Abstract

Better understanding of particle–particle and particle–fluid interactions requires accurate 3D measurements of particle distributions and motions. We introduce the application of in-line digital holographic microscopy as a viable tool for measuring distributions of dense micrometer (3.2  μm) and submicrometer (0.75  μm) particles in a liquid solution with large depths of 110  mm. By recording a magnified hologram, we obtain a depth of field of 1000 times the object diameter and a reduced depth of focus of approximately 10 particle diameters, both representing substantial improvements compared to a conventional microscope and in-line holography. Quantitative information on depth of field, depth of focus, and axial resolution is provided. We demonstrate that digital holographic microscopy can resolve the locations of several thousand particles and can measure their motions and trajectories using cinematographic holography. A sample trajectory and detailed morphological information of a free-swimming copepod nauplius are presented.

© 2006 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. D. DeAngelis and L. Gross, Individual-Based Models and Approaches in Ecology (Chapman & Hall, 1992).
  2. W. Fennel and T. Neumann, Introduction to the Modelling of Marine Ecosystems, Vol. 72 of the Elsevier Oceanography Series (Elsevier, 2004).
  3. W. Fennel and T. Osborn, A Unifying Framework for Marine Ecological Model Comparison Deep Sea Research II, Rep. DSR-WTD-06, 2004.
  4. E. Malkiel, J. Sheng, J. Katz, and J. R. Strickler, "The three-dimensional flow field generated by a feeding calanoid copepod measured using digital holography," J. Exp. Biol. 206, 3657-3666 (2003).
    [CrossRef] [PubMed]
  5. G. Pan and H. Meng, "Digital holography of particle fields: reconstruction by use of complex amplitude," Appl. Opt. 42, 827-833 (2003).
    [CrossRef] [PubMed]
  6. U. Schnars and W. P. O. Juptner, "Digital recording and numerical reconstruction of holograms," Meas. Sci. Technol. 13, R85-R101 (2002).
    [CrossRef]
  7. J. H. Milgram and W. C. Li, "Computational reconstruction of images from holograms," Appl. Opt. 41, 853-864 (2002).
    [CrossRef] [PubMed]
  8. W. B. Xu, M. H. Jericho, I. A. Meinertzhagen, and H. J. Kreuzer, "Digital in-line holography for biological applications," Proc. Natl. Acad. Sci. U.S.A. 98, 11301-11305 (2001).
    [CrossRef] [PubMed]
  9. L. Xu, X. Peng, J. Miao, and A. K. Asundi, "Studies of digital microscopic holography with applications to microstructure testing," Appl. Opt. 40, 5046-5051 (2001).
    [CrossRef]
  10. W. Xu, M. H. Jericho, H. J. Kreuzer, and I. A. Meinertzhagen, "Tracking particles in four dimensions with in-line holographic microscopy," Opt. Lett. 28, 164-166 (2003).
    [CrossRef] [PubMed]
  11. D. Carl, B. Kemper, G. Wernicke, and G. von Bally, "Parameter-optimized digital holographic microscope for high-resolution living-cell analysis," Appl. Opt. 43, 6536-6544 (2004).
    [CrossRef]
  12. T. Colomb, F. Durr, E. Cuche, P. Marquet, H. G. Limberger, R. P. Salathe, and C. Depeursinge, "Polarization microscopy by use of digital holography: application to optical-fiber birefringence measurements," Appl. Opt. 44, 4461-4469 (2005).
    [CrossRef] [PubMed]
  13. G. Coppola, P. Ferraro, M. Iodice, S. De Nicola, A. Finizio, and S. Grilli, "A digital holographic microscope for complete characterization of microelectromechanical systems," Meas. Sci. Technol. 15, 529-539 (2004).
    [CrossRef]
  14. F. Dubois, M. L. N. Requena, C. Minetti, O. Monnom, and E. Istasse, "Partial spatial coherence effects in digital holographic microscopy with a laser source," Appl. Opt. 43, 1131-1139 (2004).
    [CrossRef] [PubMed]
  15. P. Ferraro, G. Coppola, S. De Nicola, A. Finizio, and G. Pierattini, "Digital holographic microscope with automatic focus tracking by detecting sample displacement in real time," Opt. Lett. 28, 1257-1259 (2003).
    [CrossRef] [PubMed]
  16. P. Marquet, B. Rappaz, P. J. Magistretti, E. Cuche, Y. Emery, T. Colomb, and C. Depeursinge, "Digital holographic microscopy: a noninvasive contrast imaging technique allowing quantitative visualization of living cells with subwavelength axial accuracy," Opt. Lett. , 30468-470 (2005).
    [CrossRef] [PubMed]
  17. L. Repetto, E. Piano, and C. Pontiggia, "Lensless digital holographic microscope with light-emitting diode illumination," Opt. Lett. 29, 1132-1134 (2004).
    [CrossRef] [PubMed]
  18. F. Dubois, C. Minetti, O. Monnom, C. Yourassowsky, J. C. Legros, and P. Kischel, "Pattern recognition with a digital holographic microscope working in partially coherent illumination," Appl. Opt. 41, 4108-4119 (2002).
    [CrossRef] [PubMed]
  19. J. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).
  20. J. Sheng, E. Malkiel, and J. Katz, "Single beam two-views holographic particle image velocimetry," Appl. Opt. 42, 235-250 (2003).
    [CrossRef] [PubMed]
  21. H. Meng and F. Hussain, "In-line recording and off-axis viewing technique for holographic particle velocimetry," Appl. Opt. 34, 1827-1840 (1995).
    [CrossRef] [PubMed]
  22. J. Zhang, B. Tao, and J. Katz, "Turbulent flow measurement in a square duct with hybrid holographic PIV," Exp. Fluids 23, 373-381 (1997).
    [CrossRef]
  23. C. S. Vikram, Particle Field Holography (Cambridge U. Press, 1992).
    [CrossRef]
  24. W. D. Yang, A. B. Kostinski, and R. A. Shaw, "Depth-of-focus reduction for digital in-line holography of particle fields," Opt. Lett. 30, 1303-1305 (2005).
    [CrossRef] [PubMed]
  25. S. Inoué and K. R. Spring, Video Microscopy: the Fundamentals (Plenum, 1997).
    [CrossRef]

2005 (3)

2004 (4)

2003 (5)

2002 (3)

2001 (2)

L. Xu, X. Peng, J. Miao, and A. K. Asundi, "Studies of digital microscopic holography with applications to microstructure testing," Appl. Opt. 40, 5046-5051 (2001).
[CrossRef]

W. B. Xu, M. H. Jericho, I. A. Meinertzhagen, and H. J. Kreuzer, "Digital in-line holography for biological applications," Proc. Natl. Acad. Sci. U.S.A. 98, 11301-11305 (2001).
[CrossRef] [PubMed]

1997 (1)

J. Zhang, B. Tao, and J. Katz, "Turbulent flow measurement in a square duct with hybrid holographic PIV," Exp. Fluids 23, 373-381 (1997).
[CrossRef]

1995 (1)

Asundi, A. K.

Carl, D.

Colomb, T.

Coppola, G.

G. Coppola, P. Ferraro, M. Iodice, S. De Nicola, A. Finizio, and S. Grilli, "A digital holographic microscope for complete characterization of microelectromechanical systems," Meas. Sci. Technol. 15, 529-539 (2004).
[CrossRef]

P. Ferraro, G. Coppola, S. De Nicola, A. Finizio, and G. Pierattini, "Digital holographic microscope with automatic focus tracking by detecting sample displacement in real time," Opt. Lett. 28, 1257-1259 (2003).
[CrossRef] [PubMed]

Cuche, E.

De Nicola, S.

G. Coppola, P. Ferraro, M. Iodice, S. De Nicola, A. Finizio, and S. Grilli, "A digital holographic microscope for complete characterization of microelectromechanical systems," Meas. Sci. Technol. 15, 529-539 (2004).
[CrossRef]

P. Ferraro, G. Coppola, S. De Nicola, A. Finizio, and G. Pierattini, "Digital holographic microscope with automatic focus tracking by detecting sample displacement in real time," Opt. Lett. 28, 1257-1259 (2003).
[CrossRef] [PubMed]

DeAngelis, D.

D. DeAngelis and L. Gross, Individual-Based Models and Approaches in Ecology (Chapman & Hall, 1992).

Depeursinge, C.

Dubois, F.

Durr, F.

Emery, Y.

Fennel, W.

W. Fennel and T. Neumann, Introduction to the Modelling of Marine Ecosystems, Vol. 72 of the Elsevier Oceanography Series (Elsevier, 2004).

W. Fennel and T. Osborn, A Unifying Framework for Marine Ecological Model Comparison Deep Sea Research II, Rep. DSR-WTD-06, 2004.

Ferraro, P.

G. Coppola, P. Ferraro, M. Iodice, S. De Nicola, A. Finizio, and S. Grilli, "A digital holographic microscope for complete characterization of microelectromechanical systems," Meas. Sci. Technol. 15, 529-539 (2004).
[CrossRef]

P. Ferraro, G. Coppola, S. De Nicola, A. Finizio, and G. Pierattini, "Digital holographic microscope with automatic focus tracking by detecting sample displacement in real time," Opt. Lett. 28, 1257-1259 (2003).
[CrossRef] [PubMed]

Finizio, A.

G. Coppola, P. Ferraro, M. Iodice, S. De Nicola, A. Finizio, and S. Grilli, "A digital holographic microscope for complete characterization of microelectromechanical systems," Meas. Sci. Technol. 15, 529-539 (2004).
[CrossRef]

P. Ferraro, G. Coppola, S. De Nicola, A. Finizio, and G. Pierattini, "Digital holographic microscope with automatic focus tracking by detecting sample displacement in real time," Opt. Lett. 28, 1257-1259 (2003).
[CrossRef] [PubMed]

Goodman, J.

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

Grilli, S.

G. Coppola, P. Ferraro, M. Iodice, S. De Nicola, A. Finizio, and S. Grilli, "A digital holographic microscope for complete characterization of microelectromechanical systems," Meas. Sci. Technol. 15, 529-539 (2004).
[CrossRef]

Gross, L.

D. DeAngelis and L. Gross, Individual-Based Models and Approaches in Ecology (Chapman & Hall, 1992).

Hussain, F.

Inoué, S.

S. Inoué and K. R. Spring, Video Microscopy: the Fundamentals (Plenum, 1997).
[CrossRef]

Iodice, M.

G. Coppola, P. Ferraro, M. Iodice, S. De Nicola, A. Finizio, and S. Grilli, "A digital holographic microscope for complete characterization of microelectromechanical systems," Meas. Sci. Technol. 15, 529-539 (2004).
[CrossRef]

Istasse, E.

Jericho, M. H.

W. Xu, M. H. Jericho, H. J. Kreuzer, and I. A. Meinertzhagen, "Tracking particles in four dimensions with in-line holographic microscopy," Opt. Lett. 28, 164-166 (2003).
[CrossRef] [PubMed]

W. B. Xu, M. H. Jericho, I. A. Meinertzhagen, and H. J. Kreuzer, "Digital in-line holography for biological applications," Proc. Natl. Acad. Sci. U.S.A. 98, 11301-11305 (2001).
[CrossRef] [PubMed]

Juptner, W. P. O.

U. Schnars and W. P. O. Juptner, "Digital recording and numerical reconstruction of holograms," Meas. Sci. Technol. 13, R85-R101 (2002).
[CrossRef]

Katz, J.

E. Malkiel, J. Sheng, J. Katz, and J. R. Strickler, "The three-dimensional flow field generated by a feeding calanoid copepod measured using digital holography," J. Exp. Biol. 206, 3657-3666 (2003).
[CrossRef] [PubMed]

J. Sheng, E. Malkiel, and J. Katz, "Single beam two-views holographic particle image velocimetry," Appl. Opt. 42, 235-250 (2003).
[CrossRef] [PubMed]

J. Zhang, B. Tao, and J. Katz, "Turbulent flow measurement in a square duct with hybrid holographic PIV," Exp. Fluids 23, 373-381 (1997).
[CrossRef]

Kemper, B.

Kischel, P.

Kostinski, A. B.

Kreuzer, H. J.

W. Xu, M. H. Jericho, H. J. Kreuzer, and I. A. Meinertzhagen, "Tracking particles in four dimensions with in-line holographic microscopy," Opt. Lett. 28, 164-166 (2003).
[CrossRef] [PubMed]

W. B. Xu, M. H. Jericho, I. A. Meinertzhagen, and H. J. Kreuzer, "Digital in-line holography for biological applications," Proc. Natl. Acad. Sci. U.S.A. 98, 11301-11305 (2001).
[CrossRef] [PubMed]

Legros, J. C.

Li, W. C.

Limberger, H. G.

Magistretti, P. J.

Malkiel, E.

J. Sheng, E. Malkiel, and J. Katz, "Single beam two-views holographic particle image velocimetry," Appl. Opt. 42, 235-250 (2003).
[CrossRef] [PubMed]

E. Malkiel, J. Sheng, J. Katz, and J. R. Strickler, "The three-dimensional flow field generated by a feeding calanoid copepod measured using digital holography," J. Exp. Biol. 206, 3657-3666 (2003).
[CrossRef] [PubMed]

Marquet, P.

Meinertzhagen, I. A.

W. Xu, M. H. Jericho, H. J. Kreuzer, and I. A. Meinertzhagen, "Tracking particles in four dimensions with in-line holographic microscopy," Opt. Lett. 28, 164-166 (2003).
[CrossRef] [PubMed]

W. B. Xu, M. H. Jericho, I. A. Meinertzhagen, and H. J. Kreuzer, "Digital in-line holography for biological applications," Proc. Natl. Acad. Sci. U.S.A. 98, 11301-11305 (2001).
[CrossRef] [PubMed]

Meng, H.

Miao, J.

Milgram, J. H.

Minetti, C.

Monnom, O.

Neumann, T.

W. Fennel and T. Neumann, Introduction to the Modelling of Marine Ecosystems, Vol. 72 of the Elsevier Oceanography Series (Elsevier, 2004).

Osborn, T.

W. Fennel and T. Osborn, A Unifying Framework for Marine Ecological Model Comparison Deep Sea Research II, Rep. DSR-WTD-06, 2004.

Pan, G.

Peng, X.

Piano, E.

Pierattini, G.

Pontiggia, C.

Rappaz, B.

Repetto, L.

Requena, M. L. N.

Salathe, R. P.

Schnars, U.

U. Schnars and W. P. O. Juptner, "Digital recording and numerical reconstruction of holograms," Meas. Sci. Technol. 13, R85-R101 (2002).
[CrossRef]

Shaw, R. A.

Sheng, J.

E. Malkiel, J. Sheng, J. Katz, and J. R. Strickler, "The three-dimensional flow field generated by a feeding calanoid copepod measured using digital holography," J. Exp. Biol. 206, 3657-3666 (2003).
[CrossRef] [PubMed]

J. Sheng, E. Malkiel, and J. Katz, "Single beam two-views holographic particle image velocimetry," Appl. Opt. 42, 235-250 (2003).
[CrossRef] [PubMed]

Spring, K. R.

S. Inoué and K. R. Spring, Video Microscopy: the Fundamentals (Plenum, 1997).
[CrossRef]

Strickler, J. R.

E. Malkiel, J. Sheng, J. Katz, and J. R. Strickler, "The three-dimensional flow field generated by a feeding calanoid copepod measured using digital holography," J. Exp. Biol. 206, 3657-3666 (2003).
[CrossRef] [PubMed]

Tao, B.

J. Zhang, B. Tao, and J. Katz, "Turbulent flow measurement in a square duct with hybrid holographic PIV," Exp. Fluids 23, 373-381 (1997).
[CrossRef]

Vikram, C. S.

C. S. Vikram, Particle Field Holography (Cambridge U. Press, 1992).
[CrossRef]

von Bally, G.

Wernicke, G.

Xu, L.

Xu, W.

Xu, W. B.

W. B. Xu, M. H. Jericho, I. A. Meinertzhagen, and H. J. Kreuzer, "Digital in-line holography for biological applications," Proc. Natl. Acad. Sci. U.S.A. 98, 11301-11305 (2001).
[CrossRef] [PubMed]

Yang, W. D.

Yourassowsky, C.

Zhang, J.

J. Zhang, B. Tao, and J. Katz, "Turbulent flow measurement in a square duct with hybrid holographic PIV," Exp. Fluids 23, 373-381 (1997).
[CrossRef]

Appl. Opt. (9)

H. Meng and F. Hussain, "In-line recording and off-axis viewing technique for holographic particle velocimetry," Appl. Opt. 34, 1827-1840 (1995).
[CrossRef] [PubMed]

L. Xu, X. Peng, J. Miao, and A. K. Asundi, "Studies of digital microscopic holography with applications to microstructure testing," Appl. Opt. 40, 5046-5051 (2001).
[CrossRef]

J. H. Milgram and W. C. Li, "Computational reconstruction of images from holograms," Appl. Opt. 41, 853-864 (2002).
[CrossRef] [PubMed]

F. Dubois, C. Minetti, O. Monnom, C. Yourassowsky, J. C. Legros, and P. Kischel, "Pattern recognition with a digital holographic microscope working in partially coherent illumination," Appl. Opt. 41, 4108-4119 (2002).
[CrossRef] [PubMed]

J. Sheng, E. Malkiel, and J. Katz, "Single beam two-views holographic particle image velocimetry," Appl. Opt. 42, 235-250 (2003).
[CrossRef] [PubMed]

G. Pan and H. Meng, "Digital holography of particle fields: reconstruction by use of complex amplitude," Appl. Opt. 42, 827-833 (2003).
[CrossRef] [PubMed]

F. Dubois, M. L. N. Requena, C. Minetti, O. Monnom, and E. Istasse, "Partial spatial coherence effects in digital holographic microscopy with a laser source," Appl. Opt. 43, 1131-1139 (2004).
[CrossRef] [PubMed]

D. Carl, B. Kemper, G. Wernicke, and G. von Bally, "Parameter-optimized digital holographic microscope for high-resolution living-cell analysis," Appl. Opt. 43, 6536-6544 (2004).
[CrossRef]

T. Colomb, F. Durr, E. Cuche, P. Marquet, H. G. Limberger, R. P. Salathe, and C. Depeursinge, "Polarization microscopy by use of digital holography: application to optical-fiber birefringence measurements," Appl. Opt. 44, 4461-4469 (2005).
[CrossRef] [PubMed]

Exp. Fluids (1)

J. Zhang, B. Tao, and J. Katz, "Turbulent flow measurement in a square duct with hybrid holographic PIV," Exp. Fluids 23, 373-381 (1997).
[CrossRef]

J. Exp. Biol. (1)

E. Malkiel, J. Sheng, J. Katz, and J. R. Strickler, "The three-dimensional flow field generated by a feeding calanoid copepod measured using digital holography," J. Exp. Biol. 206, 3657-3666 (2003).
[CrossRef] [PubMed]

Meas. Sci. Technol. (2)

U. Schnars and W. P. O. Juptner, "Digital recording and numerical reconstruction of holograms," Meas. Sci. Technol. 13, R85-R101 (2002).
[CrossRef]

G. Coppola, P. Ferraro, M. Iodice, S. De Nicola, A. Finizio, and S. Grilli, "A digital holographic microscope for complete characterization of microelectromechanical systems," Meas. Sci. Technol. 15, 529-539 (2004).
[CrossRef]

Opt. Lett. (5)

Proc. Natl. Acad. Sci. U.S.A. (1)

W. B. Xu, M. H. Jericho, I. A. Meinertzhagen, and H. J. Kreuzer, "Digital in-line holography for biological applications," Proc. Natl. Acad. Sci. U.S.A. 98, 11301-11305 (2001).
[CrossRef] [PubMed]

Other (6)

D. DeAngelis and L. Gross, Individual-Based Models and Approaches in Ecology (Chapman & Hall, 1992).

W. Fennel and T. Neumann, Introduction to the Modelling of Marine Ecosystems, Vol. 72 of the Elsevier Oceanography Series (Elsevier, 2004).

W. Fennel and T. Osborn, A Unifying Framework for Marine Ecological Model Comparison Deep Sea Research II, Rep. DSR-WTD-06, 2004.

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

C. S. Vikram, Particle Field Holography (Cambridge U. Press, 1992).
[CrossRef]

S. Inoué and K. R. Spring, Video Microscopy: the Fundamentals (Plenum, 1997).
[CrossRef]

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (10)

Fig. 1
Fig. 1

Theoretical depth of field as a function of resolution or field of view.[25] M refers to the power of a microscope objective.

Fig. 2
Fig. 2

Optical setup of the digital holographic microscope.

Fig. 3
Fig. 3

(a) Part of a recorded hologram using a 10× objective, containing 3.189 μm diameter particles in a 1 mm deep solution. (b)–(d) Reconstruction of planes located 120, 580, and 800 μm from the hologram plane. In-focus particles appear as dark spots on the bright background. (e) A combined and∕or compressed image containing all the particles covered by the hologram section shown in (a). (f) Location of all the particles detected within the entire 1.5 × 1.5 × 1 mm3 volume, totaling 5769 particles.

Fig. 4
Fig. 4

Ensemble-averaged intensity distribution along the depth direction. For definitions of terms, see Eq. (9). The inset is an isointensity surface plot of a typical reconstructed particle at 75% of its peak intensity. The depths of the sample are 1 mm for the 3 μm particles and 0.1 mm for the 0.75 μm particle.

Fig. 5
Fig. 5

Statistics on the properties of reconstructed particle traces. Rows are arranged as experimental conditions A–D (Table 1), with the following magnifications, particle diameters, and sample depth. A, 10×, 3.189 μm, 1 mm; B, 18×, 3.189 μm, 1 mm; C, 40×, 3.189 μm, 1 mm; and D, 40×, 0.75 μm, 0.1 mm. The columns are (from left to right) Dd , Lz∕Dd, Lz , where Dd is the detected particle diameter and Lz is the depth of focus, both based on 75% of the peak intensity. The numbers in 〈 〉 above the distributions indicate mean values.

Fig. 6
Fig. 6

Summary of the present test conditions, including largest successfully tested sample depth (Zd ) and depth of focus normalized by the nominal particle diameter and theoretical depth of field (Φ). Letters indicate test conditions (Table 1 and Fig. 5).

Fig. 7
Fig. 7

A demonstration of a cinematographic digital holographic microscope. (a) Combined and∕or compressed tracks, consisting of five exposures, of a 3.189 μm particle located within a 1 mm deep sample. (b) Sample tracks, consisting of seven consecutive exposures, of 0.75 μm particles, combined over a depth of 100 μm.

Fig. 8
Fig. 8

A 3D, 4 s trajectory, and behavior of a free-swimming copepod nauplius, stage VI, obtained using a cinematographic digital holographic microscope. The images in A–E show reconstructions of planes dissecting the center of the organism at different times, as indicated in each frame.

Fig. 9
Fig. 9

Intensity distribution of a hologram generated by a single 15 μm wide, 1D slit, located 1 mm away from the hologram plane. The solid curve represents intensity distribution of fringes in the hologram, while the dashed curve outlines their envelope.

Fig. 10
Fig. 10

Detected particle size distribution of Case C (3 μm particles, 40× objective) conditionally sampled based on distance from the hologram plane.

Tables (1)

Tables Icon

Table 1 Parameters of Present Tests, and Resulting Mean Diameters and Depth of Focus of Reconstructed Particle Traces

Equations (12)

Equations on this page are rendered with MathJax. Learn more.

U H ( x , y ) = A e j k r n H + i a i ( x , y ; z i ) h z ( x x , y y ; z i ) d x d y ,
h z ( x x , y y ; z i ) = 1 j λ z i exp { j k 2 z i [ ( x x ) 2 + ( y y ) 2 ] } .
I z ( x , y ) = U H U H *
=  A 2 A a * ( x , y ) h z * ( x , y ) A a ( x , y ) h z ( x , y ) + | a ( x , y ) h z ( x , y ) | 2 ,
U i ( x i , y i ; d i ) = h l ( x i , y i ; x o , y o ) U o ( x o , y o ) d x o d y o ,
h l ( x i , y i ; x o , y o ) = 1 M δ ( x i M + x o , y i M + y o ) exp [ j k 2 M 2 d 0 ( x i     2 + y i     2 ) ] exp [ j k 2 d i ( x o     2 + y o     2 ) ] ,
U i ( x i , y i , d i ) = 1 M U H ( x i M , y i M ) exp [ j k 2 M 2 d 0 ( x i     2 + y i     2 ) ] exp [ j k 2 d i ( x o     2 + y o     2 ) ] .
I i ( x i , y i ) = 1 M 2 U H ( x i M , y i M ) U H * ( x i M , y i M ) ,
φ p ( x , y ; z ) = I i ( x , y ) h z ( x , y ; z ) ,
I p ( x , y ; z ) = φ p φ p * .
I combined ( x , y ) = min z I ( x , y , z ) .
Δ I center ( z ) / Δ I center ( z min ) = [ I ¯ x V ( z ) I center ( z ) ] / [ I ¯ x V ( z min ) I center ( z min ) ]

Metrics