Abstract

We describe an active millimeter-wave holographic imaging system that uses compressive measurements for three-dimensional (3D) tomographic object estimation. Our system records a two-dimensional (2D) digitized Gabor hologram by translating a single pixel incoherent receiver. Two approaches for compressive measurement are undertaken: nonlinear inversion of a 2D Gabor hologram for 3D object estimation and nonlinear inversion of a randomly subsampled Gabor hologram for 3D object estimation. The object estimation algorithm minimizes a convex quadratic problem using total variation (TV) regularization for 3D object estimation. We compare object reconstructions using linear backpropagation and TV minimization, and we present simulated and experimental reconstructions from both compressive measurement strategies. In contrast with backpropagation, which estimates the 3D electromagnetic field, TV minimization estimates the 3D object that produces the field. Despite undersampling, range resolution is consistent with the extent of the 3D object band volume.

© 2010 Optical Society of America

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2009 (3)

H. Choo, J. Woo, D. Kim, S. Shin, and Y. Yu, “DC suppression in in-line digital holographic microscopes on the basis of an intensity-averaging method using variable pixel numbers,” Opt. Laser Technol. 41, 741–745 (2009).
[CrossRef]

D. J. Brady, K. Choi, D. L. Marks, R. Horisaki, and S. Lim, “Compressive holography,” Opt. Express 17, 13040–13049 (2009).
[CrossRef] [PubMed]

L. Denis, D. Lorenz, E. Thiébaut, C. Fournier, and D. Trede, “Inline hologram reconstruction with sparsity constraints,” Opt. Lett. 34, 3475–3477 (2009).
[CrossRef] [PubMed]

2008 (5)

2007 (2)

J. Bioucas-Dias and M. Figueiredo, “A new TwIST: two-step iterative shrinkage/thresholding algorithms for image restoration,” IEEE Trans. Image Process. 16, 2992–3004 (2007).
[CrossRef] [PubMed]

T. Latychevskaia and H. Fink, “Solution to the twin image problem in holography,” Phys. Rev. Lett. 98, 233901(2007).
[CrossRef] [PubMed]

2006 (3)

S. Stanko, F. Kloppel, J. Huck, D. Notel, M. Hagelen, G. Briese, A. Gregor, S. Erukulla, H.-H. Fuchs, H. Essen, and A. Pagels, “Remote concealed weapon detection in millimeter-wave region: active and passive,” Proc. SPIE 6396, 639606 (2006).
[CrossRef]

R. N. Anderton, R. Appleby, J. E. Beale, P. R. Coward, and S. Price, “Security scanning at 94GHz,” Proc. SPIE 6211, 62110C (2006).
[CrossRef]

E. Candes, J. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Commun. Pure Appl. Math. 59, 1207–1223 (2006).
[CrossRef]

2005 (1)

2004 (1)

R. Appleby, R. N. Anderton, N. H. Thomson, and J. W. Jack, “The design of a real-time 94GHz passive millimetre-wave imager for helicopter operations,” Proc. SPIE 5619, 38–46 (2004).
[CrossRef]

2001 (2)

D. Sheen, D. McMakin, and T. Hall, “Three-dimensional millimeter-wave imaging for concealed weapon detection,” IEEE Trans. Microwave Theory Techn. 49, 1581–1592 (2001).
[CrossRef]

A. Abubaker and P. Van Den Berg, “Total variation as a multiplicative constraint for solving inverse problems,” IEEE Trans. Image Process. 10, 1384–1392 (2001).
[CrossRef]

1997 (1)

T. M. Kreis and W. P. O. Jüptner, “Suppression of the dc term in digital holography,” Opt. Eng. 36, 2357–2360 (1997).
[CrossRef]

1980 (1)

R. Mueller, M. Kaveh, and R. Inversion, “A new approach to acoustic tomography using diffraction techniques,” Acoust. Imaging 8, 615–628 (1980).
[CrossRef]

1962 (1)

Abubaker, A.

A. Abubaker and P. Van Den Berg, “Total variation as a multiplicative constraint for solving inverse problems,” IEEE Trans. Image Process. 10, 1384–1392 (2001).
[CrossRef]

Anderton, R. N.

R. N. Anderton, R. Appleby, J. E. Beale, P. R. Coward, and S. Price, “Security scanning at 94GHz,” Proc. SPIE 6211, 62110C (2006).
[CrossRef]

R. Appleby, R. N. Anderton, N. H. Thomson, and J. W. Jack, “The design of a real-time 94GHz passive millimetre-wave imager for helicopter operations,” Proc. SPIE 5619, 38–46 (2004).
[CrossRef]

Appleby, R.

R. N. Anderton, R. Appleby, J. E. Beale, P. R. Coward, and S. Price, “Security scanning at 94GHz,” Proc. SPIE 6211, 62110C (2006).
[CrossRef]

R. Appleby, R. N. Anderton, N. H. Thomson, and J. W. Jack, “The design of a real-time 94GHz passive millimetre-wave imager for helicopter operations,” Proc. SPIE 5619, 38–46 (2004).
[CrossRef]

Baraniuk, R.

W. L. Chan, M. Moravec, R. Baraniuk, and D. Mittleman, “A single-pixel terahertz imaging system based on compressed sensing,” Appl. Phys. Lett. 93, 121105 (2008).
[CrossRef]

Baraniuk, R. G.

Beale, J. E.

R. N. Anderton, R. Appleby, J. E. Beale, P. R. Coward, and S. Price, “Security scanning at 94GHz,” Proc. SPIE 6211, 62110C (2006).
[CrossRef]

Bioucas-Dias, J.

J. Bioucas-Dias and M. Figueiredo, “A new TwIST: two-step iterative shrinkage/thresholding algorithms for image restoration,” IEEE Trans. Image Process. 16, 2992–3004 (2007).
[CrossRef] [PubMed]

Brady, D. J.

Briese, G.

S. Stanko, F. Kloppel, J. Huck, D. Notel, M. Hagelen, G. Briese, A. Gregor, S. Erukulla, H.-H. Fuchs, H. Essen, and A. Pagels, “Remote concealed weapon detection in millimeter-wave region: active and passive,” Proc. SPIE 6396, 639606 (2006).
[CrossRef]

Candes, E.

E. Candes, J. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Commun. Pure Appl. Math. 59, 1207–1223 (2006).
[CrossRef]

Chan, W. L.

W. L. Chan, M. L. Moravec, R. G. Baraniuk, and D. M. Mittleman, “Terahertz imaging with compressed sensing and phase retrieval,” Opt. Lett. 33, 974–976 (2008).
[CrossRef] [PubMed]

W. L. Chan, M. Moravec, R. Baraniuk, and D. Mittleman, “A single-pixel terahertz imaging system based on compressed sensing,” Appl. Phys. Lett. 93, 121105 (2008).
[CrossRef]

Choi, K.

Choo, H.

H. Choo, J. Woo, D. Kim, S. Shin, and Y. Yu, “DC suppression in in-line digital holographic microscopes on the basis of an intensity-averaging method using variable pixel numbers,” Opt. Laser Technol. 41, 741–745 (2009).
[CrossRef]

Costianes, P. J.

P. J. Costianes, “An overview of concealed weapons detection for homeland security,” in Applied Imagery and Pattern Recognition Workshop, 2005 (IEEE, 2005), pp. 2–6.

Coward, P. R.

R. N. Anderton, R. Appleby, J. E. Beale, P. R. Coward, and S. Price, “Security scanning at 94GHz,” Proc. SPIE 6211, 62110C (2006).
[CrossRef]

Currie, N.

R. McMillan, N. Currie, D. Ferris, Jr., and M. Wicks, “Concealed weapon detection using microwave and millimeter wave sensors,” in Microwave and Millimeter Wave Technology Proceedings, 1998 (IEEE, 1998), pp. 1–4 .

Denis, L.

Devaney, A.

Erukulla, S.

S. Stanko, F. Kloppel, J. Huck, D. Notel, M. Hagelen, G. Briese, A. Gregor, S. Erukulla, H.-H. Fuchs, H. Essen, and A. Pagels, “Remote concealed weapon detection in millimeter-wave region: active and passive,” Proc. SPIE 6396, 639606 (2006).
[CrossRef]

Essen, H.

S. Stanko, F. Kloppel, J. Huck, D. Notel, M. Hagelen, G. Briese, A. Gregor, S. Erukulla, H.-H. Fuchs, H. Essen, and A. Pagels, “Remote concealed weapon detection in millimeter-wave region: active and passive,” Proc. SPIE 6396, 639606 (2006).
[CrossRef]

Farhat, N. H.

N. H. Farhat and W. R. Guard, “Millimeter wave holographic imaging of concealed weapons,” in Proceedings of the Institute of Electrical and Electronics Engineers (IEEE, 1971), pp. 1383–1384.

Ferris, D.

R. McMillan, N. Currie, D. Ferris, Jr., and M. Wicks, “Concealed weapon detection using microwave and millimeter wave sensors,” in Microwave and Millimeter Wave Technology Proceedings, 1998 (IEEE, 1998), pp. 1–4 .

Fetterman, M. R.

Figueiredo, M.

J. Bioucas-Dias and M. Figueiredo, “A new TwIST: two-step iterative shrinkage/thresholding algorithms for image restoration,” IEEE Trans. Image Process. 16, 2992–3004 (2007).
[CrossRef] [PubMed]

Fink, H.

T. Latychevskaia and H. Fink, “Solution to the twin image problem in holography,” Phys. Rev. Lett. 98, 233901(2007).
[CrossRef] [PubMed]

Fournier, C.

Fuchs, H.-H.

S. Stanko, F. Kloppel, J. Huck, D. Notel, M. Hagelen, G. Briese, A. Gregor, S. Erukulla, H.-H. Fuchs, H. Essen, and A. Pagels, “Remote concealed weapon detection in millimeter-wave region: active and passive,” Proc. SPIE 6396, 639606 (2006).
[CrossRef]

Gehm, M. E.

Goodman, J.

J. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

Grata, J.

Gregor, A.

S. Stanko, F. Kloppel, J. Huck, D. Notel, M. Hagelen, G. Briese, A. Gregor, S. Erukulla, H.-H. Fuchs, H. Essen, and A. Pagels, “Remote concealed weapon detection in millimeter-wave region: active and passive,” Proc. SPIE 6396, 639606 (2006).
[CrossRef]

Guard, W. R.

N. H. Farhat and W. R. Guard, “Millimeter wave holographic imaging of concealed weapons,” in Proceedings of the Institute of Electrical and Electronics Engineers (IEEE, 1971), pp. 1383–1384.

Guo, P.

Hagelen, M.

S. Stanko, F. Kloppel, J. Huck, D. Notel, M. Hagelen, G. Briese, A. Gregor, S. Erukulla, H.-H. Fuchs, H. Essen, and A. Pagels, “Remote concealed weapon detection in millimeter-wave region: active and passive,” Proc. SPIE 6396, 639606 (2006).
[CrossRef]

Hall, T.

D. Sheen, D. McMakin, and T. Hall, “Three-dimensional millimeter-wave imaging for concealed weapon detection,” IEEE Trans. Microwave Theory Techn. 49, 1581–1592 (2001).
[CrossRef]

Horisaki, R.

Huck, J.

S. Stanko, F. Kloppel, J. Huck, D. Notel, M. Hagelen, G. Briese, A. Gregor, S. Erukulla, H.-H. Fuchs, H. Essen, and A. Pagels, “Remote concealed weapon detection in millimeter-wave region: active and passive,” Proc. SPIE 6396, 639606 (2006).
[CrossRef]

Inversion, R.

R. Mueller, M. Kaveh, and R. Inversion, “A new approach to acoustic tomography using diffraction techniques,” Acoust. Imaging 8, 615–628 (1980).
[CrossRef]

Jack, J. W.

R. Appleby, R. N. Anderton, N. H. Thomson, and J. W. Jack, “The design of a real-time 94GHz passive millimetre-wave imager for helicopter operations,” Proc. SPIE 5619, 38–46 (2004).
[CrossRef]

Jubic, G.

Juptner, W.

U. Schnars and W. Juptner, Digital Holography, Digital Hologram Recording, Numerical Reconstruction and Related techniques (Springer-Verlag, 2005).

Jüptner, W. P. O.

T. M. Kreis and W. P. O. Jüptner, “Suppression of the dc term in digital holography,” Opt. Eng. 36, 2357–2360 (1997).
[CrossRef]

Kak, A.

A. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE, 1988).

Kaveh, M.

R. Mueller, M. Kaveh, and R. Inversion, “A new approach to acoustic tomography using diffraction techniques,” Acoust. Imaging 8, 615–628 (1980).
[CrossRef]

Kemp, M.

M. Kemp, “Millimetre wave and terahertz technology for detection of concealed threats—a review,” in Infrared and Millimeter Waves, 2007 and the 2007 15th International Conference on Terahertz Electronics (IEEE, 2007), pp. 647–648.

Kim, D.

H. Choo, J. Woo, D. Kim, S. Shin, and Y. Yu, “DC suppression in in-line digital holographic microscopes on the basis of an intensity-averaging method using variable pixel numbers,” Opt. Laser Technol. 41, 741–745 (2009).
[CrossRef]

Kinast, J.

Kiser, J. W. L.

Kloppel, F.

S. Stanko, F. Kloppel, J. Huck, D. Notel, M. Hagelen, G. Briese, A. Gregor, S. Erukulla, H.-H. Fuchs, H. Essen, and A. Pagels, “Remote concealed weapon detection in millimeter-wave region: active and passive,” Proc. SPIE 6396, 639606 (2006).
[CrossRef]

Kreis, T. M.

T. M. Kreis and W. P. O. Jüptner, “Suppression of the dc term in digital holography,” Opt. Eng. 36, 2357–2360 (1997).
[CrossRef]

LaRoque, S. J.

Latychevskaia, T.

T. Latychevskaia and H. Fink, “Solution to the twin image problem in holography,” Phys. Rev. Lett. 98, 233901(2007).
[CrossRef] [PubMed]

Leith, E. N.

Lim, S.

Lorenz, D.

Marks, D. L.

McMakin, D.

D. Sheen, D. McMakin, and T. Hall, “Three-dimensional millimeter-wave imaging for concealed weapon detection,” IEEE Trans. Microwave Theory Techn. 49, 1581–1592 (2001).
[CrossRef]

D. McMakin, “Remote concealed weapons and explosive detection on people using millimeter-wave holography,” in 30th Annual 1996 International Carnahan Conference on Security Technology, 1996 (IEEE, 1996) pp. 19–25.

McMillan, R.

R. McMillan, N. Currie, D. Ferris, Jr., and M. Wicks, “Concealed weapon detection using microwave and millimeter wave sensors,” in Microwave and Millimeter Wave Technology Proceedings, 1998 (IEEE, 1998), pp. 1–4 .

Mersereau, R.

R. Mersereau and A. Oppenheim, “Digital reconstruction of multidimensional signals from their projections,” in Proceedings of the IEEE (IEEE, 1974), pp. 1319–1338.
[CrossRef]

Mittleman, D.

W. L. Chan, M. Moravec, R. Baraniuk, and D. Mittleman, “A single-pixel terahertz imaging system based on compressed sensing,” Appl. Phys. Lett. 93, 121105 (2008).
[CrossRef]

Mittleman, D. M.

Moravec, M.

W. L. Chan, M. Moravec, R. Baraniuk, and D. Mittleman, “A single-pixel terahertz imaging system based on compressed sensing,” Appl. Phys. Lett. 93, 121105 (2008).
[CrossRef]

Moravec, M. L.

Mueller, R.

R. Mueller, M. Kaveh, and R. Inversion, “A new approach to acoustic tomography using diffraction techniques,” Acoust. Imaging 8, 615–628 (1980).
[CrossRef]

Notel, D.

S. Stanko, F. Kloppel, J. Huck, D. Notel, M. Hagelen, G. Briese, A. Gregor, S. Erukulla, H.-H. Fuchs, H. Essen, and A. Pagels, “Remote concealed weapon detection in millimeter-wave region: active and passive,” Proc. SPIE 6396, 639606 (2006).
[CrossRef]

Oppenheim, A.

R. Mersereau and A. Oppenheim, “Digital reconstruction of multidimensional signals from their projections,” in Proceedings of the IEEE (IEEE, 1974), pp. 1319–1338.
[CrossRef]

Pagels, A.

S. Stanko, F. Kloppel, J. Huck, D. Notel, M. Hagelen, G. Briese, A. Gregor, S. Erukulla, H.-H. Fuchs, H. Essen, and A. Pagels, “Remote concealed weapon detection in millimeter-wave region: active and passive,” Proc. SPIE 6396, 639606 (2006).
[CrossRef]

Pan, X.

Price, S.

R. N. Anderton, R. Appleby, J. E. Beale, P. R. Coward, and S. Price, “Security scanning at 94GHz,” Proc. SPIE 6211, 62110C (2006).
[CrossRef]

Richards, M. A.

M. A. Richards, Fundamentals of Radar Signal Processing (McGraw-Hill, 2005).

Romberg, J.

E. Candes, J. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Commun. Pure Appl. Math. 59, 1207–1223 (2006).
[CrossRef]

Schnars, U.

U. Schnars and W. Juptner, Digital Holography, Digital Hologram Recording, Numerical Reconstruction and Related techniques (Springer-Verlag, 2005).

Sheen, D.

D. Sheen, D. McMakin, and T. Hall, “Three-dimensional millimeter-wave imaging for concealed weapon detection,” IEEE Trans. Microwave Theory Techn. 49, 1581–1592 (2001).
[CrossRef]

Shin, S.

H. Choo, J. Woo, D. Kim, S. Shin, and Y. Yu, “DC suppression in in-line digital holographic microscopes on the basis of an intensity-averaging method using variable pixel numbers,” Opt. Laser Technol. 41, 741–745 (2009).
[CrossRef]

Sidky, E. Y.

Slaney, M.

A. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE, 1988).

Stanko, S.

S. Stanko, F. Kloppel, J. Huck, D. Notel, M. Hagelen, G. Briese, A. Gregor, S. Erukulla, H.-H. Fuchs, H. Essen, and A. Pagels, “Remote concealed weapon detection in millimeter-wave region: active and passive,” Proc. SPIE 6396, 639606 (2006).
[CrossRef]

Tao, T.

E. Candes, J. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Commun. Pure Appl. Math. 59, 1207–1223 (2006).
[CrossRef]

Thiébaut, E.

Thomson, N. H.

R. Appleby, R. N. Anderton, N. H. Thomson, and J. W. Jack, “The design of a real-time 94GHz passive millimetre-wave imager for helicopter operations,” Proc. SPIE 5619, 38–46 (2004).
[CrossRef]

Trede, D.

Upatnieks, J.

Van Den Berg, P.

A. Abubaker and P. Van Den Berg, “Total variation as a multiplicative constraint for solving inverse problems,” IEEE Trans. Image Process. 10, 1384–1392 (2001).
[CrossRef]

Visnansky, A.

Wicks, M.

R. McMillan, N. Currie, D. Ferris, Jr., and M. Wicks, “Concealed weapon detection using microwave and millimeter wave sensors,” in Microwave and Millimeter Wave Technology Proceedings, 1998 (IEEE, 1998), pp. 1–4 .

Woo, J.

H. Choo, J. Woo, D. Kim, S. Shin, and Y. Yu, “DC suppression in in-line digital holographic microscopes on the basis of an intensity-averaging method using variable pixel numbers,” Opt. Laser Technol. 41, 741–745 (2009).
[CrossRef]

Wu, Z.

Xin, H.

Yu, Y.

H. Choo, J. Woo, D. Kim, S. Shin, and Y. Yu, “DC suppression in in-line digital holographic microscopes on the basis of an intensity-averaging method using variable pixel numbers,” Opt. Laser Technol. 41, 741–745 (2009).
[CrossRef]

Acoust. Imaging (1)

R. Mueller, M. Kaveh, and R. Inversion, “A new approach to acoustic tomography using diffraction techniques,” Acoust. Imaging 8, 615–628 (1980).
[CrossRef]

Appl. Phys. Lett. (1)

W. L. Chan, M. Moravec, R. Baraniuk, and D. Mittleman, “A single-pixel terahertz imaging system based on compressed sensing,” Appl. Phys. Lett. 93, 121105 (2008).
[CrossRef]

Commun. Pure Appl. Math. (1)

E. Candes, J. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Commun. Pure Appl. Math. 59, 1207–1223 (2006).
[CrossRef]

IEEE Trans. Image Process. (2)

J. Bioucas-Dias and M. Figueiredo, “A new TwIST: two-step iterative shrinkage/thresholding algorithms for image restoration,” IEEE Trans. Image Process. 16, 2992–3004 (2007).
[CrossRef] [PubMed]

A. Abubaker and P. Van Den Berg, “Total variation as a multiplicative constraint for solving inverse problems,” IEEE Trans. Image Process. 10, 1384–1392 (2001).
[CrossRef]

IEEE Trans. Microwave Theory Techn. (1)

D. Sheen, D. McMakin, and T. Hall, “Three-dimensional millimeter-wave imaging for concealed weapon detection,” IEEE Trans. Microwave Theory Techn. 49, 1581–1592 (2001).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

Opt. Eng. (1)

T. M. Kreis and W. P. O. Jüptner, “Suppression of the dc term in digital holography,” Opt. Eng. 36, 2357–2360 (1997).
[CrossRef]

Opt. Express (3)

Opt. Laser Technol. (1)

H. Choo, J. Woo, D. Kim, S. Shin, and Y. Yu, “DC suppression in in-line digital holographic microscopes on the basis of an intensity-averaging method using variable pixel numbers,” Opt. Laser Technol. 41, 741–745 (2009).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. Lett. (1)

T. Latychevskaia and H. Fink, “Solution to the twin image problem in holography,” Phys. Rev. Lett. 98, 233901(2007).
[CrossRef] [PubMed]

Proc. SPIE (3)

S. Stanko, F. Kloppel, J. Huck, D. Notel, M. Hagelen, G. Briese, A. Gregor, S. Erukulla, H.-H. Fuchs, H. Essen, and A. Pagels, “Remote concealed weapon detection in millimeter-wave region: active and passive,” Proc. SPIE 6396, 639606 (2006).
[CrossRef]

R. N. Anderton, R. Appleby, J. E. Beale, P. R. Coward, and S. Price, “Security scanning at 94GHz,” Proc. SPIE 6211, 62110C (2006).
[CrossRef]

R. Appleby, R. N. Anderton, N. H. Thomson, and J. W. Jack, “The design of a real-time 94GHz passive millimetre-wave imager for helicopter operations,” Proc. SPIE 5619, 38–46 (2004).
[CrossRef]

Other (11)

M. Kemp, “Millimetre wave and terahertz technology for detection of concealed threats—a review,” in Infrared and Millimeter Waves, 2007 and the 2007 15th International Conference on Terahertz Electronics (IEEE, 2007), pp. 647–648.

P. J. Costianes, “An overview of concealed weapons detection for homeland security,” in Applied Imagery and Pattern Recognition Workshop, 2005 (IEEE, 2005), pp. 2–6.

R. McMillan, N. Currie, D. Ferris, Jr., and M. Wicks, “Concealed weapon detection using microwave and millimeter wave sensors,” in Microwave and Millimeter Wave Technology Proceedings, 1998 (IEEE, 1998), pp. 1–4 .

D. McMakin, “Remote concealed weapons and explosive detection on people using millimeter-wave holography,” in 30th Annual 1996 International Carnahan Conference on Security Technology, 1996 (IEEE, 1996) pp. 19–25.

National Research Council Committee on Assessment of Security Technologies for Transportation, Assessment of Millimeter-Wave and Terahertz Technology for Detection and Identification of Concealed Explosives and Weapons (National Academies Press, 2007).
[PubMed]

N. H. Farhat and W. R. Guard, “Millimeter wave holographic imaging of concealed weapons,” in Proceedings of the Institute of Electrical and Electronics Engineers (IEEE, 1971), pp. 1383–1384.

U. Schnars and W. Juptner, Digital Holography, Digital Hologram Recording, Numerical Reconstruction and Related techniques (Springer-Verlag, 2005).

R. Mersereau and A. Oppenheim, “Digital reconstruction of multidimensional signals from their projections,” in Proceedings of the IEEE (IEEE, 1974), pp. 1319–1338.
[CrossRef]

J. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

A. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE, 1988).

M. A. Richards, Fundamentals of Radar Signal Processing (McGraw-Hill, 2005).

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Figures (14)

Fig. 1
Fig. 1

Fourier-transform domain sampling of the object band volume in a transmission geometry. (a) 2D slice of a 3D sphere where the dotted curve represents the measurement from single plane wave illumination. (b) Rectilinear pattern represents wave vectors sampled by the hologram due to a finite detector plane sampling. (c) Wave normal sphere cross section for spatial and axial resolution analysis.

Fig. 2
Fig. 2

(a) Spectrum for an off-axis hologram recording depicting an inherent increase in bandwidth for adequate object separation from undiffracted terms. (b) Spectrum for a Gabor hologram recording, depicting the overlay of undiffracted, object, and conjugate terms. (c) Transverse slices from linear inverse propagation results at various z planes.

Fig. 3
Fig. 3

Transverse slices from TV minimization reconstructions at different z planes. A dominant squared-field term is confined to the z = 0 plane.

Fig. 4
Fig. 4

Sampling windows for sparse measurement where (a) 3.9%, (b) 9.77%, (c) 23.83%, (d) 44.56%, and (e) 54.68% of the measurements are removed.

Fig. 5
Fig. 5

Synthetic 3D slit object results with an applied transmittance function and corrupted by AWGN at a 30 dB measurement SNR using (a) backpropagation and (b) TV minimization for 3D tomographic object estimation. Various values for τ ( 0.2 1.0 ) are used for sparsely sampled (0.0–54.68%) TV reconstructions (see Table 1).

Fig. 6
Fig. 6

Synthetic 3D dagger and gun object results with an applied transmittance function and corrupted by AWGN at a 30 dB measurement SNR using (a) backpropagation and (b) TV minimization for 3D tomographic object estimation. A τ value of 0.2 is used for TV minimization reconstructions from sparsely sampled detector measurements corrupted by AWGN (see Table 1).

Fig. 7
Fig. 7

Plot of reconstruction PSNR (in dB) versus measurement SNR (in dB) from MMW holography detector measurements corrupted by AWGN. TV minimization reconstruction results with 0.0–54.7% measurement reduction are shown for the (a) synthetic slit target and (b) synthetic gun and dagger target. Backpropagation reconstruction results with 0.0–54.7% measurement reduction are shown for the (c) synthetic slit target and (d) synthetic gun and dagger target.

Fig. 8
Fig. 8

Optical schematic for MMW Gabor holography containing a waveguide (WG), object extent ( L x ), detector plane sampling with number of pixels (N) and pixel pitch (dx), waveguide to object distance ( z 1 ), and object to receiver distance ( z 3 ).

Fig. 9
Fig. 9

Superheterodyne receiver (a) circuit diagram and (b) experimental layout where incident energy (RF in) is mixed with a local oscillator (LO), down converted to an intermediate frequency (IF), amplified by both an LNA and a second amplifier, filtered with a bandpass filter (BPF), and detected with a Schottky diode.

Fig. 10
Fig. 10

Object scale of a semitransparent polymer (a) wrench, (b) dagger, and (c) gun.

Fig. 11
Fig. 11

Experimental holographic recording of a (a) model dagger and a model gun and (b) model dagger, model gun, and model wrench located in different z planes.

Fig. 12
Fig. 12

Experiment with a polymer model gun and dagger placed at two different distances along the axial plane. (a) Photograph of the experiment. Transverse slices in four different z planes of the (b) backpropagated and (c) TV minimization reconstructions. Amplitude pixel ( x , y ) as a function of z, in 10 mm increments, where TV minimization and backpropagation for a central point on the (d) barrel of the gun and (e) on the blade edge of the dagger are plotted.

Fig. 13
Fig. 13

Experiment with a polymer model wrench, gun, and dagger placed at three different distances along the axial plane. (a) Photograph of the experiment. Transverse slices in four different z planes of the (b) backpropagated and (c) TV minimization reconstructions. Amplitude pixel ( x , y ) as a function of z, in 5 mm increments, where TV minimization reconstructions and backpropagation estimates for a center point on the (d) wrench, (e) gun, and (f) dagger are plotted.

Fig. 14
Fig. 14

Sparse measurement reconstruction of experimental data using (a) linear backpropagation and (b) TV minimization for 3D object estimation. Amplitude of a central pixel ( x , y ) on the blade edge of the dagger as a function of z, plotted in 10 nm increments, from (c) 3.9% holographic measurement removal and (d) 54.68% holographic measurement removal.

Tables (1)

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Table 1 Synthetic 3D Slit (ST) and 3D Gun and Dagger (GD) Sparsity

Equations (36)

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g ( r h ) = E r ( r h ) + E o ( r h ) 2 = E r ( r h ) 2 + E o ( r h ) 2 + 2 E r ( r h ) E o ( r h ) cos [ θ r ( r h ) θ o ( r h ) ] ,
E o ( r h ) = π λ 2 E r ( r ) f o ( r ) h ( r h r ) d r ,
h ( r h r ) = exp ( 2 π i r h r / λ ) r h r .
E 0 ^ ( u x , u y , z h ) = 1 i π λ f o ^ ( u x u x o , u y u y o , 1 λ 2 u x 2 u x 2 u z 0 ) G 2 D ( u x , u y , z ) ,
G 2 D ( u x , u y , z ) = exp ( 2 π i z 1 λ 2 u x 2 u y 2 ) 1 λ 2 u x 2 u y 2 ,
E o ^ ( u x , u y , z ) = 1 i π λ f o ^ ( u x u x o , u y u y o , λ 2 ( u x 2 + u y 2 ) ) exp ( 2 π i z 1 λ 2 u x 2 u y 2 ) .
u max = 1 2 d x .
θ c min θ c z θ c max ,
θ c max = sin 1 [ ( λ 2 ) ( 1 d x u B ) ] ,
θ c min = sin 1 ( 3 λ u B 2 ) ,
g ˜ offaxis = E r ( r h ) * E o ( r h ) .
g ˜ onaxis = E o ( r h ) 2 + E r * ( r h ) E o ( r h ) + E r ( r h ) E o * ( r h ) .
N = n x n y ,
n x = W x d x ,
W x = λ z Δ x o ,
n sin θ u = W x 2 z = λ 2 Δ x o ,
sin ( θ u ) = Δ u x u ,
u θ u = Δ u x .
Δ x = λ NA .
Δ u z = Δ u z , max Δ u z , min = u ( 1 cos ( θ u ) ) = u θ u 2 .
Δ z = λ NA 2 .
g = H f 2 + H f + H * f + n ,
g n 1 , n 2 = l 2 D 1 { f ^ m 1 , m 2 , l exp ( ı l Δ z 1 λ 2 m 1 2 Δ u 2 m 2 2 Δ u 2 ) } n 1 , n 2 ,
H = 2 D 1 2 D .
f = H g ,
H = 2 D 2 D 1 .
Q l = 1 , m 1 , m 2 = exp ( ı l Δ z 1 λ 2 m 1 2 Δ u 2 m 2 2 Δ u 2 ) .
g = 2 { H [ f ] } + n + e .
f * = argmin f g H f 2 2 + τ Φ T V ( f ) ,
Φ T V = l i , j ( f i + 1 , j , l f i , j , l ) 2 + ( f i , j + 1 , l f i , j , l ) 2 ,
t ( x , y ) = p , q t p , q rect [ x p Δ T Δ T , y q Δ T Δ T ] .
n 1 , n 2 = t n 1 , n 2 g n 1 , n 2 .
g = W F 2 D 1 Q F 2 D f
f = F 2 D Q [ F 2 D 1 ] W g .
PSNR ( d B ) = def 20 log 10 { MAX A 1 n x n y d i = 1 n x j = 1 n y l = 1 d [ A i , j , l B i , j , l ] 2 } ,
f i , j , l = ( f i + 1 , j , l f i , j , l ) 2 + ( f i , j + 1 , l f i , j , l ) 2 .

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