edge-preserving sectional image reconstruction in optical scanning holography

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1630 J. Opt. Soc. Am. A/Vol. 27, No. 7 /July 2010 X. Zhang and E. Y. Lam

Edge-preserving sectional image reconstruction inoptical scanning holography

Xin Zhang and Edmund Y. Lam*

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

*Corresponding author: [email protected]

Received March 25, 2010; revised May 19, 2010; accepted May 21, 2010;posted May 21, 2010 (Doc. ID 126004); published June 16, 2010

Optical scanning holography (OSH) enables us to capture the three-dimensional information of an object, anda post-processing step known as sectional image reconstruction allows us to view its two-dimensional cross-section. Previous methods often produce reconstructed images that have blurry edges. In this paper, we arguethat the hologram’s two-dimensional Fourier transform maps into a semi-spherical surface in the three-dimensional frequency domain of the object, a relationship akin to the Fourier diffraction theorem used in dif-fraction tomography. Thus, the sectional image reconstruction task is an ill-posed inverse problem, and herewe make use of the total variation regularization with a nonnegative constraint and solve it with a gradientprojection algorithm. Both simulated and experimental holograms are used to verify that edge-preserving re-construction is achieved, and the axial distance between sections is reduced compared with previous regular-ization methods. © 2010 Optical Society of America

OCIS codes: 090.1760, 090.1995, 100.3020, 100.3190, 180.6900, 110.1758, 170.3880.

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. INTRODUCTIONptical scanning holography (OSH) is a digital holo-raphic technique that preserves phase information byptical heterodyne detection and records the three-imensional (3D) diffracted propagation of an object aswo-dimensional (2D) holograms [1,2]. Computationalethods are then employed to reconstruct individual 2D

ectional images, and recent techniques aim to enhancehe quality of reconstructed sections with higher reso-ution [3,4]. This includes an inverse imaging method,here the OSH system is modeled by an operation matrixnd sectional image reconstruction is cast as an ill-posednverse problem [5]. Tikhonov regularization is then useds a smoothness constraint. While this algorithm hashown a very good reconstruction quality in terms of sup-ressing defocus noise compared with other recent meth-ds [6,7], sharp edges tend to be blurred as a result ofinimizing the total energy (or mean square error) in the

econstruction algorithm [8].It may seem straightforward, then, to experiment with

eplacing the Tikhonov regularization by other nonlinearechniques. However, we argue that one should first un-erstand the imaging process and investigate how theata are undersampled. This can help us appreciate howrior information can be needed, and how it may be for-ulated in the nonlinear regularization. As we will see in

he third section, with encoding the 3D information of anbject into a 2D hologram by optical coherent detection9], OSH effectively captures an object’s diffracted propa-ation. Therefore, the sectional image reconstruction isquivalent to retrieving sections from the diffraction,hich is known as diffraction tomography in a variety ofpplications, ranging from ultrasound imaging to geo-hysical exploration [10,11]. Proposed algorithms have

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volved from interpolation based on Born and Rytov ap-roximations to filtered backpropagation (FBP) [12], ando inverse scattering algorithms and total variation (TV)inimization more recently [13,14]. The amount of data

apture is gradually decreasing with more advanced algo-ithms. Thus, in relating OSH to diffraction tomography,e can take advantage of these more recent techniques in

econstructing sectional images.This paper is organized as follows. Section 2 describes

he OSH system. Then, in Section 3, we analyze the rela-ionship between diffraction tomography and the OSHystem. A sampling of 3D Fourier transform result is pre-ented in the mechanism of generating the hologram. Sec-ion 4 presents the inverse problem of representing a re-onstruction of sectional images from a hologram. Bothhe total variation (TV) penalty and the nonnegativity aresed as constraints to regularize a solution. In Section 5,xperimental results are presented to demonstrate theerformance of the reconstruction method. Section 6 con-ludes the paper.

. OPTICAL SCANNING HOLOGRAPHYYSTEM

n OSH, an object is scanned by a system with a complexresnel-zone-plate (FZP) impulse response [15]. Theneterodyne detection collects diffracted and scattered

ight from the object and records it as electronic holo-rams. The holograms encode the 3D position informationf the diffracted propagation through the object.

Fig. 1 shows a schematic diagram of an OSH system. Aelium-neon (HeNe) laser is used to scan an object. Aeam splitter BS splits the laser beam. One beam works
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X. Zhang and E. Y. Lam Vol. 27, No. 7 /July 2010 /J. Opt. Soc. Am. A 1631

s a reference light, and the other is an object light tocan the object. The former goes through an acousto-opticrequency shifter (AOFS) to alter its temporal frequencyrom �0 to �0+�, which is then reflected by a mirror M2 tolluminate a pupil, p2�x ,y�. Meanwhile, the object light iseflected by M1 and controlled by another pupil, p1�x ,y�.he two controlled light beams go through lenses L2 and1, respectively, and are then combined by another beamplitter BS2. The combined light is used by a 2D scannero scan an object. A photodetector PD accepts all lightrom the object and sends an electric signal to the de-odulation part, which includes a band-pass filter BPF

uned at the frequency �, electronic multipliers, and low-ass filters LPFs. Finally, quadrature electronic holo-rams containing the complete information of the objectre recorded [16,17]. Holograms we measure by the OSHystem are complex and we often separate them into realnd imaginary parts in viewing.The optical transfer function (OTF) in OSH is given by

H�kx,ky;z� = exp�−jz

2k0�kx

2 + ky2�� , �1�

here z is the distance between the scanner and a par-icular section of interest of the object, k0 is the wavenum-er, and kx and ky are the transverse spatial frequency co-rdinates. Derivations of this formula can be found in16,17].

. OSH AS DIFFRACTION TOMOGRAPHY. OSH in Fourier Basiso express the OSH data capture using Fourier bases, werst note that the wavenumber k0 is related to kx, ky, andz, which are the x, y, and z components of the propaga-ion vector, by k0=�kx

2+ky2+kz

2. Making use of the paraxialpproximation [18], we have

Fig. 1. (Color online) Schematic diagra

kz = �k02 − kx

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he requirement of kx2+ky

2�k02 is reasonable in OSH be-

ause we assume that waves are forward propagatinglong the positive z direction. Thus, the x and y compo-ents are much smaller in comparison.Substituting the approximation into Eq. (1), we can ex-

ress the OTF as [19]

H�kx,ky;z� = exp�−jz

2k0�kx

2 + ky2�� exp− jz�k0 − kz�.

�3�

ow consider an object g�x ,y ;z� with the Fourier trans-orm of its squared intensity G�kx ,ky ;z�=Fxy�g�x ,y ;z��2.he resulting hologram, denoted as q�x ,y�, is given by

q�x,y� =�−�

�

Fxy−1G�kx,ky;z�H�kx,ky;z�dz. �4�

aking use of the approximation in Eq. (3), we get

q�x,y� �� −�

�

G�kx,ky;z�exp− jz�k0 − kz�

�expj�xkx + yky�dkxdkydz

=��−�

�

�G�kx,ky;z�exp− jzk0�

�expj�xkx + yky + kzz�dkxdkydz. �5�

n words, this states that the hologram from OSH equalshe 3D inverse Fourier transform of the 2D Fourier trans-

he optical scanning holography system.

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orm of (the squared intensity of) the object multiplied bylinear phase factor.

. Fourier Diffraction Theorem in OSHsimpler statement can in fact be made if we consider,

nstead, the Fourier transform of the hologram, which weenote as Q�kx ,ky�. From Eq. (4), we have

Q�kx,ky� =�−�

�

G�kx,ky;z�H�kx,ky;z�dz

��−�

�

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=��−�

�

�g�x,y;z��2 exp− j xkx + yky − z�kz

− k0��dxdydz. �6�

etting G�kx , ky , kz�=Fxyz�g�x ,y ;z��2, i.e., the 3D Fourierransform of the squared magnitude of the object, we canurther simplify Q�kx ,ky� as

Q�kx,ky� = G�kx,ky,− �kz − k0�� = G�kx,ky,k0 − �k02 − kx

2 − ky2�.

�7�

n words, the samples of Q�kx ,ky� comprise points equidis-ant from the point �0,0,k0� in the 3D spatial frequencyomain. This is a semi-spherical surface depicted inig. 2.We can interpret the relationship in Eq. (7) in light of

he Fourier diffraction theorem commonly used in diffrac-ion tomography. OSH records a diffracted propagationhrough the object by a collimated light. The theoremtates that when an object is illuminated by a plane wave,he Fourier transform of the diffracted field produces theourier transform of the object along the semi-sphericalurface in the 3D spatial frequency domain [20,21]. In aense, this is a generalization of the projection-slice theo-em [22], applicable to cases such as computed tomogra-hy (CT), by taking into account the wave nature of lightropagation and therefore incorporating the diffraction

Fig. 2. (Color online) 2D slice of the 2D Fourier tr

ffect [23]. The acquired data also form a “slice” in thepatial frequency domain, albeit the slice is bent to aemi-spherical surface.

Our interest is in the reconstruction of �g�x ,y ;z��2 fromhe acquired slice. In CT, multiple slices are acquired, andethods such as filtered back-projection can be employed

o reconstruct the volumetric data [24]. This is analyti-ally “perfect” in the absence of noise. On the other hand,n OSH we effectively receive only a single semi-sphericallice of data in the spatial frequency domain. The recon-truction problem is therefore severely underdetermined,hich explains why regularization is necessary to copeith its ill-posed nature [5]. In addition, many recent al-orithms in the reconstruction of CT and other signals in-olve making use of prior information of the object, par-icularly the sparsity of its gradient [25,26], to ensurehat the reconstructed image contains sharp boundaries.e make use of this to produce our edge-preserving OSH

ectional image reconstruction, as detailed in Section 4.

. EDGE-PRESERVING RECONSTRUCTIONe develop our edge-preserving reconstruction in the spa-

ial domain as follows. Referring back to Eq. (4), we canxpress the capturing of the hologram as

q�x,y� =�−�

�

�g�x,y;z��2 � h�x,y;z�dz, �8�

here h�x ,y ;z�=Fxy−1H�kx ,ky ;z�=−jk0 /2�z expjk0�x2

y2� /2z as derived in [5,17]. We then discretize z as1,z2 , . . .zn, and represent the lexicographical ordering of�x ,y� and �g�x ,y ;zi��2 by q and gi, respectively. The con-olution operation is written as a matrix-vector multipli-ation by forming the matrix Hi from h�x ,y ;zi� [27], sohat Eq. (8) becomes

q = �i=1

n

Higi = Hg, �9�

here H= H1¯Hn� and g= g1T¯gn

T�T. Our sectional im-ge reconstruction problem is then an inverse problem,here we seek to estimate g from the observation q and anown H.

m of a diffraction in the spatial frequency domain.
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To solve this, one can formulate a regularized leastquares problem by minimizing the function

J�g� = �q − Hg�2 + ��Cg�2, �10�

here the C matrix is chosen to be a high-pass filter, suchs Laplacian of Gaussian, and � is the regularization pa-ameter, which balances between smoothness and fidelityo observed data. The �2 norm is used in both instances inhis equation. This is the Tikhonov regularization algo-ithm developed in [5]. Note that in this method, smooth-ess is imposed on the whole reconstructed image at thexpense of reconstructing sharp edges. Hence, althoughhe reconstruction quality and the flexibility of this algo-ithm show advantage over other recent schemes [6,7],ne noticeable disadvantage of this algorithm is that theeconstructed objects often show blurred edges. This notnly affects the visual appearance, but also presents chal-enges in possible further image analysis algorithms, suchs identifying the size of an object.To improve on the edge quality, we can instead mini-ize the function

JTV�g� = �q − Hg�2 + ��g�TV, �11�

here � · �TV stands for the total variation (TV) regulariza-ion given by

�v�x,y��TV =��−�

�

��v�x,y��dxdy.

n addition, �v�x ,y�= ��v /�x ,�v /�y� stands for the gradi-nt of v�x ,y�.

Such TV regularization has shown to be capable ofaintaining edge sharpness in various image and video

estoration problems (for example, in [28–30]). TV norms also essentially �1 norm on the gradient of an image,

Initialization PDN's Method GP Method

Iteration No. < 30Yes

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End

ig. 3. Flowchart of edge-preserving reconstruction method“PDN’s Method” stands for the primal-dual Newton’s methodnd “GP Method” stands for the gradient projection method).

FZ

(a)

ig. 4. Object and FZPs in the first experiment. Shown in (a) issections. Shown in (b) are the real parts of FZPs of a point so
2
nd it has been shown in recent years that this is a pow-rful tool to reconstruct images from very limited data ife can assume sparsity in signal transition (i.e., edges) inn image [31,32]. In our OSH sectional image reconstruc-ion problem, we indeed have only very limited data asointed out in Section 3. To the best of our knowledge, theresent paper is the first report on a TV regularization-ased sectional image reconstruction algorithm to tacklehis problem.

Moreover, the computational of Hg also differentiateshe present problem from other image restoration algo-ithms. It does not represent a convolution, but a sum ofonvolutions; as such, we cannot directly apply the fastourier transform to implement the matrix-vector prod-ct efficiently. Instead, we developed a method earlierwhich is reported in [33]) for this specific purpose. In ad-ition, a further refinement to Eq. (4) is also possible.ince gi contains entries in �g�x ,y ;zi��2, it must be nonne-ative for all pixels. Thus, we should seek to minimize

JTV�g� = �q − Hg�2 + ��g�TV, subject to g � 0, �12�

here the inequality is taken to be componentwise. Toolve this, we can make use of the primal-dual Newton’sethod [34] and combine it with a gradient projectionethod to regularize the solution. In each iteration, the

rimal-dual Newton’s method gives a preliminary guessf the solution, which should be regularized by the TVenalty. However, the solution may be negative in certainntries, which does not agree with the nonnegative inten-ity captured by the scanning holography system [35].his is fixed by the gradient projection method, which up-ates the solution to make it nonnegative. A flowchart ofhe reconstruction is shown in Fig. 3. Here the regulariza-ion parameter is not very sensitive, and we set it to 100n general. Iterating these steps allows us to obtain a so-ution that satisfies the TV regularization and nonnega-ivity at the same time.

. EXPERIMENTSn experiment is conducted to demonstrate the perfor-ance of this edge-preserving reconstruction algorithm.

o be consistent with previous work in order to allow con-enient comparison, we use a set of simulated data thatnclude two distinct sections in the z direction, whereach section contains a rectangular bar. The two bars aret z1=14 mm and z2=15 mm away from the scanner, as

z1=14mm FZP at z

2=15mm

(b)

ject, in which two elements (i.e., rectangular bars) are at z1 andz and z .

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hown in Fig. 4(a). The object is then imported into Eq. (8)o generate a simulated hologram for the experiment.hown in Fig. 4(b) are the real parts of the Fresnel zonelates [16] of a point source at z1 and z2, respectively. Theeal and imaginary parts of the hologram are shown inig. 5, where (a) is the sine hologram (the real part) and

b) is the cosine hologram (the imaginary part).The reconstruction of the two sections from the holo-

ram is shown in Fig. 6. As can be seen, the two sectionsre well separated and are reconstructed at the correct lo-ation and with sharp edges. Defocus noise, which is aroblem with conventional techniques [16], does not haveny visible effect on the reconstructed sections. Moreover,e can compare this reconstruction with the one using

he Tikhonov regularization, as shown in Fig. 7. With thisethod, the two reconstructed sections also show the cor-

esponding bars clearly, but with more defocus noise. Thentensity in each section is also recovered nonuniformly,.e., corners and edges are recovered with higher inten-ity, while the middle parts have lower intensity in com-arison.To look into the quality of edge preservations of our pro-

osed method, we extract a cross-section of the middleow of the section at z1, which is shown in Fig. 8. The blueashed line shows the row in the original object, while theed solid curve is from the reconstructed section. We canbserve that the two agree quite well. In particular, whilehe ideal object exhibits an abrupt change at the edgeithin one pixel, in our reconstruction the change can

ake place within two pixels. Another region of notice ishe flat area with high intensity values, which corre-ponds to the area within the object. In our reconstruc-

(a) (b)

ig. 5. Holograms containing two-sectional images of the objectn the first experiment. Sine hologram in (a) is the real part andosine hologram in (b) is the imaginary part.

(a) (b)

ig. 6. Reconstructed sections by the TV regularization method.a) Section at z =14 mm and (b) at z =15 mm.
1 2
ion, the intensities are not flat, but contain no more than% variations. This gives rise to a rather uniform visualrightness of the object.Another comparison between the edge-preserving re-

onstruction and the Tikhonov-regularized inverse imag-ng methods is the axial distance �z between the two barseeded for the reconstruction to show two distinct objectst different depths. In this experiment, the hologram is ofize 2 mm by 2 mm, which translates to 500�500 pixels.he lateral resolution of the hologram is then given byx=�y=4 m. As shown in Fig. 6, the edge-preserving re-onstruction separates the two sections with �z=1 mmompletely, while the reconstruction with Tikhonov regu-arization fails to separate them while suppressing defo-us noise, as shown in Fig. 7. (Note that in [5], Tikhonovegularization was successfully applied to separate twoars with �z=1 mm, but in that experiment �x=�y1 m, which translates to greater constraints if we were

o realize this in a physical system.) If the section at z2 isoved to 24 mm and a new hologram is measured, the re-

onstruction by Tikhonov regularization shows compa-able performance as our method here in terms of defocusoise suppression. Thus, it can be argued that with the

(a) (b)

ig. 7. Reconstructed sections by the Tikhonov regularizationethod. (a) Section at z1=14 mm and (b) at z2=15 mm.

Fig. 8. (Color online) Cross-section of the middle row at z .
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ame lateral resolution of a hologram, the edge-reserving reconstruction is more competent in separat-ng close sections, by reducing the axial separation from0 mm to 1 mm in this experiment.Finally, we evaluate the current algorithm using ex-

erimental data that have been used earlier in [36]. Theologram is of size 1.5 cm by 1.5 cm, and the two sectionsre at z1=87 cm and z2=107 cm. Note that this system is100 times larger than that used in the previous simula-

ions. The diameter of the collimated beam is c=25 mmnd the focal length of the lens is f=500 mm. Thus, the

(a)(c

(cm

)

−0.5 0 0.5

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

ig. 9. Holograms measured by a physical OSH system [36]. (a)ologram.

ig. 10. Object in the second experiment scanned by a physicalystem.

Fig. 11. Reconstructed sectional images by TV regular

umerical aperture (NA) of the recorded full-parallaxomplex hologram is NA�c /2f=0.025. Shown in Fig. 9re the real and imaginary parts of the complex holo-ram, where the object consists of two transparenciesith the letters “S” at z1 and “H” at z2, as shown in Fig.0. Note that the two letters are hollow inside, which isnlike the two solid bars discussed above, and thereforehe reconstruction should preserve such hollow appear-nce. Our proposed reconstruction method is applied tohe hologram, and the extracted sectional images arehown in Fig. 11, where (a) presents the reconstructedection at z1 and (b) is the section at z2. In both cases, theetters are recovered and the edges are well preserved. Weompare the reconstruction with that using Tikhonovegularization, as shown in Fig. 12. Although the inten-ity of the recovered images is larger, this actually devi-tes more from the object, which consists of letters withutlines only.

. CONCLUSIONn this paper we have analyzed the data collection in OSHnd developed an edge-preserving sectional image recon-truction method that improves upon the inverse imagingethod using Tikhonov regularization in an earlier work

5]. The new technique makes use of total-variation regu-arization and a nonnegativity constraint, and is imple-

(b)(cm)−0.5 0 0.5

−0.6

−0.4

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0.2

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art of the recorded hologram. (b) Imaginary part of the recorded

and nonnegative constraint are shown in (a) and (b).

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ented using a primal-dual Newton’s method combinedith gradient projection. Simulation results show good re-

onstruction quality with sharp edges and fairly uniformntensities within the object.

CKNOWLEDGMENTShe authors wish to express their appreciation for helpfuliscussions with Dr. Kerkil Choi regarding compressiveolograms, with Prof. Terence Tao about the Fourier mea-urements in compressive sensing, and with Prof. Tae-eun Kim about the resolution of the experimental OSHystem. This work was supported in part by the Univer-ity Research Committee of the University of Hong Kongnder Projects 10208291, 10208648, and10400399.

EFERENCES1. T.-C. Poon, “Scanning holography and two-dimensional im-

age processing by acousto-optic two-pupil synthesis,” J.Opt. Soc. Am. A 2, 521–527 (1985).

2. J. Swoger, M. Martínez-Corral, J. Huisken, and E. Stelzer,“Optical scanning holography as a technique for high-resolution three-dimensional biological microscopy,” J. Opt.Soc. Am. A 19, 1910–1918 (2002).

3. U. Schnars and W. Jueptner, Digital Holography: DigitalHologram Recording, Numerical Reconstruction, and Re-lated Techniques (Springer-Verlag, 2005).

4. T.-C. Poon, Digital Holography and Three-Dimensional Dis-play: Principles and Applications (Springer, 2006).

5. X. Zhang, E. Y. Lam, and T.-C. Poon, “Reconstruction of sec-tional images in holography using inverse imaging,” Opt.Express 16, 17215–17226 (2008).

6. T. Kim, “Optical sectioning by optical scanning holographyand a Wiener filter,” Appl. Opt. 45, 872–879 (2006).

7. H. Kim, S.-W. Min, B. Lee, and T.-C. Poon, “Optical section-ing for optical scanning holography using phase-space fil-tering with Wigner distribution functions,” Appl. Opt. 47,D164–D175 (2008).

8. S.-J. Kim, K. Koh, M. Lustig, S. Boyd, and D. Gorinevsky,“An interior-point method for large-scale l1-regularizedleast squares,” IEEE J. Sel. Top. Signal Process. 1, 606–617(2007).

9. G. Indebetouw, “Properties of a scanning holographic mi-croscopy: Improved resolution, extended depth-of-focus,and/or optical sectioning,” J. Mod. Opt. 49, 1479–1500(2002).

0. R. K. Mueller, M. Kaveh, and G. Wade, “Reconstructive to-mography and applications to ultrasonics,” Proc. IEEE 67,567–587 (1979).

Fig. 12. Reconstructed sectional images by inverse im

1. A. J. Devaney, “Geophysical diffraction tomography,” IEEETrans. Geosci. Remote Sens. GE-22, 3–13 (1984).

2. A. J. Devaney, “A filtered backpropagation algorithm fordiffraction tomography,” Ultrason. Imaging 4, 336–350(1982).

3. P. Guo and A. J. Devaney, “Comparison of reconstruction al-gorithms for optical diffraction tomography,” J. Opt. Soc.Am. A 22, 2338–2347 (2005).

4. S. J. LaRoque, E. Y. Sidky, and X. Pan, “Accurate image re-construction from few-view and limited-angle data in dif-fraction tomography,” J. Opt. Soc. Am. A 25, 1772–1782(2008).

5. G. Indebetouw, P. Klysubun, T. Kim, and T.-C. Poon, “Imag-ing properties of scanning holographic microscopy,” J. Opt.Soc. Am. A 17, 380–390 (2000).

6. T.-C. Poon, Optical Scanning Holography with MATLAB(Springer-Verlag, 2007).

7. E. Y. Lam, X. Zhang, H. Vo, T.-C. Poon, and G. Indebetouw,“Three-dimensional microscopy and sectional image recon-struction using optical scanning holography,” Appl. Opt. 48,H113–H119 (2009).

8. J. W. Goodman, Introduction to Fourier Optics, 3rd ed.(Roberts, 2005),

9. G. Indebetouw and W. Zhong, “Scanning holographic mi-croscopy of three-dimensional fluorescent specimens,” J.Opt. Soc. Am. A 23, 1699–1707 (2006).

0. E. Wolf, “Three-dimensional structure determination ofsemi-transparent objects from holographic data,” Opt. Com-mun. 1, 153–156 (1969).

1. C. L. Matson and H. Liu, “The Fourier diffraction theoremfor turbid media,” in “Advances in Optical Imaging andPhoton Migration,” J. Fujimoto and M. Patterson, eds., Vol.21 of OSA Trends in Optics and Photonics (Optical Societyof America, 1998) p. ATuD14.

2. R. Bracewell, The Fourier Transform and Its Applications,3rd ed. (McGraw-Hill, 1999),

3. A. J. Devaney, “A computer simulation study of diffractiontomography,” IEEE Trans. Biomed. Eng. BME-30, 377–386(1983).

4. Y. Zou and X. Pan, “Image reconstruction on PI-lines by useof filtered backprojection in helical cone-beam CT,” Phys.Med. Biol. 49, 2717–2731 (2004).

5. T. Chan and J. Shen, Image Processing and Analysis: Varia-tional, PDE, Wavelet, and Stochastic Methods (SIAM,2005).

6. E. Candès, J. Romberg, and T. Tao, “Robust uncertaintyprinciples: Exact signal reconstruction from highly incom-plete frequency information,” IEEE Trans. Inf. Theory 52,489–509 (2006).

7. R. Gonzalez and R. Woods, Digital Image Processing, 3rded. (Pearson Prentice Hall, 2008).

8. Y. Li and F. Santosa, “A computational algorithm for mini-mizing total variation in image restoration,” IEEE Trans.Image Process. 5, 987–995 (1996).

9. D. Strong and T. Chan, “Edge-preserving and scale-

sing Tikhonov regularization are shown in (a) and (b).
aging u

3

3

3

3

3

3

3


dependent properties of total variation regularization,” In-verse Probl. 19, S165–S187 (2003).

0. M. K. Ng, H. Shen, E. Lam, and L. Zhang, “A total variationregularization based super-resolution reconstruction algo-rithm for digital video,” EURASIP Journal on Advances inSignal Processing 2007, 1–16 (2007). Article ID 74585.

1. E. Candès and T. Tao, “Near-optimal signal recovery fromrandom projections: Universal encoding strategies?” IEEETrans. Inf. Theory 52, 5406–5425 (2006).

2. E. Candès and M. B. Wakin, “An introduction to compres-sive sampling,” IEEE Signal Process. Mag. 25, 21–30(2008).

3. X. Zhang, E. Y. Lam, and T.-C. Poon, “Fast iterative sec-

tional image reconstruction in optical scanning hologra-phy,” in Digital Holography and Three-Dimensional Imag-ing (DH), OSA Technical Digest (CD) (Optical Society ofAmerica, 2009), paper DMA3.

4. S. Boyd and L. Vandenberghe, Convex Optimization (Cam-bridge Univ. Press, Cambridge, 2004).

5. G. Indebetouw, “A posteriori quasi-sectioning of the three-dimensional reconstructions of scanning holographic mi-croscopy,” J. Opt. Soc. Am. A 23, 2657–2661 (2006).

6. X. Zhang, E. Y. Lam, T. Kim, Y. S. Kim, and T.-C. Poon,“Blind sectional image reconstruction for optical scanningholography,” Opt. Lett. 34, 3098–3100 (2009).

edge-preserving sectional image reconstruction in optical scanning holography

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