Abstract

An irreducible support is one that ensures a unique solution to the phase problem without requiring that it be explicitly imposed as a reconstruction constraint. However, the only nontrivial support that has been shown to be irreducible is Eisenstein’s support. Herein a general description of support irreducibility is developed for discrete functions, both over positive, real and over complex numbers. In addition, a method of testing an arbitrary support of finite extent for irreducibility is introduced, and a number of simple irreducible supports are presented that are not of the Eisenstein type.

© 1987 Optical Society of America

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References

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  1. R. P. Boas, Entire Functions (Academic, New York, 1954).
  2. G. Ross, M. A. Fiddy, M. Nieto-Vesperinas, M. W. L. Wheeler, “The phase problem in scattering phenomena: the zeros of entire functions and their significance,” Proc. R. Soc. London Ser. A. 360, 25–45 (1978).
    [CrossRef]
  3. A. H. Greenaway, “Proposal for phase recovery from a single intensity distribution,” Opt. Lett. 1, 10–12 (1977).
    [CrossRef] [PubMed]
  4. R. J. Sault, “Uniqueness condition for the phase problem in one dimension,” Opt. Lett. 9, 325–326 (1984).
    [CrossRef] [PubMed]
  5. T. R. Crimmins, J. R. Fienup, “Uniqueness of phase retrieval for functions with sufficiently disconnected support,”J. Opt. Soc. Am. 73, 218–221 (1983).
    [CrossRef]
  6. R. H. T. Bates, “Uniqueness of solutions to two-dimensional Fourier phase problems for localized and positive images,” Comput. Vision Graphics Image Process. 25, 205–217 (1984).
    [CrossRef]
  7. J. R. Fienup, “Reconstruction of objects having latent reference points,”J. Opt. Soc. Am. 73, 1421–1426 (1983).
    [CrossRef]
  8. H. A. Arsenault, K. Chalasinska-Macukow, “The solution to the phase retrieval problem using the sampling theorem,” Opt. Commun. 47, 380–386 (1983).
    [CrossRef]
  9. R. H. T. Bates, “Fourier phase problems are uniquely solvable in more than one dimension. I: Underlying theory,” Optik 61, 247–262 (1982).
  10. M. H. Hayes, T. F. Quatieri, “Recursive phase retrieval using boundary conditions,”J. Opt. Soc. Am. 73, 1427–1433 (1983).
    [CrossRef]
  11. J. R. Fienup, “Phase retrieval using boundary conditions,” J. Opt. Soc. Am. A 2, 284–288 (1986).
    [CrossRef]
  12. J. R. Fienup, T. R. Crimmins, W. Holsztynski, “Reconstruction of the support of an object from the support of its autocorrelation,”J. Opt. Soc. Am. 72, 610–624 (1982).
    [CrossRef]
  13. M. A. Fiddy, B. J. Brames, J. C. Dainty, “Enforcing irreducibility for phase retrieval in two dimensions,” Opt. Lett. 8, 96–98 (1983).
    [CrossRef] [PubMed]
  14. B. L. van der Waerden, Modern Algebra, 5th ed. (Ungar, New York, 1970), Vol. I.
  15. Yu. Bruck, L. G. Sodin, “On the ambiguity of the image reconstruction problem,” Opt. Commun. 30, 304–308 (1979).
    [CrossRef]
  16. M. H. Hayes, J. H. McClellan, “Reducible polynomials in more than one variable,” Proc. IEEE 70, 197–198 (1982).
    [CrossRef]
  17. N. K. Bose, “Sum and product separabilities of multivariable functions and applications,”J. Franklin Inst. 299, 53–66 (1975).
    [CrossRef]
  18. M. A. Fiddy, J. C. Dainty, “The essential role of prior knowledge in phase retrieval,” Opt. Acta 31, 325–330 (1984).
    [CrossRef]
  19. B. J. Brames, “Uniqueness and other aspects of the optical phase problem,” Ph.D. dissertation (University of Rochester, Rochester, N.Y., 1985).
  20. B. J. Brames, “Unique phase retrieval with explicit support information,” Opt. Lett. 11, 61–63 (1986).
    [CrossRef] [PubMed]
  21. B. Grünbaum, Convex Polytopes (Wiley, London, 1967).
  22. W. Prenowitz, J. Janosciak, Join Geometries (Springer-Verlag, New York, 1979).
    [CrossRef]
  23. M. Nieto-Vesperinas, J. C. Dainty, “A note on Eisenstein’s irreducibility criterion for two-dimensional sampled objects,” Opt. Commun. 54, 333–334 (1985).
    [CrossRef]

1986 (2)

1985 (1)

M. Nieto-Vesperinas, J. C. Dainty, “A note on Eisenstein’s irreducibility criterion for two-dimensional sampled objects,” Opt. Commun. 54, 333–334 (1985).
[CrossRef]

1984 (3)

M. A. Fiddy, J. C. Dainty, “The essential role of prior knowledge in phase retrieval,” Opt. Acta 31, 325–330 (1984).
[CrossRef]

R. J. Sault, “Uniqueness condition for the phase problem in one dimension,” Opt. Lett. 9, 325–326 (1984).
[CrossRef] [PubMed]

R. H. T. Bates, “Uniqueness of solutions to two-dimensional Fourier phase problems for localized and positive images,” Comput. Vision Graphics Image Process. 25, 205–217 (1984).
[CrossRef]

1983 (5)

1982 (3)

J. R. Fienup, T. R. Crimmins, W. Holsztynski, “Reconstruction of the support of an object from the support of its autocorrelation,”J. Opt. Soc. Am. 72, 610–624 (1982).
[CrossRef]

M. H. Hayes, J. H. McClellan, “Reducible polynomials in more than one variable,” Proc. IEEE 70, 197–198 (1982).
[CrossRef]

R. H. T. Bates, “Fourier phase problems are uniquely solvable in more than one dimension. I: Underlying theory,” Optik 61, 247–262 (1982).

1979 (1)

Yu. Bruck, L. G. Sodin, “On the ambiguity of the image reconstruction problem,” Opt. Commun. 30, 304–308 (1979).
[CrossRef]

1978 (1)

G. Ross, M. A. Fiddy, M. Nieto-Vesperinas, M. W. L. Wheeler, “The phase problem in scattering phenomena: the zeros of entire functions and their significance,” Proc. R. Soc. London Ser. A. 360, 25–45 (1978).
[CrossRef]

1977 (1)

1975 (1)

N. K. Bose, “Sum and product separabilities of multivariable functions and applications,”J. Franklin Inst. 299, 53–66 (1975).
[CrossRef]

Arsenault, H. A.

H. A. Arsenault, K. Chalasinska-Macukow, “The solution to the phase retrieval problem using the sampling theorem,” Opt. Commun. 47, 380–386 (1983).
[CrossRef]

Bates, R. H. T.

R. H. T. Bates, “Uniqueness of solutions to two-dimensional Fourier phase problems for localized and positive images,” Comput. Vision Graphics Image Process. 25, 205–217 (1984).
[CrossRef]

R. H. T. Bates, “Fourier phase problems are uniquely solvable in more than one dimension. I: Underlying theory,” Optik 61, 247–262 (1982).

Boas, R. P.

R. P. Boas, Entire Functions (Academic, New York, 1954).

Bose, N. K.

N. K. Bose, “Sum and product separabilities of multivariable functions and applications,”J. Franklin Inst. 299, 53–66 (1975).
[CrossRef]

Brames, B. J.

Bruck, Yu.

Yu. Bruck, L. G. Sodin, “On the ambiguity of the image reconstruction problem,” Opt. Commun. 30, 304–308 (1979).
[CrossRef]

Chalasinska-Macukow, K.

H. A. Arsenault, K. Chalasinska-Macukow, “The solution to the phase retrieval problem using the sampling theorem,” Opt. Commun. 47, 380–386 (1983).
[CrossRef]

Crimmins, T. R.

Dainty, J. C.

M. Nieto-Vesperinas, J. C. Dainty, “A note on Eisenstein’s irreducibility criterion for two-dimensional sampled objects,” Opt. Commun. 54, 333–334 (1985).
[CrossRef]

M. A. Fiddy, J. C. Dainty, “The essential role of prior knowledge in phase retrieval,” Opt. Acta 31, 325–330 (1984).
[CrossRef]

M. A. Fiddy, B. J. Brames, J. C. Dainty, “Enforcing irreducibility for phase retrieval in two dimensions,” Opt. Lett. 8, 96–98 (1983).
[CrossRef] [PubMed]

Fiddy, M. A.

M. A. Fiddy, J. C. Dainty, “The essential role of prior knowledge in phase retrieval,” Opt. Acta 31, 325–330 (1984).
[CrossRef]

M. A. Fiddy, B. J. Brames, J. C. Dainty, “Enforcing irreducibility for phase retrieval in two dimensions,” Opt. Lett. 8, 96–98 (1983).
[CrossRef] [PubMed]

G. Ross, M. A. Fiddy, M. Nieto-Vesperinas, M. W. L. Wheeler, “The phase problem in scattering phenomena: the zeros of entire functions and their significance,” Proc. R. Soc. London Ser. A. 360, 25–45 (1978).
[CrossRef]

Fienup, J. R.

Greenaway, A. H.

Grünbaum, B.

B. Grünbaum, Convex Polytopes (Wiley, London, 1967).

Hayes, M. H.

M. H. Hayes, T. F. Quatieri, “Recursive phase retrieval using boundary conditions,”J. Opt. Soc. Am. 73, 1427–1433 (1983).
[CrossRef]

M. H. Hayes, J. H. McClellan, “Reducible polynomials in more than one variable,” Proc. IEEE 70, 197–198 (1982).
[CrossRef]

Holsztynski, W.

Janosciak, J.

W. Prenowitz, J. Janosciak, Join Geometries (Springer-Verlag, New York, 1979).
[CrossRef]

McClellan, J. H.

M. H. Hayes, J. H. McClellan, “Reducible polynomials in more than one variable,” Proc. IEEE 70, 197–198 (1982).
[CrossRef]

Nieto-Vesperinas, M.

M. Nieto-Vesperinas, J. C. Dainty, “A note on Eisenstein’s irreducibility criterion for two-dimensional sampled objects,” Opt. Commun. 54, 333–334 (1985).
[CrossRef]

G. Ross, M. A. Fiddy, M. Nieto-Vesperinas, M. W. L. Wheeler, “The phase problem in scattering phenomena: the zeros of entire functions and their significance,” Proc. R. Soc. London Ser. A. 360, 25–45 (1978).
[CrossRef]

Prenowitz, W.

W. Prenowitz, J. Janosciak, Join Geometries (Springer-Verlag, New York, 1979).
[CrossRef]

Quatieri, T. F.

Ross, G.

G. Ross, M. A. Fiddy, M. Nieto-Vesperinas, M. W. L. Wheeler, “The phase problem in scattering phenomena: the zeros of entire functions and their significance,” Proc. R. Soc. London Ser. A. 360, 25–45 (1978).
[CrossRef]

Sault, R. J.

Sodin, L. G.

Yu. Bruck, L. G. Sodin, “On the ambiguity of the image reconstruction problem,” Opt. Commun. 30, 304–308 (1979).
[CrossRef]

van der Waerden, B. L.

B. L. van der Waerden, Modern Algebra, 5th ed. (Ungar, New York, 1970), Vol. I.

Wheeler, M. W. L.

G. Ross, M. A. Fiddy, M. Nieto-Vesperinas, M. W. L. Wheeler, “The phase problem in scattering phenomena: the zeros of entire functions and their significance,” Proc. R. Soc. London Ser. A. 360, 25–45 (1978).
[CrossRef]

Comput. Vision Graphics Image Process. (1)

R. H. T. Bates, “Uniqueness of solutions to two-dimensional Fourier phase problems for localized and positive images,” Comput. Vision Graphics Image Process. 25, 205–217 (1984).
[CrossRef]

J. Franklin Inst. (1)

N. K. Bose, “Sum and product separabilities of multivariable functions and applications,”J. Franklin Inst. 299, 53–66 (1975).
[CrossRef]

J. Opt. Soc. Am. (4)

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

Opt. Acta (1)

M. A. Fiddy, J. C. Dainty, “The essential role of prior knowledge in phase retrieval,” Opt. Acta 31, 325–330 (1984).
[CrossRef]

Opt. Commun. (3)

Yu. Bruck, L. G. Sodin, “On the ambiguity of the image reconstruction problem,” Opt. Commun. 30, 304–308 (1979).
[CrossRef]

H. A. Arsenault, K. Chalasinska-Macukow, “The solution to the phase retrieval problem using the sampling theorem,” Opt. Commun. 47, 380–386 (1983).
[CrossRef]

M. Nieto-Vesperinas, J. C. Dainty, “A note on Eisenstein’s irreducibility criterion for two-dimensional sampled objects,” Opt. Commun. 54, 333–334 (1985).
[CrossRef]

Opt. Lett. (4)

Optik (1)

R. H. T. Bates, “Fourier phase problems are uniquely solvable in more than one dimension. I: Underlying theory,” Optik 61, 247–262 (1982).

Proc. IEEE (1)

M. H. Hayes, J. H. McClellan, “Reducible polynomials in more than one variable,” Proc. IEEE 70, 197–198 (1982).
[CrossRef]

Proc. R. Soc. London Ser. A. (1)

G. Ross, M. A. Fiddy, M. Nieto-Vesperinas, M. W. L. Wheeler, “The phase problem in scattering phenomena: the zeros of entire functions and their significance,” Proc. R. Soc. London Ser. A. 360, 25–45 (1978).
[CrossRef]

Other (5)

R. P. Boas, Entire Functions (Academic, New York, 1954).

B. J. Brames, “Uniqueness and other aspects of the optical phase problem,” Ph.D. dissertation (University of Rochester, Rochester, N.Y., 1985).

B. L. van der Waerden, Modern Algebra, 5th ed. (Ungar, New York, 1970), Vol. I.

B. Grünbaum, Convex Polytopes (Wiley, London, 1967).

W. Prenowitz, J. Janosciak, Join Geometries (Springer-Verlag, New York, 1979).
[CrossRef]

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Figures (9)

Fig. 1
Fig. 1

Eisenstein’s support consists of the hatched rectangular region plus point B. It is irreducible for discrete functions if the function is nonzero at points A and B.

Fig. 2
Fig. 2

An example of indistinguishable components. The view is from above the s plane, with nonzero (unit) values of a characteristic function set at black or gray circles representing points in a support. Observe that Sh and Sh′ are components of Sf in (a) and (b) and that S f / / S h and S f / / S h yield Sg in (c) and (d); therefore Sh and Sh′ are indistinguishable.

Fig. 3
Fig. 3

Testing a support for P irreducibility. Nonzero values of the characteristic function are at points of the support displayed as filled circles. The function in (a) is F reducible, but its characteristic function is P irreducible. Note that the correlation of Sf and Sg′ is shown, not the normalized correlation, because of space limitations.

Fig. 4
Fig. 4

The function depicted in (a) is F reducible, although its characteristic function is P irreducible. A nonequivalent solution is shown in (b).

Fig. 5
Fig. 5

The characteristic function defined on the support in (a) is P irreducible. Adding five points gives rise to a P-reducible function with components Sg and Sh, which correspond to those of an F-reducible function defined in (a).

Fig. 6
Fig. 6

Several examples of a support A and its convex hull CH(A). All points in the respective sets are indicated by shaded regions or filled circles.

Fig. 7
Fig. 7

The vertex points of a convex set are sufficient to describe it. Points in a set are indicated by shaded regions or filled circles.

Fig. 8
Fig. 8

Nonunique sets of kernels. The characteristic function defined on the support in (a) or (b) may be reduced to two mutually exclusive sets (indicated by 1 and 2) of C-irreducible components (kernels).

Fig. 9
Fig. 9

Increasingly complex examples of C-irreducible supports are shown in (a)–(e). Those in (f) are C reducible. Points in a support are represented by filled circles, whereas the corresponding convex hull is indicated by the shaded regions.

Equations (55)

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F 0 ( z ) = exp ( A + B z ) z k n = 1 ( 1 - z z n ) exp ( z / z n ) ,
f ( s ) = j γ j δ ( s - a j ) ,             γ j 0 ,
δ ( s ) = { 1 for s = 0 0 otherwise .
P ( w 1 , w 2 ) = n = 0 N m = 0 M f ( n , m ) w 1 n w 2 m = k = 1 K p k ( w 1 , w 2 ) .
f 0 ( s ) = g ( s ) * h ( s ) ,
f 1 ( s ) = g * ( - s ) * h ( s )
A Sup [ f ( s ) ] = { a j } ,
S f ( s ) = { 1 for f ( s ) > 0 0 for f ( s ) = 0 .
g ( s ) = k = 1 K α k δ ( s - a k )
h ( s ) = l = 1 L β l δ ( s - b l ) ,
f ( s ) = k , l α k β l δ [ s - ( a k + b l ) ] .
f ˜ ( s ) = k , l α k β l δ [ s - ( a k + b l ) ] = g ( s ) * h ( s ) .
S f ˜ = S ( g * h )
= S ( S g * S h ) ,
f ( s 0 ) = 0
f ˜ ( s 0 ) > 0
θ 0 θ c < θ 0 + π 2
f ( s ) = l β l { k α k δ [ s - ( a k + b l ) ] }
= l β l g ( s - b l ) .
T ( s ) = [ S f ( s ) ] [ S g ( s ) ]
= d t S { k , l α k β l δ [ t - ( a k + b l ) ] } × S { i α i δ ( t - s - a i ) }
= l S { k , l δ [ s + a i - ( a k + b l ) ] }
= i S { l , k = i δ [ s + a i - ( a k + b l ) ] } + i S { l , k = i δ [ s + a i - ( a k + b l ) ] } + i S { l , k i δ [ s + a i - ( a k + b l ) ] } × { m [ 1 - δ ( s - d m ) ] } ,
T 1 ( s ) = i S { l , k = i [ s + a i - ( a k + b l ) ] }
= K l δ ( s - b l ) = K S h ( s ) .
S h = S h + i δ ( s - u i ) ,
o ( s ) = S o ( s ) = δ { T ( s ) t S g ( t ) - 1 } ,
f ( s 1 , s 2 ) = δ ( s 2 ) [ δ ( s 2 ) + 3 δ ( s 1 - 1 ) + 3 δ ( s 1 - 2 ) + δ ( s 1 - 3 ) ] + δ ( s 1 , s 2 - 3 ) .
f ( s 1 , s 2 ) = [ δ ( s 1 , s 2 ) + δ ( s 1 - 1 , s 2 ) + δ ( s 1 , s 2 - 1 ) ] * { δ ( s 2 ) [ 1 + 2 δ ( s 1 - 1 ) + δ ( s 1 - 2 ) ] - δ ( s 2 - 1 ) [ 1 + δ ( s 1 - 1 ) ] + δ ( s 1 , s 2 - 2 ) } .
r = λ z i + ( 1 - λ ) z j ,             0 λ 1
Z = CH [ CI ( Z ) ] .
C = A + B { a k + b l a k A ; b l B } .
CH ( C ) = CH ( A + B )
= CH [ CH ( A ) + CH ( B ) ]
= CH ( A ) + CH ( B ) ,
f ( c j v ) = g ( a k v ) h ( b l v ) .
CI ( C ) = CI [ CH ( C ) ]
= CI [ CH ( A ) + CH ( B ) ]
= CI [ CI ( A ) + CI ( B ) ] ,
CH ( h ) CH [ CI ( h ) U ] CH ( h ) + r
CH ( f ) = CH { [ CI ( h ) U ] + CI ( g ) }
= CH ( h ) + CH ( g ) .
CH ( f ) = CH ( h ) + CH ( g ) .
q = λ z 1 + ( 1 - λ ) z 2 ,             0 λ 1
q = λ ( x 1 + y 1 ) + ( 1 - λ ) ( x 2 + y 2 ) ,
q = { λ x 1 + ( 1 - λ ) x 2 } + { λ y 1 + ( 1 - λ ) y 2 } .
q = x + y Z ;
z v = x 1 + y 1 = x 2 + y 2 ,
z v = ½ { 2 z v } = ½ { x 1 + y 1 + x 2 + y 2 }
= λ ( x 1 + y 1 ) + ( 1 - λ ) ( y 1 + x 2 ) λ = ½
= λ q + ( 1 - λ ) r λ = ½ .
x 1 = λ s + ( 1 - λ ) t ,             s , t X ,             0 < λ < 1
z v = [ λ s + ( 1 - λ ) t ] + y 1
= λ ( s + y 1 ) + ( 1 - λ ) ( t + y 1 ) ,             0 < λ < 1.
P ( w 1 , w 2 ) = a 0 ( w 2 ) + a 1 ( w 2 ) w 1 + a 2 ( w 2 ) w 1 2 + + a n ( w 2 ) w 1 n .

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