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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 17, Iss. 15 — Jul. 20, 2009
  • pp: 13040–13049
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Compressive Holography

David J. Brady, Kerkil Choi, Daniel L. Marks, Ryoichi Horisaki, and Sehoon Lim  »View Author Affiliations


Optics Express, Vol. 17, Issue 15, pp. 13040-13049 (2009)
http://dx.doi.org/10.1364/OE.17.013040


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Abstract

Compressive sampling enables signal reconstruction using less than one measurement per reconstructed signal value. Compressive measurement is particularly useful in generating multidimensional images from lower dimensional data. We demonstrate single frame 3D tomography from 2D holographic data.

© 2009 Optical Society of America

1. Introduction

Studies presented in 1949 by Shannon and Golay have recently collided in a major revision to signal and image measurement theory. Shannon is traditionally cited to support the claim that the sampling rate in a high fidelity imaging system must be greater than twice the spatial bandwidth [1

1. C. E. Shannon, “Communications in the presence of noise,” in Proc.of the IREv 37, 10–21 (1949). [CrossRef]

]. Golay introduced the idea of artificial discrete multiplex coding in optical measurements [2

2. M. Golay, “Multislit spectroscopy,” J. Opt. Soc. Am. 39, 437–444 (1949). [CrossRef] [PubMed]

]. Candés, Tao and Romberg [3

3. E. J. Candes, J. K. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Communications on Pure and Applied Mathematics 59, 1207–1223 (2006). [CrossRef]

, 4

4. E. J. Candes and T. Tao, “Near-optimal signal recovery from random projections: Universal encoding strategies?,” IEEE Transactions on Information Theory 52(12), 5406–5425 (2006). [CrossRef]

] and Donoho [5

5. D. L. Donoho, “Compressed sensing,” IEEE Transactions on Information Theory 52(4), 1289–1306 (2006). [CrossRef]

] have demonstrated that signals, assumed to be sparse in some basis, sampled by multiplex encodings may be accurately infered with high probability using many fewer measurements than suggested by Shannon’s sampling theorem. Takhar et al. applied compressive sampling algorithms in demonstrating a single pixel camera [6

6. D. Takhar, J. N. Laska, M. B. Wakin, M. F. Duarte, D. Baron, S. Sarvotham, K. F. Kelly, and R. G. Baraniuk, “A new compressive imaging camera architecture using optical-domain compression,” in Computational Imaging IV, 6065, (San Jose, CA, USA), p. 606509, SPIE, 2006.

]. We have previously applied these methods with Golay-inspired coded aperture spectroscopy to demonstrate compressive measurement of 3D spectral datacubes from 2D measurements [7

7. M. E. Gehm, R. John, D. J. Brady, R. M. Willett, and T. J. Schulz, “Single-shot compressive spectral imaging with a dual-disperser architecture,” Opt. Express 15(21), 14013–14027 (2007). [CrossRef]

, 8

8. A. Wagadarikar, R. John, R. Willett, and D. J. Brady, “Single disperser design for coded aperture snapshot spectral imaging,” Appl. Opt. 47(10), B44–B51 (2008). [CrossRef]

].

This paper explores the relationship between compressive sampling and a third paper from the late 1940’s, Gabor’s invention of holography [9

9. D. Gabor, “A new microscopic principle,” Nature 161, 777–778 (1948). [CrossRef] [PubMed]

]. Gabor holography is an amazingly simple and effective encoder for compressive sampling. Decompressive inference improves holographic systems by increasing the number of pixels or voxels one can infer from a single hologram and by resolving reconstruction ambiguities. Holography is a comparatively effective encoder for compressive imaging because holographic multiplex measurement weights are complex valued.

After the invention of the laser, Gabor holography was generally supplanted by off-axis Leith-Upatnieks holograpy[10

10. E. N. Leith and J. Upatnieks, “Reconstructed wavefronts and communication theory,” J. Opt. Soc. Am. 52, 1123–1130 (1962). [CrossRef]

]. Signal reconstruction from off-axis holograms may be understood simply using Shannon sampling and linear filtering theory. As illustrated by recent studies of holographic cytometry [11

11. S. Seo, T. W. Su, D. K. Tseng, A. Erlinger, and A. Ozcan, “Lensfree holographic imaging for on-chip cytometry and diagnostics,” Lab on a Chip 9(6), 777–787 (2009). [CrossRef]

], however, the simplicity of the Gabor geometry remains attractive and competitive. We demonstrate in this paper that interference terms that may not be removed from Gabor holograms by linear filtering may be removed by imposing a sparsity constraint.

Our main result is that decompressive inference enables 3D tomography from a single 2D monochromatic digital hologram. While this result is in apparent conflict with Devaney’s proof that holographic tomography is ill-posed, the overthrow of Devaney’s objection using a sparsity prior and convex optimization parallels the overthrow of Shannon sampling by similar methods.

Our results also suggest that holography may have general advantages in compressive optical imaging. Since the irradiance is the only observable in conventional optical imaging, multiplex measurement codes are constrained to nonnegative weights. This results in relatively poor measurement conditioning. Holography, in contrast, is an interferometric modality in which both the amplitude and the phase of a field can be obtained as in radar or MRI. The complex-valued encodings may provide a more direct application of compressive sensing. Chan et al. [19

19. W. L. Chan, M. L. Moravec, R. G. Baraniuk, and D. M. Mittleman, “Terahertz imaging with compressed sensing and phase retrieval,” Opt. Lett. 33(9), 974–976 (2008). [CrossRef]

] have previously described phase-sensitive terahertz compressive imaging.

2. Compressive sensing background

Compressive sensing theory ensures highly accurate reconstruction for multiplex encoders that satisfy a sufficient condition called the restricted isometry property (RIP) [3

3. E. J. Candes, J. K. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Communications on Pure and Applied Mathematics 59, 1207–1223 (2006). [CrossRef]

, 4

4. E. J. Candes and T. Tao, “Near-optimal signal recovery from random projections: Universal encoding strategies?,” IEEE Transactions on Information Theory 52(12), 5406–5425 (2006). [CrossRef]

]. Let a S-sparse signal be defined by a signal that has only S nonzero components and (NS) coefficients that are exactly zero. A matrix H∈RM×N is said to satisfy S-RIP with constant δS∈(0,1) if, for any S-sparse f,

(1δS)fT22HTfT22(1+δS)fT22,
(1)

where T denotes the set of indices on which the S-sparse signal is supported, and ‖·‖2 denotes the Euclidean norm. This condition implies that for any S-sparse object to be reconstructed accurately and reliably, the corresponding sub-matrix HT of H composed of S columns of H has to form a nearly isometry transformation. Note that the condition also implies that all the eigenvalues of the Gram matrix of any S-column sub-matrix HT are distributed near 1 (in fact in the range of [1-δS 1+δS]), which consequently ensures that any S-column sub-matrix HT is well-conditioned.

Let µ 1(H,Ψ) be defined by

μ1(H,Ψ)=Nmaxhm,ψn,
1mM
1nN
(2)

where hm and ψn denote the m-th row of H and the n-th column of Ψ, respectively. Candés et al. [3

3. E. J. Candes, J. K. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Communications on Pure and Applied Mathematics 59, 1207–1223 (2006). [CrossRef]

] and Donoho [5

5. D. L. Donoho, “Compressed sensing,” IEEE Transactions on Information Theory 52(4), 1289–1306 (2006). [CrossRef]

] showed that if f is S-sparse in Ψ, and M satisfies

MCμ12(H,Ψ)SlogN,
(3)

then an accurate reconstruction can be obtained with high probability by solving

θe=argminθθ1suchthatg=Hfe=HΨθe,
(4)

where ‖θ ‖1=∑i|θi|. S denotes the number of nonzero (or significant) coefficients of f in the Ψ domain. As clear from Eqn. (3), the smaller µ 1 is, the more accurate the reconstruction would be for the same M.

An alternative definition of coherence is given by

μ2(D)=maxij,1i,jN{diTdjdi2dj2},
(5)

where di is the i-th column of the sensing matrix D=HΨ. This can also be interpreted as the maximum off-diagonal element of the Gram matrix of D, whose columns are normalized. When µ 2(D) is minimal such that

S<12(1+1μ2(D)),
(6)

then the S-sparse is necessarily the sparsest solution that satisfies minθθ0, and hence can be obtained by solving Eqn. (4) [23

23. D. L. Donoho and M. Elad, “Optimally sparse representation in general (nonorthogonal) dictionaries via 𡄓1 minimization,” Proceedings of the National Academy of Sciences of the United States of America , 100(5), 2197–2202 (2003). [CrossRef]

].

Note that for Ψ=I with I being an identity matrix (i.e., the canonical basis), µ 1(H, I)=1 for a discrete Fourier transform (DFT) matrix H. Hence, the DFT matrix that generates Fourier samples distributed uniformly at random over the frequency domain satisfies the so-called restricted isometry property with high probability given that MCSlogN is satisfied. Our Gabor hologram multiplex encoder may be considered as 3D Fourier transform encoder. However, our Fourier samples are limited to a certain band volume, which may produce a relatively large µ1, larger than 1 in general. Holographic measurements using illumination with multiple wavelengths and/or angles can provide the Fourier samples over a larger band volume improving µ 1 and, consequently, the RIP for our Gabor hologram multiplex encoder.

3. Theory and Methods

As illustrated in Fig. 1, a Gabor hologram is formed in interference between a plane wave A and a 3D object with scattering density η(x′,y′, z′). A 2D detector array records the irradiance

I(x,y)=A+E(x,y)2
=A2+E(x,y)2+A*E(x,y)+AE*(x,y),
(7)

where the scattered field E is defined under the Born approximation as

E(x,y)=dxdydzη(x,y,z)h(xx,yy,zz),
(8)

where h is the product of exp(ikz) representing the phase delay at a distance z [24

24. R. E. Blahut, Theory of Remote Image Formation, (Cambridge University Press, 2004). [CrossRef]

] and the inverse Fourier transform of the propagation transfer function (izk2kx2ky2) [25

25. J. W. Goodman, Introduction to Fourier optics, 3rd Ed., (Roberts and Company Publishers, 2005).

, Eqn. (3–74)]. Note that the squared field term |E(x,y)|2 produces the autocorrelation of the Fourier transform of the field E in the Fourier domain. In Eqn. (7), the term |A|2 is simply a constant, and hence the effect of |A|2 can be removed by eliminating the DC term (the term at the origin) from the Fourier transform of the interference irradiance measurements I(x,y). Also, we may assume that A is 1 without loss of generality. Then, we may proceed with A*E(x,y)+AE*(x,y)+|E(x,y)|2=2Re{E(x,y)}+|E(x,y)|2=2Re{E(x,y)}+e(x,y). If we neglect the nonlinearity caused by |E(x,y)|2 and regard e(x,y) as model error, then Eqn. (7) represents a linear mapping between the object scattering density and measurement data.

Fig. 1. Gabor hologram geometry.

The 2D sampled field at the detector plane with the sample spacings defined above can be expressed by

En1n2=E(n1Δ,n2Δ)
=1(2π)2dzdxdydkxdkydxdyη(x,y,z)eikzei(kxx+kyy)Σm1Σm2δ(xm1Δ)
δ(ym2Δ)eizk2kx2ky2δ(zz)ei(kxx+kyy)δ(xn1Δ)δ(yn2Δ)Σm1Σm2δ(kxm1Δk)
δ(kym2Δk)Σlδ(zlΔz)
=1N2ΣlΣm1Σm2[Σm'2Σm2ηm1m2lei2πm1m1+m2m2N]eiklΔzeilΔzk2m12Δk2m22Δk2ei2πn1m1+n2m2N,
(9)

where ηm1 m2 l=η(m1Δ,m2Δ, lΔz). Note that the terms in the square bracket form the 2D discrete Fourier transform of η. Also note that the last exponential term in the last line of Eqn. (9) also forms the inverse 2D Fourier transform in conjunction with the summations over m 1 and m 2. Hence, this equation can simply be written as

En1n2=𝓕2D1{Σlη̂m1m2leiklΔzeilΔzk2m12Δk2m22Δk2},
(10)

where η̂ denotes the Fourier transform of η, and 𝓕 −1 denotes the inverse Fourier transform operator. Equation (10) is called a multislice approximation [26

26. D. M. Paganin, Coherent X-ray Optics, (Oxford Science Publications, 2006). [CrossRef]

]. Eqn. (9) may also be viewed as a 2D slice of the 3D Fourier transform of η. The 2D slice can be interpreted as a surface patch of the Ewald k-sphere as in traditional diffraction tomography [27

27. A. C. Kak and M. Slaney, Principle of Computerized Tomographic Imaging, (Society for Industrial and Applied Mathematics, 2001). [CrossRef]

]. Note that while the 3D Fourier transform interpretation requires 3D interpolation of either the 3D Fourier transform of η or the 3D spatial scattering density η, the multiscale approximation requires no such interpolation and is less sensitive to errors that would result from 3D interpolation.

Define gˉ(n21)×Nx+n1=En1,n2 and f(l1)×(Nx×Ny)+(m21)×Nx+m1=ηm1,m2,l,, where Nx and Ny denote the numbers of pixels of the detector in the x-direction and y-direction, respectively. With these definitions, Eqn. (10) may be rewritten as

gˉ=G2DQBf,
(11)

where B=bldiag(F2D,F2D, ⋯,F2D) with F2D being the matrix representing the 2D DFT whose size is (Nx×Ny)×(Nx×Ny) and “bldiag” denoting the block diagonal matrix, Q=[P 1 P 2PNz] with [Pl]m1m2=eiklΔzeilΔzk2m12Δ22m22Δk2; represents the element of the matrix Pl at the intersection of the row m1 and the column m 2, and G2D represents the 2D inverse DFT matrix. As discussed earlier, ignoring the non-linearity of Eqn. (10) caused by the squared field term e, the Gabor hologram measurement may be algebraically written as

g=2Re{gˉ}=2Re{G2DQBf}=2Re{Hf}+e+n,
(12)

where Nx×Ny represents the Gabor hologram from which the DC component is removed, Hij=[G2DQB]i j, and e and n denote vectorized |E(x,y)|2 and additive noise, respectively. The nonlinear term is traditionally removed using off-axis holography. As demonstrated below, however, this term may be eliminated algorithmically from Gabor data using decompressive inference. We discuss the effects of e in Sec. 5.

We remark that the elements of the measurement matrix H (e.g., see Eqn. (12)) in holography are complex valued. In contrast, the typical optical imaging modality provides only nonnegative values because the only observable are the intensities. Such nonnegativity constraints confine the measurement basis space to be only the nonnegative orthant, which limits the angles between the measurement basis vectors (the columns of H) and effectively makes the coherence values large. On the other hand, interferometric imaging modalities such as holography allow the measurement vectors to reside in the entire space as opposed to the nonnegative orthant by allowing for negative values in the measurement basis vectors. Such negative values may enable us to design a sensing system with a small coherence value. This advantage with holography for compressive sensing remains valid in Gabor holography because although the f and g may be assumed to be real, the elements of the multiplex encoder H are still complex valued. We also note that in other holography such as the Leith-Upatnieks holography [25

25. J. W. Goodman, Introduction to Fourier optics, 3rd Ed., (Roberts and Company Publishers, 2005).

], the f and g as well as H are, in general, all complex valued.

Prior to discussing experimental estimation of f from Eqn. (12), we briefly consider the spatial resolution expected in the reconstruction and a sparsity constraint to enable decompressive inference. Optical measurement over a finite aperture is bandlimited. The band volume is the support in the 3D Fourier space of η(x′,y′, z′) for sampling E(x,y) over a finite aperture. Spatial resolution in imaging systems is assumed to be inversely proportional to the limits of the band volume, which yields transverse resolution Δx=λ z/D for objects at range z observed over aperture D and the axial resolution is Δz=λ (2z)2/D 2 [28

28. D. J. Brady, Optical Imaging and Spectroscopy, (Wiley, 2009). [CrossRef]

]. Our objective is to achieve this resolution over the 3D datacube using decompressive inference of a 2D hologram.

If the entire diffraction pattern is captured by the hologram, the effective aperture size D for a Gabor hologram observed in the Fresnel diffraction zone is determined by size of the object feature observed. A feature of cross section w produces a diffraction pattern with cross section D ≈λz/w. This implies that Δxw and Δz≈4w 2/λ. The dependence of axial resolution on feature size is the result of the “missing cone” in the paraxial band volume [28

28. D. J. Brady, Optical Imaging and Spectroscopy, (Wiley, 2009). [CrossRef]

]. The missing cone may be removed by recording holograms along multiple axes. In the experiments described here, however, we are satisfied with the single axis band limited resolution. We anticipate, for example, that object features of approximately 100µ cross section will be observed under visible light with approximately 1 cm axial resolution.

Forward models such as Eqn. (12) have been inverted by decompressive inference by either selecting a basis, typically a particular wavelet basis, on which f may be assumed to be sparse or by enforcing a sparsity constraint on the total variation, as defined by Rudin et al [29

29. L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D 60(1–4), 259–268 (1992). [CrossRef]

], of f. We choose the second approach here and estimate f as

f̂=argminffTVsuchthatg=Hf,
(13)

where ‖fTV is defined by

fkTV=ΣkΣn1Σn2(fk)n1,n2,
(14)

where fk denotes a 2D plane of the 3D object datacube. We adapt the two-step iterative shrinkage/thresholding algorithm (TwIST) [30

30. J. M. Bioucas-Dias and M. A. T. Figueiredo, “A new twist: Two-step iterative shrinkage/thresholding algorithms for image restoration,”, IEEE Transactions on Image Processing 16, 2992–3004 (2007). [CrossRef] [PubMed]

] to solve this optimization problem.

4. Experimental results

As illustrated in Fig. 2, we illuminated two seed parachutes of common dandelions (taraxacum) with a collimated, spatially filtered Helium-Neon laser of 632.8 nm wavelength. One object is placed 1.5 cm away from the detector array, and the other dandelion is placed 5.5 cm away from the detector array. Figures 3(b) and 3(c) are photographs of the two seed parachutes. The illumination and scattered fields were captured in the Gabor hologram shown in Fig. 3(a). Figure 3(d) is the 3D datacube estimated from the Gabor recording by the TV-minimization algorithm. As the reconstruction shows, the stem and the petals, representing the high-frequency features in the image, are reconstructed well. In addition, the distance between the detector plane and the first parachute and the distance between the two parachutes are also accurately estimated. We notice that there are still some errors in some reconstruction planes including the planes in which the two seed parachutes. We conjecture that the errors are mainly reconstruction errors and the effects of noise, and they occur because of the rather insufficient number of measurements. The reconstruction errors can be suppressed by exploiting the phase-shifting holography that can increase the effective SNR and the effective number of measurements [31

31. V. Micó, J. García, Z. Zalevsky, and B. Javidi, “Phase-shifting Gabor holography,” Opt. Express 17, 1492–1494 (2009).

]. The reconstruction error in the plane of z=0 is explained in Sec. 5. Figure 3(e) shows the backpropagated (or digital refocusing) field that is obtained by digitally backpropagating the hologram using the propagation kernel h in Eqn. (8) [25

25. J. W. Goodman, Introduction to Fourier optics, 3rd Ed., (Roberts and Company Publishers, 2005).

]. In contrast to the reconstruction in Fig. 3(d), the backpropagated field shows messy planes full of out-of-focus features obscuring the object features of the two parachutes.

Fig. 2. Experimental apparatus.

5. Discussion

To understand the effects of the squared field e(x,y) on the reconstruction, we consider the behavior of the algorithm. The algorithm determines how much portion of the measurement (or measurement estimate) should be placed in which object plane based on the amount of correlation that the diffraction patterns of the measurement and the diffraction pattern that the estimate at a particular object plane would produce. The squared field e is inclined to produce the diffraction patterns that bear little correlation with all of the object planes. Consequently, most of the reconstruction error caused by e tends to remain in the plane of z=0. In this way, we may effectively isolate most of the errors that result from the squared field term. Since these errors are concentrated in the plane of z=0 (measurement plane), such reconstruction values are not part of the object scattering density η by definition. Therefore, we may effectively remove most of the errors that would result from the squared field (zero-order) term in the reconstructions.

Fig. 3. (a) Raw Gabor hologram for seed parachutes of taraxacum arranged as in Fig. 2, (b) and (c) photographs of the individual objects, (d) transverse slices at various ranges of the tomographic reconstruction of the 3D data volume containing both objects, and (e) transverse slices at various ranges of the backpropagated (numerical refocusing) field.
Fig. 4. Simulations showing the effects of squared field term e: the 3D datacube estimates from the squared field with no diffraction (a) and the squared field with diffraction (b).

Now, we consider the effects of the conjugate term E*(x,y) on the reconstruction. In conventional digital holography, the effect of the conjugate term on reconstructions is called the twin image problem [25

25. J. W. Goodman, Introduction to Fourier optics, 3rd Ed., (Roberts and Company Publishers, 2005).

]. Considering that e would behave as discussed above, our forward model including the conjugated field E*(x,y) is approximated well to be linear. In our approach, the effect of the conjugate term on the reconstruction can be numerically eliminated by confining our estimate domain to only the one side of the measurement plane (i.e., z≥0). Our decompressive inference method directly reconstructs the 3D volume object η rather than the field at a distance which is a superposition of all the object planes each of which is at a different focus. This implies that the virtual object η* is placed to the other side of the measurement plane (i.e., z≤0) by our inference method. This aspect of our decompressive inference allows a separation of the real and the virtual objects. Hence, by confining the reconstruction domain to be the region in which z≥0, we can readily resolve the twin image problem in the reconstruction. A similar philosophy has successfully been applied to 2D holographic reconstruction to remove the twin image problem [32

32. T. Latychevskaia and H. W. Fink, “Solution to the Twin Image Problem in Holography,” Phys. Rev. Lett. 98, 233901 (2007). [CrossRef] [PubMed]

].

In conclusion, our results show that we can obtain a 3D (512×512×10, or 712×712×10 after preprocessing to avoid the circular convolution effect) datacube of voxels with 5.2 µm transverse resolution and 0.8 cm axial resolution reconstructed from a single 2D (512×512) hologram. This demonstrates the main advantages of compressive holography, i.e. that holograms naturally encode high quality multiplex data and that decompressive inference can infer multidimensional objects from lower dimensional data. Extensions of compressive holography may use off-axis encoding to filter nonlinear terms and multispectral illumination to increase the band volume and improve axial resolution. It would be also useful to combine our approach with phase-shifting digital holography.

Acknowledgments

This research was supported by a DARPA under AFOSR contract FA9550-06-1-0230. The authors also thank the anonymous reviewers for their insightful comments and suggestions that significantly improve and help clarify this paper.

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J. M. Bioucas-Dias and M. A. T. Figueiredo, “A new twist: Two-step iterative shrinkage/thresholding algorithms for image restoration,”, IEEE Transactions on Image Processing 16, 2992–3004 (2007). [CrossRef] [PubMed]

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V. Micó, J. García, Z. Zalevsky, and B. Javidi, “Phase-shifting Gabor holography,” Opt. Express 17, 1492–1494 (2009).

32.

T. Latychevskaia and H. W. Fink, “Solution to the Twin Image Problem in Holography,” Phys. Rev. Lett. 98, 233901 (2007). [CrossRef] [PubMed]

OCIS Codes
(070.0070) Fourier optics and signal processing : Fourier optics and signal processing
(090.1995) Holography : Digital holography

ToC Category:
Holography

History
Original Manuscript: May 18, 2009
Revised Manuscript: July 2, 2009
Manuscript Accepted: July 5, 2009
Published: July 15, 2009

Citation
David J. Brady, Kerkil Choi, Daniel L. Marks, Ryoichi Horisaki, and Sehoon Lim, "Compressive Holography," Opt. Express 17, 13040-13049 (2009)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-15-13040


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