Index matching computerized tomography
IIndex matching computerized tomography
Vincent Daley and Mark Paetkau
Physical Sciences Department, Thompson Rivers University
Owen Paetkau
Department of Physics and Astronomy, University of Calgary
Abstract
Computerized tomography (CT) has been used for decades by medical professionals to detectand diagnose injuries and ailments. CT scanners are based on interesting physics, but due totheir bulk, cost, and safety, hands on experience with a medical CT scanner is unrealistic forundergraduate students. Therefore, operationally similar, yet small, safe, and inexpensive CTscanners are desirable teaching tools. This project details the development of a novel modelCT scanning apparatus. The experimental setup presented utilizes visible light, has short dataacquisition time, and operates on the same physics as its X-ray counterpart. The apparatusemploys a laser and photodiode to image a transparent material, while avoiding loss of transmittedintensity through index of refraction matching. A simple back-projection algorithm results in a 2Dcross section of the scan object. We found we could collect data and reliably image samples in 15minutes. a r X i v : . [ phy s i c s . e d - ph ] J a n . INTRODUCTION Computerized tomography (CT) imaging techniques have become common place in manyfields, especially in medical imaging and diagnosis. The implementation of CT imaging hasallowed for multislice X-rays to produce a 3D image of the relative densities within an X-raytransparent object. Data acquisition for a CT image begins by collecting X-ray projectiondata across a single 2D slice of the object of interest. A simple or filtered back-projectionalgorithm may then be employed to recreate this slice. Translating the object through an X-ray field will allow for several slices to be combined to produce a 3D image.
Understandingthe method of both data acquisition and reconstruction algorithms is important to studentsintent on working with CT imaging systems.CT imaging systems, which are specified for clinical use, have high clinical demand andmay not be available for teaching purposes. Simple and safe model CT imaging systemsare valuable teaching tools. Mylott et al. produced a model system employing a laser,photogate, and floral foam cylinders. This system was able to create a 2D projection of thecylinder outlines with an imaging time of around 20 minutes. Several other studies haveproduced experimental setups employing CT scanning techniques employing light from bothinfrared and visible spectra.
Another study completed by Paetkau et al. employed a Srbeta source and Geiger counter to produce and detect beta rays through a floral foam objectwith offset cutout. The beta-ray system reproduced the physics of an X-ray system and wasable to accurately image a 1-cm wide, bean-shaped hole within the foam. The drawback ofthe beta-ray system was the imaging time of 1.25 hours.The ideal setup employs the use of a laser and photodetector for fast scanning time, whileimaging a transparent material and avoiding reflection and refraction. Glass is a transparentmaterial in the visible spectrum but, in air, a glass rod redirects light by reflection andrefraction. However, a glass rod immersed in a liquid of identical index of refraction willeliminate reflection and refraction, causing the glass rod to “disappear.” This effect maybe leveraged to employ the fast scanning laser and photogate setup shown by Mylott etal. Furthermore, the use of colored glass will cause absorption, dependent on the color ofglass, which simulates material with spatially variant density. This paper presents a novelmodel CT imaging setup, which is able to simulate an X-ray transparent system. Our goalin designing this setup was to provide a short imaging time demonstration of the CT scan2xperimental technique and reconstruction methods for use in the undergraduate physicsinstructional laboratory.
II. THEORY
X-rays used in clinical CT imaging systems are neither reflected or refracted from thematerial being imaged, so the imaging is based on the relative attenuation of tissues. Thisattenuation is reported as the mass attenuation coefficient µ , which is dependent on thedensity and effective atomic number of the imaged material. The mass attenuation coeffi-cient is energy dependent and a known quantity for most organs. As X-rays pass througha single organ, the attenuation is exponential in form. In our system, visible light replacesX-rays. Below, we show that by matching refractive indices, reflections and refractions areeliminated. Furthermore materials may have different attenuation coefficients, thus with thecorrect choice of materials, visible light can replace X-rays in a model CT imaging setup.Light incident upon a material boundary is refracted according to the angle of incidence θ i and the refractive indices n and n of the two adjacent media. The relation is given bySnell’s Law: n sin θ i = n sin θ t . (1)When the refractive indices are matched (i.e., equal), Eq. (1) shows the light’s incidentangle θ i and the transmitted angle θ t are equal. Thus, light will travel in a straight linethrough the interface.Light incident on a boundary is also reflected and follows the reflection law: θ i = θ r .The amount of light reflected is governed by the reflection coefficient r , which is the ratioof the reflected electric field amplitude to the incident. Reflection coefficients are usuallywritten for p -polarization (in plane of incidence) and s -polarization (perpendicular to planeof incidence). Using this general approach, the reflection coefficients can be written asFresnel’s sine and tangent laws: r p = tan( θ i − θ t )tan( θ i + θ r ) (2) r s = − sin( θ i − θ t )sin( θ i + θ r ) . (3)3hen indices are matched, the incident and transmitted angles are equal and we see r p = r s = 0, i.e., no light is reflected.When light is incident on two different materials with matched indices of refraction, thelight is not reflected and is not refracted, and so propagates along a straight line throughthe materials. However, while two materials may have matched indices, they may absorbthe light differently. The absorption is modeled as an exponential decay: I = I e − µ ∆ x (4)where I is the final light intensity, I is the initial light intensity, ∆ x is the thickness of theabsorbing material, and µ is the attenuation coefficient of the material.The simple mechanism of absorption becomes more complex as light passes through manydifferent materials, each contributing intensity loss. In this case, the intensity expressionbecomes an exponential sum of the coefficients weighted by the distance through each ma-terial I = I o e − ( µ ∆ x + µ ∆ x + ... + µ i ∆ x i ) . (5)This extension of the attenuation expression is the basis for clinical CT reconstruction, wherethe goal is to determine the unknown attenuation coefficients at each pixel throughout theimage. From a medical perspective, different internal structures have different densities,hence different attenuation coefficients. Our index-matched visible-light system uses Pyrextubes immersed in mineral oil to provide index-matching. By varying the color of these Pyrextubes, a spatial variation of the attenuation coefficient is produced. Since mineral oil andPyrex have refractive indices near 1.48 ± Th attenuation effect may be leveraged to produce the CT projection data required for imagereconstruction. The Pyrex vials are hollow and cylindrical producing a difficult, non-uniformthickness material to image. The thickness of the Pyrex affects the total attenuation andtherefore the reconstruction data as shown in Eq. 5. An expression to describe the expectedsignal after the laser has passed through the Pyrex vials may be used to determine theattenuation coefficients from the measured data. The expected signal expressing, f ( x ),becomes complex as the laser is translated across the face of the cylinder due to the varyingthicknesses. The location of x = 0 is taken to be the center of the cylinder, with a piece-wisefunction describing the attenuation as seen below:4 ( x ) = e − µ (cid:104) √ r − x − √ r i − x (cid:105) ≤ x < r i ,e − µ √ r − x r i ≤ x < r o , x ≥ r o . (6)In Eq. (6), µ is the attenuation coefficient, r o is the outer radius of the Pyrex tube, r i is the inner radius of the Pyrex tube and x is the distance from the center of the tube.The piece-wise function describes a region through which the laser passes through the entirewidth of both sides, through the thick edges and once the position is outside the outer radiususing the equation of a circle as the basis for derivation. The radii can be measured directly,and therefore the attenuation coefficient can be determined from a best fit of the data tothis function. III. BACK-PROJECTION ALGORITHM
A simplified back-projection, described in great detail by Delaney and Rodriguez, isemployed in our experiment, allowing the projection data collected from successive rotationsand translations to be formed into a CT image. The algorithm is a straightforward methodof generating an image based on absorption measurements. Filtered back-projection canalso be applied, offering improved image quality. In the Delaney and Rodriguez paper, a graphical explanation is used, so we will lookat a simple numerical explanation. We start with a sample represented by a 3 × A ( x, y ) = ln | I | − ln | I ( s, θ ) | (7) I is the intensity from the source without any interfering medium (maximum or average),and I ( s, θ ) is the intensity transmitted through the material at translation s and rotation θ . A is actually µ ∆ x , but since the ∆ x ’s are the about the same, we call this the absorption.When radiation is directed through the material I ( s, θ ) < I , the absorption A is positive.Regions of little absorption mean I ≈ I and result in A ≈
0. When plotted as a grayscaleimage, the image appears as an classic X-ray image: regions of low absorption are black andregions of high absorption are white. The back-projection algorithm described above wascoded in the R programming language using the pseudo code below to produce the desiredresults.Since the image is made of discrete points P ij = ( x, y ), Delaney et al. note, for a sampletranslated s and rotated θ , the image pixels fall on the line within a distance D of thefollowing line: y cos θ − x sin θ − s = 0 (8)The pixels are square, so to determine the pixels that the line intersects, an inequality isused: y cos θ − x sin θ − s ≤ D (9)where D is taken as half the translation increment.The process of constructing an image of the sample proceeds as follows. The absorptionat ( x, y ) is initially set to zero. Then, for each translation and rotation ( s, θ ), if the inequality[Eq. (9)] is satisfied, then the absorption A ( x, y ) is calculated and incremented. The processis repeated for all pixels ( x, y ) in the image. Initialize the image array, A (81x81), to zeroInitialize spatial coordinates array X and Y nitialize Angle_array (0 to 359.1 in steps of 0.9)Initialize the translation array, S (-2 to 2 in steps of 0.1)D=0.05Read data into array N, sizeof(S) rows x sizeof(Array_angle) columnsUse average of unobstructed transmission data to determine Nofor all image points (i,j)for all data points(k,l)if abs(Y[j]cos(Angle_array[l])-X[i]sin(Angle_array[l])-S[k])<=D thenincrement A[i,j] by ln|No| - ln|N[k,l]|Plot the array, A, as an image using coordinates X and YApply filter to plot a binary image as needed IV. METHOD
Data collection was automated using an Arduino Uno microcontroller board, steppermotors, laser pointer, and photodiode. The photodiode used was a Hinds InstrumentsDET-90 silicon photodiode detector. The motor drivers were SparkFun ROB-12859 BigEasy boards. For the linear rail and stepper, OpenBuilds V-Slot Mini V Linear ActuatorBundle with a NEMA 17 stepper were used and the rotary stage was comprised of a KurokesuRSA1 also paired with a NEMA 17 stepper. The laser was battery powered pen laser-pointerwith a 5 mW maximum output and 405 nm wavelength.Samples were colored Pyrex glassblowing tubes. The tubes were bathed in heavy mineraloil (a mild laxative found at most drug stores). Corn or vegetable oil also would work, butmineral oil has the advantage of being colorless. The index of refraction of the mineral oilwas measured using a hollow prism and the minimum deviation method. The index ofrefraction of the mineral oil was measured to be 1.489 ± ± ± ± ± × Image analysis was used to measure the diameters, thicknesses, and other geometriccharacteristics of the tubes. Lower and upper window limits were provided to isolate specificrelative absorption levels. This provided a mask on which the inner and outer diametersmight be measured. The inner diameter was determined from the area enclosed by the maskand the outer was determined from the limits of the mask. These circles were plotted aroundthe centroid determined by a center-of-mass calculation for each tube. These centroids couldbe used to verify distance metrics as compared to the experimental setup.To test the quality of the index matching, a linear scan was made of three tubes in airand in oil. In this case, the laser was translated across the width of the tube (no rotation),while the microcontroller measured the voltage from the photo-diode. The linear step sizein these measurements was about 0.03mm. 8 . RESULTSA. Index Matching
Figure 3 is a twist on the familiar “vanishing stir rod” experiment. The top half of thePyrex tubes are in air and the bottom halves are in mineral oil. The oil surface can be seenat the midway point of the image, just below the thick white line. The yellow lines in thetop half of the image is the voltage of the photodiode (intensity) as the laser is scannedacross the three tubes. The data have been normalized. The white line represents zero. Asthe laser encounters a tube, the intensity goes to zero as the light is reflected/refracted awayfrom the detector. At the center of the tube (normal incidence), some light is detected, butclearly much of it is still reflected away.The bottom halves of the tubes are immersed in mineral oil and the intensity plot is verydifferent. Again the data are normalized and the broad white line is zero. In this case, as thelaser reaches the edge of a tube, the intensity drops, but not to zero. As the laser reaches theinner edge of the tube, the intensity is minimized as this is the thickest part of the glass. Theintensity recovers as the laser traverses the inner edge, reaching a local maximum (minimumthickness of colored glass) at the midpoint. The data are symmetric about the tube. We seethat the different colors of glass have different absorption coefficients. This behavior can bemodeled using Eq. (6). The least-squares best fit is shown by the orange line. The fittingis based only on attenuation properties of the glass, so we see the index matching has donean excellent job of removing reflection/refraction effects. The attenuation coefficients weremeasured to be 2.30 ± − , 1.78 ± − and 0.82 ± − for the lightgreen, dark green, and blue Pyrex tubes respectively. B. CT Images
A CT scan of the blue Pyrex tube using the purple 405 nm laser with 1 mm translation and0.9 degree rotation steps was completed. Each scan produces two useful images, as shown inFig. 4, the sinogram of the data and the reconstructed image. The sinogram displays the rawcollected data for each translation and rotation. If we follow the sinogram from left to rightwe see where the tube is at any degree of rotation. We see that at zero degrees, the centerof the sinusoid is at the same position on the y -axis as the center of the tube. So we can9ee that by following the center of the tube as it rotates through 360 degrees, the trajectoryis sinusoidal. Additionally, the sinogram displays raw data so regions of high absorption(i.e., low detected intensity) appear dark. Conversely, in the CT images such regions havehigh intensity, white coloring, consistent with their higher attenuation coefficients. Theglass tubes have much higher attenuation than the surrounding oil, thus they appear white.Inner and outer diameters of both tubes were measured from the reconstructed images usingwindowing techniques. The window levels were selected using a value relative to maximumpixel value observed of the vials. They were adjusted manually using trial and error untilthey fit on the images and the diameters were subsequently measured. The blue tube hada measured inner diameter of 6.4 ± ± ± ± ± ± ± ± ± ± ± I. DISCUSSION
Teaching students the basics of computed tomography scans and the related reconstruc-tion algorithms is a valuable component of an undergraduate education. CT techniques areemployed in many scientific and medical fields and access to X-ray CT apparatus is notoften feasible. The availability of materials, low scan times and semi-transparent samplesare the limitations of model CT scanners. Previous work accomplished similar results usingeither a laser or a beta-ray source, but had failed to combine low scan time, transparentobject imaging, and low-cost as we have here.
Our setup is easily attainable, equipmentcosts amounted to roughly $
500 CAD. The index matching to produce a semi-transparentmaterial has made for an effective analog for varied attenuation coefficients. Figures 4 and5 provide sinograms and reconstructed CT scans of Pyrex tubes. In each case, both theinner and outer radii of the semi-transparent material were resolvable. These results wereachieved while requiring about 15 minutes for data acquisition and image reconstruction.The accuracy of the CT images shown in Figs. 4 and 5 was first evaluated using severalgeometric measurement. The outer and inner radii of each Pyrex tube were measured firstusing vernier calipers to provide a reference value. The same radii were then measured inthe one tube and two tube scans to produce quantitative values to compare to evaluate theaccuracy of the model CT scans with results displayed in millimeters in Table I. The modelCT platform was able to measure the inner radius to within 0.7 mm for the one and two tubescans. Additionally, the outer radius was measured to within 0.25 mm of the reference value.Accurate measurements using this model CT platform were possible within the uncertaintiesprovided.The distance between the two tubes in the double scan center was measured from Fig. 5to be 19.8 ± ± ± Furthermore there was evidence the vessel holding the mineral oil contributed to11mperfect images. The vessel was a plastic box and there is evidence the sides of the boxare slightly bowed, leading to reflections. Future improvements will use a container of flatPyrex glass (likely biological specimen slides). Fortunately, it was not imperative that therefractive indices be matched exactly provided the refraction did not obscure the change inabsorption through different points in the tubes. It was also important the levels recordedby the photodiode be significantly different in material and in air. The plots seen in Fig. 3indicate the difference in attenuation between air and oil scans.Ensuring the origin of the reconstruction corresponded to the center of rotation of thesample was also important to maximize image quality. If the center was off by as littleas 1 mm, image quality was severely deteriorated. The sinograms seen in Figs. 4 and 5were useful to verify centering. We ensured the peaks and troughs of the sinogram wereequally spaced from the zero line to confirm the centering. If off-center, the scan could bere-centered in image reconstruction although it was preferred the scan center was knownduring data collection. Confirming the image was centered prior to image reconstructiongreatly improved image quality.The sophistication of our image reconstruction algorithm is another major considera-tion of image quality. We employed simple back-projection, but it is known that filteredback-projection offers superior image reconstruction.
Use of a more sophisticated recon-struction algorithm may remove the elevated intensities inside of the tubes noticed in thereconstructed images. However, the implementation of filtered back-projection is beyondthe scope of this paper.In addition to providing an introduction to the physics of X-ray CT scanning in itscurrent configuration, the apparatus is ideal for further student investigations. The effectsof laser wavelength and tubes colors may be investigated, or the effects of using a differentoil. The index of refraction might be subtly altered by mixing another fluid. More complexshapes composed of Pyrex could be scanned to determine the level of detail the scannercan resolve. One of the improved image reconstruction algorithms alluded to previously can be employed and compared to the results presented here. Finally, the data collectionsequence and intervals are not necessarily optimal, time savings may remain available indata collection.The results presented indicate our index-matching CT scanner is an excellent, low cost,fast, and safe apparatus for introducing students to the physics of CT scanners.12
II. CONCLUSIONS
We have assembled a novel laser-sourced CT scanning apparatus, which allows for fastscanning time while resolving the inner and outer boundaries of a transparent object. Thesystem makes use of a laser pointer, photodiode detector, stepper motors, and colored Pyrexglass tubes. The equipment is available in most undergraduate physics departments, makingthe apparatus an ideal platform for university physics students to explore CT scanning. Thedevice presents a powerful and multifaceted learning experience, allowing students to explorethe absorption of radiation by different objects and optics as well as computer based dataacquisition and analysis.
VIII. ACKNOWLEDGEMENTS
We are very appreciative of the excellent work by Dave Pouw in setting up the steppermotors and controllers. J. T. Bushberg, J. A. Seibert, A. M. Leidholdt, and J. M. Boone,