Optical imaging techniques may be separated intotwo categories based on whether image formationis performed by optical components or computation-ally with the help of a computer. Confocal microscopyis the simplest example that straddles these two cat-egories by using a computer to assemble the three-dimensional image, whereas each pixel (or voxel)is formed purely optically by scanning the source re-lative to the sample [1].Computational approaches for optical image
formation based on tomographic reconstructionwere developed, along with tomography techniques,during the past decades. The confocal scanning mi-croscopy method is sometimes referred to as tomo-graphy, as it, indeed, forms slices of the image. Inthis manuscript, however, we use the term “tomogra-phy” to refer to more computationally intense andnovel reconstruction methods that we called “compu-tational confocal tomography.” The key to computedtomography is the collection of projections of the data
over a range of angles, to produce a Radon transformof the image [2]. The image of the data is then recon-structed using an inverse Radon transform tech-nique, such as filtered backprojection. The vast ma-jority of development of this method originated in thefield of medical imaging, utilizing x-ray fields [3].Various optical systems that use collimated lightbeams have been developed to produce an opticalanalog of the x-ray projection approach, employingthe approximation that the light rays follow straightlines paths. However, the large amount of scatteringencountered at optical frequencies, as opposed to xrays, caused difficulty with this assumption. Tech-niques to more accurately satisfy the small-angle as-sumption for validity of the Born or Rytov approxi-mations [4], as well as diffuse optical tomography[5] methods (at the large scattering-angle extreme)have been developed and demonstrated.
Of course, scattering properties of the objectvary, depending on the specific application. Forexample, for the microscopic imaging of cells, theypose an opposite problem, as the refractive index var-iations between many cellular structures are verysmall [6]. Fortunately, advances in interferometryhave enabled the measurement of very small phase
variations [7], and this, in turn, enables poten-tial imaging of these structures via their refractiveindex. Furthermore, Wedberg et al. [8] investigatedsome basic examples that show negligible improve-ment provided by using the Born approximation overthe straight-ray approximation for phase objects,which lends further credence to this straight-rayapproximation.Of course, imaging the sample attenuation coeffi-
cient is itself possible by measurement of intensityinstead of phase. Katawa et al. [9] produced imagesof spirogyra in a system that essentially achieves ro-tation of the sample within a collimated beam. Thisfairly direct application of computed tomography tooptics is the most common approach. Zysk et al. [10]demonstrated image formation of a purely refractivesample, where the data was collected in this fashion.Their “projected index computed tomography” meth-od uses standard backprojection techniques to recon-struct the image under the straight-ray assumption.Renaud et al. [11] demonstrated a conceptually simi-lar technique that is called “confocal axial tomogra-phy,” employing microfluidics to rotate cells withinthe light beam. Sharpe et al. [12] also employed rota-tion of a large size object such that the depth of focusof the illumination/detection beam had to be takeninto consideration.Less commonly, systems employ a focused beam.
Kikuchi et al. [13] considered so-called multiple axisimaging systems, where multiple microscopes withfoci at different angles were intersected, effectivelytreating the source near the focus as a very small col-limated beam. Vishnyakov et al. [14] constructed asystem in which the sample was placed on a mirrorin an interferometer, while the focal point wasscanned along the mirror by displacing the pointsource in the conjugate plane, in order to reconstructthe refractive index. This is done by using the angu-lar spread of the rays themselves, more akin to cone-beam tomography. This system is also interesting inthat, unlike those previously discussed, Vishnya-kov’s system operates in a reflective mode. Lue et al.[15] described a similar concept that employs a fo-cused line beam scanned through a sample flowingin a microfluidic channel, with a cylindrical objec-tive lens. Marks et al. [16] derive and simulate anapproach to the estimate group refractive index,wherein they derive a solution for a tomographicmeasurement with a high numerical aperture.These systems also demonstrate the ability to pro-
duce tomographic images of microscopic structures,such as cellular structures, though the straight-rayapproximation, and for that matter ray optics itself,becomes less applicable in this microscopic regime.For example, Vishnyakov et al. [14] produced imagesof a lymphocyte. Renaud et al. [11] produced imagesof SW13 cells. The approaches used apply, or could beadapted to apply, to the attenuation and refractiveindex of the volume illuminated by the light. Butthe physical effects at the focus itself, for the systemsthat use a focused beam, are not exploited.
As is the case in confocal microscopy, the idealmode for such a system is often reflective, wherebydepth sections of thick samples may be collected. Inconfocal microscopy, the focus (of both source and de-tection objective) is reimaged onto a pinhole that re-jects light scattered from elsewhere in the sample.This forms the basis for a variety of imaging tech-niques. In this paper, we further consider the pro-blem of performing computational reconstruction ofthe sample at the same time as the imaging of theobject at focus. Hence we describe this techniqueas “computational confocal,” as we are computation-ally reproducing the signal at the pinhole. Howeverinstead of rejecting the scattered light outside thepinhole, we assume that this scatter will be negligi-ble (except from the focus itself) and instead use theentire aperture of light, to also reconstruct the sam-ple volume between the focus and the objective with-out need for depth scanning. We view the methoddescribed here to be a superset of conventional con-focal imaging for low scattering situations, which webelieve provides a novel and useful perspective.
This is also the first demonstration, to our knowl-edge, of the combination of scanned object imag-ing with computed tomography reconstruction. Wedemonstrate the ability to collect data “around” anocclusion, in order to image the object behind it,which we believe is a potentially very useful advancein microscopy.
In this manuscript we show how to use a series oftranslational steps during the data collection processwhere we collect a high-numerical aperture signal,rather than collect a series of projections over succes-sive angles with a collimated beam. In effect, ratherthan serially filling out a desired angular range witha fixed field of view, we fill out a desired field ofview with a fixed angular range. The total collectedangular bandwidth range within each measurementis determined by the numerical aperture of the mi-croscope objective, suggesting use of modern high-numerical aperture objectives to cover large angularbandwidth.
A diagram of the computational imaging systemthat we investigate here is provided in Fig. 1(a),shown in comparison with a similar confocal three-dimensional scanning system. It is evident thatour system requires only limited modifications bemade to the conventional confocal imaging system.
Both systems use a collimated laser source that isfocused and is being scanned within the volume ofthe object, but in contrast to conventional confocalmicroscopy that relies on three-dimensional volumescanning, our system performs a single scan in thetransverse direction. In our approach the three-dimensional information is reconstructed from thecollected data. Furthermore, we detect the entirecomplex signal of the Fourier transform of the re-flected field, rather than reimaging the object on apinhole and only detecting the intensity of a singlepoint, as is typically performed in confocal imaging(this is equivalent to detecting the “DC” spatial fre-
quency of the object). In effect, while the conven-tional confocal system samples a single pixel of thesample volume at each point in the scan, the compu-tational system collects a range of data over the sam-ple volume at each step. In Section 2, we will startfrom the analysis of our approach by describing howattenuation and refractive index variation may becombined with conventional confocal measurementsat the objective focus in a complex attenuation para-meter, which can be imaged. Section 2 will also showhow the complex confocal image data may be viewedas a rearrangement of projection data to which tomo-graphic reconstruction can be applied. In Section 3we describe the experimental system and applythe new imaging technique to the experimentallymeasured results for a specially made sample to de-monstrate the simultaneous imaging of an occlusionand an object with varying reflectivity beyond it. Theconclusions and discussions will be described inSection 4.
2. Analysis of the Computational ConfocalTomography System
To analyze the confocal tomography system intro-duced in Fig. 1(a), first we describe how attenuation,phase aberrations, and even the scattering or reflec-tivity at the focal point may all be combined into acomplex attenuation parameter that can be imagedas discussed and quantified below.
A. Complex Attenuation Parameter
We simplify the computational system of Fig. 1(a)further by projecting the pixel locations of the coher-ent detector array to the entrance pupil (i.e., aper-ture) of the microscope objective [see Fig. 2]. Wealso neglect the transmission (i.e., illumination) pathand assume that we start from a point source locatedat the focal point of the microscope objective.For an isotropic scalar spherical wave radiated
from the focus at distance d from the observationplane, the spatial component of the optical field sig-nal at any observation point r would be
uðrs; rÞ ¼ A1
jr − rsjexpfj½kjr − rsj�g; ð1Þ
where A is the value of the amplitude associated withthe strength of the particular scattering point (e.g.,due to the transmitted amplitude and scatteringcross section), rs is the point-source location, k is 2π=λfor a monochromatic source with wavelength λ invacuum, and j is
ffiffiffiffiffiffi−1
p. So A is the value we desire
in normal confocal imaging.Generally, the sample volume between the focus
and the detector will induce a phase delay and am-plitude attenuation (i.e., phase and amplitude mod-ulation) at the detector pixel located at r ¼ rp (seeFig. 2), yielding
uðrs; rpÞ ¼ aðrs; rpÞA1
jrp − rsjexpfj½kjrp − rsj
þΔϕðrs; rpÞ�g; ð2Þ
where aðrs; rpÞ is the amplitude attenuation andΔϕðrs; rpÞ is the phase delay.
Next we combine all the terms into a complexattenuation coefficient such that Eq. (2) may berewritten as
The subscript t refers to the fact that this term con-tains the tomographic information of the sample.
Fig. 2. Geometry of locations of a virtual detector pixel and itsphysical relationship to the point source, which is assumed tobe fixed relative to the detector pixels.
Fig. 1. (a) Computational imaging system versus (b) conventionalscanning confocal system. The dashed lines on the sample repre-sent the scanning paths of the focal point as the sample is movedon a translation stage.
Furthermore, we can describe the phase delay asthe result of the refractive index variation integratedalong the optical path followed by the light “ray” be-tween rp and rs:
Δϕðrs; rpÞ ¼Zrp
rs
kΔnðrÞds; ð8Þ
where ΔnðrÞ is nðrÞ − n0, nðrÞ is the varying index ofthe sample, and n0 is the medium index for which theoptics is corrected (e.g., air, oil, glass). The integralRds refers to the path integral between the points
at the integration limits; the specifics of the followedpaths will be described below. The attenuation re-sulting from the integrated attenuation coefficientalong the same path is
aðrs; rpÞ ¼ exp�−
Zrp
rs
αðrÞds�: ð9Þ
So, the complex attenuation function given in Eq. (7)can be rewritten as
μtðrs; rpÞ ¼ −jZrp
rs
kΔnðrÞdsþZrp
rs
αðrÞds − lnA: ð10Þ
Moreover, if we form the impulse function δðrÞ loca-lized at the point source and employ the integral ex-pression
Zrp
rs
δðr − rsÞds ¼ 1; ð11Þ
we can define the point-source amplitude A in anintegral form as well:
lnA ¼ ðlnAÞZrp
rs
δðr − rsÞds: ð12Þ
We use Eq. (12) in Eq. (10) and combine all the termsinto a single integral:
μtðrs; rpÞ ¼Zrp
rs
ηAðr; rsÞds; ð13Þ
ηAðr; rsÞ ¼ −jkΔnðrÞ þ αðrÞ − δðr − rsÞ lnA: ð14Þ
The complex attenuation coefficient describes the at-tenuation resulting from the sample as an integral
over the path between the point source and the de-tector pixel.
Our final result combines Eqs. (3), (5), and (13),yielding the observed complex amplitude of the opti-cal field given by
uðrs; rpÞ ¼ u0ðrs; rpÞ exp8<:−
Zrp
rs
ηAðr; rsÞds9=;; ð15Þ
where u0ðrs; rpÞ is the sample-independent signalfrom a unit-amplitude source, expf−μ0ðrs; rpÞg. Thisis the deterministic component of the signal, whichmay be partially or completely eliminated by the mi-croscope objective itself (particularly the sphericalphase component). Alternatively, it may be detectedand computed during the calibration process of thesystem.
Now we reconsider the difference between theamplitude A, and the attenuation in the sample,aðrs; rpÞ. We note that ηAðr; rsÞ is a function of rspurely due to the term δðr − rsÞ lnA, describing thesource location. If we are scanning the sample, thenthe value of A, which we recall incorporates thesource amplitude as well as the reflection or scatter-ing coefficient at the focal point, corresponds to a par-ticular point in the scan. First we assume that thefocal point locations are known for every point inthe scan (for example, along an object plane). Andsecond we assume we scan in such a way that a pointat focus is not illuminated when the system scans toa different focus. Then we may combine the sourceamplitudes into a complex attenuation function
ηðrÞ ¼ −jkΔnðrÞ þ αðrÞ −XMm¼0
δðr − rsðmÞÞ lnAðmÞ;
ð16Þ
where rsðmÞ andAðmÞ are the focal point location andthe corresponding amplitude for the mth step in thescan (see Fig. 3). We can treat the complex attenua-tion function plus the object-plane information asindependent of the focal point under the conditionthat none of the rays captured by the detector crossthrough multiple focal points. Then projections ofηAðr; rsÞ and ηðr; rsÞ will be consistent. For transversescanning, this requirement would hold, whereas foraxial scanning it would not. Therefore, in the case oftransverse scanning (the technique to be employed insubsequent sections) we may treat the point-sourceamplitude as an attenuation and assume that thecomplex attenuation (with the amplitude included)is independent of the focal point. This will be usefulwhen we can reconstruct the complex attenuationand get the amplitude values as part of that result.
B. Scanning Tomography
Next we describe the method for processing the com-putational confocal scanning data to reconstruct and
view tomographic projection data. For simplicity, weassume a one-dimensional detector array and aone-dimensional scan procedure. We also assumethat during the scanning process, the confocal sourceand detector relative to the sample are moved (weassume the sample is stationary in our analysis, andthe optics is translated) by a fixed distance incrementequal to the separation between pixels in the detec-tor array, Δx. The vector describing this step is de-fined by Δx ¼ ðΔx; 0Þ. Thus at the mth step in thescan, the focal point is at a location
rsðmÞ ¼ rð0Þs þmΔx; ð17Þ
where rð0Þs is the location of the initial focal point. Alsoat the mth step in the scan, the nth pixel in the de-tector array has a coordinate at location
rpðm;nÞ ¼ rð0Þp þmΔxþ nΔx; ð18Þ
and rð0Þp is the coordinate of the location of the initialpixel at the initial step.If we consider the complex data collected at each
pixel for each scan step, we can enumerate them asuðrsðmÞ; rpðm;nÞÞ, or simply um;n [see Fig. 4 and thematrix representation shown in Fig. 5(a)]. Assumingthat the ray paths are all straight lines, and the sig-nal is only affected by a phase delay and attenuation,we can view the data of Fig. 5(a) as a version of theslant stack of tomographic projection data of thesample. Since the rays are assumed to follow straightlines, the focal point has a fixed location relative tothe detectors in the array; hence, the angle betweenthe focal point and a given detector pixel will be con-stant over the scan. For example, if we observe therays corresponding to the column [n ¼ 2 in Fig. 5(a)],they collectively form a projection of the sample at afixed direction angle, as shown in Fig. 5(b).By combining Eqs. (17) and (18) with Eq. (15), we
obtain
um;n ¼ u0;m;n exp
8>><>>:−
Zrð0Þp þmΔxþnΔx
rð0Þs þmΔx
ηðrÞds
9>>=>>;: ð19Þ
Next we define p0ðxÞ, the projection of ηðrÞ:
p0ðxÞ ¼Z∞
−∞
ηðx; zÞdz: ð20Þ
Note that, since the detector pixels are always in thesame plane, the projections we form at different an-gles are technically not rotated, but sheared. Figure 6shows how a sheared version of the sample providesa zero-angle projection. The sheared projectionwould then be
pθðxÞ ¼Z∞
−∞
ηðx − z tan θ; zÞdz: ð21Þ
Fig. 3. Definition of the sample as a complex attenuation functionηðrÞ, which includes the point-source amplitudes in the focal planeof the microscope objective.
Fig. 4. Schematic diagram describing the detector pixels and thefocal point locations at three different steps in a one-dimensionalconfocal scan of a two-dimensional sample. This example showsfour pixels of the photodetector array and three scanning pointsof the object.
Fig. 5. (a) Matrix representation of the complex data from all pix-els over all steps in the scanning process for the example of Fig. 4for a four-pixel, three-step scan. (b) Rays corresponding to the ðn ¼2Þ column of the slant-stack data.
Assuming that the sample is transparent outsidethe region bounded in z by the focal point and detec-tor (since we are not trying to image those outer re-gions and they are not illuminated), we may combineEq. (19) with Eq. (21) to get
um;n ¼ ðu0Þm;n expf−pθðnÞðmΔxÞg; ð22Þ
where
θðnÞ ¼ tan−1
�zð0Þp − zð0Þs
xð0Þp − xð0Þs þ nΔx
�; rð0Þs ¼ ðxð0Þs ; zð0Þs Þ;
rð0Þp ¼ ðxð0Þp ; zð0Þp Þ: ð23Þ
The complex attenuation is a sampled version of theprojection of the complex attenuation coefficient.Furthermore, we see that by collecting data over arange of n and m, we get a range of projection anglesover a volume of the sample, forming a subset of theRadon transform of the attenuation coefficient ηðrÞ.More details are provided in Appendix A, which alsoaddresses the three-dimensional case.
3. Experimental Validation of Computational ConfocalTomography
A. System Implementation
In the experimental validation of the technique, weconsider a slightly modified configuration where asample is located on a planar mirror surface (seeFig. 7). The “source amplitude” is determined bythe laser amplitude and the reflectivity of the mirror.Each ray is attenuated by the sample projection inthe forward path during illumination, the reflectivityof the mirror, and attenuated by the sample projec-tion in the backscattered path (see Fig. 7)]. We fur-ther incorporated large refractive index changes atthe air–sample interface, by refracting the rays viaSnell’s law (see Fig. 7). Also, the air–sample interfacemay not be a concern if the sample is only expectedto vary beneath the coverslip, as long as the projec-tion angles are properly adjusted. This only re-quires knowledge of the sample thickness, which weestimate using the quadratic term in the measuredphase error.
To collect the complex amplitude data, we em-ployed a standard Michelson interferometer, shownschematically in Fig. 8. A replica of the source andthe signal returning from the sample interfere,and the interference pattern is detected by a digitalcamera. The detected phase must be unwrapped; forthis experiment, the MATLAB built-in unwrap func-tion performed the task sufficiently.
In the experiments we use a Nikon plano apochro-mat objective with a numerical aperture of 0.75,which provides about 97° of angular bandwidthspread in the air (i.e., above the coverslip). Thesource was a 10mW He–Ne laser at 632:8nm, anda SBIG ST-402 camera was used to record the inter-ference pattern magnified using a standard tele-centric imaging system to a pixel spacing of 3:0 μm.The sample was placed on the mirror at the work-ing distance of the objective, and the mirror wasmounted on a translation stage, which was movedtransversely to the optical beam. The camera pixelspacing effectively achieves 3:0 μm sampling at theentrance aperture of the objective. The sample wasmoved by a translation stage in steps of 3:0 μm be-tween image captures, using a Physik InstrumenteC-844 controller and M224 motors. We performeda one-dimensional scan, and, hence, the slant stackand final image are two dimensional. As describedearlier, each complex data set recorded at the detec-
Fig. 6. Rays corresponding to the ðn ¼ 2Þ column of the slant-stack data with sheared coordinates to obtain vertical projection.
Fig. 7. Description of a sample-on-mirror system.
Fig. 8. Schematic diagram of a spatial heterodyne system for de-tection of the complex amplitude of the signal from the object.
tor at a point in the scan provides a new row for theslant stack (see Fig. 5).
B. Reconstruction of the Sample Image
To demonstrate the capability of our method to re-construct the image in the presence of occlusion andvarying reflectivity and to validate the performanceof the method experimentally, we prepared a specialsample. The sample consists of nonreflective stripesprinted on the mirror. We also placed a coverslip overthe mirror and placed an opaque fiber onto the cover-slip, parallel to the stripes, as shown in Fig. 9.The sample (see Fig. 9) was scanned in one di-
mension along the direction perpendicular to the fi-ber and the opaque stripes (x direction in Fig. 9);thus, the reconstructed image would provide a two-dimensional cross section of the sample similar tothat shown in Fig. 9(b). The slant-stack data col-lected for this sample are shown in Fig. 10, both withand without the correction of the deterministicterm u0ðrs; rpÞ.The camera image was averaged over the perpen-
dicular direction (y direction) to produce the row ofslant-stack data. Note that the final image will bein the x − z plane. As the image shows, there wereapproximately 750 pixels across the camera, and theimage was captured for 400 scanning steps. The darkhorizontal bands of the slant stack correspond to theimages where the focus was on the dark stripes onthe mirror; hence, there was no return signal at all.The diagonal bands correspond to the partial occlu-sion caused by the fiber, as it moves across the beam.There are two such stripes because it occluded boththe converging signal before reflecting on the mirror,and the diverging signal after reflection.The inverse Radon transform of this data was
computed using filtered backprojection (using the
piecewise-linear rays described earlier) to producethe image shown in Fig. 11. As the attenuation isdefined to be positive everywhere, we were able toreduce artifacts by clamping the data values at zero.Furthermore, the stripes were much darker than thefiber, so saturation of the pixel values was performedto allow the weaker fiber image to be more visible.The experimental results in Fig. 11 show that the im-age of the stripes and the fiber can be distinguished.
4. Discussion
We show how the data from the aperture of a scan-ning confocal system can be used to perform a tomo-graphic reconstruction of the sample attenuation andrefractive index, as well as the reflectivity at the focalpoint. We demonstrated this method experimentallyin two dimensions, using a sample consisting of avarying reflectivity mirror with an occluding objectabove it. The approach assumes minimal scatteringand that ray paths follow straight lines within thesample.
The axial resolution within the straight-ray ap-proximation for the tomographic reconstruction cor-responds to approximately 2:6 μm, assuming that it isbased on collected spatial bandwidth representing apoint. Of course, this is an incomplete description ofthe imaging performance, as this bandwidth is only
Fig. 9. Description of the measured sample: (a) three-dimen-sional view and (b) cross-sectional view.
Fig. 10. Amplitude of slant stack for the sample of Fig. 9: (a) rawand (b) corrected for deterministic variation.
partly filled in the k space (see Appendix A), lead-ing to artifacts associated with limited-angle tomo-graphy. Moreover, within the sample the resolutionwould be scaled by the inverse of the index, as therange of angles is reduced. Also, in the presence ofocclusion, as in the experiment we have conducted,the actual range of rays collected is obviously re-duced further, as some rays may be blocked. Thiswould also manifest as a shadow artifact extendingaxially from the occlusion, covering the region of theimage from which no rays can be collected by the ob-jective. If the occlusion is larger than the width of theconverging beam, the shadow will reach the object,covering the completely occluded region.As noted in Section 1, conventional computed to-
mography involves the collection of some angularrange of projections, with a fixed field of view deter-mined by the size of the detector, while here we havea fixed angular range determined by the objective,and we collect data over the field of view serially.As a result, while our angular range is limited bythe hardware, the field of view imaged is essentiallyunlimited in the transverse direction. This trade-offcould be useful in applications where rotating or cir-cumnavigating the desired region of the sample isnot possible, for example, in imaging near the surfaceof the skin.The technique could also be applicable to imaging
biological samples, which contain mostly small indexvariations in addition to highly attenuating regions(e.g., mitochondria and nuclei). It also should benoted that the method can be easily integrated withstandard laser scanning confocal microscopy sys-tems. Also, the method could be adapted to other sys-tem geometries, such as transmissive or reflective, byadapting the reconstruction algorithm using knowl-edge of the ray paths.
Here we extend the idea to three dimensions andreview the k-space description of the Radon trans-form data. First we note how the rearrangementof rays from Fig. 4 to Fig. 5(b) can be directly ex-tended to three dimensions. The detector pixelsand focal plane locations fall on planes parallel to the
y − x plane, as in Fig. 12, and the collection of slant-stack data now forms a three-dimensional cubeof data.
The collection of data corresponding to a given de-tector pixel now describes a projection at some anglein both x and y, corresponding to the pixel’s relation-ship to the focal point. In three dimensions, we haveshears both in x and y. So the coordinate transformfor both shears would be
ðx; y; zÞ → ðx − z tan θxz; y − z tan θyz; zÞ; ðA1Þ
which we can write in matrix form as
x0 ¼ Ax; ðA2Þ
where x ¼ ðx; y; zÞT, as usual [we will also inter-changeably use notation such as f ðxÞ ¼ f ðx; y; zÞwhen convenient], and we similarly will use k ¼ðkx; ky; kzÞT for the frequency domain. The shearingmatrix is
Aθxz;θzx;θyz;θzy ¼0@ 1 0 − tan θzx
0 1 − tan θzy− tan θxz − tan θyz 1
1A;
ðA3Þ
where the x and y coordinates are only sheared withrespect to z, or vice versa. In the case of Eq. (A2) wehave A0;θzx;0;θzy . Then the shearing of the k space thatresults from the shearing of the coordinate system is
FðkÞ ¼ FTff ðxÞg;FðA
−θzx;0;−θzy;0kÞ ¼ FTff ðA0;θzx;0;θzyxÞg: ðA4Þ
So when the x and y coordinates are sheared with re-spect to z, the Fourier transform has its z coordinatesheared with respect to x and y by the negative of theshearing angles.
Now we reconsider the projection-slice theoremwith shearing transformations. From the usual zero-angle starting point:
Fig. 11. Inverse Radon transform of slant-stack data.
Fig. 12. Ray between focal point and detector pixel in 3D.
So the unsheared projection is the horizontal slice ofthe Fourier transform. The projection through asheared version of the data will be the horizontalslice of the sheared data’s Fourier transform, whichis itself a sheared version of the original Fouriertransform from Eq. (A4).Consider sheared data:
gðxÞ ¼ f ðA0;θzx;0;θzyxÞ;¼ f ðx − z tan θzx; y − z tan θzx; zÞ;
GðkÞ ¼ FðA−θzx;0;−θzy;0kÞ;
¼ Fðkx; ky; kz þ kx tan θzx þ ky tan θzyÞ: ðA6Þ
The projection of these data,
pθðx; yÞ ¼Z
gðx; y; zÞdz;
Pθðkx; kyÞ ¼ Gðkx; ky; 0Þ;¼ Fðkx; ky; 0þ kx tan θzx þ ky tan θzyÞ; ðA7Þ
is a slice through a sheared plane of the Fouriertransform. Also we can see that the kx and ky dimen-sions have not been scaled (as they are in the case ofrotation). Hence, the sample spacing in those dimen-sions remains unchanged with shearing. So interpo-lation in kx and ky is not needed.Now we consider sampled k-space data. If the pro-
jection is sampled with spacing as Δ in x and y, thenthe sampling frequency is Δ−1. An N-point discreteFourier transform (DFT) of the zeroth projection willproduce samples in k space with spacing ðNΔÞ−1 inboth kx and ky. The k-space samples will, therefore,be at locations ððNΔÞ−1s; ðNΔÞ−1t; 0Þ, where s and tare integers. For a sheared projection, the locationsof k-space samples for the DFT will fall on spatialfrequencies (see Fig. 13):
ðkx; ky; kzÞ ¼ ððNΔÞ−1s; ðNΔÞ−1t; ðNΔÞ−1s tan θzxþ ðNΔÞ−1t tan θzyÞ: ðA8Þ
The volume corresponding to each pixel in kspace is
ðA10Þwhere ΔθðmxÞ and ΔθðmyÞ are the angle spacingbetween successive detector pixels and the focus atpixel numbers mx and my. If we assume these arethe constant Δθ, we get
ΔkxΔkyΔkz ¼ ðNΔxÞ−3 tanΔθðsþ tÞ: ðA11Þ
The density of pixels is, therefore, the inverse ofthis, and the amplitude of the filter needed tonormalize the pixel density in a backprojectionalgorithm would be the inverse again of the pixeldensity. So the appropriate backprojection filter isa linear high-pass filter similar to the rotational case:
HðkÞ ∝ jkx þ kyj: ðA12ÞHowever, with very large numerical apertures, thechange in angle between successive detector pixelsbecomes significantly smaller at large angles, anda filter that more accurately fits Eq. (A10) mustbe used.
We are thankful to Fabio Bonomelli and Lin Pangfor helping in the assembly and operation of the ex-periments performed in this research. We gratefullyacknowledge support from the National ScienceFoundation (NSF) and the Defense Advanced Re-search Projects Agency (DARPA).
References1. S. W. Paddock, “Principles and practices of laser scanning
confocal microscopy,” Mol. Biotechnol. 16, 127–149 (2000).2. S. R. Deans, The Radon Transform and Some of Its Applica-
tions (Wiley, 1983).3. T. M. Buzug, Computed Tomography: From Photon Statistics
to Modern Cone-Beam CT (Springer, 2008).4. J. Devaney, “Reconstructive tomography with diffracting
wavefields,” Inverse Probl. 2, 161–183 (1986).
Fig. 13. Locations of discrete samples in k space on the kx − kzplane.
5. D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio,M. Kilmer, R. J. Gaudette, and Q. Zhang, “Imaging the bodywith diffuse optical tomography,” IEEE Signal Process. Mag.18, 57–75 (2001).
6. J. Beuthan, O. Minet, J. Helfmann, M. Herrig, and G. Muller,“The spatial variation of the refractive index in biologicalcells,” Phys. Med. Biol. 41, 369–382 (1996).
7. P. Hariharan, Optical Interferometry, 2nd ed. (Elsevier,2003).
8. T. C. Wedberg, J. J. Stamnes, and W. Singer, “Comparison ofthe filtered backpropagation and the filtered backprojectionalgorithms for quantitative tomography,” Appl. Opt. 34,6575–6581 (1995).
9. S. Kawata, O. Nakamura, T. Noda, H. Ooki, K. Ogino,Y. Kuroiwa, and S. Minami, “Laser computed-tomography mi-croscope,” Appl. Opt. 29, 3805–3809 (1990).
10. A. M. Zysk, J. J. Reynolds, D. L. Marks, P. S. Carney, andS. A. Boppart, “Projected index computed tomography,” Opt.Lett. 28, 701–703 (2003).
11. O. Renaud, J. Viña, Y. Yu, C. Machu, A. Trouvé, H. Van derVoort, B. Chalmond, and S. L. Shorte, “High-resolution 3-D
imaging of living cells in suspension using confocal axial tomo-graphy,” Biotechnol. J. 3, 53–62 (2008).
12. J. Sharpe, U. Ahlgren, P. Perry, B. Hill, A. Ross, J. Hecksher-Sorensen, R. Baldock, and D. Davidson, “Optical projectiontomography as a tool for 3D microscopy and gene expressionstudies,” Science 296, 541–545 (2002).
13. S. Kikuchi, K. Sonobe, and N. Ohyama, “Three-dimensionalmicroscopic computed tomography based on generalizedRadon transform for optical imaging systems,” Opt. Commun.123, 725–733 (1996).
14. G. N. Vishnyakov, G. G Levin, V. L. Minaev, V. V. Pickalov,and A. V. Likhachev, “Tomographic interference microscopyof living cells,” Microscopy Anal. 18, 15–17 (2004).
15. N. Lue, W. Choi, G. Popescu, K. Badizadegan, R. R. Dasari,and M. S. Feld, “Synthetic aperture tomographic phase micro-scopy for 3D imaging of live cells in translational motion,”Opt.Express 16, 16240–16246 (2008).
16. D. L. Marks, S. C. Schlachter, A. M. Zysk, and S. A. Boppart,“Group refractive index reconstruction with broadbandinterferometric confocal microscopy,” J. Opt. Soc. Am. A 25,1156–1164 (2008).
