Abstract

In general, the reconstructed image in coded aperture imaging is affected by the source configuration. Fenimore’s balanced convolution method in conjunction with the uniformly redundant array can remove the interference due to the source configuration. As an extension of Fenimore’s balanced convolution method, we present general conditions for designing an unbiased mean estimator for a far-field coded aperture imaging system with a random binary mask. As part of the general conditions, we propose decoding arrays whose elements are variable with respect to source directions. We also show that the unbiased mean estimator from Fenimore’s balanced convolution method is a special case of the general conditions. We also present a practical example of designing restoring arrays for a coded aperture system with a random mask.

© 2013 Optical Society of America

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References

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  1. E. Caroli, J. B. Stephen, G. Di Gocco, L. Natalucci, and A. Spizzichino, “Coded aperture imaging in x- and gamma-ray astronomy,” Space Sci. Rev. 45349–403 (1987).
    [CrossRef]
  2. E. E. Fenimore, “Coded aperture imaging: predicted performance of uniformly redundant arrays,” Appl. Opt. 17, 3562–3570 (1978).
    [CrossRef]
  3. C. Brown, “Multiplex imaging with multi-pinhole cameras,” J. Appl. Phys. 45, 1806–1811 (1974).
    [CrossRef]
  4. E. F. Fenimore and T. M. Cannon, “Coded aperture imaging with uniformly redundant arrays,” Appl. Opt. 17, 337–347 (1978).
    [CrossRef]
  5. E. Caroli, R. C. Butler, G. Di Cocco, P. P. Maggili, L. Natalucci, and A. Spizzichino, “Coded Masks in X- and gamma-ray astronomy: the problem of the signal-to-noise ratio evaluation,” Il Nuovo Cimento 7, 786–804 (1984).
    [CrossRef]
  6. S. R. Gottesman and E. E. Fenimore, “New family of binary arrays for coded aperture imaging,” Appl. Opt. 28, 4344–4352 (1989).
    [CrossRef]

1989 (1)

1987 (1)

E. Caroli, J. B. Stephen, G. Di Gocco, L. Natalucci, and A. Spizzichino, “Coded aperture imaging in x- and gamma-ray astronomy,” Space Sci. Rev. 45349–403 (1987).
[CrossRef]

1984 (1)

E. Caroli, R. C. Butler, G. Di Cocco, P. P. Maggili, L. Natalucci, and A. Spizzichino, “Coded Masks in X- and gamma-ray astronomy: the problem of the signal-to-noise ratio evaluation,” Il Nuovo Cimento 7, 786–804 (1984).
[CrossRef]

1978 (2)

1974 (1)

C. Brown, “Multiplex imaging with multi-pinhole cameras,” J. Appl. Phys. 45, 1806–1811 (1974).
[CrossRef]

Brown, C.

C. Brown, “Multiplex imaging with multi-pinhole cameras,” J. Appl. Phys. 45, 1806–1811 (1974).
[CrossRef]

Butler, R. C.

E. Caroli, R. C. Butler, G. Di Cocco, P. P. Maggili, L. Natalucci, and A. Spizzichino, “Coded Masks in X- and gamma-ray astronomy: the problem of the signal-to-noise ratio evaluation,” Il Nuovo Cimento 7, 786–804 (1984).
[CrossRef]

Cannon, T. M.

Caroli, E.

E. Caroli, J. B. Stephen, G. Di Gocco, L. Natalucci, and A. Spizzichino, “Coded aperture imaging in x- and gamma-ray astronomy,” Space Sci. Rev. 45349–403 (1987).
[CrossRef]

E. Caroli, R. C. Butler, G. Di Cocco, P. P. Maggili, L. Natalucci, and A. Spizzichino, “Coded Masks in X- and gamma-ray astronomy: the problem of the signal-to-noise ratio evaluation,” Il Nuovo Cimento 7, 786–804 (1984).
[CrossRef]

Di Cocco, G.

E. Caroli, R. C. Butler, G. Di Cocco, P. P. Maggili, L. Natalucci, and A. Spizzichino, “Coded Masks in X- and gamma-ray astronomy: the problem of the signal-to-noise ratio evaluation,” Il Nuovo Cimento 7, 786–804 (1984).
[CrossRef]

Di Gocco, G.

E. Caroli, J. B. Stephen, G. Di Gocco, L. Natalucci, and A. Spizzichino, “Coded aperture imaging in x- and gamma-ray astronomy,” Space Sci. Rev. 45349–403 (1987).
[CrossRef]

Fenimore, E. E.

Fenimore, E. F.

Gottesman, S. R.

Maggili, P. P.

E. Caroli, R. C. Butler, G. Di Cocco, P. P. Maggili, L. Natalucci, and A. Spizzichino, “Coded Masks in X- and gamma-ray astronomy: the problem of the signal-to-noise ratio evaluation,” Il Nuovo Cimento 7, 786–804 (1984).
[CrossRef]

Natalucci, L.

E. Caroli, J. B. Stephen, G. Di Gocco, L. Natalucci, and A. Spizzichino, “Coded aperture imaging in x- and gamma-ray astronomy,” Space Sci. Rev. 45349–403 (1987).
[CrossRef]

E. Caroli, R. C. Butler, G. Di Cocco, P. P. Maggili, L. Natalucci, and A. Spizzichino, “Coded Masks in X- and gamma-ray astronomy: the problem of the signal-to-noise ratio evaluation,” Il Nuovo Cimento 7, 786–804 (1984).
[CrossRef]

Spizzichino, A.

E. Caroli, J. B. Stephen, G. Di Gocco, L. Natalucci, and A. Spizzichino, “Coded aperture imaging in x- and gamma-ray astronomy,” Space Sci. Rev. 45349–403 (1987).
[CrossRef]

E. Caroli, R. C. Butler, G. Di Cocco, P. P. Maggili, L. Natalucci, and A. Spizzichino, “Coded Masks in X- and gamma-ray astronomy: the problem of the signal-to-noise ratio evaluation,” Il Nuovo Cimento 7, 786–804 (1984).
[CrossRef]

Stephen, J. B.

E. Caroli, J. B. Stephen, G. Di Gocco, L. Natalucci, and A. Spizzichino, “Coded aperture imaging in x- and gamma-ray astronomy,” Space Sci. Rev. 45349–403 (1987).
[CrossRef]

Appl. Opt. (3)

Il Nuovo Cimento (1)

E. Caroli, R. C. Butler, G. Di Cocco, P. P. Maggili, L. Natalucci, and A. Spizzichino, “Coded Masks in X- and gamma-ray astronomy: the problem of the signal-to-noise ratio evaluation,” Il Nuovo Cimento 7, 786–804 (1984).
[CrossRef]

J. Appl. Phys. (1)

C. Brown, “Multiplex imaging with multi-pinhole cameras,” J. Appl. Phys. 45, 1806–1811 (1974).
[CrossRef]

Space Sci. Rev. (1)

E. Caroli, J. B. Stephen, G. Di Gocco, L. Natalucci, and A. Spizzichino, “Coded aperture imaging in x- and gamma-ray astronomy,” Space Sci. Rev. 45349–403 (1987).
[CrossRef]

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Figures (29)

Fig. 1.
Fig. 1.

Illustration of the coded aperture imaging system.

Fig. 2.
Fig. 2.

Conversion of the 2D representation of arrays to the vector representation.

Fig. 3.
Fig. 3.

Decoding arrays based on a look-up table.

Fig. 4.
Fig. 4.

Encoding mask based on a 3-by-5 URA.

Fig. 5.
Fig. 5.

3-by-5 submasks for decoding.

Fig. 6.
Fig. 6.

Superposition of submasks to construct the whole decoding array.

Fig. 7.
Fig. 7.

Illustration of the setup of a binary mask and detector.

Fig. 8.
Fig. 8.

10-by-18 random mask.

Fig. 9.
Fig. 9.

Detector was projected on the mask plane to show the alignment.

Fig. 10.
Fig. 10.

Location of the first restoring array with respect to the encoding binary mask.

Fig. 11.
Fig. 11.

Location of the ninth restoring array with respect to the binary mask.

Fig. 12.
Fig. 12.

Encoding mask for the first direction.

Fig. 13.
Fig. 13.

Design of a restoring array based on a binary encoding array.

Fig. 14.
Fig. 14.

Restoring array with α’s [ρ(i)=αi] in place of the opaque elements.

Fig. 15.
Fig. 15.

Implementation of a restoring array with varying α’s.

Fig. 16.
Fig. 16.

Illustration of a ray-tracing setup.

Fig. 17.
Fig. 17.

(a) Detected counts of direction 1: the density of incoming rays is 16 rays/square, and the detected average is about 14.7 rays/square due to the oblique incoming angle. (b) Detected counts of direction 3.

Fig. 18.
Fig. 18.

Poisson sampling of the detected counts: (a) direction 1, (b) direction 3, (c) sum of directions 1 and 3.

Fig. 19.
Fig. 19.

Comparison of estimates from two restoring arrays for 10 consecutive realizations; dots indicate the estimates from the proposed method and stars indicate the estimates from Fenimore’s (+1/1) restoring array.

Fig. 20.
Fig. 20.

13-by-20 binary mask.

Fig. 21.
Fig. 21.

Direction 1 in the 13-by-20 mask. The upper left 10-by-10 submask is aligned for direction 1.

Fig. 22.
Fig. 22.

Interference coefficients of Fenimore’s (+1,1) restoring array for direction 1.

Fig. 23.
Fig. 23.

Interference coefficients of the proposed method for direction 1.

Fig. 24.
Fig. 24.

Restoring array for direction 1.

Fig. 25.
Fig. 25.

Comparison of sums of squares of the interference coefficients with respect to the flux directions; dots are for the proposed method and stars are for Fenimore’s (+1,1) method.

Fig. 26.
Fig. 26.

Illustration of a coded aperture system. This picture was adapted from [2].

Fig. 27.
Fig. 27.

Overlapped shadow from two sources. This picture was adapted from [2].

Fig. 28.
Fig. 28.

Correlation between the effective decoding array and the recorded shadow. This picture was adapted from [2].

Fig. 29.
Fig. 29.

Conversion of the 2D representation of arrays to the 1D representation.

Tables (2)

Tables Icon

Table 1. Coefficients for Interference from Varying α’s

Tables Icon

Table 2. Coefficients for Interference from Constant α’s

Equations (56)

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dt=i=1Mϕzatz,
dt0=ϕt0a0+z0Mϕzȃt0z,
E[Φ̑u,v]=E{[αβ(Mα+i,β+jΦα,β)]Gu+i,v+j}=E(ij{[αβ(Mα+i,β+jΦα,β)]Gu+i,v+j})=ij[(Mu+i,v+jE[Φu,v])Gu+i,v+j]+E[ij(αuβv[(Mα+i,β+jΦα,β)Gu+i,v+j])]=CN×E[Φu,v]+0,
Var[Φ̑u,v]=Var{ij[(Mu+i,v+jΦu,v)Gu+i,v+j]}+αuβv{Var[Φαu,βv]CMCN(CNCM)2},
Kαβ=Φαβ+Bαβ.
d=F·(ϕz+bz).
κ^=H·d,
E[κ^z]=CN×E[κz]+0.
Gα+i,β+j={1ifMα+i,β+j=1αα+i,β+jifMα+i,β+j=0.
E[ϕ^z]=(t(1+αt)·(wtz·atz)E[ϕz])tαt·(atz·E[ϕz])+{t((1+αt)·zz(wtz·atz)E[ϕz])}tzzαt·(atz·E[ϕz]),
E[Φ̑u,v]=E{ij[(Mu+i,v+jΦu,v)Gu+i,v+j]}+E[ij(αuβv[(Mα+i,β+jΦα,β)Gu+i,v+j])].
t((1+αt)·zz(wtz·atz)·E[ϕz])tαt·(zz(atz·E[ϕz]))=0,
ij(αuβv[(Mα+i,β+jE[Φα,β])·Gu+i,v+j])=0.
zz{(·t[(1+αt)·(wtz·atz)]tαt·(atz))·E[ϕz]}=0.
(t{(1+αt)(wtz·atz)}t(αt·atz))zz=0.
t((1+α)·zz(wtz·atz)·E[ϕz])tα·(zz(atz·E[ϕz]))=0.
α=t(wtz·atz)t(wtz·atz)t(atz)=CMCMCN.
Rα+i,β+j={1ifMα+i,β+j=1αα+i,β+jifMα+i,β+j=0.
E[ϕz]1st halfterm=(t(1+αt)·(wtz·atz)E[ϕz])tαt·(atz·E[ϕz]),=t(wtz·atz)E[ϕz]+tαt(wtz·atzatz)E[ϕz],=t(wtz·atz)E[ϕz]+tαt((atz)T·atzatz)E[ϕz],=t(wtz·atz)E[ϕz]+0,
(t{(1+αt)(wtzatz)}=t{αtatz})zz.
(tαt(wtzatzatz)=twtzatz)zz,
(tαt(wtzatzatz)+twtzatz=εz)zz,
ε=zz(εz)2.
Di,j=Mα+i,β+j·Φα,β,
Di,j=αβ(Mα+i,β+j·Φα,β),
Φ̑u,v=[αβ(Mα+i,β+jΦα,β)]Gu+i,v+j0i<p,0j<q0α<p,0β<q=ij{[αβ(Mα+i,β+jΦα,β)]Gu+i,v+j},
Gu+i,v+j={1ifMu+i,v+j=1CM/(CMCN)ifMu+i,v+j=0,
E[Φ̑u,v]=E{[αβ(Mα+i,β+jΦα,β)]Gu+i,v+j},=E(ij{[αβ(Mα+i,β+jΦα,β)]Gu+i,v+j}),=E{ij[(Mu+i,v+jΦu,v)Gu+i,v+j]},+E[ij(αuβv[(Mα+i,β+jΦα,β)Gu+i,v+j])].
E[ij{αuβv[(Mα+i,β+jΦα,β)·Gu+i,v+j]}],=αuβv(E{ij[(Mα+i,β+jΦα,β)·Gu+i,v+j]}).
αuβv(E{ij[(Mα+i,β+jΦα,β)·Gu+i,v+j]}),=αuβvE[Φα,β]{ij[(Mα+i,β+j)·Gu+i,v+j]}.
E[Φα,β]·{ij[(Mα+i,β+j)·Gu+i,v+j]},=E[Φα,β]·[CM×1+(CNCM)CMCMCN],=E[Φα,β]×0.
E[Φ̑u,v]=E{ij[(Mu+i,v+jΦu,v)Gu+i,v+j]}+0,=E[Φu,v]ij[(Mu+i,v+j)Gu+i,v+j],=CNE[Φu,v].
Var[Φ̑u,v]=Var[ij{[αβ(Mα+i,β+jΦα,β)]Gu+i,v+j}],=Var{ij[(Mu+i,v+jΦu,v)Gu+i,v+j]},+Var[ij{αuβv[(Mα+i,β+jΦα,β)Gu+i,v+j]}].
Var[ij{αuβv[(Mα+i,β+jΦα,β)Gu+i,v+j]}],=Var[Φαu,βv]ij(Mα+i,β+j×Gu+i,v+j)2,=Var[Φαu,βv][CM×12+(CNCM)(CMCMCN)2].
Var[Φ̑α,β]=Var{ij[(Mu+i,v+jΦu,v)Gu+i,v+j]},+αuβv{Var[Φαu,βv]CMCN(CNCM)2}.
Ku,v=Φu,v+Bu,v,
E[κ^u,v]=CN·E[Φu,v+Bu,v].
E[κ^u,v]CN=E[Φu,v+Bu,v].
Kα,β=Φα,β+Bα,β,
E[κ^u,v]=E[ij({αβ[Mα+i,β+j·(Φα,β+Bα,β)]}·Ru+i,v+j)],
F(z,t)=[a11a12···a1p×q··at1at2atp×q··ap×q1ap×q2ap×qp×q],
d=F·(ϕz+bz)=F·κz.
d=[a11a12···a1p×q··at1at2atp×q··ap×q1ap×q2ap×qp×q][ϕ1+b1ϕ2+b2·ϕz+bz··ϕp×q+bp×q].
k^=H·d,
κ^=[h11h12···h1p×qh21h22h2p×q··hz1hz2hzp×qhp×q1hp×q2hp×qp×q][d1d2··dz·dp×q].
κ^=[a11a21ap×q1a12ap×q2a1p×qap×qp×q][a11a12a1p×qa21a22a2p×qap×q1ap×q2ap×qp×q][κ1κ2κp×q].
hzt=(1+CNp×qCN)(atz)TCNp×qCN,=(1+CNp×qCN)wztCNp×qCN,
z=t[hztz(atzbz)]=t{[(1+CNp×qCN)(wzt)CNp×qCN]z(atzbz)}.
E[z]=E{t[hztz(atzbz)]}=t{[(1+CNp×qCN)(wzt)CNp×qCN]z(atzE[bz])}.
E[z]=E[b1]t{[(1+CNp×qCN)(wzt)CNp×qCN]z(atz)},=E[b1]t{[(1+CNp×qCN)(wzt)CNp×qCN]CN},=E[b1][(1+CNp×qCN)CNCN2p×qCN(p×q)],=0.
E[Φ̑z]=E{htzz[atzϕz)]},=t{[(1+CNp×qCN)(wzt)CNp×qCN]z(atzE[ϕz])},=t[(1+CNp×qCN)(wzt)z(atzE[ϕz])],+t[CNp×qCNz(atzE[ϕz])],=(p×qp×qCN)z{[t(wztatz)]ϕz},(CNp×qCN)tz(atzE[ϕz]).
twztatz={CNifz=zCMifzz.
z[(twztatz)·ϕz]=CN·ϕz+zzCM·ϕz.
tz(atz·E[ϕz])=z(CN·E[ϕz])=z=z(CN·E[ϕz])+zz(CN·E[ϕz]).
E[Φ̑z]=E{thztz[atzϕz]},=(1+CNp×qCN)(CNE[ϕz]+zzCME[ϕz]),(CNp×qCN)[(CNE[ϕz]+zzCNE[ϕz])],=CNE[ϕz]+zz{[(1+CNp×qCN)CM(CNp×qCN)CN]E[ϕz]}.
E[ϕ^z]/CN=E[ϕz]+1CN·zz{[(1+CNp×qCN)·CM(CNp×qCN)·CN]·E[ϕz]}.

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