Auto calibration of a cone-beam-CTDaniel Grossa) and Ulrich HeilDepartment of Design, Computer Science and Media, RheinMain University of Applied Sciences,65195 Wiesbaden, Germany and Institute of Computer Science, Johannes Gutenberg University Mainz,55128 Mainz, Germany

Ralf SchulzeDepartment of Oral Surgery (and Oral Radiology), University Medical Center of the Johannes GutenbergUniversity Mainz, 55131 Mainz, Germany

Elmar SchoemerInstitute of Computer Science, Johannes Gutenberg University Mainz, 55128 Mainz, Germany

Ulrich SchwaneckeDepartment of Design, Computer Science and Media, RheinMain University of Applied Sciences,65195 Wiesbaden, Germany

(Received 19 July 2011; revised 9 July 2012; accepted for publication 11 July 2012; published 12September 2012)

Purpose: This paper introduces a novel autocalibration method for cone-beam-CTs (CBCT) or flat-panel CTs, assuming a perfect rotation. The method is based on ellipse-fitting. Autocalibration refersto accurate recovery of the geometric alignment of a CBCT device from projection images alone,without any manual measurements.Methods: The authors use test objects containing small arbitrarily positioned radio-opaque markers.No information regarding the relative positions of the markers is used. In practice, the authors usethree to eight metal ball bearings (diameter of 1 mm), e.g., positioned roughly in a vertical line suchthat their projection image curves on the detector preferably form large ellipses over the circularorbit. From this ellipse-to-curve mapping and also from its inversion the authors derive an explicitformula. Nonlinear optimization based on this mapping enables them to determine the six relevantparameters of the system up to the device rotation angle, which is sufficient to define the geometryof a CBCT-machine assuming a perfect rotational movement. These parameters also include out-of-plane rotations. The authors evaluate their method by simulation based on data used in two similarapproaches [L. Smekal, M. Kachelriess, S. E, and K. Wa, “Geometric misalignment and calibrationin cone-beam tomography,” Med. Phys. 31(12), 3242–3266 (2004); K. Yang, A. L. C. Kwan, D. F.Miller, and J. M. Boone, “A geometric calibration method for cone beam CT systems,” Med. Phys.33(6), 1695–1706 (2006)]. This allows a direct comparison of accuracy. Furthermore, the authorspresent real-world 3D reconstructions of a dry human spine segment and an electronic device. Thereconstructions were computed from projections taken with a commercial dental CBCT device havingtwo different focus-to-detector distances that were both calibrated with their method. The authorscompare their reconstruction with a reconstruction computed by the manufacturer of the CBCT deviceto demonstrate the achievable spatial resolution of their calibration procedure.Results: Compared to the results published in the most closely related work [K. Yang, A. L. C. Kwan,D. F. Miller, and J. M. Boone, “A geometric calibration method for cone beam CT systems,” Med.Phys. 33(6), 1695–1706 (2006)], the simulation proved the greater accuracy of their method, as wellas a lower standard deviation of roughly 1 order of magnitude. When compared to another similarapproach [L. Smekal, M. Kachelriess, S. E, and K. Wa, “Geometric misalignment and calibration incone-beam tomography,” Med. Phys. 31(12), 3242–3266 (2004)], their results were roughly of thesame order of accuracy. Their analysis revealed that the method is capable of sufficiently calibratingout-of-plane angles in cases of larger cone angles when neglecting these angles negatively affects thereconstruction. Fine details in the 3D reconstruction of the spine segment and an electronic deviceindicate a high geometric calibration accuracy and the capability to produce state-of-the-art recon-structions.Conclusions: The method introduced here makes no requirements on the accuracy of the test ob-ject. In contrast to many previous autocalibration methods their approach also includes out-of-planerotations of the detector. Although assuming a perfect rotation, the method seems to be sufficientlyaccurate for a commercial CBCT scanner. For devices which require higher dimensional geometrymodels, the method could be used as a initial calibration procedure. © 2012 American Association ofPhysicists in Medicine. [http://dx.doi.org/10.1118/1.4739247]

Key words: CBCT, cone-beam, flat panel, tomography, calibration, geometry, reconstruction

5959 Med. Phys. 39 (10), October 2012 © 2012 Am. Assoc. Phys. Med. 59590094-2405/2012/39(10)/5959/12/$30.00

5960 Gross et al.: Auto calibration of a cone-beam-CT 5960

I. INTRODUCTION

Flat-panel cone-beam CTs (CBCTs) are widely used inmedicine for three-dimensional reconstructions, but also haveapplications in industry and science. The data basis is formedby a large number of x-ray projection images which are uni-formly distributed around the object of interest. In most casesthere is a rotating C-arm composed of an x-ray tube and aflat rectangular detector. To permit a volumetric reconstruc-tion the precise knowledge of the geometric alignment of thedetector and the x-ray tube in relation to the rotational axisis an indispensable precondition. Otherwise various artifactscan be observed. We simplify our assumptions by restrictingthe movement to a true circular rotation (360◦) without grav-itational or other technical disturbance. While this may seemunrealistic in a real-world set-up, we will show that our devicemodel is appropriate for a commercial dental CBCT-scannerwhich we tested. Additionally, we assume mechanical stabil-ity of the device in the sense that geometry parameters do notchange between several scans. Since no metric information isrequired for the reconstruction alone, we also neglect spatialscaling of the reconstructed volume.

There are eight parameters that define a CBCT-devicewhich rotates about a fixed axis. These are, the distance ofthe focus (i.e., the x-ray source) to the rotational axis (fod),the pixel size (ps), the position of the detector (ox, oy, oz), andits orientation (φ, σ , ψ), both relative to a cartesian coordi-nate system defined by the rotational axis and the focus. Tobe precise, we consider the orientation of the detector as threeEuler-angles which correspond to an in-plane rotation ψ , alsoknown as skew, and two out-of-plane angles φ and σ , alsoknown as tilt and slant, respectively. From these parametersthe focus-to-detector distance (fdd) can be derived as the sumof fod and oz.

To provide a reconstruction up to spatial scaling we willshow in the first section that it is sufficient to know exactlysix “ratios” of these parameters. In this paper we present anautocalibration method which can determine these six param-eters (=ratios) needed for volume reconstruction, assuminga perfect rotation of the device. To clarify this, Rizo et al.showed that for a perfect circular rotational movement sevenparameters are sufficient to calibrate a CBCT scanner whichcorrespond to the just addressed six ratios plus the measuringunit inducing a spatial scaling of the reconstructed volume.1

We use a test object containing several dot like metal mark-ers. During image acquisition the circular trajectory of themarkers around the rotational axis will be projected to el-lipses. Our method relies on the fact that the geometric con-figuration of the CBCT device needed for reconstruction canbe identified from these ellipses alone. It must be emphasizedthat no information about the relative position of the markersis required, nor are manual measurements required, and eventhe real pixel size can be neglected. We formulate the problemas a nonlinear optimization problem based on the geometricrestrictions described above and the ellipse parameters as in-put data and solve it iteratively.

There are numerous methods present in literature for cal-ibrating a CBCT with different restrictions on the device

geometry or the calibration phantom. Only some take out-of-plane rotations into account.1–8 Common to all of thesemethods is the need for a priori information about the usedcalibration phantom. It has been demonstrated that very pre-cisely fabricated phantoms allow for robust calibration evenof in-lab C-arm devices yielding a high spatial resolution ofthe reconstruction.7 An analytic method based on ellipse pa-rameters obtained from two small ball-shaped calibration ob-jects for circular rotation orbits has been introduced by Nooet al.9 Their direct method, however, requires the detector tobe oriented parallel to the rotation axis and hence only ac-commodates one of two out-of-plane rotations (here denotedby σ , θ ). Other approaches in Refs. 3, 4, 8, 10–15 rely onthe knowledge of the relative positions of pointlike markerswithin the phantom object. Although using different solvingstrategies, the calibration parameters are determined basicallyby comparing expected positions of the markers with thoseobserved. For example, Beque et al.10 match estimated andobserved projected positions of point sources by minimizinga suitable quadratic term. While such methods usually allowsomeone to calibrate each projection separately, the calibra-tion accuracy is additionally limited by knowledge of the pre-cise adjustment of the pointlike markers. Other methods takemore sophisticated phantoms into account, for example, twometal rings of known radius and distance,5 several steel ballbearings in two plane-parallel circles,11, 12 or a special tubu-lar object container.2 The method of Yang et al.,16 which isone of the most closely related to the one presented here, alsouses a phantom consisting of arbitrarily positioned markers,the relative positions of which may be unknown. Yang et al.only need a rough measurement of the distance between twoof the markers to adjust the focus-to-object distance, but incontrast to our method they assumed out-of-plane rotations tobe negligible. Very similar to our approach, Smekal et al. alsoconsider out-of-plane angles, and do not utilize knowledgeabout the marker positions in their phantom.15 They provevery accurate results based on analytic expressions derivedfrom Fourier analysis of the projection-orbit of the point-markers.15 For the micro-CT geometry they consider for theirsimulation, they also show that out-of-plane (tilt and slant)angles of realistic magnitude have very minor effects on re-construction accuracy.

We present a novel calibration method for CBCT-devices,that uses a phantom of pointlike metal markers. No knowl-edge of their position is required. Under a few assumptionsthat seem to sufficiently approximate real-world CBCT scan-ner geometries, the advantage of this novel approach lies in itssimplicity. Although no metric information about the phan-tom has to be known, all parameters can be determined withan accuracy that is roughly 1 order of magnitude higher thanthat obtained by the method proposed by Yang et al.16 Com-pared to the paper of Smekal et al.,15 our results are slightlyless accurate. We will demonstrate, that for real-world den-tal CBCT machines implementing larger cone-angles out-of-plane angles of realistic magnitude do negatively affect thecalibration when neglected.

In Sec. II, we introduce the device model followed by asection with the main concept and the theory of the approach.

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5961 Gross et al.: Auto calibration of a cone-beam-CT 5961

((

(( ((

((

FIG. 1. Our device model.

Sec IV provides implementation details. Finally, the methodis evaluated by some analytical considerations and simula-tions using data from the literature to allow for comparability.Real-world reconstructions from a commercial dental CBCT-machine calibrated with the method are also included.

II. THE DEVICE MODEL

Our device model (see Fig. 1) is based on the followingthree assumptions: First, the x-ray source and the detectorhave to be statically coupled to one another, such that bothhave a static position and orientation in a common local coor-dinate frame. Often, this part of the device is called the C-armof the CBCT. Second, we assume a flat-panel detector.Finally, the focus-detector-unit rotates about an arbitraryaxis during image acquisition. Note that this axis does nothave to be aligned with the detector in any special way.These assumptions are nearly fulfilled for many dental C-arm CBCTs or micro CTs. Despite this simplification, wewill show that our model allows us to generate state-of-the-artreconstructions.

The assumptions made above lead to a device model withfewer degrees of freedom compared to the general case, whereeach projection is calibrated individually. In fact, a rotationalCBCT can be described by a set of eight parameters

qreal = [φ, σ,ψ, fod, ps, ox, oy, oz], (1)

which are the distance of the focus to the rotational axis (fod),the pixel-size (ps), the position of the detector (ox, oy, oz),and its orientation (φ, σ , ψ). Assuming a right-handed co-ordinate system in which the rotational axis corresponds tothe y-axis and the focus is placed on the negative z-axis theseeight parameters apply as follows. The focus f of the system isat f = (0, 0,−fod)T . Then, the local coordinate frame of the

detector is given by a corner o = (ox, oy, oz)T (the position ofthe detector) and two adjacent vectors

dx = ps R

⎛⎜⎝ 1

0

0

⎞⎟⎠ dy = ps R

⎛⎜⎝ 0

1

0

⎞⎟⎠ (2)

with

R = Rx(φ) Ry(σ ) Rz(ψ), (3)

which describes the orientation of the detector. Here, Rx(φ),Ry(σ ), Rz(ψ) are rotation matrices about the x-, y-, and z-axes, respectively. From Eqs. (2) and (3) it follows that bothvectors dx and dy are orthogonal to each other with the lengthof one pixel.

To introduce a projection matrix framework, the vectorsdx, dy, o, f which represent the complete geometry informa-tion can be combined into the homogeneous calibration ma-trix D ∈ R4×4 given by

D =[

dx dy f o

0 0 1 1

]=

⎛⎜⎜⎜⎝dx

x dyx 0 ox

dxy d

yy 0 oy

dxz d

yz −fod oz

0 0 1 1

⎞⎟⎟⎟⎠. (4)

From D the projection matrix P ∈ R3×4 can be derived,which projects a point in real-world coordinates onto the de-tector given by o, dx, and dy and provides the point within thedetector’s local coordinate system. It is given as

P = ZD−1 with Z =

⎛⎜⎝ 1 0 0 0

0 1 0 0

0 0 0 1

⎞⎟⎠. (5)

Here, Z is a simple orthogonal projection in z-direction.During image acquisition the focus-detector-unit rotates

about the y-axis. If we assume that at a fixed time t, i.e., imagenumber t, the device is rotated about the angle α(t) then the fo-

cus has the position Rα(t)(f1 ) and the detector unit is given by

Rα(t)(o1 ), Rα(t)(

dx

1 ), and Rα(t)(dy

1 ). Thereby, Rα ∈ R4×4 per-

forms a simple rotation about the y-axis through the angle α ina mathematically positive sense. As a consequence the detec-tor matrix Dt , as well as the corresponding projection matrixPt at time t are given by

Dt = Rα(t)D and Pt = ZD−1t = ZD−1RT

α(t). (6)

Now the projection of a fixed point x ∈ R4 at time t is givenby

pt = Pt x = ZD−1RTα(t)x . (7)

II.A. Normalizing the geometry

From projection images themselves, the eight parametersof qreal defining the vectors dx, dy, o, f are only determinablewith two ambiguities which lead to a system of equivalentgeometry configurations with six degrees of freedom (DOFs).

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5962 Gross et al.: Auto calibration of a cone-beam-CT 5962

First, we can freely choose the unit in which we measure thelast five parameters of qreal as in Eq. (1). Thus, a change ofthe measurement unit corresponds to a uniform scaling

q = [q1, q2, q3, q4, q5, q6, q7, q8]

�→Uγ

[q1, q2, q3, γ q4, γ q5, γ q6, γ q7, γ q8]. (8)

While scaling the geometry of the system the spatial size ofthe reconstructed volume also gets scaled with the same fac-tor γ . Vice versa, if one knows the real distance of two re-constructed points one can easily compute this scaling factor.Reconstructions provided by q and Uγ (q) are equal exceptfor spatial scaling. Second, since we can only observe 2D-projections of 3D-points one can freely scale the size of thedetector and its distance with respect to the position of thefocus. More exactly this transformation Sμ looks like

q = [q1, q2, q3, q4, q5, q6, q7, q8]

�→Sμ

[q1, q2, q3, q4, μq5, μq6, μq7, μq8 + (μ − 1)q4].

(9)

Thereby the position of the focus and the size of volumeremain fixed. Reconstructions provided by q and Sμ(q) areidentical.

If we apply these transformations to the original geometryconfiguration qreal with γ = 1/fod and μ = 2 fod/(oz + fod)we get a normalized version qnorm of the geometry:

qreal = [φ, σ,ψ, fod, ps, ox, oy, oz]

�→Sμ◦Uγ

[φ, σ,ψ, 1,

2 ps

oz + fod,

2 ox

oz + fod,

2 oy

oz + fod, 1

]

= qnorm. (10)

Two parameters (qnorm4 , qnorm

8 ) of the normalized geometryqnorm are fixed to one. The remaining six entries, respectively,ratios of the original geometry, can be determined just fromthe projection images without any a priori information.

Now, if we form the vectors dxnorm, dy

norm, onorm, fnorm de-fined by qnorm in analogy to Eqs. (1)–(3) and combine theseinto a normalized calibration matrix Dnorm it holds that

Dnorm =[

dxnorm dy

norm fnorm onorm

0 0 1 1

]

=[

μγ dx μγ dy γ f μγ o + (1 − μ)γ f

0 0 1 1

](11)

=

⎛⎜⎜⎝γ 0 0 00 γ 0 00 0 γ 00 0 0 1

⎞⎟⎟⎠[dx dy f o0 0 1 1

]⎛⎜⎜⎝μ 0 0 00 μ 0 00 0 1 (1 − μ)0 0 0 μ

⎞⎟⎟⎠(12)

with γ = 1/fod and μ = 2 fod/(oz + fod). For the normal-ized projection matrix Pnorm it holds that

Pnorm = Z D−1norm

= Z

⎛⎜⎜⎜⎝1μ

0 0 00 1

μ0 0

0 0 1 μ−1μ

0 0 0 1μ

⎞⎟⎟⎟⎠D−1

⎛⎜⎜⎜⎝1γ

0 0 00 1

γ0 0

0 0 1γ

00 0 0 1

⎞⎟⎟⎟⎠ (13)

= 1

μZD−1

⎛⎜⎜⎜⎝1γ

0 0 00 1

γ0 0

0 0 1γ

00 0 0 1

⎞⎟⎟⎟⎠ � ZD−1

⎛⎜⎜⎜⎝1γ

0 0 00 1

γ0 0

0 0 1γ

00 0 0 1

⎞⎟⎟⎟⎠(14)

from which one can easily see that the projection matricesbelonging to qreal and qnorm only differ in a uniform spatialscaling of the volume space. The symbol � means projectiveequivalence.

To summarize, the normalized geometry qnorm has only sixDOFs, while providing a reconstruction which only differs ina spatial scaling compared to a reconstruction based on thereal geometry qreal. As demonstrated in this paragraph, it suf-fices to consider normalized CBCT geometries for both cali-bration and reconstruction without loss of generality. Hence,in the rest of this paper, we omit the superscript (·)norm and as-sume a normalized geometry with a focus-to-object distanceof fod ≡ 1 and a z-translation of oz ≡ 1 which correspond tothe entries q4 and q8 of the geometry-defining parameter vec-tor q [compare with equation (1)].

III. MATHEMATICAL MODEL

In the following, we present all theoretical aspects of thecalibration method. Section III.A gives an overview followedby a detailed derivation in Secs. III.B–III.C.

III.A. Main concept

From Eq. (7) it is easy to see that the projection-curve of afixed point in dependency of time t is identical to the projec-tion of an orbit around the y-axis at time t = 0. As a conse-quence we can drop the time component by considering cir-cular orbits around the y-axis (called y-orbit in the following)instead of single points.

These y-orbits get projected to ellipses. In Sec. III.B wewill describe how to obtain these ellipses. Our approach isbased on the fact that the ellipses determined by the y-orbitsof the radio-opaque markers can be measured directly withinthe projections. This observation allows us to determine theunknown calibration matrix D and the y-orbits of the markers.In the following, we represent the ellipses as homogeneoussymmetric matrices Ci ∈ R3×3, i = 1, . . . , n. An observedellipse Ci depends on the geometric configuration representedby the calibration matrix D, the radius ri, and the height hi ofthe y-orbit of a marker.

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5963 Gross et al.: Auto calibration of a cone-beam-CT 5963

The main contribution of this paper is a decomposition ofthe conic section equation describing the ellipses which al-lows for a direct computation of the pair (ri, hi) when Ci andD are given. More precisely, if one assumes a fixed calibra-tion matrix D there is a bijection, mapping y-orbits definedby (ri, hi) onto observable ellipses Ci in the image domain.We derive an explicit formula for this bijective mapping andmuch more important for its inversion. This explicit formulawill be used to reduce the complexity (6 variables instead of6 + 2n) of our optimization algorithm which determines theCBCT geometry.

In Secs. III.B and III.C we prove that the resulting problemcan be stated as follows: Given n ellipses Ci ∈ R3×3, i = 1,. . . , n measured from the projection images. Find a homogra-phy (i.e., a bijective projective mapping)

G =

⎛⎜⎝ dxx d

yx ox

dxy d

yy oy

12dx

z12d

yz 1

⎞⎟⎠ (15)

between the real detector plane and a canonical detector planesuch that for some arbitrary (ri, hi) ∈ (R+,R) defining thecanonical ellipses

Cic =

⎛⎜⎜⎝1 0 0

0 1−r2i

h2i

− 2hi

0 − 2hi

4

⎞⎟⎟⎠ (16)

the following equation holds

Ci � GT CicG, i = 1, . . . , n. (17)

This simple algebraic representation of the relationship ofCBCT-geometry, y-orbit and observed ellipse can be achievedby temporarily adding a canonical detector plane (given by thematrix Dc, see Sec. III.C) and afterward mapping the canon-ical detector to the real detector (see Fig. 2). As mentionedpreviously Eq. (17) can be solved explicitly for (ri, hi). Thematrix G contains the complete geometric information of theCBCT required for reconstruction.

FIG. 2. Decomposition of the whole projective mapping into two steps: 1.Mapping of all input orbits onto a canonically defined common plane. 2.The homography G transforming the fixed canonical plane onto the arbitrarydetector plane.

III.B. Identifying the trajectory of a fixed point with aconic section

Let x ∈ R4 be a homogeneous representation of a point onan orbit with radius r and height h. Then x has the representa-tion

x = ( rx1, h, rx2, 1 )T

(18)

with x21 + x2

2 = 1. Now define W ∈ R4×3 and y ∈ R3 with

W =

⎛⎜⎜⎜⎜⎝r 0 0

0 0 h

0 r 0

0 0 1

⎞⎟⎟⎟⎟⎠ and y =

⎛⎜⎝x1

x2

1

⎞⎟⎠. (19)

That means

x = Wy (20)

with y being a point on the two-dimensional unit circle. Theprojection P maps the point x to image coordinates

p = Px = PWy. (21)

Note that both P ∈ R3×4 and W ∈ R4×3 are not square matri-ces, in contrast to PW which is square and invertible (exceptfor unrealistic detector geometries).

Since y is a point on a unit circle the following conic sec-tion equation holds

0 = yT Ky with K =

⎛⎜⎝ 1 0 0

0 1 0

0 0 −1

⎞⎟⎠. (22)

With y = (PW)−1

p we derive the following equation for theimage point p :

0 = yT Ky = pT Cp, (23)

where

C = (PW)−T

K(PW)−1

. (24)

In summary, this means that, through the perspective projec-tion, the orbit with radius r and height h around the y-axiswill be mapped to the conic section C. In our case these conicsections are ellipses. Furthermore with Eq. (5) we find

C = (ZD−1W)−T

K(ZD−1W)−1

(25)

with nonsquare matrices Z ∈ R3×4, W ∈ R4×3 and squarematrices D ∈ R4×4, K ∈ R3×3.

III.C. Decomposition of the conic section equation

Let us define a canonical calibration matrix Dc ∈ R4×4

with

Dc =

⎛⎜⎜⎜⎜⎝1 0 0 0

0 1 0 0

0 0 −1 1

0 0 1 1

⎞⎟⎟⎟⎟⎠ (26)

and a projective mapping G′ ∈ R4×4 with

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5964 Gross et al.: Auto calibration of a cone-beam-CT 5964

G′ =

⎛⎜⎜⎜⎜⎝dx

x dyx 0 ox

dxy d

yy 0 oy

− 12dx

z − 12d

yz 1 0

12dx

z12d

yz 0 1

⎞⎟⎟⎟⎟⎠. (27)

Then D = DcG′ and the conic section equation (25) can bedecomposed into

C = (ZG′−1D−1

c W)−T

K(ZG′−1D−1

c W)−1

. (28)

Using Eq. (15) and the fact that ZG′−1 = G−1Z we get

C = (G−1ZD−1c W)

−TK(G−1ZD−1

c W)−1

. (29)

Since G is square and invertible it can be factored out and itfollows that

C = GT (ZD−1c W)

−TK(ZD−1

c W)−1

G. (30)

With

Cc = (ZD−1c W)

−TK(ZD−1

c W)−1

, (31)

Eq. (30) simplifies to

C = GT CcG. (32)

Note that Cc only depends on the radius and the height of theorbit. In particular, it is independent of the device geometry.Substituting Eqs. (5), (19), (22) and (26) into Eq. (31) we getthe explicit representation

Cc = 1

r2

⎛⎜⎝ 1 0 0

0 1−r2

h2 − 2h

0 − 2h

4

⎞⎟⎠ �

⎛⎜⎝ 1 0 0

0 1−r2

h2 − 2h

0 − 2h

4

⎞⎟⎠ (33)

of the canonical conic section defined by a y-orbit with highth and radius r.

IV. OPTIMIZATION PROCESS

Before solving the conic section equation (17) in the op-timization process, we have to consider some preprocessingsteps to obtain the elliptic projection trajectories Ci in thegiven x-ray images. In our approach any kind of pointlike,radio-opaque markers can be used. For all recovering process-ing steps standard methods in imaging science exist.17 We ex-tract the midpoints of our metal ball bearings using a bordersegmentation followed by an elliptical Hough transformation,both with subpixel precision. Subsequently or along the way,each trajectory can be tracked, e.g., by a Kalman filter and op-tical flow procedures.18 To fit ellipses to the point set we usea method similar to the standard approach by Fitzgibbon andco-workers.19

Given n ellipses Ci one has to find a normalized ge-ometry vector qnorm (defining the homography G) such thatEq. (17) holds for some arbitrary (ri, hi) ∈ (R+,R) defin-ing Ci

c as in Eq. (16). The implemented optimization process(compare to algorithm 1) is a simple random search in thegeometry vector qnorm (6 DOF) combined with an annealingprocess which minimizes an objective function f that will bedescribed in the next paragraph. The annealing process itself

ALGORITHM I. Local optimization process.

Input: ellipses C1, . . . , Cn, search window qmin ≤ qmax

K Number of iterations, J Number of shrinkage steps, I Number of randomsamples, δ shrinkage factorOutput: calibration vector qopt (local optimum)

qopt = qmin+qmax

2for k ← 1, . . .,K do

qcmin = qmin

qcmax = qmax

for j ← 1, . . .,J do// Random samplesfor i ← 1, . . .,I do

qr = randomVector (qcmin,qc

max)qopt = argmin

x∈{qopt,qr}f(x, C1, . . . , Cn)

end// Shrinkagew = qc

max − qcmin

qcmin = qopt − δ w

2qc

max = qopt + δ w2

// Adjusting the search window if neededif qc

min < qmin thenqc

max = qcmax + (qmin − qc

min)qc

min = qmin

endif qc

max > qmax thenqc

min = qcmin − (qc

max − qmax)qc

max = qmax

endend

endreturn qopt

is implemented by shrinking (by a factor 0 < δ < 1, J times)a boxlike search window centered around the current opti-mum qopt after a fixed number I of random samples withinthis box. To cope with local minima we restart the search Ktimes. The local optimization process is illustrated in detail inalgorithm 1.

The boxlike search window qmin ≤ q ≤ qmax (pointwise)is defined by six degrees of freedom of the normalized ge-ometry vector qnorm. Throughout this paper, we use the sameinitial conditions for all calibrations of simulated as well asreal data sets. These are

K = 100, J = 1000, I = 10, δ = 0.99, (34)

qmin = (−10◦, −10◦, −10◦, 1, 2 × 10−4, −3000 × 10−4,

−3000 × 10−4, 1), (35)

qmax = (+10◦, +10◦, +10◦, 1, 2 × 10−3, 0, 0, 1). (36)

The search window qmin ≤ q ≤ qmax is chosen such that itcovers a large class of real CBCT devices. This includes de-tector rotations up to ±10◦, a focus-to-detector distance be-tween 103 and 104 pixel ((q4 + q8)/q5) as well as detectortranslations between −1500 and 0 pixel (q6/q5 and q7/q5), in-dependent from the absolute dimensions of the device. Givena pixel spacing of 0.05 mm, as a micro-CT example, qmin

and qmax admits of feasible focus-to-detector distances rang-ing from 50 mm to 500 mm.

The global objective function f is the sum of individualobjective functions

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5965 Gross et al.: Auto calibration of a cone-beam-CT 5965

f (q, C1, . . . , Cn) =n∑

i=0

f (q, Ci) (37)

with

f (q, Ci) = match(Ci , GT

q correct(G−T

q CiG−1q

)Gq

)(38)

being the objective function for a single ellipse. The matrixGq ∈ R3×3 is uniquely defined by the geometry vector q inanalogy to Eqs. (1)–(3) and (15). Note that the matrix Gq isinvertible if and only if the focus does not lie in the detec-tor plane. This will be guaranteed by the initial search win-dow. Thereby the match(·) function is a heuristic that quanti-fies the match of two ellipses. It is implemented as the sum ofthe absolute differences of four corresponding points whichare given as the intersections of the ellipse curve with bothprincipal axes. These four points can be determined from thedefining conic section matrix C by simple algebra. The func-tion correct (C) normalizes the matrix C by division with itstop-left entry and forces the structure⎛⎝ 1 0 0

0 ∗ ∗0 ∗ 4

⎞⎠. (39)

Taking this into account, the term GTq correct

(G−Tq CG−1

q )G in Eq. (38) is an approximation for the

projection of the y-orbit (r, h) which matches the ellipse Cbest (for a fixed candidate Gq). As a consequence of the in-vertibility of Gq the matched ellipse pairs are nondegeneratedif and only if the input ellipses are nondegenerated. The realbest-fitting y-orbit is given by

argmin match(r, h)

⎛⎜⎝C, GTq

⎛⎜⎝ 1 0 0

0 1−r2

h2 − 2h

0 − 2h

4

⎞⎟⎠ Gq

⎞⎟⎠ . (40)

From Eq. (17), it follows that for the correct Gq the conicsection G−T

q CG−1q (after normalization) is of the form (39).

Consequently, we can do the correction step (and so the ap-proximation) by forcing the known components of G−T

q CG−1q

to the correct values [see also Eqs. (32) and (33) in Sec. III.C].Note that this approximation step reduces the number of vari-ables in the optimization process dramatically. The approx-imated objective function acts as an upper bound to the de-sired objective function but with the same optimum and thesame optimal value in an error-free setting. In such a casethe objective function f (q, C1, . . . , Cn) is zero for the correctgeometry q, otherwise it is greater than zero. Nevertheless,in the case of corrupted input ellipses Ci , there is no proofthat the match(·) function is optimal in the sense that the ap-proximated objective function [Eq. (38)] and the desired one[Eq. (40)] share the optimum.

V. EVALUATION

The evaluation section concentrates on the calibration ofmicro-CT systems introduced in the most closely relatedwork15, 16 as well as dental CBCT systems available to the

authors. The relevant parameters of all calibrated devicesare listed in Table I, including quantities such as magnifica-tion (fdd/fod) and cone angle. For a first CBCT geometryclassification we illustrate by Fig. 3 how an out-of-plane errorpropagates to object space with respect to the geometry. InSec. V.A we give simulation results on the accuracy of ourcalibration method. More precisely, we consider five set-tings with different constraints on the geometry, the numberof markers, and the noise in the observed positions of themarkers. In Sec. V.B we show real-world reconstructions oftwo projection data sets with different geometry configura-tions which were calibrated by the proposed method.

Figure 3(a) introduces a function h(φ) to estimate the ef-fect of an out-of-plane rotation error (tilt φ) to a point nearthe rotational axis. Here, h(φ) is the offset of the intersec-tion point of the rotational axis with a virtual ray which strikesthe detector margin. Figure 3(b) evaluates the maximum re-construction error from 0◦ to 5◦. If we assume that the voxelspacing in the reconstruction is equal to or below the pixelspacing, the intersection of the error plot with the horizontalpixel spacing line indicates when the error exceeds one voxel.This can be evaluated for each geometry. Looking at geome-try II this point is about 1◦, for geometry III between 2◦ and 3◦

and within geometry I it is above 5◦. This demonstrates thatthe influence of out-of-plane rotation errors depends to a greatextent on the device geometry. Consequently, the out-of-planeangle precision of the calibration method must be evaluatedwith respect to this influence.

V.A. Simulation

For the first simulation we take the device geometry I (seeTable I) from Smekal et al.15 The projected trajectories of12 vertically aligned markers are computed and perturbed byGaussian noise with a standard deviation of 0.01, 0.1, and0.2 pixel, respectively. We use exactly the same positionsof the markers as listed in Ref. 15 (see Table II Column 3on p. 3252). The out-of-plane rotations for this geometry areφ = −1.2◦ and σ ◦ = 1.5. Table II shows our calibration re-sults in mean and standard deviation for 1000 simulations. Itshows clearly that the input geometry can be reconstructed ifthe noise level tends to zero. For noise 0.01 the errors as wellas the standard deviations for all rotations angles (φ, σ , ψ) areless than 0.1◦. For higher noise levels the standard deviationsof the calibrated parameters increases roughly by the samefactor as the noise level increases. The highest errors and thehighest standard deviation arise in the tilt rotation angle φ forall three noise levels. In comparison with Smekal et al.15 weobtain slightly higher standard deviations. However, all meanvalues roughly equal the values in this paper.15 As alreadymentioned according to Fig. 3, the out-of-plane rotation er-rors introduced by both calibration methods are nonsevere.

Second we compare our method with a simulation of Yanget al.16 whose approach is also closely related to the onepresented here. Their approach also relies on elliptical pro-jected trajectories and estimates all calibration parametersmentioned except both out-of-plane rotations (φ, σ ). Theysimulated the projection trajectories of eight virtual metal

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5966 Gross et al.: Auto calibration of a cone-beam-CT 5966

TABLE I. Overview of all applied CBCT geometries. Our real values geometry vector qreal supplemented by some meta information as well as the normalizedgeometry vector qnorm is given.

Geometry II Geometry IIIGeometry I micro CBCT dental CBCT

Simulation Our micro CBCT (Yang et al. (Ref. 16) (J. Morita Corp.)parameters nomenclature (Smekal et al. (Ref. 15) (IIa)/(IIb) (IIIa) / (IIIb)

φ (degree) qreal1 −1.2 0/−2.5 1.16 / 1.25

σ (degree) qreal2 1.5 0/1.5 0.37 / 0.27

ψ (degree) qreal3 −1.0 −1.0 −0.26 / −0.13

fod (mm) qreal4 280 150 500

ps (mm) qreal5 0.117 0.048 0.2

ox (mm) qreal6 −28.5 −47.86 −60.0 / −58.40

oy [mm] qreal7 −28.7 −26.11 −63.9 / −62.72

oz (mm) qreal8 421.2 250 210.48 / 300

Focus-detector-d. fdd (mm) 701.2 400 710.48 / 800Detector size (mm) 60 × 60 98.3 × 49.15 116.8 × 123.2Detector size (px) 512 × 512 2048 × 1024 584 × 616Magnification fdd/fod ≈2.5 ≈2.66 ≈1.42 / ≈1.60Cone-angle (maximal) ≈4.9◦ ≈14.01◦ ≈9.84◦ / ≈8.81◦

φ qnorm1 −1.2 0 / −2.5 1.16 / 1.25

σ qnorm2 1.5 0 / 1.5 0.37 / 0.27

ψ qnorm3 −1.0 −1.0 −0.26 / −0.13

1 qnorm4 1 1 1

2 ps/(oz + fod) qnorm5 3.34 × 10−4 2.4 × 10−4 5.63 × 10−4/5.0 × 10−4

2 ox/(oz + fod) qnorm6 −8.13 × 10−2 −2.39 × 10−1 −1.69 × 10−1/−1.46 × 10−1

2 oy/(oz + fod) qnorm7 −8.19 × 10−2 −1.31 × 10−1 −1.80 × 10−1/−1.56 × 10−1

1 qnorm8 1 1 1

balls in a given CBCT geometry (see Table I, geometry IIa)and compared the output of their calibration process with theoriginal geometry. Prior to the calibration, the projected posi-tions of the makers were perturbed by Gaussian noise with astandard deviation of 0.4 pixel. We repeated exactly the sameexperiment with our new method. Since they use a slightlydifferent notation we introduce a new vector

f′ = 1

psRT (f − o) = − 1

psRT

⎛⎜⎝ ox

oy

fod + oz

⎞⎟⎠

= − 1

qnorm5

RT

⎛⎜⎝qnorm6

qnorm7

2

⎞⎟⎠ (41)

to enable direct comparison of the accuracy of both methodsin mean and standard deviation. The vector f′ is the positionof the focus f relative to the detector frame (compare withSec. II and Fig. 1]. Note that the information contained in thepair [(R, f′) is identical with the information contained in thenormalized parameter vector qnorm. A summary of the com-parison is given in Table III including all relevant simulationparameters. It shows slight improvements in all estimated val-ues (both mean and standard deviation) and a significant im-provement in the estimation of f ′

z .In the next two simulations (Table IV) we also consider

out-of-plane rotations and reduce the number of markers fromeight to four to demonstrate robustness. Furthermore, we in-crease the noise in the positions of the markers (last columnof Table IV). Table IV shows the results of 1000 simulationsand calibrations, respectively.

TABLE II. Simulation results based on the geometry I in Table I from Smekal et al. (Ref. 15) with out-of-plane rotations (φ, σ ) = ( − 1.2, 1.5). The positionsof the 12 markers were perturbed by Gaussian noise with a standard deviation of 0.01, 0.1, and 0.2 pixel. For the simulations the mean of 1000 simulations isgiven along with their standard deviations in parentheses.

Simulation parameters Real values Estimated values (Noise: 0.01) Estimated values (Noise: 0.1) Estimated values (Noise: 0.2)

qnorm1 (tilt φ) −1.2◦ −1.20◦ (±0.079◦) −1.18◦ (±0.754◦) −1.26◦ (±1.237◦)

qnorm2 (slant σ ) 1.5◦ 1.50◦ (±0.023◦) 1.51◦ (±0.178◦) 1.54◦ (±0.472◦)

qnorm3 (skew ψ) −1.0◦ −0.99◦ (±0.002◦) −1.00◦ (±0.019◦) −1.00◦ (±0.033◦)

qnorm5 (2 ps/(oz + fod)) 3.34 × 10−4 3.34 × 10−4 (±7.45 × 10−8) 3.34 × 10−4 (±4.19 × 10−7) 3.35 × 10−4 (±1.06 × 10−6)

qnorm6 (2 ox/(oz + fod)) −8.13 × 10−2 −8.13 × 10−2 (±5.23 × 10−6) −8.13 × 10−2 (±5.77 × 10−5) −8.15 × 10−2 (±1.11 × 10−4)

qnorm7 (2 oy/(oz + fod)) −8.19 × 10−2 −8.19 × 10−2 (±4.74 × 10−6) −8.19 × 10−2 (±3.94 × 10−5) −8.21 × 10−2 (±7.46 × 10−5)

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5967 Gross et al.: Auto calibration of a cone-beam-CT 5967

(a) (b)

FIG. 3. Relevance of the out-of-plane rotation error within the CBCT calibration: (a) definition sketch of an out-of-plane rotation error approximation h inobject space. (b) Out-of-plane rotation error propagation h into the object reconstruction for increasing tilt angles φ relating to the geometries listed in Table I.

To summarize, for the geometry IIa/b with or without out-of-plane rotations all simulations confirm the high accuracy ofthe method presented here. As expected the mean values andstandard deviation are more accurate than calibrations withgeometry I [compare Fig. 3(b)].

V.B. Real-world reconstruction example

To illustrate the applicability of our method we calibrateda dental CBCT device (3DAccuitomo, J. Morita Corp., Ky-oto, Japan). Most dental CBCT devices have a rotating C-armwhile the patient is sitting or standing near to the rotationalaxis of the system. In this particular case the device has twodata acquisition modes with different focal-lengths and dif-ferent in-plane translations. When switching between modesthe detector moves while the x-ray source remains in its initialposition. Both geometries of the device are given in Table I.

To calibrate the first mode (fod = 800 mm) we used aphantom containing 17 metal ball bearings of 1 mm in diam-eter manually arranged more or less vertically on a woodenplate (Fig. 4). In Fig. 5 one can see a projection image of thisphantom along with an overlay of several images showing theellipse trajectories generated by the rotation. From this weextracted six ellipses [Fig. 5(c)] as input for our optimizationprocess without using a priori information about the geom-etry of the calibration phantom. To extract the best possible

ellipse information, the lower and upper three ellipses wereused [see Fig. 5(c)] for the calibration. All initial conditionsof the optimization process are given in Sec. IV.

Based on this geometry we produced a reconstruction ofa mobile phone (Fig. 6). As one can see, even fine-graineddetails of the circuit board can be observed without any geo-metrical distortions.

For the second mode of the device (fod = 710.5 mm) weattached three markers to a dry human spine segment, seeFig. 7 for a projection image. This setting also generatedsuitable ellipse trajectories for the calibration process, seeFigs. 7(b) and 7(c).

Though we have no general proof that three markers aresufficient for solving the equation system uniquely we candemonstrate that in this particular setting the resulting geom-etry is capable of perfectly reconstructing very sharp featuresof the examined object (Fig. 8).

Also, the device-manufacturers reconstruction is given toconfirm that state-of-the-art reconstructions can be obtainedbased on the proposed calibration method. By visual inspec-tion our calibration method facilitates a higher spatial resolu-tion.

The reconstructions themselves were obtained by the well-known Feldkamp-David-Kress algorithm (FDK).20 Typicalmisalignment artifacts such as double structures, blurring, andgeometric distortions are not visible.

TABLE III. Comparison of the simulation results based on the geometry IIa in Table I from Yang et al. (Ref. 16)without out-of-plane rotations (φ, σ ) = (0, 0). The positions of the eight markers were perturbed by Gaussiannoise with a standard deviation of 0.4 pixel. For the simulations the mean of 1000 simulations is given along withtheir standard deviations in parentheses.

Simulation parameters Real values Yang et al. (Ref. 22) Our approach

qnorm1 (tilt φ) 0◦ 0◦ (constraint) 0◦ (constraint)

qnorm2 (slant σ ) 0◦ 0◦ (constraint) 0◦ (constraint)

qnorm3 (skew ψ) −1◦ −0.99◦ (±0.03◦) −1.001◦ (±0.001◦)

f ′x 1005 1005.1 (±0.4) 1005.001 (±0.05)

f ′y 480 480 (±1) 479.956 (±0.153)

f ′z −8333.33 −8354.16 (±41.66) −8329.58 (±2.148)

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5968 Gross et al.: Auto calibration of a cone-beam-CT 5968

TABLE IV. Simulation results based on a geometry IIb in Table I with out-of-plane rotations (φ, σ ) = (−2.5,1.5). The positions of the makers (number: 4) were perturbed by Gaussian noise with a standard deviation of0.4, respectively, 0.8 pixel. For both simulations the mean of 1000 simulations is given along with their standarddeviations in parentheses.

Simulation parameters Real values Estimated values (Noise: 0.4) Estimated values (Noise: 0.8)

qnorm1 (tilt φ) −2.5◦ −2.36◦ (±0.188◦) −2.41◦ (±0.514◦)

qnorm2 (slant σ ) 1.5◦ 1.5◦ (±0.017◦) 1.5◦ (±0.055◦)

qnorm3 (skew ψ) −1◦ −1.003◦ (±0.005◦) −1.003◦ (±0.015◦)

f ′x 1005 1005.02 (±0.06) 1004.96 (±0.177)

f ′y 480 480.02 (±0.29) 480.04 (±0.443)

f ′z −8333.33 −8329.82 (±2.57) −8326.13 (±4.902)

VI. DISCUSSION AND CONCLUSION

Geometric calibration refers to knowing the exact scan ge-ometry of the acquisition geometry with very high precision.Geometric accuracy is fundamental in image reconstructionto avoid typical misalignment errors.1, 21 Visible artifacts oc-cur even from minute deviation of the true geometry from thedesired geometry.22 On the other hand, it is well known thatmechanical stability, which permits off-line calibration andrepeatability, is very important for typical C-arm-based sys-tems such as CBCTs,22 since the systems are not as stable asgantry-based CTs. Hence, it is obvious that simple calibrationprocedures are a very important prerequisite to obtain high-quality 3D reconstructions.

We introduced a novel calibration method for CBCT flat-panel machines with a circular image acquisition orbit (360◦).It is an off-line procedure, i.e., the geometric parameters aredetermined in a scan before the system is operated on pa-tients. This assumption seems reasonable for sufficiently sta-ble machines and is also made by other authors.3, 9, 11, 15 Ourmethod is based on a simple phantom, that does not require

FIG. 4. A photograph of the calibration phantom with manually more or lessvertically arranged ball bearings.

accurate fabrication with only minute tolerances as describedin literature2, 5, 11, 12 or distinct spatial alignment of markers.3

Any pointlike highly-dense objects can be used. However, thetiny pointlike radio-opaque markers in our phantom shouldbe distributed such that they produce as large as possible el-lipses on the detector over the circular orbit. In order to obtainaccurate calibration results, it is always desirable to positionthe markers in such a way that their projection orbits extendover the full detector width.15 Already, a more or less verticalline of markers which is positioned some distance away fromthe rotation axis (which in many devices is indicated by a laserbeam crossing) fulfills this requirement. A possible configu-ration used in our experiments is shown in Fig. 4, however,this preferred setting is not an absolute prerequisite. Such aphantom may easily be produced manually within a few min-utes, e.g., by placing metal balls taken out of a ballpoint penin a wax-plate or acrylic plate. If the resulting ellipses do notfulfill the conditions explained above, the objects can easilybe replaced in other positions until the observed ellipses arelarge enough and have sufficiently long minor axes. Such aphantom is very affordable to produce and also very flexible.Calibration can be performed time-efficiently in 1–5 min onan up-to-date laptop computer. Thus, if fully implemented insoftware, it could be used for repeated recalibration by theuser. Although in our real-world calibrations we selected theellipses manually, this procedure could easily be automated,e.g., by using a threshold ratio between major and minor axisas the criterion. From the ellipses, seven parameters that com-pletely describe a CBCT scanner with truly circular acquisi-tion geometry1 can be determined. By combining two param-eters into a ratio and normalizing this ratio, we reduce these

FIG. 5. (a) Single projection of the calibration phantom. (b) Overlay of 104projections of the calibration phantom. (c) Upper and lower three ellipsesused to calibrate the device geometry.

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5969 Gross et al.: Auto calibration of a cone-beam-CT 5969

FIG. 6. Reconstruction of a mobile phone with a voxel resolution of about76 μm. Left: Front-view slice. Right: Top-view slice. No geometric distor-tions are visible (apart from metal reconstruction artifacts).

seven parameters to six. This step is a fundamental prerequi-site for our mathematical solution to determine the unknowncalibration matrix Dt from the observed ellipses Ci . The maincontribution of this work is the decomposition of the conicsection equations of the ellipses. From our empirical obser-vations we observed that it is better to have few (>4) clearlydefined ellipses rather than many ellipses which also includesome degenerated ones. General results on the stability withrespect to the number of markers and their positions must beleft to further investigations.

Similar to the method of Smekal et al.,15 our approach iscapable of calibrating detector tilt and slant, i.e., the two out-of-plane angles φ and σ . Other authors neglect these errorscompletely.16 According to Smekal et al.,15 detector tilt hasthe weakest effect of all geometric misalignments. We demon-strated, however, that for at least the geometric parameters oftwo typical dental CBCT machines on the market, an error of,e.g., 2◦ (tilt) yields an error in the back projection process ofvoxel size in the periphery of the reconstructed volume. Er-rors of this magnitude seem reasonable for CBCT machinesof this type, as the lower manufacturing cost may limit theaccuracy. Our theoretical results demonstrate that the largerthe cone angle, the larger is the effect of the out-of-planeangles. The cone angles of the machines we calibrated rangebetween 4.9 and 14◦. Future applications of large size CBCTdevices, e.g., for thoracal imaging have to cope with largercone angles, where accurate out-of-plane angle calibrationbecomes more important. Smekal et al. reported double con-tours for unrealistically large slant-errors (up to 20◦), i.e., the

FIG. 7. (a) Single projection of the spine segment containing three metalmarkers. (b) Overlay of 104 projections of spine segment. (c) Three extractedellipses used to calibrate the device geometry.

FIG. 8. Comparison with the manufacturers reconstruction of a spine seg-ment with a voxel resolution of about 156 μm. (a) Coronal, (b) axial, and(c) sagital slices through our reconstruction based on a geometry determinedwith the proposed calibration method. No geometric distortions are visible.(d) Coronal, (e) axial, and (f) sagital slices through the manufacturers re-construction. By visual inspection our calibration method facilitates a higherspatial resolution.

out-of-plane error occurring due to a detector rotation aboutthe y−axis in our model. Regardless of the overall effectof these two out-of-plane errors on the reconstruction, ourtheoretical results indicate, that the method introduced hereis capable of substantially reducing the error. An importantfinding is, that the proposed method is capable of calibratingthe out-of-plane angles with increasing accuracy in caseswhere their effect also increases. In other words, for largercone angles when neglecting out-of-plane angles negativelyaffects reconstruction accuracy, our method becomes moreeffective and accurate.

On the negative side, our approach uses nonlinear op-timization with six unknowns, yielding the risk of gettingtrapped in a nonglobal minimum. Well known numerical diffi-culties of nonlinear optimization routines require a good start-ing estimate and a careful selection of the sequence of pa-rameters that are optimized. To deal with these difficultieswe need to define a reasonable search bounding box basedon manufacturer specifications and approximate estimations.Although we need some approximate a priori knowledge ofthe range of geometric parameters to obtain stable results,this is not very restrictive. As demonstrated, we were ableto calibrate devices such as a micro CBCT as well as a den-tal CBCT with the same initial conditions of the optimizationprocess. Also, our method only provides a calibration exceptfor scaling. Similar to the method describe in Ref. 3 if thepixel size is unknown, we determine all parameters in unitsu, i.e., the focus-to-detector distance. Scaling could be deter-mined either by knowledge of the true distance of details inan object23 or by knowing, e.g., the focus-to-detector distanceplus pixel size. It should be noted, however, that this disad-vantage also produces an advantage: that is, fabrication errorsin the phantom cannot propagate into calibration errors. Theunknown distribution of the point-markers in our phantomalso makes it impossible to provide information on angular

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5970 Gross et al.: Auto calibration of a cone-beam-CT 5970

spacing between the projections. Therefore, we simply esti-mate the angle by dividing the rotation angle (2π in our cases)by the number of projections. This simple estimation is basedon the assumption of a rather uniform circular movement ofthe source-detector unit. At least for the two machines cali-brated, this assumption seems to be justified. It cannot, how-ever, be generalized for all machines on the market. Scale-invariant calibration suitable for a large class of CBCTs andlow restrictions on initial conditions of the optimization pro-cess are the major reasons which qualify the method for be-ing a starting point for more complex calibration procedures,e.g., when each image is calibrated separately.10 Future workwill focus on fully automated selection of the ellipses and ona further investigation of the difficult-to-handle out-of-planeangles that seem to play an increasing role the larger the coneangles become.

ACKNOWLEDGMENTS

The authors would like to thank S. Langer-Grey for help-ful comments on the paper. This work was supported by theGerman Federal Ministry of Education and Research (FKZ1748X09).

a)Author to whom correspondence should be addressed. Electronic mail:grossdaniel@web.de

1P. Rizo, P. Grangeat, and R. Guillemaud, “Geometric calibration methodfor multiple-head cone-beam SPECT system,” IEEE Trans. Nucl. Sci. 41,2748–2757 (1994).

2I. Kyprianou, S. Paquerault, B. Gallas, A. Badano, S. Park, and K. Myers,“Framework for determination of geometric parameters of a cone beam CTscanner for measuring the system response function and improved objectreconstruction,” Proceedings of the Third IEEE International Symposiumon Biomedical Imaging: Nano to Macro, Arlington, VA (IEEE, New York,2006), pp. 1248–1251.

3C. Mennessier and R. Clackdoyle, “Automated geometric calibration andreconstruction in circular cone-beam tomography,” in Nuclear ScienceSymposium Conference Record (NSS), Dresden, Germany (IEEE, NewYork, 2008), pp. 5081–5085.

4P. Rizo, P. Grangeat, and R. Guillemaud, “Geometric calibration methodfor multiple heads cone-beam SPECT system,” in Nuclear Science Sym-posium and Medical Imaging Conference, 1993 IEEE Conference Record,San Francisco, CA (IEEE, New York, 1993), Vol. 3, pp. 1764–1768.

5N. Robert, K. N. Watt, X. Wang, and J. G. Mainprize, “The geometric cal-ibration of cone-beam systems with arbitrary geometry,” Phys. Med. Biol.54, 7239–7261 (2009).

6G. Strubel, R. Clackdoyle, C. Mennessier, and F. Noo, “Analytic calibra-tion of cone-beam scanners,” in Nuclear Science Symposium ConferenceRecord, 2005 IEEE, Fajardo, Puerto Rico (IEEE, New York, 2005), Vol. 5,pp. 2731–2735.

7K. Wiesent, K. Barth, N. Navab, P. Durlak, O. Schuetz, and W. Seissler,“Enhanced 3D-reconstruction algorithm for C-arm systems suitable forinterventional procedures,” IEEE Trans. Med. Imaging 19, 391–403(2000).

8X. Zhang, F. Chen, Y. Li, Q. Wei, H. Zhang, and Y. Qi, “Unified geometriccalibration and image registration for detached small animal SPECT/CT,”in Nuclear Science Symposium Conference Record (NSS/MIC), 2009 IEEE,Orlando, FL (IEEE, New York, 2009), pp. 3318–3321.

9F. Noo, R. Clackdoyle, C. Mennessier, T. A. White, and T. J. Roney, “An-alytic method based on identification of ellipse parameters for scannercalibration in cone-beam tomography,” Phys. Med. Biol. 45, 3489–3508(2000).

10D. Beque, J. Nuyts, G. Bormans, and P. Suetens, “PatrickDupont: Char-acterization of pinhole SPECT acquisition geometry,” IEEE Trans. Med.Imaging 22(5), 599–612 (2003).

11Y. Cho, D. J. Moseley, J. H. Siewerdsen, and D. A. Jaffray, “Accurate tech-nique for complete geometric calibration of cone-beam computed tomog-raphy systems,” Med. Phys. 32, 968–983 (2005).

12M. J. Daly, J. H. Siewerdsen, Y. B. Cho, D. A. Jaffray, and J. C. Irish, “Ge-ometric calibration of a mobile c-arm for intraoperative cone-beam CT,”Med. Phys. 35, 2124–2136 (2008).

13M. Defrise, C. Vanhove, and J. Nuyts, “Perturbative refinement of the ge-ometric calibration in pinhole SPECT,” IEEE Trans. Med. Imaging 27(2),204–214 (2008).

14C. Mennessier, R. Clackdoyle, and F. Noo, “Direct determination of ge-ometric alignment parameters for cone-beam scanners,” Phys. Med. Biol.54(6), 1633–1660 (2009).

15L. Smekal, M. Kachelriess, E. Stepina, and W. A. Kalender, “Geomet-ric misalignment and calibration in cone-beam tomography,” Med. Phys.31(12), 3242–3266 (2004).

16K. Yang, A. L. C. Kwan, D. F. Miller, and J. M. Boone, “A geometriccalibration method for cone beam CT systems,” Med. Phys. 33(6), 1695–1706 (2006).

17E. R. Davies, Machine Vision: Theory, Algorithms, Practicalities, 3rd ed.(Morgan Kaufmann Publishers Inc., San Francisco, CA, 2005).

18R. E. Kalman, “A new approach to linear filtering and prediction prob-lems,” ASME J. Basic Eng. 82(Series D), 35–45 (1960).

19M. Pilu, A. Fitzgibbon, and R. Fisher, “Ellipse-specific direct least-squarefitting,” IEEE Image Proc. 3, 599–602 (1996).

20L. A. Feldkamp, L. C. Davis, and J. W. Kress, “Practical cone-beam algo-rithm,” J. Opt. Soc. Am. A 1(6), 612–619 (1984).

21R. Schulze, U. Heil, D. Gross, D. Bruellmann, E. Dranischnikow,U. Schwanecke, and E. Schoemer, “Artefacts in CBCT: A review,” Den-tomaxillofac Radiol. 40(11), 265–273 (2011).

22W. A. Kalender and Y. Kyriakou, “Flat-detector computed tomography(FD-CT),” Eur. Radiol. 17(11), 2767–2779 (2007).

23G. Gullberg, B. Tsui, J. Ballard, J. Hagius, and C. Crawford, “Estimationof geometrical parameters and collimator evaluation for cone beam tomog-raphy,” Med. Phys. 17, 264–272 (1990).

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