Digital tomographic compressive holographic reconstruction of three-dimensional objects in transmissive and reflective geometries Logan Williams, Georges Nehmetallah,* and Partha P. Banerjee Electro-Optics Program, University of Dayton, Dayton, Ohio 45469, USA *Corresponding author:
[email protected] Received 3 December 2012; revised 8 February 2013; accepted 9 February 2013; posted 11 February 2013 (Doc. ID 180999); published 8 March 2013
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 OCIS codes: 090.1995, 100.3190, 100.6950.
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]. In the same year, Golay introduced the idea of artificial discrete multiplex coding in optical measurements [2]. More than 50 years later, Candes et al. [3], Candes and Tao [4], and Donoho [5] 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]. 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]. 1559-128X/13/081702-09$15.00/0 © 2013 Optical Society of America 1702
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Compressive holography (CH) is based on CS. A detailed description of the CH theory can be found in [6]. In CH, the “samples” of a function (or object) f are voxels in a three-dimensional (3D) volume. Gabor (or in-line) holography is a means to record the 3D object onto a 2D focal plane array, where the measurement is related to the Fourier transform of the object. Hence, Gabor holography is regarded as an effective encoder for CH [6,7]. In principle, decompression enables 3D reconstruction from a single 2D digital hologram without the need for multiple angles. However, as we will show, multiple angles are needed to improve the axial resolution. CH directly collects a smaller number of measurements (M) than the number of voxels (N) in the reconstruction (underdetermined problem). The word “compressive” means that our holographic sampling or sensing process encodes and compresses 3D datacube information into 2D holographic measurements. This encoding is then inverted using the compressive sampling theory [4,5,8]. 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 Gabortype 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 P are sparsity and the l1 norm defined as ‖c‖1 i jci j, whereas the usual Euclidean l2 norm adds thePsquare of the components of a vector: ‖c‖2 i jci j2 1∕2 . Thus, we can find the vector cj (a N × 1 matrix) by solving the underdetermined problem Akj cj bk ;
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Fig. 1. (Color online) Schematic of the different matrices used for CS.
cˆ arg min ‖c‖1 ; c
such that b Φf ΦΨc Ac: (2)
The use of the l1 norm helps in finding sparse solutions. Based on CS, CH using a Gabor (or in-line) holography setup is a means to record the 3D object on a 2D focal plane array, where the measurement is related to the Fourier transform of the object. A Gabor configuration is shown in Fig. 2 [6]. The irradiance recorded on the CCD has the following form: Ix; y jER EO x; yj2 jER j2 jEO x; yj2 ER EO x; y ER EO x; y ∝ 2 RefEO x; yg ex; y;
(3)
where ER and EO are the complex reference and object waves, respectively. The recorded intensity can then be regarded as the object wave of interest plus some “error” term, e, which encompasses the remaining terms in Eq. (3). The assumption in writing the last equality is that the reference beam and the diffracted field EO from the object(s) have approximately equal path lengths, which is true for in-line holograms. If the objects are in a 3D volume, the composite scattered field from the object(s) can be generically expressed as ZZZ EO x; y; z
Oξ; η; ςgPSF x − ξ; y − η; z − ςdξdηdς; (4)
(1)
where Φ ϕki is a rectangular M × N matrix called the projection or measurement matrix (K < M ≪ N), Ψ ψ ij 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 f i of a function f using f i ψ ij cj . Figure 1 shows a schematic of the different matrices involved in CS [9,10]. Once these conditions are satisfied, a valid reconstruction can be accomplished with high probability if we solve [8,9]
Fig. 2. (Color online) Typical Gabor-type setup using the transmissive geometry of an object surrounded by small objects. FPA, focal plane array. 10 March 2013 / Vol. 52, No. 8 / APPLIED OPTICS
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where Oξ; η; ς is the 3D object scattering density. gPSF can be written as p x2 y2 z2 exp −jk 0 jk p gPSF x; y; z ≈ 0 ; 2π x2 y2 z2
(5)
‖b − Φf ‖2 , and its degree of undesirability, given by ‖Ψf ‖l1. (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] fˆ arg min Of
where gPSF is the spatial impulse response. 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]
f
1 ‖b − 2 ReΦf ‖2l2 λΓf 2 f 1 arg min ‖b − 2 ReΦf ‖2l2 λ‖f ‖TV 2 f
arg min
¯ 2 RefI −1 PIf g e ≡ 2 RefΦf g e; (6) b 2 Refbg where I, I −1 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 2 RefEO x; yg ex; 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 λ2z∕D2 . The relationship between the transverse and longitudinal resolutions is identical to the relationship between the transverse and longitudinal magnifications during holographic reconstruction [11]. 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 Of 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 fˆ arg min Of f
1 2 arg min ‖b − 2 ReΦf ‖l2 λΓf 2 f 1 2 arg min ‖b − 2 ReΦf ‖l2 λ‖Ψf ‖l1 ; 2 f
(7)
where Γf 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 1704
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(8)
with ‖f ‖TV defined as ‖f ‖TV XXX j∇f l n1 ;n2 j l
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X X X q f l;n1 1;n2 −f l;n1 ;n2 2 f l;n1 ;n2 1 −f l;n1 ;n2 2 ; (9) where f l is a 2D plane of the 3D object datacube. The two-step iterative shrinkage/thresholding (TwIST) algorithm [12] 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 [13–15]. Notably, high reference-to-object wave ratios (RORs) can significantly improve the interplane interference rejection ratio, which is a measure of axial resolution [16]. 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] [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 I j 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]: I
M Y
Ij
(10)
1
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 BS Mirror Mirror
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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. 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 543 nm) and a Lumenera camera with 1024 × 1024 pixels of size 6.7 μm is used. Using the single-beam/single-CCD configuration shown in Fig. 4, the beam is passed through the air bubble sample at 90° and 0° with respect to the normal to the CCD, and a composite hologram is recorded in a single shot [Fig. 5(a)]. Illumination of the bubbles from two angles represents the simplest case of tomography. Due to the double pass of the beam, each bubble present in the sample forms two holograms, seen to be side by side as in Fig. 5(a). The larger hologram represents a longer path (approximately 61.8 cm) after scattering from the bubble to the CCD, while the smaller hologram represents a shorter path (approximately 20.6 cm) after scattering from the bubble to the CCD. Figure 5(b) shows the 3D reconstruction. Figures 5(c) and 5(e) show the reconstructed holograms using TwIST at distances of 20.6 and 61.8 cm, respectively. Since the reconstruction for 61.8 cm gives the locations of the bubbles in the y-z plane and the 20.6 cm reconstruction gives their locations in the x-y plane, the 3D coordinates of the bubbles are uniquely determined through this tomographic process, which is used to generate Fig. 5(b). The axial separation of the bubble
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(b) Fig. 3. (Color online) (a) Experimental setup of typical SHOT-MT recording scheme and (b) schematic showing the principle of SHOT-MT reconstruction.
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centers along the z axis is 0.76 mm and can be measured directly from the recorded hologram (i.e., diffraction pattern center-to-center). The two-angle tomographic reconstruction shows the axial separation to be 0.78 mm with an axial resolution on the order of 200 μm. However, far-field reconstruction via CH alone at a single angle (i.e., without tomography) produced an axial resolution on the order of 3–4 cm. This poor axial resolution is expected, in accordance with Rivenson and Stern [15], and Rivenson et al. [16], due to the recording geometry and low ROR (∼1) provided by these objects. Under ideal
near-field imaging conditions for similar objects, CH has been shown by Tian et al. to yield an axial resolution on the order of 1 mm [18]. The y-z and x-y projections of the 3D reconstruction in Fig. 5(b) are again shown in Figs. 5(d) and 5(f), next to Figs. 5(c) and 5(e). These are identical, as expected. 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
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Fig. 5. (Color online) (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 543 nm, 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). 1706
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Fig. 6. (Color online) (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. 10 March 2013 / Vol. 52, No. 8 / APPLIED OPTICS
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two hologram reconstructions from two directions (90° and 180° at 33 cm). A 480 × 508, 9.8 μm Spiricon camera was used with λ 632.8 nm 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. 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]. 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
λd ; NΔx
(11)
where d is the reconstruction distance at wavelength λ and N is the total number of camera pixels of size Δx [19]. The excess height, h, is given by h
C − S; 2 tanα∕2
(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. 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 1708
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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 72 μm∕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. 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
Fig. 7. (Color online) 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 2R sinαR ∕2, S is the sag height of p the arc, S R − R2 − C∕22 , and h is the excess height (tomographic measurement error).
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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. 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
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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.
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Fig. 10. (Color online) (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 633 nm, d 31 cm, and demagnification M 0.315. 10 March 2013 / Vol. 52, No. 8 / APPLIED OPTICS
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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 633 nm, d 31 cm, and demagnification M 0.315. 4. Conclusion
In conclusion, the proposed DiTCH technique used is a combination of CH and tomography using the SHOT-MT algorithm to improve the axial resolution of CH. This technique can be effectively used for weakly scattering objects, because we need digital holography to retain the depth information, and tomography requires as few as two projections to improve axial resolution. This technique allows us to determine the position, velocity, and shape of weakly scattering 3D objects. The technique is next applied to a more strongly scattering object, by employing multiple projections to unambiguously recover the 3D shape. As a modification to the original Gabortype two-projection setup, we propose a new configuration for shadow elimination for improved results in the case of nonweakly scattering objects. Finally, we extend the CH technique to the reconstruction of a partially diffusive object in a reflection-type geometry. References 1. C. E. Shannon, “Communications in the presence of noise,” Proc. IRE 37, 10–21 (1949). 2. M. Golay, “Multislit spectroscopy,” J. Opt. Soc. Am. 39, 437–444 (1949).
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