A Graphical Method for Determining the In-Plane Rotation Angle in Geometric Calibration of Circular...

IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 31, NO. 3, MARCH 2012 825

A Graphical Method for Determining the In-PlaneRotation Angle in Geometric Calibration of

Circular Cone-Beam CT SystemsJingyan Xu* and Benjamin M. W. Tsui

Abstract—It is well known that seven parameters completelydescribe a circular cone-beam geometry in either flat-panel X-raycomputed tomography (CT) or single pinhole SPECT imaging.This paper considers the problem of determining one of the sevenparameters only, the detector in-plane rotation or twist angle .We describe a graphical procedure that can determine indepen-dently of all other six parameters from a geometric calibrationscan of point objects. Our method is exact in the ideal noise-freecase and is general in that the other two out-of-plane detectorrotation angles and can be nonzero. The calibration scantypically needs at least two point objects and an even number ofprojection views over a full 360 data acquisition. Under certainconditions, projection data truncation or a short scan acquisitionof 180 fan angle can be accommodated without affecting theaccuracy of the calibration result. The graphical method is equallyapplicable to rotational multipinhole SPECT geometry. In thiscase, the final result is averaged from the individual estimatesconsidering each pinhole separately. We use computer simula-tions and a multipinhole SPECT experiment to demonstrate theaccuracy and precision of the proposed method.

Index Terms—Ellipse fitting, flat-panel X-ray computed to-mography (CT), geometric calibration, multipinhole, perspectivegeometry, pinhole single-photon emission computed tomography(SPECT), vanishing point.

I. INTRODUCTION

G EOMETRIC calibration in cone-beam X-ray computedtomography (CT) and single-photon emission computed

tomography (SPECT) or pinhole SPECT1 aims to determinethe complete imaging parameters that are crucial in artifact-free image reconstruction. The importance of accurate geomet-rical information is evidenced by the extensive literature on this

Manuscript received July 18, 2011; revised December 21, 2011; accepted De-cember 30, 2011. Date of publication January 05, 2012; date of current versionMarch 02, 2012. This work was supported by the National Institutes of Healthunder Grant R44 EB006712 and Grant R01 EB008730. Asterisk indicates cor-responding author.*J. Xu is with the Division of Medical Imaging Physics, Department of

Radiology, Johns Hopkins University, Baltimore, MD 21287 USA (e-mail:[email protected]).B. M. W. Tsui is with the Division of Medical Imaging Physics, Department

of Radiology, Johns Hopkins University, Baltimore, MD 21287 USA (e-mail:[email protected]).Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TMI.2012.2183003

1Subsequently we use cone-beam CT to include all these modalities and maynot mention them individually each time.

topic, not only in computed tomography [1]–[3] but also in di-verse fields such as robotics [4] and computer vision [5].Throughout this work we adopt the common assumption that

the 2-D detector is flat with square pixels and known pixel sizes.It is well known in this case that seven parameters completelydescribe the geometry of a circular cone-beam imaging system[3], [6]. Different but equivalent seven-parameter sets have beenused in the literature [6]–[8]. One definition used in [6] is thefollowing.• , the focal length or the perpendicular distance from theX-ray source or pinhole aperture to the detector.

• , the radius of rotation or the distance from the focal pointto the rotational axis .

• , , the projected position of the focal point on thedetector.

• , , , the three rotation angles that define the detectororientation. When acting individually, means thatthe rotational axis is not parallel to the detector pixelscolumn direction; means the rotational axis isnot parallel to the detector plane; and the remaining isa rotation of the detector plane around . The angles(detector tilt) and (detector slant) are also referred to asthe two out-of-plane angles.

These parameters can be determined using either iterative pa-rameter fitting or direct analytical inversion. Direct analyticalinversion methods avoid the problems in iterative parameter fit-ting such as sensitivity to parameter initialization or potentiallocal optimal solutions. However, they sometimes involve as-sumptions to simplify the analytical derivations. In [9], bothout-of-plane angles and are assumed to be zero. In [6], onlyone of the two out-of-plane angles is assumed to be zero, andall other six parameters are determined analytically. One ex-ceptional work is that of [10], in which six out of seven pa-rameters (including ) are determined completely analyticallywithout assuming or . The remaining undeter-mined parameter is the radius of rotation that the authors as-sume known a priori. Another interesting work is that of [11],fromwhichmany geometrical concepts, e.g., parallel lines in theobject space intersect on a common “converging point” on theplanar detector, will be adopted in this work. However the cal-ibration method in [11] is not accurate when or is nonzero,and the inaccuracy increases as or deviates away from zero[11].In this work, we borrow concepts from perspective geometry

and demonstrate that the in-plane rotation angle can be deter-mined independently of the other six parameters in the circular

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826 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 31, NO. 3, MARCH 2012

Fig. 1. Simulated projection orbits of nine point objects in a cone-beam geom-etry. Each point traces out an ellipse. The interesting case is when the ellipsedegrades to a line segment as in the fifth point. The vertical and horizontal axeshave units of millimeters.

cone-beam geometry. This same observation was also made in[10]. The method there however was analytical in nature; on theother hand, our method is a graphical procedure based on anal-ysis of the cone-beam geometry.The organization of the paper is as follows. In Section II,

we first explain the general idea of our approach using anexample geometric calibration scan. The geometrical prop-erties embedded in the projection data lead to the proposedgraphical procedure. In Section III we present computer sim-ulation studies and one experimental multipinhole SPECTstudy to demonstrate the accuracy and precision of our method.In Section IV, we discuss some implications and possibleextensions of our approach, e.g., dealing with projection datatruncation, and relate to other works in the literature. We finishin Section V with conclusions.

II. METHOD

A. Overview

We use Fig. 1 to illustrate the basic idea. Similar figures canbe found in many publications, e.g., Fig. 9 in [9] and [10]. InFig. 1, ellipses are fitted to the projection data (the cross marks)of nine point objects rotating around a common axis in thecone-beam geometry. As each point rotates along a circle in theobject space, their cone-beam projection traces out an ellipse onthe detector. The interesting case is the fifth point whose projec-tion locations traces out a degraded ellipse, i.e., a line segment[Fig. 1]. Obviously this only happens when the circle plane, de-noted by , on which the point rotates contains the focal point(the X-ray source or the pinhole aperture). The straight line(the degraded ellipse) we see on the detector is the intersec-tion of two planes, and the detector plane . It is shown inSection II-C that the slope of the line on the detector planeis regardless of all other six parameters. In other words,if such a degraded ellipse is observed, then the in-plane rotationangle can be determined.Very often, however, this favorable observation cannot be

made, in which case we can still use other regular ellipses todeduce the location of . This is where this work significantly

Fig. 2. The coordinate system definition used in [6]. The detector orientation,defined by the axes , , and , are specified by three rotation angles, , ,, with respect to the object space. The other four parameters to be determinedare , , and .

departs from previous methods. We borrow concepts from per-spective geometry and provide a graphical procedure of locating.

B. The Coordinate System and Notations

We use the definition in [6] (see Fig. 2) and denote the rota-tional axis as the axis in the object space. The axis containsthe focal point, and the axis is defined using the right-handrule. Three angles , , define the detector orientation in theobject space. Let be the detector normal, and , be thetwo axes on the detector surface that are parallel to the row andthe column direction of the detector, respectively. Their expres-sions in the object space can be written in the matrix decompo-sition form

(1)

where , , are rotation matrices, and , , and areunit vectors in the object space. The focal point is at ,an alternative representation in the detector coordinate system is

. Here for consistency, all distance or lengthquantities, such as , , , and are measured in physicalunits, e.g., millimeters or centimeters. The seven unknown pa-rameters , , , , , , are to be determined from a geo-metrical calibration scan of point-like objects (metal ball bear-ings for X-ray CT and point sources for SPECT). As defined in[6], we assume the unit vectors , , , and , , are allrow vectors.Our graphical procedure for determining requires at least

two point objects. As the two points rotate around the axis,projection data are taken in a step-and-shoot manner. Let theunknown locations of the points be described by , ,, 2, which are respectively the distance of the point to therotational axis and to the plane. We also introduce the

XU AND TSUI: A GRAPHICAL METHOD FOR DETERMINING THE IN-PLANE ROTATION ANGLE 827

Fig. 3. Illustration of the circle on which the point object traverses , thecircumscribing square of the circle , and the lines defined in (4)–(6). Thesubscript specifies each point object.

notion of circumscribing squares of the circles on whichthe point object traverses. Let

(2)

(3)

where are the trajectory of the points in the object space,and circumscribes . Note that there exist infinite numbersof circumscribing squares given the point circles . The def-inition in (3) is a special choice since have edges that areparallel to the unit vectors and in the object coordinatesystem. We divide these edges into two groups, those that areparallel to the axis, , and those that are parallel to the

axis,

(4)

(5)

(6)

Our graphical procedure finds and by identifying the pro-jection of and from a calibration scan. The defini-tions in (2)–(6) are illustrated in Fig. 3.

C. The Slope of the Line

As we described earlier, the ellipse on the detector plane de-grades to a line when the circle plane on which the pointobject rotates contains the focal point. Using the coordinate def-inition in Fig. 2, this plane is simply the - plane in the objectspace, i.e., . The detector plane is given by

The line of intersection of the two planes and can beparametrized by , where

(7)

Using the following detector coordinate system to convert (7)

(8)

(9)

after some algebraic manipulation, we have

(10)

(11)

The slope of is given by .

D. Preparation Steps of Finding

When no degraded ellipse is observed from a geometric cali-bration scan, we can still deduce the location of and its slopefrom other regular ellipses. Assume we have available the pro-jection data from a calibration scan using two point objects. De-note by , , , 2, the estimatedcentroid locations of the th point at projection view index .We also assume that there are an even number of projectionviews evenly distributed over 360 data acquisition range. Thefollowing two preliminary steps, (1) determining the cone-beamprojection of the rotational axis on the detector and (2) ellipsefitting to point projection , are standard techniques ingeometric calibration. These are the starting point of the graph-ical procedure that locates . For completeness, we briefly de-scribe them and refer the readers to the cited publications fordetails.1) The Cone-Beam Projection of the Rotational Axis : As

described in [6], the projected rotation axis is determinedby utilizing the “radial pairs” [9] of point projection data. Foreach point object , the common intersection of all linesegments connecting with those that are 180 apart,

, , determines one point on therotational axis. When the line segments do not intersect at acommon point due to noise, a least-squares solution can be used[6]. The two intersection points from , two will determinethe projected rotation axis on the detector.2) Ellipse Fitting: Using the method in [6], for each point ob-

ject , fit an ellipse to the point projection ensemble ,. We denote the parametric form of the fitted

ellipses by , , 2, where ,, are the coefficients of the quadratic terms in the ellipse

expressions, and , is the geometric center. These para-metric forms are useful in calculating the line-ellipse intersec-tion points or the line-ellipse tangent points introduced shortly.The following summarizes the outcome of the preparatory

steps before we start the graphical procedure.• The fitted ellipses , , 2.• The projection of the rotational axis .• The two points that determine , , , 2.• The four intersection points between the ellipses and ,

, , 2, calculated by using the ellipse expres-sions Fig. 4.

E. A Graphical Procedure to Determine

Nowwe provide the step-by-step instructions of the graphicalprocedure. On a few occasions we will mention “multiple (morethan 2) lines converge to (or intersect at) the same point.” This


Fig. 4. Step 1. The four lines either converge to orare parallel. In the latter case, it will be shown later that the slope of any of

is also . The object space line segments corresponding toare highlighted in red in the right panel.

Fig. 5. Step 2. Connecting with , the two lines intersect with theellipses at . They are the cone-beam projection of the -paralleldiagonal lines in the object space in the right panel.

should be understood in the noise-free case, when ideal point ob-jects are used in the calibration scan, and there is no error in cal-culating the point projection locations . We will de-scribe approaches to calculating “common intersection points”in presence of noise in the numerical studies (Section III).Figs. 4–6 provide the graphical illustrations corresponding

to steps 1–3, respectively. The left and right panels are relatedby a cone-beam projection or perspective mapping that will befurther explained in Section II-F.1) Draw lines that are tangent to the ellipse at .

The four lines, denoted by , in Fig. 4, ,2, should converge to the same point . Thisis the “converging point” in [11] due to .

2) Connect with , , 2, on the rotational axis. These two lines, denoted by , intersect with the

two ellipses at four points [Fig. 5].3) Draw lines that are tangent to the ellipse at .The four resulting lines, denoted by in Fig. 6, andthe rotational axis have a common intersection point. The point is the projection of the origin in the

cone-beam geometry [Fig. 6].4) Connect with from step 1. This is and its slopeis .

When , the four lines in step 1 are parallel anddoes not exist. Wemodify the above steps 2 and 4 as follows.

Fig. 6. Step 3. The tangent lines to the ellipse at are .They correspond to the cone-beam projection of the -parallel edges inthe object space shown in the right panel.

• 2m) In case are parallel with each other, thenpasses and is parallel to . The two lines

, , 2, intersect with the two ellipses at fourpoints .

• 4m) is parallel with and contains . In this casewe could skip step 2–4 since the slope of is also

. This only happens in the noise-free case and when.

F. Analysis of the Graphical Procedure

The following analysis follows the steps of the graphical pro-cedure in Section II-E. In Figs. 4–6, we have side-by-side thecorresponding entities between the object space and the projec-tion space. We use the notation to represent that ge-ometrical entities (points or lines) and are related by thecone-beam projection, or equivalently a perspective mapping.Between and , which is in the object space and which is theprojection should be clear from the context. When more thanone entity appears in either side of , the order in whichthey appear is irrelevant to our analysis.Some fundamental concepts in perspective geometry [12]

will be helpful in understanding the analysis. The conceptsmostly involve preservation of incidence (tangency and in-tersections) under perspective transform. One example is thatparallel lines in the object space either remain parallel orconverge to a common intersection point, the vanishing point,under perspective mapping. For readers who are unfamiliarwith these concepts, it is equally adequate to take them asaxioms.1) We claim that the lines are the cone-beam pro-jection of the y-parallel edges (5) of the circumscribingsquares (3) of the point object circles (2). In other words,

. To see why this is true, we useas one example. Since a circumscribing square of acircle projects to a circumscribing quadrilateral of thecorresponding ellipse, to establish the correspondencebetween and , we only need to establishthe perspective mapping between the point of tangency,i.e., . This is obvious since

is on the - plane which projects to . Thepoint is also on the point object circlewhich projects to . Therefore,


and . Similarly for the second point pro-jection we have . Since are theperspective projection of parallel lines in the object space,they either converge to a common point ,or they remain parallel on the detector. Using the analogyin [11] of the “spine” and pages of a book, we see that the“spine,” the line connecting the focal point and , is alsoparallel to the group or equivalently . Since thefocal point is on the - plane, so is . This last fact willbe used in step 4.

2) Since , (the common intersection ofradial pairs maps to points on ,) then the cone-beam pro-jection of must have the same vanishing point as

. This is exactly how we draw , which impliesand .

3) Using the correspondence between the point of tangencyfrom step 2, we see that

by a similar argument as in step 1. Thevanishing point that the four lines converge to mustalso lie on the axis [11] which contains the X-ray focalpoint and is parallel to the group . In other words,the four lines converge to the intersection of axiswith the detector, i.e., . And certainly is on sincethe plane contains . Therefore, we have establishedthat converge to on .

4) Since all three points , , and the X-ray focal point, areon the plane, the line connecting andis the intersection of the plane with the detector. Asshown in Section II-C, this line has slope .

III. NUMERICAL STUDIES

The motivation of this work was to develop geometrical cal-ibration and image reconstruction methods for a novel asym-metric multipinhole (MPH) collimator set in a four-head smallanimal SPECT system [13]. The calibration scan can be per-formed either before or after the experimental scan using theidentical data acquisition. The estimated geometric parametersare an integral part of the iterative ML-EM or OS-EM imagereconstruction algorithm to account for the geometric misalign-ment [13].To apply the graphical method to the MPH geometry, the co-

ordinate system in Fig. 2 will need to be defined for each pin-hole in the MPH collimator, and this will result in seven un-known parameters for each pinhole. We first explain that thereare common parameters shared by all pinholes and is one ofthem. This allows us to apply the graphical estimation proce-dure to each pinhole and average the results.Our numerical evaluations proceed in two steps. First we

study the accuracy and precision of the estimated in a singlepinhole SPECT simulation study. The imaging parameters aremodeled after our standard small animal SPECT studies. Wethen use one experimental study to further demonstrate the ap-plication of our method in a MPH SPECT system. In all numer-ical studies, we adhere to the steps described in Section II. Thetwo groups of tangent lines, and , are calculatedfrom the fitted ellipse expressions. Their “common” intersec-tion points, namely and , may not exist due to noise, in

Fig. 7. Five-pinhole collimator. The parameters of pinhole #2 was use in thesingle pinhole SPECT computer simulations. This collimator was also used inthe experimental evaluations. The point source calibration data of pinholes # 2,3, 4 are shown in Fig. 9.

which case we approximate them by least squares method as in[6].

A. Multipinhole Geometry

As shown in Fig. 7, in a MPH collimator design, multipleholes are drilled in a tungsten or lead plate that is mounted on theSPECT detector. The multiple pinholes share a common axis ofrotation during data acquisition. Since the focal point definesthe axis in Fig. 2, each pinhole defines a different objectspace coordinate system. If we choose one of the pinholes and itsassociated coordinate system definition as the “reference,” anddenote the reference focal point by , and its seven-parameterset by , , , , , , , then the objectspace coordinate system from another pinhole is related by atranslation in the axis 2 and a rotation around the axis

(12)

where the rotation angle depends on (1) the radius of ro-tation of the reference pinhole and (2) the pinhole posi-tion offset between the th pinhole and the reference pinhole,

.By concatenating (1) with (12), we see immediately that the

same detector shared by all pinholes have orientation angles, , that only differ in . In other words, we have

and for all pinholes . For the otherparameters, apparently for ,

and the difference can be determined from the pin-hole pattern. Depending on the MPH design, the focal length

of different pinholes may not be the same. For the interestof this work, we only need to know that all pinholes in a MPHcollimator have the same in-plane rotation .

B. Simulation Studies in Single-Pinhole SPECT

For the single-pinhole simulations, the data acquisition pa-rameters are summarized in Table I. These parameters wereselected to cover the typical range of parameter values in ourfive-pinhole collimator [13]. The SPECT detector had 80 80pixels with pixel size of 1.6 mm. The nominal pinhole colli-mator focal length was 50 mm for all pinholes. The radiusof rotation for the central pinhole (#3) was around 25 mm. The

2This translation is irrelevant to our discussion and is not mentioned further.


TABLE IPARAMETERS USED IN COMPUTER SIMULATIONS

TABLE IIENSEMBLE MEAN AND STANDARD DEVIATION OF THE ESTIMATEDAT DIFFERENT , VALUES. THE TRUE VALUE IS

detector slant angle varies from 0 (pinhole #3), to(pinhole #1, 5), and (pinhole #2, 4).We generated ideal point projection data of two point sources

over a full 360 data acquisition range, i.e., ignoring the finitepinhole aperture size, the point source dimension, and the finitedistribution of pinhole point response functions. The effects ofthese finite dimensions on calibration will be studied shortly.These ideal point projection locations were then perturbed byGaussian distributed random errors with standard deviation 1/10of a pixel size (0.16 mm). The perturbed projection locations,i.e., , , 2, and , were used toestimate . The random perturbations, to some extent, couldaccount for the nonideal estimation of the projection centroidlocations in experiments.The simulation studies were performed using different true

values of , , , and different projection view numbersTable I. For each parameter setting, we used 50 independentnoise realizations to generate and to calculate the en-semble mean and standard deviation of the estimated .Table II is the ensemble mean and standard deviation of the

estimated at different and values using projec-tion views. We observe that the bias and standard deviation donot change much with and , and should be related to the per-turbation noise (1/10 pixel size), and the number of projectionviews .When the projection noise is lower or there are more pro-

jection views in the data acquisition, the ellipse fitting can beperformed to a better accuracy which improves the subsequentestimate. When is small, all horizontal tangent lines

are nearly parallel to each other and approaches infinity. Theestimate in this case still remains rather stable. One explana-

tion is that may be subject to large variations, the resultingline (that has slope ) should nevertheless be almostparallel to those lines , the variation of is bounded bythe slope variations of .Table III is the standard deviation of the estimated at, , and at different values and using different num-

bers of projection views for calibration. For each value, thestandard deviation of remains stationary regardless of value.When more projection views are available , the standard

TABLE IIISTANDARD DEVIATION OF THE ESTIMATED AT VARIOUS

VALUE AND PROJECTION VIEW NUMBERS

deviation of decreases as the estimation of the ellipses, theellipse-line intersection and tangent points, all improve in accu-racy.In the previous simulations, the effect of noise was modeled

by perturbing the ideal point projection (the center projection)by Gaussian noise. In experimental calibration scans, the pointobject is usually approximated by small metal ball bearings forX-ray CT or point sources in SPECT. The finite size of thesenonideal “point” objects generates a finite projection image. Asthe center projection is difficult to estimate directly, the centroidof the projection image or the center of the projection image’selliptical support are usually used as substitutes [14]. Either sub-stitute causes a systematic bias in the estimate of the center pro-jection [14]–[16]. Whether this bias is negligible in the downstream geometric parameter estimation depends on the dimen-sion of these “point” objects and the cone-beam imaging geom-etry.Calibration scans on our small animal SPECT scanner are

routinely performed using standard Co57 point sources of1 mm diameter. The pinhole aperture size of the scanner variesfrom 0.5 mm diameter for the high-resolution collimator to1.3 mm diameter for the low resolution collimator. Given thedefault magnification ( and 25:50) under the twoimaging conditions, the effects of the point source dimensionand the pinhole aperture size can be lumped together as

mm for the highresolution and

mm for the low resolution MPH collimator.We consider the effect of a finite spherical object of radius

mm by generating the elliptical support of its projectionusing a typical calibration scan configuration and in the defaultcone-beam imaging geometry. The point object is located 5 mmaway from the rotational axis and is 10.5 mm above the pinholeaperture plane. At each projection view, the ellipse centerwas extracted from the elliptical support, and the offset betweenthe ellipse center and the center projection was numer-ically calculated.We compared our numerical evaluation with the analytical

expression in [14], where , , are, respec-tively, the area, the eccentricity, and the semi-short axis of theelliptical support. The offset distance is plotted in Fig. 8 ateach projection view. We notice the maximum is around0.0515 mm, less than 4% of the pixel size (1.6 mm). The othersubstitute for , the projection centroid is located betweenand but 20% closer to [14]. Therefore, the 4% bias can beconsidered an upper bound. If the estimation of ellipse center orthe projection centroid can be achieved to within 10% of a pixel[16], the 4% bias on top of the 10% random errors is arguablynegligible.


Fig. 8. The solid line is the numerical evaluation of the offset distanceat each projection. The “ ” marks the analytical calculation using

in [14]. The maximum offset is around 0.0515 mm,less than 4% of the pixel size (1.6 mm). The horizontal axis is the view index.The vertical axis is distance in millimeters.

C. A Multipinhole SPECT Experiment

We also applied the proposed graphical method to one exper-imental MPH SPECT data set. A five-pinhole collimator sim-ilar to Fig. 7 was used for data acquisition. A geometrical cali-bration scan was acquired using two point sources and 64 pro-jection views over 360 acquisition range. The data acquisi-tion geometry was similar to that in Table I. From the mechan-ical drawing of the MPH collimator, we have the nominal focallength mm, and radius of rotation mm forthe central “reference” pinhole. The common detector tilt angleis zero, and the other out-of-plane detector angle for the indi-vidual pinholes are 0 (pinhole #3), (pinhole #1, 5), and

(pinhole #2, 4). The mechanical construction error inthe manufacturing process is expected to be less than for allthese angles. The SPECT detector had 80 80 pixels with pixelsize 1.6 mm. The point source projection locationsfor , 2, , and were obtainedafter point projection segmentation and centroid calculation.For each pinhole , , an estimated was ob-

tained using the graphical procedure and the point projection

ensembles . Due to projection data noise and the fi-nite extent of the pinhole point spread function, the “common”intersection points, and , were approximated by leastsquares solutions as in the simulation studies.Fig. 9 is an illustration of the graphical procedure applied to

each pinhole. Only pinholes #2, 3, 4 are shown to avoid toomuch clutter. The two point sources used for calibration are rep-resented by ellipses of different colors, blue and green. Notethat each pinhole projects the same rotation axis to dif-ferent positions on the detector; these projected rotational axesare the prominent vertical lines in Fig. 9. The pinhole indices aremarked next to their corresponding rotational axis. The red linesthat extend from to (outside the detector surface) arethe intersection of the detector surface with the - planes de-fined for each pinhole . Their slopes should all be inthe ideal noise-free case. Using the experimental data set, the es-timated from the individual pinhole are 1.7 , 2.3 , 2.0 , 1.8 ,

Fig. 9. Applying the graphical procedure to a five-pinhole SPECT experi-mental acquisition. Results are only drawn for pinholes #2, 3, and 4. The twopoint sources are represented by ellipses of different line types. The pinholeindices are those in Fig. 7. The slope of the short thick lines should all be

. The units of the vertical and horizontal axes are in millimeters.

2.5 for pinholes 1–5, respectively. We may use their mean andstandard deviation as our final result, which is .The variations of from the different pinholes seem quite

large. We suspect two factors may contribute to this. One is non-ideal point source locations. The 5-ph experimental data wereacquired for different purposes. We observe from Fig. 9 thatsome ellipses are very elongated. This happened when the pointsources were positioned very close to the pinhole aperture plane.These elongated ellipses may affect the calculation accuracy ofline-ellipse intersections or line-ellipse tangency.A second factor could be our numerical implementation. For

the central pinhole (#3), the detector slant angle should beclose to zero. In this case, the converging point approachesinfinity and its estimation stability highly depends on the numer-ical implementation method. Our current implementation maynot be optimal for the experimental imaging geometry and thenoise level.

IV. DISCUSSIONS

The observation that the in-plane rotation angle can be esti-mated independently of all other unknown geometrical param-eters was also made in [10]. Their method is an analytical oneand only requires a minimum of one point object to estimate .Ourmethod, however, requires at least two point objects in orderto determine the rotational axis, a preliminary step for finding. Only in the rare case when the elliptical trace of point pro-jection locations degrades to a straight line, one point object issufficient.On the other hand, our proposed method can effectively deal

with projection data truncation and short scans such as using. The method in [10] utilizes the peri-

odicity of the projected point source locations and hence re-quires a full 360 data acquisition. Otherwise the needed Fouriercoefficients can only be approximated and will affect the ac-curacy of the resulting . If we examine the graphical proce-dure, the dependency on the data acquisition range is only inthe ellipse fitting and the determination of the rotational axis.


The individual point projection locations are not used after-wards. As long as a minimum of three “radial pairs” [6], [9](six views) of point projection can be located within thefan angle data acquisition,3 our graphical procedure can be car-ried out and remains exact in the ideal noise-free case.We have assumed a rotational CT system with a perfect cir-

cular acquisition geometry throughout our analysis. This geom-etry is commonly used in small animal SPECT and microCTsystems. On some pinhole SPECT systems that utilize a clinicaldetector with a custom-made pinhole collimator, it has been re-ported [17]–[19] that the scan geometry deviates from a perfectcircle, and correcting these deviations on a view-by-view basisis important to achieve the designed resolution [20]. For dedi-cated small animal pinhole SPECT systems that we mostly con-cern, the detectors are smaller; and very often multiple heads,e.g., 2 or 4, are used so the weight balancing is better. We donot observe the scan trajectory significantly departs from a per-fect circle on our system.The applicability of our method is not limited to the conven-

tional circular scan systems. For other flat-panel CT systemssuch as C-arm systems, themechanical stability is still a concernand it is preferred that the geometric calibration is performedat each projection view angle [21]. In this case, there are moreparameters to determine and the calibration procedure usuallyinvolves a more sophisticated phantom that has more point ob-jects in specifically arranged patterns [11], [21], [22]. If circularpoint object patterns are used in such calibration phantoms [11],our approach may be adapted and applied to C-arm system cal-ibration at each view angle.One implication of our work is that the point source calibra-

tion data contain rich geometrical information that can be po-tentially explored, and what we have employed in this work isonly a small amount. For example, the coordinates of the van-ishing points and (adding a superscript in the caseof multipinhole SPECT) are functions of the unknown geomet-rical parameters and their expressions can be easily derived.Moreover, in the MPH SPECT case [cf. Fig. 9], the projectedrotational axes through different pinholes also converge to acommon point ( the vanishing point due to ) whosecoordinate also reveals information about the unknown geomet-rical parameters. These coordinates, especially and , mayvary significantly even by a slight perturbation of the projectioncentroid locations. This effect may be a contributing factor in therelatively large standard deviation in estimates compared withother such measures in the literature [7]. Other factors may in-clude the magnification, the imaging geometry, and the SPECTdetector resolution.

V. CONCLUSION

We have proposed a graphical method of estimating thein-plane rotation angle in the circular cone-beam geom-etry. The method can be applied to flat-panel X-ray CTimaging, single pinhole, and multipinhole SPECT systems asdemonstrated by our computer simulations and experimentalevaluation.

3An example configuration would be three views located within 0 —fanangle, and their 180 partners are on – fan angle.

The unique nature of this work separates itself from other ana-lytical or iterative geometrical calibration methods. The noveltyof our method lies in its graphical procedure. We rely heavily onproperties of perspective geometry and identify special lines inthe cone-beam projection that can be associated with the un-known . Our method is exact in the ideal noise-free case re-gardless of other unknown geometrical parameter values; in par-ticular, the other two out-of-plane rotation angles are not as-sumed to be zero or small. After determining , the projectiondata can be correspondingly rotated [6] so that subsequent cali-bration, by either analytical or iterative methods, can proceedassuming , which will simplify the calibration proce-dure. More importantly, we are extending this work and usingthe established correspondence between the circles, the circum-scribing squares, the tangent points in the object space and theircorresponding projection locations to obtain all other six geo-metric parameters.

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