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  • Vol. 8, Iss. 4 — May. 22, 2013
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Digital tomographic compressive holographic reconstruction of three-dimensional objects in transmissive and reflective geometries

Logan Williams, Georges Nehmetallah, and Partha P. Banerjee  »View Author Affiliations


Applied Optics, Vol. 52, Issue 8, pp. 1702-1710 (2013)
http://dx.doi.org/10.1364/AO.52.001702


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Abstract

In this work compressive holography (CH) with multiple projection tomography is applied to solve the inverse ill-posed problem of reconstruction of three-dimensional (3D) objects with high axial accuracy. To visualize the 3D shape, we propose digital tomographic CH, where projections from more than one direction, as in tomographic imaging, can be employed, so that a 3D shape with improved axial resolution can be reconstructed. Also, we propose possible schemes for shadow elimination when the same object is illuminated at multiple angles using a single illuminating beam and using a single CCD. Finally, we adapt CH designed for a Gabor-type setup to a reflective geometry and apply the technique to reflective objects.

© 2013 Optical Society of America

1. Introduction

It is well known from communication theory that for a sampled signal, the sampling rate must be greater than twice the bandwidth for faithful reproduction of the original signal. The concept of sampling at the Nyquist rate was postulated by Shannon in 1949 [1

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

]. In the same year, 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]

]. More than 50 years later, Candes et al. [3

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

], Candes and Tao [4

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

], and Donoho [5

5. D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52, 1289–1306 (2006). [CrossRef]

] demonstrated that signals, which are sparse on a certain basis and sampled by multiplex encodings, may be accurately inferred with high probability using many fewer measurements than suggested by Shannon’s sampling theorem [6

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

]. This is referred to as compressive sensing (CS). CS enables signal reconstruction using less than one measurement per reconstructed signal value. In essence, compressive measurement is particularly useful in generating multidimensional images from lower dimensional data [7

7. A. J. Devaney, “Nonuniqueness in the inverse scattering problem,” J. Math. Phys. Sci. 19, 1526–1531 (1978). [CrossRef]

].

In this paper we extend a CH setup to multiple projections using tomography to solve the inverse ill-posed problem of reconstruction of 3D objects with high axial accuracy. To visualize the 3D shape, we propose digital tomographic CH (DiTCH), where projections from more than one direction, as in tomographic imaging, can be employed, so that a 3D shape with improved axial resolution can be reconstructed. This technique can also be used for a multitude of environmental applications, including determination of water contamination by bacteria, water contamination by oil spills, size and shape of water droplets in the atmosphere, etc. This technique is based on a combination of holography, tomography, and CS (CH) and is applied to two cases: first, weakly scattering water bubbles illuminated from two angles using one illuminating beam and a single CCD, and second, a scattering object like a ball-point pen spring sequentially illuminated from multiple angles and with holograms sequentially recorded on a CCD. For the latter case, we also propose techniques for shadow elimination during simultaneous illumination of the object from different directions using one illuminating beam. In such nonweakly scattering objects and, say, for a two-projection setup, the beam’s second pass through the object contains the “shadow” of the first pass. This shadow must be separately recorded and used as the illumination profile when reconstructing the second pass through the object. Finally, we adapt the CH designed for a Gabor-type setup to a reflective geometry and show reconstruction for reflective objects.

The organization of the paper is as follows. In Section 2 we discuss the theory behind the DiTCH technique. In Section 3 we discuss the four experimental setups and the results obtained. Section 4 concludes the paper.

2. Digital Tomographic Compressive Holography

The objective of CS is to recover exactly the signal from a “few” samples. The key to compressed sensing are sparsity and the l1 norm defined as (c1=i|ci|), whereas the usual Euclidean l2 norm adds the square of the components of a vector: (c2=(i|ci|2)1/2). Thus, we can find the vector [cj] (a N×1 matrix) by solving the underdetermined problem
Akjcj=bk,Akj=ϕkiψij,
(1)
where Φ=[ϕki] is a rectangular M×N matrix called the projection or measurement matrix (K<MN), Ψ=[ψij] is an N×N matrix that is the basis matrix, A=[Akj] is an M×N matrix, and K is the number of nonzero sparse elements defined in Fig. 1. Knowing [cj], we can recover samples fi of a function f using fi=ψijcj. Figure 1 shows a schematic of the different matrices involved in CS [9

9. J. Romberg and M. Wakin, “Compressed sensing: a tutorial,” in IEEE Statistical Signal Processing Workshop, Madison, Wisconsin (IEEE, 2007).

,10

10. R. Baraniuk, “Compressive sensing,” IEEE Signal Process. Mag. 24, 118–121 (2007). [CrossRef]

].

Fig. 1. Schematic of the different matrices used for CS.

Once these conditions are satisfied, a valid reconstruction can be accomplished with high probability if we solve [8

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

,9

9. J. Romberg and M. Wakin, “Compressed sensing: a tutorial,” in IEEE Statistical Signal Processing Workshop, Madison, Wisconsin (IEEE, 2007).

]
c^=argmincc1,such thatb=Φf=ΦΨc=Ac.
(2)
The use of the l1 norm helps in finding sparse solutions.

Fig. 2. Typical Gabor-type setup using the transmissive geometry of an object surrounded by small objects. FPA, focal plane array.

Equations (3)–(5) represent the relation between the 3D object scattering density and the 2D measurement data, and the CH problem can be canonically rewritten as [6

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

]
b=2Re{b¯}=2Re{I1PIf}+e2Re{Φf}+e,
(6)
where I, I1 represent the forward and inverse (discrete, 2D) Fourier transform operators, respectively, P represents the propagator or discretized transfer function that is the Fourier transform of Eq. (5), Φ is the “measurement” matrix, and f represents the “sampled” signal. To keep the same notation as in Eq. (1), b¯ represents the field EO and b represents I=2Re{EO(x,y)}+e(x,y), as shown in Eq. (3), which is the quantity recorded on the CCD.

Note that optical measurement over a finite aperture D is band-limited. The spatial resolution in an imaging systems is assumed to be inversely proportional to the limits of the band volume and yields the transverse resolution Δx,y=(λz/D) and the longitudinal resolution Δz=λ(2z/D)2. The relationship between the transverse and longitudinal resolutions is identical to the relationship between the transverse and longitudinal magnifications during holographic reconstruction [11

11. T.-C. Poon and P. P. Banerjee, Contemporary Optical Image Processing with MATLAB (Elsevier, 2001).

]. Now, an object feature of size w produces a diffracted spot size of the order of λz/w. Assuming that this diffracted field fills a detector of size D, it is clear that Δx,y=w, and hence Δz=4w2/λ. So the longitudinal resolution also depends on the feature size of the object. Equation (6) is an ill-posed optimization problem, and it can be solved by minimizing an objective function O(f) through the following:
  • (a) Selecting a basis Ψ, like a wavelet basis on which f may be assumed to be sparse. Hence f can be estimated as
    f^=argminfO(f)=argminf[12b2Re(Φf)l22+λΓ(f)]=argminf[12b2Re(Φf)l22+λΨfl1],
    (7)
    where Γ(f) is a regularizer and λ is the regularization parameter. Regularization involves introducing additional information to solve an ill-posed problem to prevent overfitting. Minimizing Eq. (7) is a compromise between the lack of fitness of a candidate estimate f to the observed data b, which is measured by bΦf2, and its degree of undesirability, given by Ψfl1.
  • (b) Enforcing a sparsity constraint on the total variation (TV) domain, which is equivalent to finding an f that minimizes the TV. For this approach f can be estimated as [6

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

    ]
    f^=argminfO(f)=argminf[12b2Re(Φf)l22+λΓ(f)]=argminf[12b2Re(Φf)l22+λfTV]
    (8)
    with fTV defined as
    fTV=ln1n2|(fl)n1,n2|=ln1n2(fl,n1+1,n2fl,n1,n2)2+(fl,n1,n2+1fl,n1,n2)2,
    (9)
    where fl is a 2D plane of the 3D object datacube. The two-step iterative shrinkage/thresholding (TwIST) algorithm [12

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

    ] is usually adopted to solve this optimization problem. The TwIST algorithm minimizes a convex quadratic problem with the addition of a sparsity constraint. The sparsity constraint is enforced on the gradient of the object estimate.

In the typical transmission setup for CH, since the illuminating beam “floods” the target, the light that gets transmitted in between the object(s) acts like a reference beam that interferes with the object field and records the Gabor hologram. In the reflection mode a Leith–Upatnieks-type holographic setup can be employed. It has been demonstrated that CH is able to provide greatly increased axial resolution compared to Fresnel backprojection using a single hologram, though the degree of improvement has been shown to depend on the object geometry, camera properties, and recording configuration used [13

13. C. Fournier, L. Denis, and T. Fournel, “On the single point resolution of on-axis digital holography,” J. Opt. Soc. Am. A 27, 1856–1862 (2010). [CrossRef]

15

15. Y. Rivenson and A. Stern, “Conditions for practicing compressive Fresnel holography,” Opt. Lett. 36, 3365–3367 (2011). [CrossRef]

]. Notably, high reference-to-object wave ratios (RORs) can significantly improve the interplane interference rejection ratio, which is a measure of axial resolution [16

16. Y. Rivenson, A. Stern, and B. Javidi, “Improved depth resolution by single-exposure in-line compressive holography,” Appl. Opt. 52, A223–A229 (2013). [CrossRef]

]. It will be shown that for the recording configurations employed here (i.e., low ROR, far-field reconstruction), the axial resolution improvement provided by CH alone is insufficient to unambiguously determine the 3D geometry of the objects tested. To improve axial resolution in CH under such conditions, we use the DiTCH technique for accurate 3D reconstruction of the targets and their distribution. A tomographic technique for recording the 3D shape reconstruction of water droplets and lenslets employing single beam holographic optical tomography using multiplicative technique (SHOT-MT) and based on Fresnel backpropagation is described in Nehmetallah and Banerjee [17

17. G. Nehmetallah and P. P. Banerjee, “SHOT: single-beam holographic tomography,” Proc. SPIE 7851, 78510I (2010). [CrossRef]

] [see Fig. 3(a)]. Specifically, digital holograms hj(x,y) corresponding to each angular orientation θj about the y axis are recorded. Thereafter, hj(x,y) are numerically reconstructed and the intensities Ij computed on multiple planes around the distance d that corresponds to the middle of the test volume where the objects are located. The numerical reconstruction involves using the discretized form of the Fresnel diffraction formula. Then, after some coordinate transformations, the 3D shape and distribution of the different objects can be reconstructed by multiplying the multiple reconstructed intensities [17

17. G. Nehmetallah and P. P. Banerjee, “SHOT: single-beam holographic tomography,” Proc. SPIE 7851, 78510I (2010). [CrossRef]

]:
I=1MIj
(10)
as shown in Fig. 3(b). In this process, the 3D volume reconstruction at a given recording angle (e.g., 0°) is directly multiplied with the corresponding 3D volume reconstruction at some other recording angle (e.g., 90°). The resulting 3D volume is then thresholded such that only the intersection terms survive, as shown in Fig. 3. This method yields good accuracy for holographic reconstructions and is computationally simpler than other tomographic reconstruction algorithms (e.g., backprojection via Fourier slice theorem, etc.). Additionally, multiangle tomography can reveal additional axial details that may not be unambiguously determined via single-angle CH reconstructions. For example, when illuminated from only one angle, the axial cross-section of a short cylinder and a sphere will yield nearly identical holographic recordings, thereby misrepresenting the true shape of the object. Thus, to visualize the 3D shape multiple projections from multiple directions, tomographic imaging systems are required. Hence, in DiTCH, CH reconstruction is used in each projection, and then the 3D shape is reconstructed using SHOT-MT by multiplying the multiple reconstructed intensities.

Fig. 3. (a) Experimental setup of typical SHOT-MT recording scheme and (b) schematic showing the principle of SHOT-MT reconstruction.

3. Experimental Results for DiTCH

As a first experimental setup of holographic tomography (see Fig. 4), the 3D reconstruction of a collection of small air bubbles (phase objects) in an aquarium, with different sizes and distances from the detector, is performed using the DiTCH-MT method. A green He–Ne source (λ=543nm) and a Lumenera camera with 1024×1024 pixels of size 6.7 μm is used.

Fig. 4. Typical DiTCH setup, transmissive geometry.

Fig. 5. (a) Single-beam hologram of two bubbles. The top two are for the first bubble and the bottom two for the second bubble. The holograms to the left are from illumination of the bubbles along the x axis (90° with respect to the normal to the CCD), while the holograms to the right are from illumination of the bubbles along z axis (0° with respect to the normal to the CCD). The left holograms look larger because the objects are farther from the CCD, while the right holograms look smaller because the objects are nearer to the CCD. (b) 3D reconstruction, λ=543nm, 6.7 μm pixels, (c) reconstructed hologram at 61.8 cm, (d) y-z projection of the 3D view in (b), (e) reconstructed hologram at 20.6 cm, and (f) x-y projection of the 3D view in (b).

As a second experiment [Fig. 6(a)] we have used a scattering object [visualize the spring of a ball-point pen shown in Fig. 6(b)], using a transmissive setup similar to the first experiment. Figure 6(c) shows multiple holograms from multiple directions recorded by rotating the object, and Fig. 6(d) shows two hologram reconstructions from two directions (90° and 180° at 33 cm). A 480×508, 9.8 μm Spiricon camera was used with λ=632.8nm illumination. Figures 6(e) and 6(f) show the 3D reconstruction resulting from 7 and 13 views, respectively, using the method given by Eq. (10). Note that instead of the sequential recordings achieved by rotating the object, the same set of holograms could be simultaneously recorded using different angles of illumination and multiple CCD cameras.

Fig. 6. (a) Schematic of lab setup, (b) ball-point pen spring 450 μm thick, (c) three representative holograms at angles 0°, 90°, and 180°, out of a total of 13 angles recorded, 0°–180° in 15° increments, (d) TwIST reconstruction from 90° to 180° recordings at 33 cm (distance of object from CCD during recording); tomographic reconstruction using (e) 7 angles, 0°–180° in 30° increments, and (f) 13 angles, 0°–180° in 15° increments.

The geometry of the spring in the Gabor recording configuration results in a ROR on the order of 1.4, which is too low to realize the needed improvement in axial resolution via CH, and therefore the 3D shape cannot be well determined by a single CH reconstruction [16

16. Y. Rivenson, A. Stern, and B. Javidi, “Improved depth resolution by single-exposure in-line compressive holography,” Appl. Opt. 52, A223–A229 (2013). [CrossRef]

]. Poor axial resolution is also explained in terms of the spring geometry, whose cross-section obscures a significant portion of the paraxial volume behind the object. Thus, to unambiguously determine the 3D shape, it is necessary to perform a tomographic reconstruction from multiple angles. In general the minimum number of recording angles for accurate reconstruction will be determined by the geometry of the object. However, an approximation can be determined from the geometry of Fig. 7 by assuming a circular cross-section and requiring the excess height, h, to be less than or equal to the resolution of the holographic reconstruction, Δη, given by
Δη=λdNΔx,
(11)
where d is the reconstruction distance at wavelength λ and N is the total number of camera pixels of size Δx [19

19. U. Schnars and W. Jueptner, Digital Holography: Digital Hologram Recording, Numerical Reconstruction, and Related Techniques (Springer, 2010).

]. The excess height, h, is given by
h=C2tan(α/2)S,
(12)
where α is the angle between adjacent beams, R is the cross-section radius, C is the chord length, and S is the sag height of the arc.

Fig. 7. Geometry of multiangle illumination of an object with a circular cross-section, where αR is the illumination angle (angle of object rotation), α is the angle between adjacent illumination beams, α=180°αR, R is the cross-section radius; C is the chord length, C=2Rsin(αR/2), S is the sag height of the arc, S=RR2(C/2)2, and h is the excess height (tomographic measurement error).

The cross-section radius of the spring object is 0.225 mm, and the hologram resolution Δη is 14.4 μm at a reconstruction distance of 33 cm. Thus, the object must ideally be rotated by 40°, or less, between successive recordings for accurate reconstruction. In practice, however, 13 projections between 0º and 180º at increments of 15º were required. These additional projections increased the SNR during tomographic reconstruction by compensating for some nonuniformity in the illumination profile and allowing for deviation from the assumption of a circular cross-section (e.g., the horizontal cross-section of the spring is actually elliptical due to the helical shape).

The tomographic reconstruction of the spring was calculated by assuming out-of-plane scattering along the z axis was small during recording, such that the 2D reconstruction for a given angle accurately reflects the object cross-section at the reconstruction distance. Each 2D reconstruction was then “lofted” to form a 3D volume and multiplied at the appropriate angles, as given by Eq. (10). Due to computational memory limitations, each 2D projection was resized to 96×96 pixels, via bicubic interpolation, prior to lofting to a 96×96×96 voxel volume. The corresponding image resolution was thus rescaled to 72μm/pixel, which causes the excess surface roughness shown in Figs. 6(e) and 6(f).

As a third experiment we describe possible fringe correction procedures for a dual-pass tomographic configuration (as in the first experiment) for a nonweakly scattering object. In this case, simultaneous recording of the ball-point pen spring is achieved using illumination from two different angles using the setup of Fig. 4; however, along the longer path, the “shadow” of the spring is the illuminating beam. This shadow must be separately recorded and used as the illumination profile when reconstructing the second pass through the object. An additional mirror and beam splitter (BS) are therefore added to the experimental setup (Fig. 8). The shadow profile is separately recorded by blocking mirror 3, and then the dual-pass hologram is recorded by blocking mirror 4. The path length from BS1 to the CCD, via mirror 4, is set equal to the path length from BS1 to the object, via mirrors 3 and 5. This ensures that the shadow recorded on the CCD is identical to the shadow impinging on the object during the second pass.

Fig. 8. Schematic of the shadow experimental setup.

As stated above, the shadow is recorded [see hologram in Fig. 8(a)], followed by the two-angle recording [see composite hologram in Fig. 8(b)]. The composite hologram is first reconstructed assuming plane wave illumination and a reconstruction distance of 23.5 cm, which equals the shortest distance from the object to the CCD [see reconstructed composite image in Fig. 9(c)]. The out-of-focus image, resulting from the first pass through the object (65.6 cm from the CCD), is clearly visible. Now, upon using the intensity of the shadow hologram of Fig. 9(a) as the illumination profile for the reconstruction, the out-of-focus image in the composite reconstruction can be partially removed [see Fig. 9(d)]. A possible reason for the spurious image remnant could be due to the fact that the intensity of the shadow hologram was used as the illumination profile rather than the appropriate complex field. For comparison, the shadow hologram [Fig. 9(a)] has been directly subtracted from the composite hologram [Fig. 9(b)] before reconstruction, and then reconstructed assuming plane wave illumination [Fig. 9(e)]. This results in superior removal of the shadow image, but at the cost of greater overall image degradation. While the reconstruction in this case has been achieved through Fresnel propagation, CS using TwIST can also be used, as shown in the two cases illustrated above.

Fig. 9. (a) Shadow recording, (b) object recording from two angles, (c) plane wave reconstruction, (d) reconstruction using shadow illumination profile, and (e) reconstruction after direct subtraction.

As a fourth example, a reflective-type setup is shown in Fig. 10(a). The divergent lens is used in a Gabor-type setup so that light from the demagnified virtual image, which acts as the effective object, writes an on-axis hologram on the CCD. Application of TwIST now ensures that the resulting reconstruction fits within the area of the zero-order reference beam, albeit at the expense of resolution. Figure 10(b) shows CH reconstruction of a dime using the TwIST algorithm in the reflective mode. Feature size in the reconstructed hologram is 28.6 μm for a CCD camera with pixel size 6.7 μm, λ=633nm, d=31cm, and demagnification M=0.315.

Fig. 10. (a) Experimental setup with diverging lens to provide demagnification and (b) CH reconstruction of a dime using the TWIST algorithm in the reflective mode. Feature size in the reconstructed hologram is 28.6 μm for a CCD camera with pixel size 6.7 μm, λ=633nm, d=31cm, and demagnification M=0.315.

4. Conclusion

References

1.

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

2.

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

3.

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

4.

E. J. Candes and T. Tao, “Near-optimal signal recovery from random projections: universal encoding strategies?,” IEEE Trans. Inf. Theory 52, 5406–5425 (2006). [CrossRef]

5.

D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52, 1289–1306 (2006). [CrossRef]

6.

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

7.

A. J. Devaney, “Nonuniqueness in the inverse scattering problem,” J. Math. Phys. Sci. 19, 1526–1531 (1978). [CrossRef]

8.

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

9.

J. Romberg and M. Wakin, “Compressed sensing: a tutorial,” in IEEE Statistical Signal Processing Workshop, Madison, Wisconsin (IEEE, 2007).

10.

R. Baraniuk, “Compressive sensing,” IEEE Signal Process. Mag. 24, 118–121 (2007). [CrossRef]

11.

T.-C. Poon and P. P. Banerjee, Contemporary Optical Image Processing with MATLAB (Elsevier, 2001).

12.

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

13.

C. Fournier, L. Denis, and T. Fournel, “On the single point resolution of on-axis digital holography,” J. Opt. Soc. Am. A 27, 1856–1862 (2010). [CrossRef]

14.

Y. Rivenson, A. Stern, and B. Javidi, “Compressive Fresnel holography,” J. Display Technol. 6, 506–509 (2010). [CrossRef]

15.

Y. Rivenson and A. Stern, “Conditions for practicing compressive Fresnel holography,” Opt. Lett. 36, 3365–3367 (2011). [CrossRef]

16.

Y. Rivenson, A. Stern, and B. Javidi, “Improved depth resolution by single-exposure in-line compressive holography,” Appl. Opt. 52, A223–A229 (2013). [CrossRef]

17.

G. Nehmetallah and P. P. Banerjee, “SHOT: single-beam holographic tomography,” Proc. SPIE 7851, 78510I (2010). [CrossRef]

18.

L. Tian, J. Lee, and G. Barbastathis, “Compressive holographic inversion of particle scattering,” in Digital Holography and Three-Dimensional Imaging, OSA Technical Digest (CD) (Optical Society of America, 2011), paper DWC27.

19.

U. Schnars and W. Jueptner, Digital Holography: Digital Hologram Recording, Numerical Reconstruction, and Related Techniques (Springer, 2010).

OCIS Codes
(100.3190) Image processing : Inverse problems
(100.6950) Image processing : Tomographic image processing
(090.1995) Holography : Digital holography

ToC Category:
Holography

History
Original Manuscript: December 3, 2012
Revised Manuscript: February 8, 2013
Manuscript Accepted: February 9, 2013
Published: March 8, 2013

Virtual Issues
Vol. 8, Iss. 4 Virtual Journal for Biomedical Optics

Citation
Logan Williams, Georges Nehmetallah, and Partha P. Banerjee, "Digital tomographic compressive holographic reconstruction of three-dimensional objects in transmissive and reflective geometries," Appl. Opt. 52, 1702-1710 (2013)
http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=ao-52-8-1702


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References

  1. C. E. Shannon, “Communications in the presence of noise,” Proc. IRE 37, 10–21 (1949). [CrossRef]
  2. M. Golay, “Multislit spectroscopy,” J. Opt. Soc. Am. 39, 437–444 (1949). [CrossRef]
  3. E. J. Candes, J. K. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Commun. Pure Appl. Math. 59, 1207–1223 (2006). [CrossRef]
  4. E. J. Candes and T. Tao, “Near-optimal signal recovery from random projections: universal encoding strategies?,” IEEE Trans. Inf. Theory 52, 5406–5425 (2006). [CrossRef]
  5. D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52, 1289–1306 (2006). [CrossRef]
  6. D. J. Brady, K. Choi, D. L. Marks, R. Horisaki, and S. Lim, “Compressive holography,” Opt. Express 17, 13040–13049 (2009). [CrossRef]
  7. A. J. Devaney, “Nonuniqueness in the inverse scattering problem,” J. Math. Phys. Sci. 19, 1526–1531 (1978). [CrossRef]
  8. E. J. Candes, J. K. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Commun. Pure Appl. Math. 59, 1207–1223 (2006). [CrossRef]
  9. J. Romberg and M. Wakin, “Compressed sensing: a tutorial,” in IEEE Statistical Signal Processing Workshop, Madison, Wisconsin (IEEE, 2007).
  10. R. Baraniuk, “Compressive sensing,” IEEE Signal Process. Mag. 24, 118–121 (2007). [CrossRef]
  11. T.-C. Poon and P. P. Banerjee, Contemporary Optical Image Processing with MATLAB (Elsevier, 2001).
  12. J. M. Bioucas-Dias and M. A. T. Figueiredo, “A new TwIST: two-step iterative shrinkage/thresholding algorithms for image restoration,” IEEE Trans. Image Process. 16, 2992–3004 (2007). [CrossRef]
  13. C. Fournier, L. Denis, and T. Fournel, “On the single point resolution of on-axis digital holography,” J. Opt. Soc. Am. A 27, 1856–1862 (2010). [CrossRef]
  14. Y. Rivenson, A. Stern, and B. Javidi, “Compressive Fresnel holography,” J. Display Technol. 6, 506–509 (2010). [CrossRef]
  15. Y. Rivenson and A. Stern, “Conditions for practicing compressive Fresnel holography,” Opt. Lett. 36, 3365–3367 (2011). [CrossRef]
  16. Y. Rivenson, A. Stern, and B. Javidi, “Improved depth resolution by single-exposure in-line compressive holography,” Appl. Opt. 52, A223–A229 (2013). [CrossRef]
  17. G. Nehmetallah and P. P. Banerjee, “SHOT: single-beam holographic tomography,” Proc. SPIE 7851, 78510I (2010). [CrossRef]
  18. L. Tian, J. Lee, and G. Barbastathis, “Compressive holographic inversion of particle scattering,” in Digital Holography and Three-Dimensional Imaging, OSA Technical Digest (CD) (Optical Society of America, 2011), paper DWC27.
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