SECTIONAL IMAGE RECONSTRUCTION IN OPTICAL SCANNING HOLOGRAPHY USINGCOMPRESSED SENSING

Xin Zhang and Edmund Y. Lam

Imaging Systems Laboratory, Department of Electrical and Electronic Engineering,University of Hong Kong, Pokfulam Road, Hong Kong

{xinzhang,elam}@eee.hku.hk

ABSTRACT

Optical scanning holography is a form of digital holo-graphic system, which allows us to capture a three-dimensional(3D) object in the two-dimensional (2D) hologram. A post-processing step known as sectional image reconstruction canthen be applied to reconstruct a 2D plane of the object. Inthis paper, we show that the Fourier transform of the holo-gram corresponds to samples along a semi-spherical surfaceof the Fourier transform of the 3D object. Since the Fourierbasis satisfies the uniform uncertainty principle, we can viewthe hologram capture as a form of compressed sensing, andreconstruction of the sectional images can then be accom-plished by solving a total variation minimization problem.

Index Terms— Image reconstruction, holography, sec-tional image, compressed sensing

1. INTRODUCTION

Three-dimensional (3D) imaging of small objects, especiallydynamic ones, is very desirable for scientists to study themicroscopic world. Currently, confocal microscopy is com-monly applied for this task. It provides two-dimensional (2D)images with high lateral resolutions but only a shallow depth,due to the inherent physics involved in the imaging process.To achieve a volumetric capture, it needs sequential acquisi-tion at different axial planes. This is, however, a slow processand is unsuitable for moving objects.

In contrast, digital holographic imaging— though cur-rently delivering images with a lower lateral resolution— cancapture the entire 3D volume during a single x-y scan. Thedata need to be “interpreted,” i.e., processed to obtain theindividual 2D sections. This is called sectional image re-construction, or simply, sectioning. High-resolution volumeimagery of dynamic objects is therefore possible with holo-graphic imaging.

In this paper we focus on a particular form of digitalholographic imaging known as optical scanning holography(OSH), which has been shown to give better than 1μm lateralresolution with fluorescent specimens [1]. We analyze theimaging process and show that the hologram is composed of

samples in the 3D spatial frequency domain. There are twomajor results. First is that the holographic data are projec-tions of the object on the 3D Fourier basis. This obeys theuniform uncertainty principle (UUP) and is incoherent withthe canonical basis [2]. Hence, we can apply compressedsensing to design and analyze the sectional image reconstruc-tion. Second, the samples fall on a semi-spherical surfacein the high frequency vicinity. This provides an alternativeway to sample the 3D spatial frequency domain, comparedwith using radial lines in computed tomography [3] and spiraltrajectory in magnetic resonance imaging.

2. COMPRESSED SENSING THEORY

Compressed sensing considers the problem of reconstructinga sparse signal (or an image) from incomplete linear mea-surements [4]. To achieve an accurate reconstruction, an op-eration matrix representing the data capture system satisfies asufficient condition referred to as the UUP [4]. The propertyessentially guarantees that columns in the operation matrixare almost orthogonal rather than globally perfectly orthogo-nal. Then, every set of column vectors with cardinality lessthan the number of nonzero entries in the original signal ap-proximately operates like an orthonormal system. Therefore,an encoding map by the operation is of an isometry, i.e., thedistance of the signal can be preserved by the orthonormalencoding system.

As a special case of the operation matrix, discrete Fouriertransform has been analyzed in the compressed sensing litera-ture and its performance on a reconstruction with incompletefrequency information is well discussed [3, 5]. The basis isincoherent with the carnonical basis and satisfies the UUP. Itis then possible to reconstruct the signal from a few measure-ments of its frequency domain by solving an �1 minimizationproblem. For the measurements based on Fourier basis, theyare not required to be sampled at random.

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Proceedings of 2010 IEEE 17th International Conference on Image Processing September 26-29, 2010, Hong Kong

BS1

p2(x, y)

L2

L1

BS2

p1(x, y)

AOFS

g(x, y; z)

L3

PD

Monitor

z

Scanner

Laser

Fig. 1: A diagram of the optical scanning holography system. AOFSstands for an acoustic-optic frequency shifter, BS a beamsplitter, andPD a photodetector.

3. OSH IMAGING SCHEME

The OSH system is depicted in Fig. 1 [6, 7]. A laser sourceis split into two beams by a beamsplitter (BS1). One beamworks as a reference light and the other scans an objectg(x, y; z). In what follows, we also denote a two-dimensionalsection of the object at depth z as g(x, y; z). The hologramswe measure by OSH system are complex, and hence we oftenseparate them into real and imaginary parts for display. Theoperating principle of OSH is described at length in [6, 7, 8].For the purposes below, it is important to state that the opticaltransfer function (OTF) for the system is

H(kx, ky; z) = exp

{−

jz

2k0(k2x + k2y)

}, (1)

where kx and ky are the spatial frequency coordinates. Thepropagation vector of a plane wave is �k, and k0 denotes themagnitude of this vector, i.e.

k0 = |�k| =√k2x + k2y + k2z . (2)

The symbol kz denotes the spatial frequency along the z-axis.

4. COMPRESSIVE OPTICAL SCANNINGHOLOGRAPHY

We can apply the paraxial approximation [9] on kz . This gives

kz =(k20 − k2x − k2y

)1/2≈ k0 −

k2x + k2y

2k0. (3)

Substituting the approximation into Eq. (1), we can representthe OTF as

H(kx, ky; z) ≈ exp {−jz(k0 − kz)}

= exp {jz(kz − k0)} . (4)

kz

ky

kx

Fxy {·}

Detector

wave

z

g(x, y)

Fig. 2: A 2D slice of the 2D Fourier transform of a diffractionin the spatial frequency domain.

The OTF is multiplied with the Fourier transform of anobject g(x, y; z) to yield the frequency domain of the digi-tal hologram for that particular section. Summing over alldepths, we can describe the Fourier transform of the digitalhologram as

Q(kx, ky) =

∞∫−∞

Fxy

{|g(x, y; z)|2

}×

exp {jz(kz − k0)} dz. (5)

We use Fxy {·} to emphasize it is a 2D Fourier transform onthe x-y plane. Writing it as a double integral, we can thendescribe the image formation with a triple integral, i.e.

Q(kx, ky) =

∞∫∫∫−∞

|g(x, y; z)|2×

exp {−j [xkx + yky − z(kz − k0)]} dxdydz. (6)

If we use Q(kx, ky, kz) to denote the three-dimensionalFourier transform of the squared magnitude of the object,we can see that the Fourier transform of the OSH hologramcan be written in terms of the 3D Fourier transform as

Q(kx, ky) = Q(kx, ky,−(kz − k0)), (7)

where kz =√k20 − k2x − k2y . Since kx = kx, ky = ky and

kz = −(kz−k0), the samples ofQ(kx, ky) comprise of pointsequidistant from the given point, (0, 0, k0), in 3D spatial fre-quency domain. This is a semi-spherical surface in the kx-ky-kz domain depicted in Fig. 2.

5. METHODOLOGY

We turn our attention to look at the reconstruction prob-

lem. Given H(kx, ky; z) = exp{− jz

2k0

(k2x + k2y)}

and

h(x, y; z) = F−1xy {H(kx, ky; z)} = − jk0

2πz exp[jk0(x

2+y2)2z

]

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[10], an OSH hologram can be expressed with respect to aconvolution [11, 12], i.e.

q(x, y) =

∫∞

−∞

g(x, y; z) ∗ h(x, y; z)dz. (8)

Let g(x, y; z) = |g(x, y; z)|2. Then

q(x, y) =

∫∞

−∞

g(x, y; z) ∗ h(x, y; z)dz. (9)

We discretize the depth z into z1, z2, . . . , zn and use alexicographical ordering of q(x, y) and g(x, y; zi) into q andgi. Making use of a multiplication to replace the convolutionin Eq. (9), h(x, y; zi) is reorganized into a matrix, Hi. Thehologram, further, is

q =

n∑i=1

Higi = Hg, (10)

where H =[H1 · · · Hn

]and g =

[gT1 · · · gT

n

]T. It

is an inverse problem to reconstruct g from a measurementvector q. In this paper, we solve it by using a total varia-tion (TV) minimization method, which preserves the finite-difference in the solution, and is shown to deliver good per-formance in compressed sensing [2]. The variational problemis of the form

JTV(g) = ‖q− Hg‖2 + α‖g‖TV, (11)

where α is the regularization parameter. The notation ‖ · ‖TV

stands for the total variation norm. Its 2D form is given by

‖v(x, y)‖TV =

∫∫|∇v(x, y)|dxdy, (12)

where ∇v(x, y) = ( ∂v∂x ,∂v∂y ) is the gradient of v(x, y) and |·|

denotes the Euclidean norm, i.e., |(x, y)| =√x2 + y2.

Note further that we need to maintain the nonnegativity ofthe solution.Hence, the overall equation should be

JTV(g) = ‖q− Hg‖2 + α‖g‖TV, such that g ≥ 0. (13)

This can be solved with the primal-dual Newton method usinga gradient projection in the iterations [13].

6. EXPERIMENTS

An experiment is conducted to demonstrate the performanceof the reconstruction method. We use an object with twosections, which has been tested with different reconstructionschemes in the past [11, 14, 15]. The two sections are at dif-ferent z positions, and each contains a bar. We simulate itsscanning by an OSH system, and obtain a complex digitalhologram as our data.

(a) The object

FZP at z1=14mm FZP at z

2=15mm

(b) FZPs at z1 and z2

Fig. 3: The object and FZPs in the experiment. Shown in (a) is theobject, in which two elements (i.e., rectangular bars) are at z1 and z2

positions, respectively. Shown in (b) are the real parts of the FZPsof a point source at z1 and z2.

(a) Sine hologram (b) Cosine hologram

Fig. 4: Holograms of the object in the experiment. Sine hologram isthe real part and cosine hologram is the imaginary part.

We aim to reconstruct sectional images at z1 and z2 fromthe hologram. Fig. 3(a) shows the simulated object, wheretwo sections are located at z1 = 14mm and z2 = 15mm awayfrom the OSH scanner. Shown in Fig. 3(b) are the real parts ofthe Fresnel zone plates (FZPs) [7] of a point source at z1 andz2 respectively. The real and imaginary parts of the complexhologram are shown in Fig. 4. Fig. 4(a) is the sine hologram,which is also known as the real hologram, and (b) is the cosinehologram, also called the imaginary hologram.

We apply the proposed method to reconstruct the sectionalimages in the hologram. As shown in Fig. 5(a) and Fig. 5(b),the two sections are well separated with sharp edges. Defo-cus noise [7] has little effect on the reconstructed sections.We can compare them with the reconstructed sections by theTikhonov regularization shown in Fig. 6 [11], which to dateis the most powerful reconstruction scheme. The two re-constructed sections show the corresponding bars clearly, butwith apparent defocus noise. The focused intensity in eachsection is recovered nonuniformly. The corners and edges arerecovered with high intensity, but the middle parts have lowerintensity.

7. CONCLUSION

In this paper we demonstrate the connection between OSHimaging and compressed sensing. We then apply the TV min-imization into the reconstruction of sectional images from theundersampled spatial frequency data, and show that our al-gorithm can preserve edges and suppress defocus noise us-

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(a) Section at z1 (b) Section at z2

Fig. 5: The reconstructed sections by the TV minimization method.(a) is the section at z1 = 14mm and (b) at z2 = 15mm.

(a) Section at z1 (b) Section at z2

Fig. 6: The reconstructed sections by the Tikhonov regularizationmethod. (a) is the section at z1 = 14mm and (b) at z2 = 15mm.

ing simulated data. Application of this method on experi-mental data, which are available at http://midas.osa.org/midaspub/midas/item/view/1149, will be in-vestigated in the future.

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[6] Ting-Chung Poon, “Scanning holography and two-dimensional image processing by acousto-optic two-pupil synthesis,” J. Opt. Soc. Am. A, vol. 2, no. 4, pp.521–527, Apr 1985.

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[9] J. W. Goodman, Introduction to Fourier Optics, Robertsand Company Publishers, 3rd edition, 2005.

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