Abstract

We examine coding strategies for coded aperture scatter imagers. Scatter imaging enables tomography of compact regions from snapshot measurements. We present coded aperture designs for pencil and fan beam geometries, and compare their singular value spectra with that of the Radon transform and selected volume tomography. We show that under dose constraints scatter imaging improves conditioning over alternative techniques, and that specially designed coded apertures enable snapshot 1D and 2D tomography.

© 2013 Optical Society of America

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References

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  1. K. MacCabe, K. Krishnamurthy, A. Chawla, D. Marks, E. Samei, and D. Brady, “Pencil beam coded aperture x-ray scatter imaging,” Opt. Express 20, 16310–16320 (2012).
    [CrossRef]
  2. K. MacCabe, A. Holmgren, M. Tornai, and D. Brady, “Snapshot 2D tomography via coded aperture x-ray scatter imaging,” Appl. Opt. 52, 4582–4589 (2013).
    [CrossRef]
  3. P. G. Lale, “The examination of internal tissues, using gamma-ray scatter with a possible extension to megavoltage radiography,” Phys. Med. Biol. 4, 159–167 (1959).
    [CrossRef]
  4. G. Harding and E. Harding, “Compton scatter imaging: a tool for historical exploration,” Appl. Radiat. Isot. 68, 993–1005 (2010).
    [CrossRef]
  5. T. T. Truong and M. K. Nguyen, “Radon transforms on generalized cormacks curves and a new compton scatter tomography modality,” Inverse Probl. 27, 125001 (2011).
    [CrossRef]
  6. F. T. Farmer and M. P. Collins, “A further appraisal of the compton scattering method for determining anatomical cross-sections of the body,” Phys. Med. Biol. 19, 808–818 (1974).
    [CrossRef]
  7. C. A. Carlsson, “Imaging modalities in x-ray computerized tomography and in selected volume tomography,” Phys. Med. Biol. 44, R23 (1999).
    [CrossRef]
  8. M. T. E. Golay, “Multi-slit spectrometry,” J. Opt. Soc. Am. 39, 437 (1949).
    [CrossRef]
  9. L. Mertz, Transformations in Optics (Wiley, 1965), Vol. 1.
  10. M. Harwit and N. J. Sloane, Hadamard Transform Optics (Academic, 1979), Vol. 1.
  11. S. R. Gottesman and E. E. Fenimore, “New family of binary arrays for coded aperture imaging,” Appl. Opt. 28, 4344–4352 (1989).
    [CrossRef]
  12. D. J. Brady, Optical Imaging and Spectroscopy (Wiley-OSA, 2009).
  13. A. Wagadarikar, R. John, R. Willett, and D. J. Brady, “Single disperser design for coded aperture snapshot spectral imaging,” Appl. Opt. 47, B44–B51 (2008).
    [CrossRef]
  14. A. Mrozack, D. L. Marks, and D. J. Brady, “Coded aperture spectroscopy with denoising through sparsity,” Opt. Express 20, 2297–2309 (2012).
    [CrossRef]
  15. B. R. Frieden, “Band-unlimited reconstruction of optical objects and spectra,” J. Opt. Soc. Am. 57, 1013–1019 (1967).
    [CrossRef]
  16. S. W. Golomb and G. Gong, Signal Design for Good Correlation: For Wireless Communication, Cryptography, and Radar (Cambridge University, 2005).
  17. P. Boccacci and M. Bertero, Introduction to Inverse Problems in Imaging (Institute of Physics, 1998).
  18. D. Slepian, “Prolate spheroidal wave functions, Fourier analysis, and uncertainty—I,” Bell Syst. Tech. J. 40, 43–63 (1961).
    [CrossRef]

2013 (1)

2012 (2)

2011 (1)

T. T. Truong and M. K. Nguyen, “Radon transforms on generalized cormacks curves and a new compton scatter tomography modality,” Inverse Probl. 27, 125001 (2011).
[CrossRef]

2010 (1)

G. Harding and E. Harding, “Compton scatter imaging: a tool for historical exploration,” Appl. Radiat. Isot. 68, 993–1005 (2010).
[CrossRef]

2008 (1)

1999 (1)

C. A. Carlsson, “Imaging modalities in x-ray computerized tomography and in selected volume tomography,” Phys. Med. Biol. 44, R23 (1999).
[CrossRef]

1989 (1)

1974 (1)

F. T. Farmer and M. P. Collins, “A further appraisal of the compton scattering method for determining anatomical cross-sections of the body,” Phys. Med. Biol. 19, 808–818 (1974).
[CrossRef]

1967 (1)

1961 (1)

D. Slepian, “Prolate spheroidal wave functions, Fourier analysis, and uncertainty—I,” Bell Syst. Tech. J. 40, 43–63 (1961).
[CrossRef]

1959 (1)

P. G. Lale, “The examination of internal tissues, using gamma-ray scatter with a possible extension to megavoltage radiography,” Phys. Med. Biol. 4, 159–167 (1959).
[CrossRef]

1949 (1)

Bertero, M.

P. Boccacci and M. Bertero, Introduction to Inverse Problems in Imaging (Institute of Physics, 1998).

Boccacci, P.

P. Boccacci and M. Bertero, Introduction to Inverse Problems in Imaging (Institute of Physics, 1998).

Brady, D.

Brady, D. J.

Carlsson, C. A.

C. A. Carlsson, “Imaging modalities in x-ray computerized tomography and in selected volume tomography,” Phys. Med. Biol. 44, R23 (1999).
[CrossRef]

Chawla, A.

Collins, M. P.

F. T. Farmer and M. P. Collins, “A further appraisal of the compton scattering method for determining anatomical cross-sections of the body,” Phys. Med. Biol. 19, 808–818 (1974).
[CrossRef]

Farmer, F. T.

F. T. Farmer and M. P. Collins, “A further appraisal of the compton scattering method for determining anatomical cross-sections of the body,” Phys. Med. Biol. 19, 808–818 (1974).
[CrossRef]

Fenimore, E. E.

Frieden, B. R.

Golay, M. T. E.

Golomb, S. W.

S. W. Golomb and G. Gong, Signal Design for Good Correlation: For Wireless Communication, Cryptography, and Radar (Cambridge University, 2005).

Gong, G.

S. W. Golomb and G. Gong, Signal Design for Good Correlation: For Wireless Communication, Cryptography, and Radar (Cambridge University, 2005).

Gottesman, S. R.

Harding, E.

G. Harding and E. Harding, “Compton scatter imaging: a tool for historical exploration,” Appl. Radiat. Isot. 68, 993–1005 (2010).
[CrossRef]

Harding, G.

G. Harding and E. Harding, “Compton scatter imaging: a tool for historical exploration,” Appl. Radiat. Isot. 68, 993–1005 (2010).
[CrossRef]

Harwit, M.

M. Harwit and N. J. Sloane, Hadamard Transform Optics (Academic, 1979), Vol. 1.

Holmgren, A.

John, R.

Krishnamurthy, K.

Lale, P. G.

P. G. Lale, “The examination of internal tissues, using gamma-ray scatter with a possible extension to megavoltage radiography,” Phys. Med. Biol. 4, 159–167 (1959).
[CrossRef]

MacCabe, K.

Marks, D.

Marks, D. L.

Mertz, L.

L. Mertz, Transformations in Optics (Wiley, 1965), Vol. 1.

Mrozack, A.

Nguyen, M. K.

T. T. Truong and M. K. Nguyen, “Radon transforms on generalized cormacks curves and a new compton scatter tomography modality,” Inverse Probl. 27, 125001 (2011).
[CrossRef]

Samei, E.

Slepian, D.

D. Slepian, “Prolate spheroidal wave functions, Fourier analysis, and uncertainty—I,” Bell Syst. Tech. J. 40, 43–63 (1961).
[CrossRef]

Sloane, N. J.

M. Harwit and N. J. Sloane, Hadamard Transform Optics (Academic, 1979), Vol. 1.

Tornai, M.

Truong, T. T.

T. T. Truong and M. K. Nguyen, “Radon transforms on generalized cormacks curves and a new compton scatter tomography modality,” Inverse Probl. 27, 125001 (2011).
[CrossRef]

Wagadarikar, A.

Willett, R.

Appl. Opt. (3)

Appl. Radiat. Isot. (1)

G. Harding and E. Harding, “Compton scatter imaging: a tool for historical exploration,” Appl. Radiat. Isot. 68, 993–1005 (2010).
[CrossRef]

Bell Syst. Tech. J. (1)

D. Slepian, “Prolate spheroidal wave functions, Fourier analysis, and uncertainty—I,” Bell Syst. Tech. J. 40, 43–63 (1961).
[CrossRef]

Inverse Probl. (1)

T. T. Truong and M. K. Nguyen, “Radon transforms on generalized cormacks curves and a new compton scatter tomography modality,” Inverse Probl. 27, 125001 (2011).
[CrossRef]

J. Opt. Soc. Am. (2)

Opt. Express (2)

Phys. Med. Biol. (3)

P. G. Lale, “The examination of internal tissues, using gamma-ray scatter with a possible extension to megavoltage radiography,” Phys. Med. Biol. 4, 159–167 (1959).
[CrossRef]

F. T. Farmer and M. P. Collins, “A further appraisal of the compton scattering method for determining anatomical cross-sections of the body,” Phys. Med. Biol. 19, 808–818 (1974).
[CrossRef]

C. A. Carlsson, “Imaging modalities in x-ray computerized tomography and in selected volume tomography,” Phys. Med. Biol. 44, R23 (1999).
[CrossRef]

Other (5)

L. Mertz, Transformations in Optics (Wiley, 1965), Vol. 1.

M. Harwit and N. J. Sloane, Hadamard Transform Optics (Academic, 1979), Vol. 1.

D. J. Brady, Optical Imaging and Spectroscopy (Wiley-OSA, 2009).

S. W. Golomb and G. Gong, Signal Design for Good Correlation: For Wireless Communication, Cryptography, and Radar (Cambridge University, 2005).

P. Boccacci and M. Bertero, Introduction to Inverse Problems in Imaging (Institute of Physics, 1998).

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Figures (10)

Fig. 1.
Fig. 1.

System geometry for planar scatter imaging.

Fig. 2.
Fig. 2.

Coded aperture based on the identity matrix. (a) Polar coordinates (r, φ). (b) Cartesian coordinates (x,y).

Fig. 3.
Fig. 3.

Coded aperture based on the DCT. (a) Polar coordinates (r, φ). (b) Cartesian coordinates (x,y).

Fig. 4.
Fig. 4.

Coded aperture based on a Hadamard matrix. (a) Polar coordinates (r, φ). (b) Cartesian coordinates (x,y).

Fig. 5.
Fig. 5.

Coded aperture based on a random binary matrix. (a) Polar coordinates (r, φ). (b) Cartesian coordinates (x,y).

Fig. 6.
Fig. 6.

Singular value spectra of the pencil beam system for each code choice.

Fig. 7.
Fig. 7.

Coded apertures based on a sinusoid in x (horizontal) and a quadratic residue in y (vertical). The number of code features in each direction (x,y) are (a) 32×29, (b) 16×29, (c) 32×17, and (d) 16×17 features.

Fig. 8.
Fig. 8.

Singular value spectra for each of the coded apertures in Fig. 7.

Fig. 9.
Fig. 9.

Coded aperture with resolution 32×29 based on uniform random values in [0, 1].

Fig. 10.
Fig. 10.

Singular value spectra for the proposed code and a random code.

Tables (1)

Tables Icon

Table 1. Scaling of Dose-Constrained Singular Values for Pencil Beam CAXSI, Fan Beam CAXSI, Radon Imaging, and SVTa

Equations (32)

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g(x)=dzmaxf(z)t[x(1dz)]dz.
g(x)=01f˜(β)t(xβ)dβ,
g(x)=1201f˜(β)[1cos(2πuxβ)]dβ.
Δz=z2Nxd.
g(φ,ρ)=dzmaxf(z,θ)t[φ,ρ(1dz)]dz
g(φ,ρ)=t(φ,r)f(r,ρ)dr.
g(x,y)=Y/2Y/2dzmaxf(x=0,y,z)×t[x(1dz),y(1dz)+ydz]dydz.
p(y)=npn[2rect(yn)1],
α=ydzβ=1dz.
g(x,y)=01F(α,β)t[xβ,βy+α]dαdβ.
FA(α,β)=Y/2Y/2X/2X/2g(x,y)t[xβ,βy+α]*dxdy.
FA(α,β)=01F(α,β)K(α,β,α,β)dαdβ
K(α,β,α,β)=X/2X/2Y/2Y/2t[xβ,βy+α]×t[xβ,βy+α]*dxdy.
KS(α,β,α,β)=X/2X/2Y/2Y/21+A(xβ)B[βy+α]2×1+A(xβ)*B[βy+α]*2dydx.
KS(α,β,α,β)=XY4+14X/2X/2A(xβ)A(xβ)*dx×Y/2Y/2B[βy+α]B[βy+α]*dy.
X/2X/2cos(2πuxβ)cos(2πuxβ)dx=12X/2X/2cos[2πux(ββ)]+cos[2πux(β+β)]dx=12πu(ββ)sin[πuX(ββ)]+12πu(β+β)sin[πuX(β+β)]X2sinc[Nx(ββ)]
KS(α,β,α,β)=XY4+X8sinc[Nx(ββ)]×Y/2Y/2B[β+β2y+α]B[β+β2y+α]*dy
KS(α,β,α,β)=XY4+X8sinc[Nx(ββ)]×Y/2Y/2[ncnexp(2πinP[β+β2y+α])]×[ncn*exp(2πinP[β+β2y+α])]dy.
KS(α,β,α,β)=XY4+X8sinc[Nx(ββ)]Y(β+β)2P×P/(β+β)P/(β+β)[ncnexp(2πinP[β+β2y+α])]×[ncn*exp(2πinP[β+β2y+α])]dy.
KS(α,β,α,β)=XY4+X8sinc[Nx(ββ)]Y(β+β)2P×P/(β+β)P/(β+β)n|cn|2exp(2πinP[αα])dy.
KS(α,β,α,β)=XY4+XY8sinc[Nx(ββ)]×n=|cn|2exp(2πinP[αα]).
KS(α,β,α,β)=XY4(1+12sinc[Nx(ββ)]BA(αα)).
λmn2F(α,β)=01Fmn(α,β)K(α,β,α,β)dαdβ
Fmn(α,β)=e2πi(αmP+βn).
λmn2e2πi(αmP+βn)=1PP/2P/2dα01dβe2πi(αmP+βn)K(α,β,α,β).
XY4PP/2P/2dαe2πiαmP01dβe2πiβn=XY4δn0δm0.
XY8PP/2P/2dαe2παmPBA(αα)01dβe2πiβnsinc[Nx(ββ)].
λmn=XY2δm0δn0+|cm|22Nxrect(nNx).
a˜(ν)=dye2πiyνa(y)=n=0N1ane2πinνP/Nyb˜(ν)=dye2πiyνb(y)=1Nysinc(νPNy)m=δ(νmP).
cm=limϵ0m/Pϵm/P+ϵdνB˜(ν)=1Nysinc(mNy)n=0Ny1ane2πinm/Ny=1Nysinc(mNy)a˜m,
1Nym=0Ny1|a˜m|2=m=0Ny1|an|2=Ny,
λmn=12XYNxNyrect(nNx)sinc(mNy).

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