Iwaguchi et al.

RESEARCH

Light path alignment for computed tomographyof scattering materialTakafumi Iwaguchi* , Takuya Funatomi, Hiroyuki Kubo and Yasuhiro Mukaigawa

Abstract

We aim to estimate internal slice of the scatteringmaterial by Computed Tomography (CT). In thescattering material,the light path is disturbed andis spread. Conventional CT cannot measure thescattering material because they rely on theassumption that rays are straight and parallel. Wepropose light path alignment to deal withscattering rays. Each path of disturbed scatteringlight is approximated with a straight line. Then thelight path in the object corresponding to a singleincident ray is modeled as straight paths spreadingfrom incident point. These spreading paths arealigned to be parallel, so that they can be useddirectly by conventional reconstruction algorithm.

1 IntroductionMeasurement of the interior of an object is a key tech-nique in various fields such as the inspection of foodfor finding foreign objects or the observation inside ahuman body.

X-ray Computed Tomography (CT) is one of thosetechniques that has been used in the inspection of theinterior. However, the conventional X-ray CT is hardto use due to its cost, invasiveness, etc. Recently, op-tical measurement using visible (VIS) / near infrared(NIR) light is spotlighted because the optical systemis inexpensive and safe compared to the X-ray.

However, optical measurement is challenging due tothe scattering property of VIS/NIR light. In the caseof the conventional CT, the interior is estimated usinga mathematical inverse transform based on an assump-tion that paths are straight and parallel. Therefore, theconventional CT is not directly applicable to optical

*Correspondence: iwaguchi.takafumi.il6@is.naist.jp

Nara Institue of Science and Technology, 8916-5, Takayama, 630-0192

Ikoma, Japan

Full list of author information is available at the end of the article

Figure 1: Schematic illustration of our method.

measurement when the scattering occurred in the tar-

get material. For the scattering material, Optical Scat-

tering Tomography (OST) [1, 2, 3] and Optical Dif-

fusion Tomography (ODT)[4, 5] have been proposed.

In these methods, the paths of the scattered light are

simulated and the interior is estimated from them. A

drawback of these methods is the high computational

cost due to simulating thousands of paths.

In this paper, we propose an optical scattering to-

mography using the mathematical inverse transform.

Figure 1 illustrates the overview of our proposed

method. First, we propose a simplified model of the

scattered light path. In this model, each path of the

scattered light is approximated by a straight line. As a

result, the paths in the object are regarded as a set of

spreading lines from the incident point. We also pro-

pose light path alignment that converts these spread-

ing paths into parallel paths. This alignment makes

it possible to estimate interior using mathematical in-

verse transform that has been used in the conventional

ray CT, although the light is scattered.

Iwaguchi et al. Page 2 of 5

Figure 2: Conventional CT measurement.(a) Paralleltransmission measurement.(b) Example of a sinogram.

2 Method2.1 Conventional CT methodTo begin with, we describe how the interior is esti-mated in the conventional CT. The conventional X-ray CT has been used for decades because it providesa clear visualization of the interior of the object. Theconventional CT relies on the important X-ray’s prop-erty that it passes straight through the object withoutreflection and refraction, but it is attenuated a certainrate depending on the material.

Figure 2(a) illustrates a setup of the conventionalCT measurement. We call this measurement paralleltransmission measurement. Parallel beam is cast ontoan object from the light source and its projection onthe plane of opposite side is observed. This measure-ment is executed from various views.

When the paths in the object are straight and par-allel, the interior is reconstructed using the inverseRadon transform. When a single ray is cast onto theobject, it travels in a straight path. Its intensity is at-tenuated according to the path length and the absorp-tion rate of the material. A projection of a single rayis regarded as the integral of the absorption over thestraight line. The inverse Radon transform will givea relationship between the projection data obtainedfrom the measurement and the spatial distribution ofthe absorption rate. Filtered Back Projection (FBP)[6] is an implementation of the inverse Radon trans-form, which is superior in terms of the accuracy andthe speed.

2.2 Model of path of scattered raysIn the case of optical measurement, the interior cannotbe estimated in the same way since the light path doesnot follow the assumption in the parallel transmissionmeasurement. Now, we describe the model of the pathsof the scattered light that our CT method relies on.

Limiting incident ray

Figure 3: Path of scattered light in the material.

In order to apply the conventional CT for recoveringthe interior, points on the surface where the ray en-tered and where the ray went out must be determinedfirst.

Figure 3 illustrates the paths of the scattered lightwhen the parallel beam is cast. When each ray entersa scattering material, the path is scattered into multi-ple and spreading paths. We cannot determine wherethe ray that reached on the specific point come frombecause the rays from all incident rays reach the samepoint on the surface. To deal with this problem, wecast a single ray on the object at the observation sothat the incident point is determined.Approximating path of scattered light

Figure 4(a) illustrates the paths of the scattered lightcorresponding to a single incident ray. In order to sim-plify the disturbed path of the scattered light, we ap-proximate each path of the scattered light by a short-est path connecting the incident point and the pointon the surface where the light reached as shown inFig. 4(b). Then the paths in the object are regardedas a set of spreading straight rays from the incidentpoint.

According to this model, we propose shortest-pathtransmission measurement to estimate the interior ofthe scattering object. The left-top figure of Fig. 1 il-lustrates the setup of our measurement. A single rayis cast toward an arbitrary point of the object andthe object’s surface is measured using a camera. Thenmeasurement is repeated with rotating the object. Asa result, sets of spreading straight rays for various an-gles of the incident rays are captured.

2.3 Light path alignmentAccording to the light path model, the paths in the ob-ject are simplified as spreading straight rays. However,it is still insufficient to apply inverse Radon transformbecause they are not parallel. Here, we introduce lightpath alignment that aligns these simplified paths toparallel paths.

Iwaguchi et al. Page 3 of 5

(a) (b) (c)

Figure 4: Model of the light path. (a) actual path ofscattered light. (b) shortest-path transmission model.(c) parallel paths obtained by light path alignment.

Schematic illustration of light path alignment isshown in Fig. 1. The light path alignment convertspaths of scattering surface measurement shown in thefirst row into the parallel paths in the second row.From a single measurement, scattered light throughthe object is observed and it is treated as a setof spreading straight paths from the incident point.Other sets of the paths are observed by casting raysfrom different directions. Then we pick up the pathsof the same direction from these sets and align themaccording to the location so as to be parallel as shownin Fig. 4(c).

Light path alignment can be implemented as a con-version of sinogram. Sinogram stores rays observed bya measurement as shown in Fig. 2(b). Suppose that auv coordinate is fixed on the object and the cameraand the light source rotates around the origin of theuv coordinate as shown in Fig. 5. Let X be the lo-cation where a ray reach on the projection plane andθ be the angle between the orientation of the cameraand v axis. In the sinogram, a ray reaches X of theprojection plane of the camera is stored at (X, θ).

We denote sinogram of shortest-path transmissionmeasurement by gs and one of parallel transmissionmeasurement by gp. Light path alignment is a conver-sion from gs to gp.

In order to formulate a conversion between both sino-gram, we discuss about the relationship between thecoordinates of the sinogram and the path of a ray.In the case of parallel transmission measurement asshown in Fig. 5(a), a ray that reaches xp of the pro-jection plane with θ = θp is stored at gp(xp, θp). Inthe case of shortest-path transmission measurement asshown in Fig. 5(b), “ray 1” that reaches xs of the pro-jection plane with θ = θs is stored at gs(xs, θs). Wefocus on the path of this ray. If this path is extendedthrough the surface, it can be regarded as a ray ofparallel transmission measurement with a virtual pro-jection plane X ′ and θ = θ′p. Let x′p be a locationthat the extended path reach then the ray is stored at

Figure 5: Geometry of measurement. (a) parallel trans-mission measurement. (b) shortest-path transmissionmeasurement.

gp(x′p, θ′p) in the sinogram of the converted sinogram.

From the discussion above, light path alignment is al-tered by moving gs(xs, θs) to gp(x′p, θ

′p) like shown in

Fig. 2. This conversion is possible when each coordi-nate in the sinogram gp represents a unique ray in theobject.

We formulate the relationship between (x′p, θ′p) and

(xs, θs) in the case of the object is a cylinder and thecamera model is orthogonal projection. Let φ be anangle between the ray from the light source and theray in the object. The relationship between (x′p, θ

′p)

and (xs, θs) is formulated as follows:

x′p =xs cosφ

sin 2φ=

xs2 sinφ

(1)

θ′p = φ− θ. (2)

Therefore, the sinogram of the shortest-path trans-mission measurement gs can be converted to one ofthe parallel transmission measurement gp by followingequation:

gp(xp, θp) = gs

(xs

2 sinφ, φ− θs

). (3)

The paths of the scattered light are converted to par-allel paths by this conversion, then the optical CT forthe scattering object using inverse Radon transform ispossible.

2.4 Optimal placement of light sourceBy light path alignment, the paths of shortest-pathtransmission measurement are converted to parallel.However, there are some unobserved paths due to theincompleteness of the measurement. For example, “ray2” in Fig. 5(b) is observed because it reaches the sur-face that can be seen from the camera. In contrast,

Iwaguchi et al. Page 4 of 5

(a) (b)

Figure 6: Observed paths corresponding to a place-ment of the light source. (a) 0◦ ≤ θl ≤ 90◦. (b)90◦ ≤ θl ≤ 180◦.

“ray 3” cannot be observed because it reaches the sur-face that is hidden from the camera. Therefore, it isnecessary to find the optimal setup of measurement tominimize the number of unobservable paths.

In the discussion so far, the light source is placed atthe opposite of the camera. However, a number of theunobserved paths changes corresponding to the anglebetween the light source and the camera. Assume thecamera is fixed, now we derive an optimal placementof the light source relative to the camera.

Figure 6 illustrates the observable area in the caseof the object is a cylinder and camera model is or-thogonal projection. We denote the angle between theorientation of the light source and the orientation ofthe camera by θl. The rays that reach the surface high-lighted by green are observable by the camera and thearea that these rays pass through is shown in blue. Itis found that there are the areas that are unobservablefrom the camera.

In order to evaluate these areas, we take a look at adistance between a ray and the center of the object.Let the slice of the cylinder is a unit circle and a dis-tance between a ray and the center of the object bed. In the case of parallel transmission measurement,d takes a range 0 ≤ d ≤ 1. In the case of shortest-path transmission measurement, the range depends onθ We denote the range of d by dmin ≤ d ≤ dmax. For0◦ ≤ θl ≤ 90◦, rays that pass near the surface areunobservable. dmin and dmax are given as follows:

dmin = 0 (4)

dmax = cos

(π

4− θl

2

). (5)

For 90◦ ≤ θl ≤ 180◦, rays that passes around the cen-ter are unobservable. dmin and dmax are given as fol-

0

not filled

1−1

(a) (b)

Figure 7: Coverage of the measurement. (a) Coverageas an area in the sinogram. (b) Coverage vs the angleof light.

lows:

dmin = sin

(θl2− π

4

)(6)

dmax = 1. (7)

We define the coverage of the observation as a ratioof the range taken by d compared to the radius. Let pdenotes the coverage of the observation, it is calculatedas follows:

p = dmax − dmin. (8)

The coverage can be comprehended as the area of sino-gram. The distance d corresponds to the x-axis of sino-gram because x-axis of sinogram is a displacement ofa ray relative to the center of the object. Thereforedmin,dmax and p mean distances shown in Fig. 7(a).The coverage will be the ratio of the area filled up inthe sinogram. Fig. 7(b) shows the coverage of observa-tion with respect to θl. It is found that the coveragetakes its maximum 1 at θl = 90◦. This means the com-plete paths are observable for this setting, thus it is theoptimal placement of the light source.

3 ExperimentWe performed the numerical simulation for the evalu-ation of the proposed method. We confirmed that ourmethod estimates the interior and confirmed the im-provement of reconstruction by using the optimal mea-surement setup.

As a synthetic data, a distribution of the absorptionrate is given as shown in the second row of Fig. 2(a)with the dimensions of 64x64. The first row of Fig. 2(a)shows the ideal sinogram obtained by the paralleltransmission measurement. It is found that traces ofthe circles appear as two sinusoidal curves.

Iwaguchi et al. Page 5 of 5

First, we show the sinogram when the scatter oc-curs and the reconstructed interior from it using FBPwithout alignment. The projection was simulated withthe assumption of the ideal shortest-path transmissionmodel where the scattering at surface is isotropic andthe path of the ray in the object is straight. We per-formed an experiment for θl = 0◦ where the coverageis not complete and for θl = 90◦ where the measure-ment setup is optimal. Fig. 8(b) shows the sinogramand the reconstructed interior for θl = 0◦. Althoughthe sinogram consists of two curves reflecting the ab-sorption corresponding to two circles, they are not per-fect sinusoids like in the sinogram of parallel transmis-sion measurement Fig. 8(a). The reconstructed interiorconsists of two blurred shapes. The circles were not re-constructed correctly because the light paths were notaligned. Fig. 8(c) shows the sinogram and the recon-structed interior for θl = 90◦. In this case, the light wascast perpendicularly to the orientation of the camerafrom the left side, thus traces appear mainly on theright side. The reconstructed interior is totally uncleardue to the corrupted shape of the sinogram.

Next, the sinogram was converted by light pathalignment. Fig. 8(d) shows the aligned sinogram andthe reconstructed interior for θl = 0◦. Two sinusoidalcurves appear after the light path alignment is exe-cuted. However, the sinogram is lacking its both sidessince the measurement setup is not optimal. In thereconstructed interior, the shapes of the circles are re-constructed. We can see that ellipse shaped artifactsappeared and the shape outside the artifacts is un-clear. These artifacts are considered to be caused bythe incompleteness of the sinogram. Fig. 8(e) showsthe aligned sinogram and the reconstructed interiorfor θl = 90◦. The aligned sinogram is complete andis substantially identical with the sinogram of paralleltransmission measurement. The complete observationof rays was possible because all the ray through thecloser half to the camera reach to the observable sur-face in this case. In the reconstructed interior, we cansee the interior is reconstructed correctly.

4 ConclusionWe have proposed an optical scattering tomographyusing inverse mathematical transform. In order to re-construct the interior using the inverse Radon trans-form, the paths in the object have to be parallel andstraight. We have proposed shortest-path transmis-sion model that approximates the paths of the scat-tered light as spreading straight lines from the incidentpoint. We have also proposed light path alignment thatconverts approximated paths to parallel and optimalplacement of the light source.

From the numerical experiments, we have presentedthe interior is reconstructed correctly by our proposed

Figure 8: Sinogram and reconstructed interior.

method. We have also shown that the reconstructionis improved by choosing the optimal placement of thelight source.

For future work, we plan to measure a real objectusing our method.

AcknowledgementsThis work was partly supported by JSPS KAKENHI Grant Numbers

JP26700013, JP15K16027.

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