ϖ(2, 3)[α(2)-α(3)]+ϖ(3, 1)ϖ(3, 1)[α(3)-α(1)]=0.
b2=1ϖ(1, 2)+1ϖ(2, 3)+1ϖ(3, 1).
S(βem-2πκ)=b  βem-2πκ bb2=(γem-2πμ) bb2,
γemβem(1, 2)+βem(2, 3)+βem(3, 1), μκ(1, 2)+κ(2, 3)+κ(3, 1)
ηbb2,
S(βem-2πκ)=(γ-2πμ)η,γγem.
(γ-2πμ)η=2π|γ/(2π)-μ|η,
γ-2πμ=γ-2πq{γ/(2π)}=arc(γ).
S(βem-2πκ)=arc(γ)η.
βs=arc(γ)η,
η(j, k)=1ϖ(j, k)1ϖ(1, 2)+1ϖ(2, 3)+1ϖ(3, 1).
p=q-(n-1).
βi(j, k)1ifj=ji,k=ki-1ifj=ki,k=ji0otherwise(i=1,, p).
ker Ct=BeF.
g1βem-2πκ,g2i=1p[γ(i)-2πμ(i)]βi.
S(βem-2πκ)=i=1p[γ(i)-2πμ(i)]ηi,
ηiSβi.
S(βem-2πκ)2i=1p[γ(i)-2πμ(i)]2ηi2.
βsa=i=1parc[γ(i)]ηi.
βsa=Sβa=βa-Rβa,βai=1parc[γ(i)]βi.
Be*Beα=Be*β,
βγi=1pγ(i)ηi,
S(βem-2πκ)=βγ-β2πμ.
βsi=1parc(γ(i))ηi.
Ω(γ, δγ)=1,
Ω(γ, δγ)φ[βs(γ+δγ), βs(γ)]βδγ.
c˜ : BeF,c˜(β)(j, k)eϖ(j, k)|ζem(j, k)-exp{iβ(j, k)}|2.
ζ˜cζe exp(-iβ˜),
βb=(βe-β˜)-βm-2πκ˜=βem-2πκ˜-β˜;
Sβb=S(βem-2πκ˜).
βsSβbβb.
βsbSβb
βsb=i=1pγb(i)ηi=i=1pγb(i)ηi.
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.
arc[γ(1)]=55°,arc[γ(2)]=40°, arc[γ(3)]=-30°.
βsa(1, 2)=3.38°,βsa(1, 3)=-47.17°, βsa(1, 4)=-22.2°,βsa(2, 3)=4.45°, βsa(2, 4)=14.46°,βsa(3, 4)=-4.99°.
arc[γ(1)]=-72°,arc[γ(2)]=54°, arc[γ(3)]=-162°.
βs(1, 2)=-2.27°,βs(1, 3)=96.35°, βs(1, 4)=-82.14°,βs(2, 3)=26.62°, βs(2, 4)=-25.87°,βs(3, 4)=19.51°.
arc[γ(1)]=-6°,arc[γ(2)]=-176°, arc[γ(3)]=13°,
βsb(1, 2)=7.02°,βsb(1, 3)=-61.12°,
βsb(1, 4)=-101.43°,βsb(2, 3)=-74.14°,
βsb(2, 4)=75.55°,βsb(3, 4)=-27.31°.
arc[γ(1)]=-8°,arc[γ(2)]=-165°, arc[γ(3)]=22°,
βsb(1, 2)=-7.08°,βsb(1, 3)=75.01°,
βsb(1, 4)=82.12°,βsb(2, 3)=74.09°,
βsb(2, 4)=-75.80°,βsb(3, 4)=29.11°.
α(j, k)ϖ(j, k)arc2{βem(j, k)-(Bα)(j, k)}.
c : ,c(θ)1nj=1narc2(θj-θ).
θ0(k)=θk+g(θk)(modulo2π),
g(θ)1nj=1narc(θj-θ).
σ0(k)[c(θk)-g2(θk)]1/2,
c˜ : ,c˜(θ)1nj=1n|exp(iθj)-exp(iθ)|2.
θ˜0=θ˜+g(θ˜)(modulo2π).
(β1 | β2)12(j, k)Beβ1(j, k)β2(j, k).
(Be*β)(j)=kAkjβ(0)(j, k),
β(0)(j, k)β(j, k)if(j, k)Be0otherwiseonB.
aj(k)1ifk=j0otherwise.
bj,k(l, m)1ifl=j,m=k-1ifl=k,m=j0otherwise.
Bi=bji,l1+bl1,l2++blr-1,ki.
Bt*Bi=(aji-al1)+(al1-al2)++(alr-1-aki)=aji-aki.
Bt*Btαi=aji-aki
β | Beα=12(j, k)Beϖ(j, k)β(j, k)[α(j)-α(k)]=12(j, k)Beα(j)[ϖ(j, k)β(j, k)]+12(j, k)Beα(k)[ϖ(k, j)β(k, j)]=(j, k)Beα(j)[ϖ(j, k)β(j, k)]=(j, k)Bα(j)[ϖ(j, k)β(0)(j, k)],
β(0)(j, k)β(j, k)if(j, k)Be0otherwiseonB.
(α1 | α2)FjAα1(j)α2(j),
(Be*β)(j)=kAϖ(j, k)β(0)(j, k)=kAkjϖ(j, k)β(0)(j, k).
ξ(j)exp{iα(j)}(αF),
δ : F(C)×F(C), δ(ξ1, ξ2)1nj=1narc2{α1(j)-α2(j)},
A : F,(Aθ)(j)θ.
(Pα)(j)=1nk=1nα(k)(jA).
Q(α-2πκ)=QjA{[α(j)-α(k)]-2π[κ(j)-κ(k)]}aj,
aj(l)1ifl=j0otherwise.
Q(α-2πκ)=jAjk[β(j, k)-2πμj]hj
β(j, k)θj-θk,
hj(l)(Qaj)(l)=1-1/nifl=j-1/notherwise.
(hi | hj)F=1-1/nifj=i-1/notherwise.
Q(α-2πκ)=j=1narc[β(j, k)]hj.
θ0(k)=α(k)-j=1narc[β(j, k)]hj(k)=θk+1nj=1narc[β(j, k)](modulo2π).
[σ0(k)]21nj=1narc[β(j, k)]hj2=1ni=1nj=1narc[β(i, k)]arc[β(j, k)](hi | hj)F.
[σ0(k)]2=1nj=1narc2[β(j, k)]-1n2j=1narc[β(j, k)]2.

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