Comparison of resection–intersection algorithms and projection geometries in radiostereometry

Comparison of resection–intersection algorithms and projection

geometries in radiostereometry

Niclas Borlin

Department of Computing Science, Umea University, Umea SE-901 87, Sweden

Received 31 October 2001; accepted 3 May 2002

Abstract

Three resection– intersection algorithms were applied to simulated projections and clinical data from radiostereometric

patients. On simulated data, the more advanced bundle-adjustment-based algorithms outperformed the classical Selvik

algorithm, even if the error reductions were small for some parameters. On clinical data, the results were inconclusive. The two

different projection geometries had a much larger influence on the error size and distribution. For the biplanar configuration, the

position and motion errors were small and almost isotropic. For the uniplanar configuration, the position errors were

comparably high and anisotropic, but still resulted in a high accuracy for some motion parameters at the expense of others. The

simplified resection– intersection algorithm by Selvik may still be considered a good and robust algorithm for radiostereometry.

More studies will have to be performed to find out how the theoretical advantages of the bundle methods can be utilized in

clinical radiostereometry.

D 2002 Elsevier Science B.V. All rights reserved.

Keywords: radiostereometry (RSA); resection; projection geometry; X-ray photogrammetry; orthopaedic measurements

1. Introduction

1.1. Radiostereometry

Radiostereometric analysis (RSA) as defined by

Selvik in 1974 (Selvik, 1989) has been widely used in

orthopaedics for studying e.g. prosthetic implant

migration and wear, joint stability and kinematics,

bone growth, and fracture healing (Karrholm, 1989;

Stokes, 1995; Karrholm et al., 1997).

In the last 10 years, RSA has seen an increasing

use and Selvik-based systems are currently installed at

all university hospitals in Sweden and numerous other

locations in Europe, North America, and Australia.

To get reliable landmarks in the skeleton, RSA

uses implanted spherical markers (F=0.5–1.0 mm)

made of Ta73, a radio-dense, biocompatible metal. The

markers are inserted into the bone during surgery. If

an implant is part of the study, markers are attached to

the implant as well.

1.1.1. RSA procedure

After initial surgery, each RSA examination con-

sists of the following steps:

(1) Dual simultaneous X-ray exposure of the patient

together with a reference cage.

0924-2716/02/$ - see front matter D 2002 Elsevier Science B.V. All rights reserved.

PII: S0924-2716 (02 )00068 -0

E-mail address: [email protected] (N. Borlin).

www.elsevier.com/locate/isprsjprs

ISPRS Journal of Photogrammetry & Remote Sensing 56 (2002) 390–400

(2) Measurements of the projected marker coor-

dinates on the two radiographs.

(3) Reconstruction of the projection geometries

(resection).

(4) Reconstruction of the 3D-coordinates of the

patient markers (intersection).

Given two or more examinations, the relative rigid-

body motion between different group of markers

(segments) may be calculated.

1.1.2. RSA applications

RSA effectively enables relative motion measure-

ments of ‘‘anything you can put markers in’’.

Implant migration. With markers in an implant and

the surrounding bone, the migration pattern of an

implant can be studied. This allows different implant

designs, bone cement, operating techniques, etc., to be

evaluated.

Polyethylene wear. With markers in the polyethy-

lene liner of a hip implant, femoral head penetration

into the liner may be measured using the femoral head

as an additional marker. This enables assessment of

creep (deformation) and wear of the liner.

Joint kinematics and stability. With markers in the

bones on both sides of a joint, e.g. the knee, the

shoulder, vertebrae in the spine, etc., joint kinematics

may be evaluated. This applies to joints with or

without implants and may be used to understand the

joint behaviour, to compare healthy vs. pathological

joints, or to evaluate joint prostheses. Joint stability

may be assessed by calculating motions between

loaded and unloaded states.

Bone growth. Inserting markers in bone growth

zones enables studies of e.g. treatments for growth

disorders in children.

Fracture healing. Inserting markers on both sides

of a bone fracture enables measurements of healing

patterns.

For further details, see e.g. Karrholm (1989),

Stokes (1995) and Karrholm et al. (1997).

1.1.3. Aim of this paper

Several improvements of the original RSA proce-

dure have been presented, e.g. algorithms for efficient

calculation of the rigid-body motion (Soderkvist and

Wedin, 1993, 1994), solutions of two correspondence

problems and loose marker detection (Nystrom et al.,

1994). In the last 10 years, the development has been

concentrated to the measurement step, which until a

few years ago was performed manually using a high-

precision measurement table (Karrholm et al., 1997).

A number of papers have been presented on measure-

ments in digital radiographs (Borlin and Karrholm,

1997; Østgaard et al., 1997; Vrooman et al., 1998;

Valstar et al., 2000; Borlin et al., 2002). Two of those

Fig. 1. The two X-ray set-ups used in RSA. In the biplanar set-up (left), the reference cage is placed around the body part of interest. In the

uniplanar set-up (right), the reference cage is placed below or behind the object of interest. Drawing not to scale.

N. Borlin / ISPRS Journal of Photogrammetry & Remote Sensing 56 (2002) 390–400 391

Fig. 2. Examples of RSA images. The fiducial markers ( f ) and control points (c) are attached to the cage. The bone markers (b) and prosthetic

markers ( p) are inside the patient. Images are negative, i.e. in bright areas, most of the radiation was blocked. In the knee patients, prosthetic

markers were also inserted into the plastic insert between the metallic components. The markers in parentheses are known to be there, but were

either obscured by the prostheses or were too dark to be seen.

N. Borlin / ISPRS Journal of Photogrammetry & Remote Sensing 56 (2002) 390–400392

(Borlin and Karrholm, 1997; Borlin et al., 2002) show

improved results on clinical images compared to

manual measurement.

In contrast, the resection–intersection algorithms

have remained unaltered since 1974. In the original

algorithm, the calibration planes of the cage are used

one at a time, and the resection is performed by sol-

ving two mathematical problems in sequence. When

studying the photogrammetric literature, it became

clear to the author that by coupling the two problems,

bundle-like algorithms could be constructed. The

assumption is that since such algorithms use more

information about the photogrammetric set-up, they

should be able to increase the precision of radio-

stereometric measurements.

The aim of this paper is to investigate whether this

assumption holds and that the Selvik algorithm should

be replaced. Furthermore, the error propagation is

studied through the whole calculation chain from

measurements via 3D positions to relative motions.

1.2. Photogrammetric set-up

RSA uses two different photogrammetric set-ups,

biplanar and uniplanar, illustrated in Fig. 1. A refer-

ence cage with markers attached at predefined posi-

tions defines a laboratory coordinate system in object

space and is used for resection.

The biplanar cage (called cage 10) is an open-

ended rectangular box with X-ray film cassettes

attached to the bottom and one side wall. The two

cage planes closest to the films each contain nine

markers, called fiducial1 markers. One of the fiducial

markers in the bottom defines the origin of the

laboratory coordinate system. The opposite planes

contain nine markers each, called control points. Both

marker types are arranged in a rectangular pattern in

their respective planes. The total volume spanned by

the cage markers is 249�160�249 mm (x�y�z). At

the examination, the object of interest, e.g. a knee,

ankle, or elbow, is placed inside the cage. The X-ray

beams are approximately orthogonal. The typical

principal distance is about 1 m.

The uniplanar cage is shaped as an inverted ‘‘T’’,

with the cassettes placed parallel at the bottom of the

cage. In one design (called cage 41), the bottom two

plates have nine fiducial markers each. The top of the

inverted ‘‘T’’ has seven control points in a straight

line, used in both images. In a more recent design

(cage 43), the number of fiducial markers have

increased to 13 and 19 control points and are arranged

in three parallel lines. The origin is defined to be at the

centre of the base plate. The total volume spanned by

the cage markers is 510�240�208 mm. At the

examination, the X-ray central beams are angled

20–40j apart, each about 10–20j off the z axis.

The patient is placed above or in front of the cage

during exposure. The typical principal distance is

1.4–1.8 m. The exposure geometry is largely dictated

by anatomical (e.g. feasible marker placements) and

geometrical (e.g. cage size) constraints.

The typical film-to-fiducial plane distance is about

10–20 mm for the biplanar cage and 25–35 mm for

the uniplanar cage.

Examples of radiostereometric images are given in

Fig. 2.

2. Reconstruction algorithms

Three resection–intersection algorithms are com-

pared in this paper; the classic algorithm by Selvik,

the ICDLT (Iterative Constrained DLT), and the

ICDLTI (Iterative Constrained DLT and Intersection).

The algorithmic details are given in Appendices A,

Appendices B, Appendices C, but briefly, the Selvik

algorithm uses a simplified resection, with the fiducial

markers acting as fiducial marks. A projective trans-

formation between the film and fiducial planes is

calculated by minimizing the error in image space.

The transformation is used to rectify all measured

image coordinates and transform them to the labora-

tory coordinate system. The object space position of

the X-ray focus (projection centre) is approximated to

the point that minimizes the distance to the bundle of

rays going through the control points. The object

marker positions are calculated by minimizing the

object space error between pairs of X-rays from the

two foci.

1 For historical reasons, the use of the term fiducial in RSA

differs slightly from its conventional use in photogrammetry, e.g.

their positions are defined in 3D object space. For a motivation on

the choice of terms, see Selvik (1989, p. 7). See also Section 2 and

Appendix A.


In the ICDLT algorithm, the DLT formulation of

the collinearity equation is used. Orthonormality con-

straints on the image space coordinate system are

imposed to reduce the number of degrees of freedom

from 11 to 9, resulting in a self-calibration bundle

adjustment problem. The problem is solved for one

focus at a time with all cage markers used as control

points. The object marker positions are calculated by

minimizing the image space error between pairs of X-

rays from the two calculated foci.

The ICDLTI algorithm formulates and solves a

dual-station self-calibration bundle adjustment prob-

lem with the object markers appearing as unknowns.

The solution to this problem thus contains the projec-

tion parameters of both foci and the positions of the

patient markers.

3. Clinical material

To get clinically relevant material, examinations of

10 total hip replacement patients (cage 41, uniplanar)

and 10 total knee replacement patients (cage 10,

biplanar) from the Borlin and Karrholm (1997) study

were analysed. The patient markers were grouped into

a reference segment (bone) and a moving segment

(prosthesis). The patients were examined twice about

15 min apart with some intermediate patient exercise,

a procedure known as a double examination. The

hypothesis is that no relative motion of the segments

should occur between the two examination, so all

calculated ‘‘motions’’ between the segments may be

attributed to measurement errors. Double examina-

tions are used clinically to evaluate the relative motion

precision (Karrholm, 1989; Karrholm et al., 1997).

The images were scanned at 300 DPI with a Sharp

JX-610 flatbed scanner (Sharp, Japan) and measured

by UmRSA Digital Measure v2.2.5 (RSA Biomedical,

Umea, Sweden).

For each patient, the projection geometries and

patient marker positions of both examinations were

recovered by the ICDLT algorithm, resulting in four

sets of projection parameters and two sets of patient

marker coordinates, P1 and P2. For the biplanar

examinations, the principal distance varied between

0.92 and 1.04 m and the angle between the central

beams was 78–83j. The corresponding values for

the uniplanar configuration was 1.47–1.74 m and

24–28j. The patient marker configurations are

given in Table 1.

The absolute rigid-body motions of both segments

between P1 and P2 and their condition numbers were

calculated as described in Soderkvist and Wedin

(1993). Absolute motion is the motion in relation to

the cage, and reflects the difference in patient place-

ment between examinations. The condition number is

related to the geometrical configuration of the markers

in a segment. A high condition number indicates that

the geometrical configuration is almost degenerate, i.e.

close to a straight line, and therefore sensitive to

measurement errors. The absolute motion statistics is

given in Table 2 with the rigid-body motion separated

Table 1

Clinical patient marker configurations

Volume range [mm] Position [mm] Markers

x y z x y z Ref. Mov.

Knee 42–71 42–83 36–52 �38–49 1–42 68–153 7–9 3–6

Hip 34–58 46–157 28–71 �33–34 �3–28 441–597 5–9 3–6

Ranges for the spanned volume, position of marker centre of gravity, and number of markers in the two segments are given.

Table 2

Absolute motions between double examinations

Euler angles [j] Translation [mm] Condition number

x y z x y z Ref. Mov.

Knee �3–4 �9–9 �2–1 �21–8 �4–8 �12–10 12–33 19–113

Hip �3–1 �5–47 �3–8 �337–42 �28–36 �24–164 20–48 38–145

Ranges for Euler angles, translation of centre of gravity, and segment condition numbers are given.


into a rotation about the segment centre of gravity

(Euler angles) and a translation of the segment centre

of gravity.

The calculated absolute motion of the reference

segment was applied to all markers in P1 to get a

synthetic position PV2 which was approximately at P2

Fig. 3. Results based on 10,000 simulated knee and hip double examinations, respectively. The position and translation errors are given along

each axis. The rotation error is given for each of the Euler angles. The mean error of rigid body fitting is given for both segments. Error bars

show meanF2 standard deviations for Appendix A (– ), Appendix B (��), Appendix C (– – ), and Appendix C (– �), except for the fitting error

where the RMS of the fitting residuals+2 standard deviations is shown.


but with a relative segment motion between P1 and PV2of exactly zero.

4. Evaluation

The synthesized projections and patient marker

positions P1 and PV2 were taken as ‘‘ground truth’’ in

simulated exposures of the patient markers.

For each clinical set-up, marker projections in the

two images were simulated by Eq. (B.1). White

Gaussian noise of a reasonable standard deviation

r=50 Am was added to simulate measurement errors.

The projection geometries and patient marker

positions were reconstructed from the ‘‘measured’’

coordinates by the Selvik algorithm (Appendix A),

the ICDLT algorithm (Appendix B), the ICDLTI

algorithm with one patient marker (Appendix C),

and the ICDLTI algorithm with all patient markers

(Appendix C). The average difference between the

reconstructed and the true marker positions were

recorded in each of n=1000 simulations of the 20

biplanar and 20 uniplanar projections. Furthermore,

the relative segment motion between the simulated

double examinations was calculated. The mean error

of rigid-body fitting, i.e. the root mean square of the

marker fitting residuals (Selvik, 1989), was also

recorded.

The evaluation of the synthetic data was supple-

mented with the corresponding relative motions cal-

Fig. 4. Results based on double examination of 10 knee and 10 hip patients. Same layout as Fig. 3, except that the position error cannot be

estimated.


culated from the clinical data (i.e. between P1 and

P2).

5. Results

The simulation results are illustrated in Fig. 3. The

biplanar position error is almost isotropic, approxi-

mately (45, 33, 49) Am ((x,y,z), two standard devia-

tions). The corresponding figures for the uniplanar

configuration is (186, 135, 612) Am, i.e. approxi-

mately 4� larger for the x and y directions, and

12� larger in the z direction.

Similar relations between the axes are seen for the

translation errors, but while the biplanar errors are

approximately doubled, the uniplanar errors are

reduced by approximately 30%, 20%, and 60% for

the three axes, respectively.

The pattern for the rotation angles is rather differ-

ent. The biplanar angle errors (0.21, 0.18, 0.23)j((x,y,z), two standard deviations) has about the same

proportions as the biplanar translation errors, but the

uniplanar angle errors (0.40, 0.50, 0.17)j are roughly

inversely proportional to the uniplanar translation

error, with the largest error about the y axis and the

smallest about the z axis.

The average fitting error is about 80 Am for the

knee segments and 90 Am for the hip segments.

A comparison of the reconstruction algorithms

consistently shows the more complex algorithms to

be equal or better than the simpler ones. However, for

the biplanar simulations, the differences between

Appendices B and C are very small. Compared to

Appendix A, the position errors are reduced by about

25% and the other errors by 2–7%.

The improvements for the uniplanar set-up are

more varied. The 28% reduction of the position error

by Appendix C is comparable to the biplanar

improvements, while the improvements for Appendi-

ces B and C (6% and 15%, respectively) are not.

Appendices B and C shows similar translation, rota-

tion, and fitting reductions (13%, 2%, and 27–29%,

respectively), with Appendix C topping the list with

19%, 5%, and 33% reductions, respectively.

For the clinical results, illustrated in Fig. 4, the

relations between axes and configurations are similar

to the simulations. However, the algorithmic differ-

ences are small and inconsistent. For the most sensi-

tive parameters in the uniplanar configuration—z

translations and y rotations—the improvements are

similar to the simulations. For some other parameters,

the smallest error is produced by Appendix A.

6. Discussion

Several comparisons between different photogram-

metric reconstruction algorithms have been perform-

ed (e.g. Hatze, 1988; Gazzani, 1993). Their applic-

ability to radiostereometry was however considered

uncertain, since they were performed in optical set-

tings including lens distortion effects, something not

appearing in X-ray photogrammetry (Selvik, 1989).

An investigation of the performance of different

reconstruction algorithms on radiostereometric set-

ups was therefore considered interesting, especially

since the Selvik algorithm may be considered a sim-

plification of a bundle adjustment problem. Yuan and

Ryd (2000) performed a similar analysis on the re-

section– intersection part of radiostereometry, but

used the Selvik algorithm ‘‘backwards’’ to simulate

the projection, and it is unclear whether the relation-

ship between the focus position and the projective

transformation parameters was retained.

The algorithmic results for the synthetic data show

that the most advanced Appendix C reduces the

translation error with 7% for the biplanar configura-

tion and 19% for the uniplanar configuration. The

corresponding reductions for the rotation angles are

2% and 5%, respectively. However, Appendix C

requires the correspondence problem between images

to be solved beforehand, which is judged less feasible

and renders an independent verification of the recon-

structed markers more difficult. A single marker

correspondence needed by Appendix C is probably

more reasonable to expect, but since the performance

of the simpler Appendix B is essentially the same, it

should be favoured. Appendix B also allows the cor-

respondence problem to be solved independently by,

e.g. the Nystrom et al. (1994) algorithm.

On the clinical data, the results are inconclusive. A

possible explanation for this is the relatively small

differences in algorithm performance together with

the low number of patients. Modelling errors is another

possibility; the formulation of the bundle algorithms

assigns the whole error budget to the 2D measure-


ments. In reality, a number of other error sources are

present, e.g. imperfect cages, film unflatness, patient

motion between nonsimultaneous exposures, and the

size of the X-ray focal spot. Furthermore, the 2D

measurement errors are assumed to be isotropic and

the image coordinate system is assumed to be orthog-

onal and without scaling errors, which may or may not

be true.

The results in this paper show that the uniplanar

configuration has an anisotropic error distribution,

which is consistent with previous findings, e.g. Wol-

tring et al. (1985). The size of the error is higher than

for the biplanar configuration. This is probably a

combined effect of the weak projection geometry

and that the measurements are performed outside the

calibrated volume. It is interesting that the relatively

high position errors—4–12� higher than the biplanar

configuration—correspond to motion errors that for

some parameters (x and y translation, z rotation) are

comparable with the biplanar set-up. The relations

between the motion parameter errors are consistent

with previous clinical RSA results (Karrholm et al.,

1997).

Algorithmic changes for medical applications

should be done conservatively, and the arguments for

a change should be strong. This is currently not the

case. Furthermore, it could be argued that the Selvik

algorithm is more robust due to its linear subproblems.

Further studies with more patients and/or phantoms

will have to be performed in order to find out how

much of the improvements seen in the simulated data

can be carried over to clinical radiostereometry.

Appendix A. The Selvik reconstruction algorithm

In the Selvik reconstruction algorithm (Selvik,

1989), the resection is performed in two steps fol-

lowed by intersection.

Consider a projective transformation between a

fiducial marker p in the fiducial plane and its projec-

tion q in the film plane

q ¼ hðpÞ ¼ ATpþ b

vTpþ 1;

where A is a 2�2 matrix and b and v are two-vectors.

Given the known fiducial plane coordinates pi and the

measured film plane coordinates qi, the parameters of

the projective transformation are calculated by solving

the problem

minA;b;v

X

i

NhðpiÞ � qiN2: ðA:1Þ

This nonlinear least squares problem is solved

iteratively with a Gauss–Newton algorithm with

line-search (see e.g. Nash and Sofer, 1996, chaps.

10 and 13).

Given the solution of Eq. (A.1), the measured

coordinates are rectified by the inverse projection

h�1. For a fiducial plane parallel to the xy-plane,

the fiducial plane coordinate si=[h�1(qi)

T,z0]T, where

z0 is the (constant) level of the fiducial plane.

The position of the focus is calculated as the

‘‘closest point of convergence’’ of the bundle of

rays S i(a)=api+(1�a)(si�pi) passing through the

true control point coordinates ci and their rectified

fiducial plane coordinates si. The focus f is thus

calculated as the solution of the linear least squares

problem

minf ;ai

X

i

Nf � S iðaiÞN2: ðA:2Þ

The 3D position pi of each patient marker is

calculated by minimizing the object space distance

to the two X-rays going from the foci f (1) and f (2) to

the projection points si(1) and si

(2), respectively. This

corresponds to the linear least squares problem

minpi;a1;a2

Npi � S 1ða1ÞN2 þ Npi � S 2ða2ÞN2; ðA:3Þ

where S j(a)=af ( j)+(1�a)(si( j)�f ( j)) are the lines con-

necting the two foci with the marker projection point

si( j) in the fiducial planes.

Appendix B. The ICDLT algorithm

In the Iterative Constrained DLT reconstruction, all

cage marker are used as control points, and the

relation between the focus, the cage, and the image

is found in one step.

To simplify the notation of the following problems,

define the central projection triplet h={A,b,v}, whereA is a 3�2 matrix, b is a two vector, and v is a three


vector. Furthermore, define the central projection

function

CPðh; pÞ ¼ ATpþ b

vTpþ 1; ðB:1Þ

which is the DLT version of the collinearity equations.

The elements in A, b, and v correspond to the L1, L2,

. . ., L11 parameters in Eqs. (4.40) and (4.41) of

McGlone (1989, p. 47) and may be used to calculate

the inner and outer orientation parameters, see e.g.

McGlone (1989, p. 47, Eqs. (4.43) and (4.44))2 or

Melen (1995).

A central projection h is called orthonormal if it

satisfies the orthonormality conditions

ðaT1a2ÞðvTvÞ ¼ ðaT1vÞðaT2vÞ;

ðaT1a1 � aT2a2ÞvTv ¼ ðaT1vÞ2 � ðaT2vÞ

2; ðB:2Þ

where a1 and a2 are the columns of A. The constraints

(Eq. (B.2)) correspond to Eq. (4.45) in McGlone

(1989, p. 48)3 and specifies that the axes of the image

coordinate system are orthogonal and have the same

scale.

With the above definitions, and given the known

cage marker coordinates pi and their measured film

plane coordinates qi, the actual problem solved is the

following constrained nonlinear least squares problem

minh;fqig

X

i

Nqi � qiN2; ðB:3aÞ

s:t: qi ¼ CPðh; piÞ; ðB:3bÞ

h orthonormal: ðB:3cÞ

Eqs. (B.3a)–(B.3c) has 9 degrees of freedom and is

mathematically equivalent to the self-calibration bun-

dle adjustment problem.

Eqs. (B.3a)–(B.3c) is formulated for each focus

independently and solved iteratively with a SQP-

based method (Nash and Sofer, 1996, chap. 15).

Given the solution h1 for focus 1 and h2 for focus2, the position of each patient marker pi is calculated

from its measured coordinates qi(1) and qi

(2) in the two

images as the solution of

minpi;q

ð1Þi;q

ð2Þi

Nqð1Þi � qð1Þi N2 þ Nqð2Þi � q

ð2Þi N2

; ðB:4aÞ

s:t: qð1Þi ¼ CPðh1; piÞ; ðB:4bÞ

qð2Þi ¼ CPðh2; piÞ; ðB:4cÞ

i.e. by minimizing the image space error rather than

the object space error as in Eq. (A.3). Eqs. (B.4a)–

(B.4c) is solved by the same algorithm as Eqs.

(B.3a)–(B.3c).

Appendix C. The ICDLTI algorithm

In the Iterative Constrained DLT and Intersection

algorithm, the two projection geometries and the

position of one or more of the patient markers are

found simultaneously.

Denote a set of patient markers P and the focus 1

and focus 2 cage markers C1 and C2, respectively.

Then the mathematical problem is

min

h1; fqð1Þi g;h2; fqð2Þi g;

P

X

i

Nqð1Þi � qð1Þi N2 þ

X

i

Nqð2Þi � qð2Þi N2

;

ðC:1aÞ

s:t: qð1Þi ¼ CPðh1; piÞ; bpia P;C1; ðC:1bÞ

h1 orthonormal; ðC:1cÞ

qð2Þi ¼ CPðh2; piÞ; bpia P;C2; ðC:1dÞ

h2 orthonormal; ðC:1eÞ

which is solved with the same algorithm as Eqs.

(B.3a)–(B.3c).

2 However, note the following typographic errors; in Eq. (4.43),

x0 should be xp and y0 should be yp in the cx and cy equations,

respectively; the kappa equation should read j=cos�1(m11/cos /),and one of the sides of Eq. (4.44) should have the opposite sign.

3 However, note that the first equal sign of Eq. (4.45) should be a

minus sign and should thus read (L12+L2

2+L32)�(L5

2+L62+L7

2)+��.


Given the solution of Eqs. (C.1a)– (C.1e), the

position of any remaining patient markers not used

in the resection is calculated as the solution of Eqs.

(B.4a)–(B.4c).

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