Abstract

A global approach to phase calibration is presented. The corresponding theoretical framework calls on elementary concepts of algebraic graph theory (spanning tree of maximal weight, cycles) and algebraic number theory (lattice, nearest lattice point). The traditional approach can thereby be better understood. In radio imaging and in optical interferometry, the self-calibration procedures must often be conducted with much care. The analysis presented should then help in finding a better compromise between the coverage of the calibration graph (which must be as complete as possible) and the quality of the solution (which must of course be reliable).

© 2005 Optical Society of America

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References

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  1. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1970).
  2. A. Lannes, E. Anterrieu, P. Maréchal, “clean and wipe,” Astron. Astrophys. Suppl. Ser. 123, 183–198 (1997).
    [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. N. Biggs, Algebraic Graph Theory (Cambridge U. Press, Cambridge, UK, 1996).
  5. A. Lannes, S. Durand, “Dual algebraic formulation of differential GPS,” J. Geod. 77, 22–29 (2003).
    [CrossRef]
  6. A. Lannes, “Integer ambiguity resolution in phase closure imaging,” J. Opt. Soc. Am. A 18, 1046–1055 (2001).
    [CrossRef]
  7. E. Agrell, T. Eriksson, A. Vardy, K. Zeger, “Closest point search in lattices,” IEEE Trans. Inf. Theory 48, 2201–2214 (2002).
    [CrossRef]
  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.

2003 (1)

A. Lannes, S. Durand, “Dual algebraic formulation of differential GPS,” J. Geod. 77, 22–29 (2003).
[CrossRef]

2002 (1)

E. Agrell, T. Eriksson, A. Vardy, K. Zeger, “Closest point search in lattices,” IEEE Trans. Inf. Theory 48, 2201–2214 (2002).
[CrossRef]

2001 (1)

1997 (1)

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

1981 (1)

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

Agrell, E.

E. Agrell, T. Eriksson, A. Vardy, K. Zeger, “Closest point search in lattices,” IEEE Trans. Inf. Theory 48, 2201–2214 (2002).
[CrossRef]

Anterrieu, E.

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

Biggs, N.

N. Biggs, Algebraic Graph Theory (Cambridge U. Press, Cambridge, UK, 1996).

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1970).

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).

Durand, S.

A. Lannes, S. Durand, “Dual algebraic formulation of differential GPS,” J. Geod. 77, 22–29 (2003).
[CrossRef]

Eriksson, T.

E. Agrell, T. Eriksson, A. Vardy, K. Zeger, “Closest point search in lattices,” IEEE Trans. Inf. Theory 48, 2201–2214 (2002).
[CrossRef]

Lannes, A.

A. Lannes, S. Durand, “Dual algebraic formulation of differential GPS,” J. Geod. 77, 22–29 (2003).
[CrossRef]

A. Lannes, “Integer ambiguity resolution in phase closure imaging,” J. Opt. Soc. Am. A 18, 1046–1055 (2001).
[CrossRef]

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

Maréchal, P.

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

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.

Vardy, A.

E. Agrell, T. Eriksson, A. Vardy, K. Zeger, “Closest point search in lattices,” IEEE Trans. Inf. Theory 48, 2201–2214 (2002).
[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).

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1970).

Zeger, K.

E. Agrell, T. Eriksson, A. Vardy, K. Zeger, “Closest point search in lattices,” IEEE Trans. Inf. Theory 48, 2201–2214 (2002).
[CrossRef]

Astron. Astrophys. Suppl. Ser. (1)

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

IEEE Trans. Inf. Theory (1)

E. Agrell, T. Eriksson, A. Vardy, K. Zeger, “Closest point search in lattices,” IEEE Trans. Inf. Theory 48, 2201–2214 (2002).
[CrossRef]

J. Geod. (1)

A. Lannes, S. Durand, “Dual algebraic formulation of differential GPS,” J. Geod. 77, 22–29 (2003).
[CrossRef]

J. Opt. Soc. Am. A (1)

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

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 (3)

N. Biggs, Algebraic Graph Theory (Cambridge U. Press, Cambridge, UK, 1996).

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1970).

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 (7)

Fig. 1
Fig. 1

Baselines on which Ψ m can be regarded as an approximation to Ψ o define the edges of the calibration graph G m . In the phase calibration operation, G m is decomposed into its connected components. The graph G shown here is an example of such a component.

Fig. 2
Fig. 2

As illustrated here, Ψ d is the phase of the calibrated data. The optimal residual discrepancy Δ a [as defined in Eq. (13)] is the restriction of arc ( Ψ d - Ψ m ) to B.

Fig. 3
Fig. 3

Spanning trees and cycles. The graph G shown here, which is the same as that presented in Fig. 1, includes six vertices (six pupil elements) and ten edges (ten baselines): n a = 6 and n b = 10 . The thick lines correspond to the selected spanning tree. Such a tree includes n a - 1 edges (i.e., five in this case). The remaining edges define as many cycles ( n = 5 ) : cycles (1, 3, 2), (1, 4, 3, 2), (2, 4, 3), (3, 5, 4), and (4, 6, 5); note that the second includes four edges.

Fig. 4
Fig. 4

Spanning tree of maximal weight (example). For the weights w ( j ,   k ) displayed on the edges of the graph G shown here, the spanning tree of maximal weight is formed by the edges (1, 2), (3, 4), and (2, 4) (thick lines).

Fig. 5
Fig. 5

Canonical decomposition of the baseline phase space G. The range of D, L, is said to be the OPD phase space; L is its orthogonal complement. The spanning-tree phase space M is isomorphic to L; its orthogonal complement, K     M , which is isomorphic to L , is the cycle-entry phase space. The oblique projection of G onto K along L is denoted by T. Likewise, U is the oblique projection of G onto L along K. For a given choice of spanning tree, any ψ lying in G ( Z ) can be uniquely decomposed in the nonorthogonal form ψ = T ψ + U ψ with T ψ in K and U ψ in L; G is therefore the direct sum of K and L.

Fig. 6
Fig. 6

Canonical decomposition of lattice G ( Z ) . The intersection of G ( Z ) with the cycle-entry phase space K, K ( Z ) , is a lattice of rank n. The intersection of G ( Z ) with the OPD phase space L, L ( Z ) , is a lattice of rank n a - 1 . For a given choice of spanning tree, any α lying in G ( Z ) can be uniquely decomposed in the nonorthogonal form α = T α + U α with T α in K ( Z ) and U α in L ( Z ) ; G ( Z ) is therefore the direct sum of K ( Z ) and L ( Z ) .

Fig. 7
Fig. 7

Geometrical representation of the discrepancy vector ( ϕ ) . According to the definition of the linear ramp function [Eq. (41)], ( ϕ ) is the discrepancy between ψ - ϕ and the nearest point of G ( Z ) , ψ - ϕ [see Eq. (16)]. As illustrated here, at the end of the initialization step we have ψ = 0 , i.e., ψ = ramp   ψ . For any α in G ( Z ) , the projections of ψ - α onto L and L are denoted by ϕ ( α ) and δ ( α ) , respectively [Eqs. (47) and (49)].

Equations (118)

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s ˆ ( u )     s ( x ) exp ( - 2 i π u     x ) d x .
u ( j ,   k ) = r ( j ) - r ( k ) λ ,
V d ( j ,   k ) = V o ( j ,   k ) exp { i Φ d ( j ,   k ) } + error terms .
( j ,   k ) B m , Ψ m ( j ,   k ) Ψ o ( j ,   k ) .
f cal ( Φ )     ( j , k ) B m w 0 ( j ,   k ) | V d ( j ,   k ) exp { - i Φ ( j ,   k ) } - V m ( j ,   k ) | 2 .
V d = ρ d exp ( i Ψ d ) , V m = ρ m exp ( i Ψ m ) ,
f cal ( Φ ) = ( j , k ) B w 0 ( j ,   k ) | ρ d ( j ,   k ) exp { i [ Ψ ( j ,   k ) - Φ ( j ,   k ) ] } - ρ m ( j ,   k ) | 2
Ψ     Ψ d - Ψ m ( on B ) .
f c ( Φ )     ( j , k ) B w ( j ,   k ) | exp { i [ Ψ ( j ,   k ) - Φ ( j ,   k ) ] } - 1 | 2 ,
w ( j ,   k )     w 0 ( j ,   k ) ρ 2 ( j ,   k ) ( j , k ) B w 0 ( j ,   k ) ρ 2 ( j ,   k ) ,
chord β     | exp ( i β ) - 1 | = 2 | sin ( β / 2 ) | ,
f c ( Φ ) = ( j , k ) B w ( j ,   k ) chord 2 { Ψ ( j ,   k ) - Φ ( j ,   k ) } ;
f c ( Φ ) = chord ( Ψ - Φ ) 2 .
f a ( Φ )     | arc ( Ψ - Φ ) | 2 .
arc β     β - 2 π β 2 π ,
Φ a ( j ,   k ) = Θ a ( j ) - Θ a ( k ) ( on B d ) .
Ψ d ( j ,   k )     Ψ d ( j ,   k ) - [ Θ a ( j ) - Θ a ( k ) ] ( on B d ) .
Δ a     arc ( Ψ - Φ a ) ( on B )
[ [ Φ a ] c ] a = Φ a
( ϑ 1   |   ϑ 2 ) F     j A ϑ 1 ( j ) ϑ 2 ( j ) ,
F 0     { ϑ F : ϑ ( j 0 ) = 0 } .
( ψ 1   |   ψ 2 ) G     ( j , k ) B w ( j ,   k ) ψ 1 ( j ,   k ) ψ 2 ( j ,   k ) = 1 2 ( j , k ) B w ( j ,   k ) ψ 1 ( j ,   k ) ψ 2 ( j ,   k ) ,
D   :   F G , ( D ϑ ) ( j ,   k )     ϑ ( j ) - ϑ ( k ) .
dim   L = n a - 1 .
D 0   :   F 0 G , D 0 ϑ     D ϑ .
D 0 * D 0 ϑ = D 0 * ψ .
( D 0 * ψ ) ( j ) = 0 if j = j 0 k A \ { j } w ( j ,   k ) ψ ( j ,   k ) otherwise .
n = n b - ( n a - 1 ) .
PD 0 ϑ 0 = P ψ ( ϑ 0 F 0 ) .
ϑ 0 ( k ) = ϑ 0 ( j ) - ψ ( j ,   k ) .
C   :   G R n , C ψ     ψ
dim ( ker   C ) = dim   G - dim   R n = n b - n ;
ker   C = L .
C ξ   :   R n K , C ξ γ     i = 1 n γ [ i ] ξ i .
T     C ξ C
T ψ = i = 1 n ψ ( i ) ξ i .
D 0 ϑ 0 = U ψ ;
ψ ( i ) = ψ ( j i ,   k i ) - [ ϑ 0 ( j i ) - ϑ 0 ( k i ) ] .
S ψ = i = 1 n ψ ( i ) η i ,
η i     S ξ i .
C η   :   R n L , C η γ     i = 1 n γ [ i ] η i .
CC η γ = i = 1 n γ [ i ] C η i = i = 1 n γ [ i ] C ξ i = i = 1 n γ [ i ] ξ i = γ .
C η = C + .
C + = C * ( CC * ) - 1 ;
C η * C η = ( CC * ) - 1
( C ψ γ ) R n = [ C ψ ] t [ γ ] = [ ψ ] t [ C ] t [ γ ] = ( ψ   |   C * γ ) G = [ ψ ] t V - 1 [ C * γ ] = [ ψ ] t V - 1 [ C * ] [ γ ] .
[ CC * ] = V c .
[ C η * C η ] = V c - 1 .
Ψ d i = 1 n d Γ d ( i ) ξ d , i .
Ψ d ( j ,   k ) Ψ d ( j ,   k ) - [ Θ 0 ( j ) - Θ 0 ( k ) ]
( for all of B d ) .
Ψ = i = 1 n Ψ ( i ) ξ i with Ψ ( i ) arc ( Ψ ( i ) ) .
ramp   β     β - β .
arc   β = 2 π   ramp β 2 π ,
f a ( Φ ) = 4 π 2 ramp ( ψ - ϕ ) 2 .
f   :   L R , f ( ϕ )     ( ϕ ) 2 ,
( ϕ )     ramp ( ψ - ϕ ) .
| ( ϕ ) ( j ,   k ) | < τ ( j ,   k )
g : G × L R , g ( α ,   ϕ )     | ψ - ϕ - α | 2 .
ϕ ( α )     R ( ψ - α ) .
δ ( α )     ψ - ϕ ( α ) - α
δ ( α ) = S ( ψ - α ) .
δ ( α ) = C η ( ψ - α ) .
( γ |   γ ) H     ( C η γ |   C η γ ) G .
( γ |   γ ) H = ( γ V c - 1 γ ) R n .
δ ( α ) G 2 = α - α ˙   H 2 ,
α ˙     ψ .
q ( γ )     ( γ V c - 1 γ ) R n ,
δ ( α ) G 2 = q ( α - α ˙ ) .
α o C ξ α     i = 1 n α [ i ] ξ i .
( ϕ ( α ) ) = ψ - ϕ ( α ) - α = δ ( α ) .
( ϕ ( α ) + ϕ 0 ) = ( ψ - ϕ ( α ) - α ) - ϕ 0 = δ ( α ) - ϕ 0 .
( ϕ + t ϕ 0 ) = ( ϕ ) - t ϕ 0 .
J ¯ τ     J τ L ,
J τ     { ψ 0 G   :   ( j ,   k ) B ,   | ψ 0 ( j ,   k ) | < τ ( j ,   k ) } .
α Ω τ     C + ( α ˙ - α ) J ¯ τ     α - α ˙ C J ¯ τ .
E ( χ )     { α   :   q ( α - α ˙ ) < χ } .
δ ( α ) 2 = ( j , k ) B w ( j ,   k ) | δ ( α ) ( j ,   k ) | 2
χ τ     ( j , k ) B w ( j ,   k ) τ 2 ( j ,   k ) .
f c   :   L R , f c ( Φ ) = 2   sin { ( Ψ - Φ ) / 2 } G 2 .
f 0 c   :   F 0 R , f 0 c ( Θ )     f c ( D 0 Θ ) .
f 0 c ( Θ ) + ( f 0 c ( Θ ) | v ) F + 1 2   ( [ f 0 c ( Θ ) ] v   |   v ) F
f 0 c ( Θ ) = D 0 * [ f c ( Φ ) ] , f 0 c ( Θ ) = D 0 * [ f c ( Φ ) ] D 0 ,
f 0 c ( Θ ) = - 2 D 0 * sin ( Ψ - Φ ) ,
f 0 c ( Θ ) = 2 D 0 * [ cos ( Ψ - Φ ) ] D 0 .
ϕ ( α ) = ( ψ - α o ) - δ ( α ) ,
Ψ - Φ ( α ) = Δ ( α ) + 2 π α o .
f 0 c ( Θ ( α ) ) = - 2 D 0 * sin   Δ ( α ) ,
f 0 c ( Θ ( α ) ) = 2 D 0 * [ cos   Δ ( α ) ] D 0 .
α c     Ψ - Φ c 2 π
Ψ ( 1 ) = - 70 ° , Ψ ( 2 ) = - 40 ° , Ψ ( 3 ) = - 15 ° .
f a ( Φ a ) 10.62 ° , f a ( Φ a ) 69.79 ° ,
f a ( Φ a ) 71.04 ° .
α = ( 0 ,   0 ,   0 ) , α = ( - 1 ,   0 ,   - 1 ) ,
α = ( - 1 ,   - 1 ,   - 1 ) .
f c ( Φ a ) 10.43 ° , f c ( Φ a ) 55.14 ° ,
f c ( Φ a ) 55.76 ° .
Δ a ( 1 ,   2 ) 3 ° , Δ a ( 1 ,   3 ) - 47 ° ,
Δ a ( 1 ,   4 ) - 22 ° ,
Δ a ( 2 ,   3 ) 5 ° , Δ a ( 2 ,   4 ) 15 ° ,
Δ a ( 3 ,   4 ) - 5 ° .
Ψ ( 1 ) = - 171 ° , Ψ ( 2 ) = 176 ° , Ψ ( 3 ) = - 177 ° .
f a ( Φ a ) 38.39 ° , f a ( Φ a ) 39.63 ° ,
f a ( Φ a ) 57.47 ° .
α = ( 0 ,   1 ,   0 ) , α = ( - 1 ,   0 ,   - 1 ) ,
α = ( 0 ,   1 ,   - 1 ) .
f c ( Φ a ) 35.82 ° , f c ( Φ a ) 36.99 ° ,
f c ( Φ a ) 40.46 ° .
Δ a ( 1 ,   2 ) 7 ° , Δ a ( 1 ,   3 ) - 61 ° ,
Δ a ( 1 ,   4 ) - 101 ° ,
Δ a ( 2 ,   3 ) - 74 ° , Δ a ( 2 ,   4 ) 76 ° ,
Δ a ( 3 ,   4 ) - 27 ° ,
Δ a ( 1 ,   2 ) - 7 ° , Δ a ( 1 ,   3 ) 74 ° ,
Δ a ( 1 ,   4 ) 91 ° ,
Δ a ( 2 ,   3 ) 76 ° , Δ a ( 2 ,   4 ) - 78 ° ,
Δ a ( 3 ,   4 ) 29 ° .
D 0 * D 0 Θ = D 0 * Ψ .
D 0 * [ cos   Ψ ] D 0 Θ = D 0 * sin   Ψ .

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