Computer Vision and Image Understanding 113 (2009) 878–890

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Computer Vision and Image Understanding

journal homepage: www.elsevier .com/ locate /cviu

A fast and automatic stereotactic registration with a single CT-slice

B. Maurin a, C. Doignon a,*, M. de Mathelin a, A. Gangi b

a LSIIT (UMR ULP-CNRS 7005) – Control, Computer Vision and Robotics Group, University of Strasbourg, Bd. Brant, 7412 Illkirch, Franceb University Hospital of Strasbourg, Radiology Department B, Strasbourg, France

a r t i c l e i n f o

Article history:Received 30 March 2008Accepted 3 March 2009Available online 13 March 2009

Keywords:Stereotactic registrationCT scanner imaging modelLine-to-point matchingComputer-assisted radiology

1077-3142/$ - see front matter � 2009 Elsevier Inc. Adoi:10.1016/j.cviu.2009.03.004

* Corresponding author. Fax: +33390244480.E-mail addresses: maurin@eavr.u-strasbg.fr (

u-strasbg.fr (C. Doignon), demath@eavr.u-strasbg.frrad6.u-strasbg.fr (A. Gangi).

a b s t r a c t

The paper presents an algorithm for the automatic registration of a stereotactic frame from a single slicecaptured by a computed tomography (CT) scanner. Such registration is needed in interventional radiologyfor CT-image guidance of a robotic assistant. Stereotactic registration is based on a set of line fiducialsthat produce a set of image points. Our objective is to achieve the automatic registration of a stereotacticframe in the presence of noisy data and outliers. To this end, a new formulation of the stereotactic reg-istration problem with a single image is proposed, for any configuration of the fiducials. With very fewfiducials, closed-form and numerical solutions are presented. This comes in useful for building a robustautomatic line to point matching algorithm based on the RANSAC statistical method. Simulation andexperimental results are presented and highlight the robustness properties of the algorithm. They showthat the registration is accurate with moderate computing requirements even for a large amount ofoutliers.

� 2009 Elsevier Inc. All rights reserved.

1. Introduction

In the last two decades, with the appearance of high-resolutionComputed Tomography (CT) and Magnetic Resonance Imaging(MRI), computer-assisted radiology has greatly expanded thecapabilities for diagnosis and therapy. It is known that computer-assisted interventions provide many benefits for the patient sincethey increase the accuracy and the safety of a broad range of med-ical procedures [1,2]. In the interventional radiology domain, radi-ologists pioneered the use of imaging devices such as CT or MRI toguide instruments like needles inside the body for percutaneousprocedures like biopsies, vertebroplasties, thermo-ablation of tu-mours with radio-frequency (RF) probes. Our research in this field,in collaboration with surgeons and radiologists, aims at expandingpotentialities of the minimally invasive surgery. Minimally inva-sive techniques are attractive as they drastically reduce pain andrecovery time for the patient. The CT scanner was early and widelyknown to be an accurate imaging device that offers good tissue dif-ferentiation. Therefore, it was the first medical imaging deviceused to guide needles [3]. CT scanners produce few disturbancesin comparison to other medical devices and allow the visualizationof electronic and metallic devices such as RF probes or vertebropl-asty needles. The outcome of medical procedures such as RF abla-tion or vertebroplasty is highly correlated with the accuracy of the

ll rights reserved.

B. Maurin), doignon@lsiit.(M. de Mathelin), gangi@

needle positioning [4]. Therefore, a repeated check of the positionwith the visual information provided by the CT scanner is required(see, e.g., [5] in the case of vertebroplasties). Moreover, whenseveral CT-slices are involved to display the trajectory of a movinginstrument, radiologists are subject to overexposition to X-ray.

Nowadays, it would be interesting to automatically performseveral ‘‘simple but repetitive gestures” in order to help the sur-geon to be focused on some critical aspects of the intervention.In order to assist radiologists, image-guided robotic devices haverecently appeared with interfaces to various medical imagingdevices such as CT/MRI scanners or ultrasound imaging probes(see, e.g., [6–15]). In the case of interventional radiology with CTscanner feedback, the X-ray energy absorbed by the radiologist issignificantly reduced when these systems are used [16]. For exam-ple, the CyberKnife system [10], a robotic radiosurgery system, isdelivering multiple beams of precisely directed radiation that con-verge upon the tumour while minimizing injury to surroundinghealthy tissue. A visual servoing can be achieved for controllingthe position of an instrument (surgical instrument, needle, probe,etc.) mounted on a robot [17]. This is particularly helpful for reduc-ing the operator tiredness and to (tele)-operate precise displace-ments. Usually, the target tissue will be seen in the scannerimage. Therefore, a fast and accurate marker-based or intensity-based registration with respect to a scanner image reference frameis a critical step in order to guide the robotic assistant.

Following this goal, the aim of this work is to provide a fast andautomatic registration with intra-operative CT imaging that can beactivated at any time during a manual procedure or trigged by arobot controller in order to follow a pre-planned trajectory. Note

Fig. 1. The stereotactic registration is used to position a needle with the ‘‘look andmove” principle, by means of a robotic assistant in the CT scanner.

B. Maurin et al. / Computer Vision and Image Understanding 113 (2009) 878–890 879

that this work is part of a collaborative medical project on robot-ized needle placement for percutaneous procedures inside a CTscanner (see Fig. 1).

1.1. Previous works

A key issue to succeed in the image guidance of a medicaldevice is to achieve its precise localization with respect to the tar-get tissue inside the intra-operative space. To do so, a device-to-scanner registration is usually performed with a set of externalor natural anatomical landmarks [18] which can be accuratelyidentified and localized. Fiducials with simple shape can be fixedonto the patient [19], implanted inside the body [20], or rigidlylinked to the robotic device (see, e.g., Fig. 2). Historically, line fidu-cials were used on a stereotactic frame rigidly screwed to thepatient’s outer skull in image-guided neurosurgery [19,21,22].For robotized medical procedures, many authors have proposedto place the fiducials on (or close to) the robot end-effector orthe instrument. For example, Kwoh et al. [3] realized a braintumour biopsy by means of an industrial robot and a set of N-shaped rods as line fiducials on three orthogonal planes. Lemieuxand Jagoe [22] have studied the effect of fiducials detection erroron the stereotactic frame localization by means of computedtomography and digitized radiography. To our knowledge, theywere the first to establish the relationship between the stabilityof the registration and the rods configuration. Susil et al. [23] haveprovided a technique which recovers the position and orientation

Fig. 2. Modelization: the intersections of line fiducials (grey rods) with the imagingdevice (dark blue plane) are used to achieve the stereotactic registration. (Forinterpretation of the references to color in this figure legend, the reader is referredto the web version of this paper.)

of line fiducials with a single CT-slice. The major contribution of[23] is the fact that accurate results can be obtained with nine linefiducials from three N-shaped rods. However, the proposed poserecovery technique is driven by the specific fiducials configurationused (three planes of three N-shaped rods) and the methodologydoes not distinguish the fiducials model and the model of theimaging device. In [24], Dai et al. present an algorithm for stereo-tactic registration for both CT and MRI. Their technique is basedon a special arrangement of rods (V-shaped) and requires 12 fea-tures. Lee et al. [25] made a comparison of several numericalapproaches for the recovery of an Euclidean transformation forthe registration with at least six line fiducials. Their line-to-pointcorrespondence problem is solved manually and similarly to [23],the authors assume that a calibrated scanner is available, i.e., thefiducial model and the imaging model are not distinguished.

Finally, note that all the methods in the literature are eitherproviding a closed-form solution or a numerical minimizationalgorithm. In the first case, every single rod is assumed to be visibleand matched to the fiducial model. In the second case, data redun-dancy is possible, but initialization and convergence problems mayarise with numerical minimization techniques.

1.2. Objectives of this work

Our purpose is to provide an efficient computer assistance dur-ing image-guided procedures for dynamic scenes. With the emer-gence of real-time CT imaging (the CT fluoroscopy herein), ourapproach for the registration involves a single image only. Whilemost current techniques are based on a manual selection of corre-spondences, a fully automated procedure is attractive, especially todrive a robotic device. Furthermore, the ability to register a set offiducials in an arbitrary configuration gives more flexibility toposition the fiducials on the medical device or the robotic assistantin the intra-operative space. Hence, the main objectives of thiswork can be summarized as follows:

� to develop a new stereotactic registration algorithm for arbitraryfiducials arrangement,

� to point out the line fiducials configurations which may lead toinconsistent registrations,

� to specify the minimum number of correspondences needed forsolving the registration problem,

� to automatically and robustly solve the registration problemwith one CT-image, in presence of outliers.

� to develop an algorithm fast enough for intra-operativeregistration.

The first objective will be achieved thanks to the use of a propermodeling of the imaging device and of the fiducials. Based on thismodeling, singular line fiducials configurations and minimumnumbers of fiducials will be determined. The algorithms developedin this work include closed-form solutions for data sets with vari-able size, as well as numerical minimization algorithms, in order toget a robust and automated registration process. While the closed-form solutions are often sensitive to noise, they provide a usefulinitialization stage for numerical minimization algorithms thatsolve the usual convergence problem. Indeed, while numericalminimization algorithms will be shown to be less sensitive tonoise, they often suffer from multiple undesirable local minima.The closed-form solutions will also be convenient to study the linefiducials arrangement which lead to inconsistent registrations.

We tackle the registration by considering it has some connec-tions with the camera resection problem with a single view, aproblem widely investigated in the field of computer vision geom-etry [26–33]. Then, we define the world-to-scanner image map-ping in a way it can be solved with similar tools including

Fig. 3. (a) The plastic cube with the line fiducials used for the experiments. (b) A CTscanner image (magnified) of the cube.

880 B. Maurin et al. / Computer Vision and Image Understanding 113 (2009) 878–890

RANSAC, and internal scanner parameters are viewed as intrinsiccamera parameters. Therefore, camera calibration approachesmay be adapted to this specific mapping [34–37]. Main differencesare in the nature of the mapping and features modeling we provideto better understand the underlying geometry from world fiducialsto CT-scan image.

The paper is organized as follows. In the next section, the imagingmodel and the geometric representation of fiducials are bothdescribed. In Section 3, a new technique for both estimating intrinsicand registration parameters is presented. In Section 4, a fast algo-rithm for solving the 3D pose reconstruction problem is presented.This algorithm is part of a general registration framework which isdescribed in Section 5 for applications with quasi real-time require-ments. Simulations and experiments in presence of noisy data andoutliers have been made to validate the proposed approach. Theresults regarding the registration accuracy and computing timesare analyzed in Section 6 and in the discussion section.

2. Image and fiducials modeling

This section describes the geometric relationships between thefiducials and the corresponding visual features in the CT-image. Asalready mentioned, most of stereotactic registration methods havebeen designed with no distinction between the fiducial model andthe imaging model. This section aims at better understanding theembedded geometrical properties while providing useful notationsand definitions.

2.1. The imaging model

Most CT imaging devices use proprietary algorithms to generateimage slices and these algorithms are usually not in the publicdomain. Therefore, the imaging device will be considered in thispaper as a black box whose internal parameters are, e.g., the thick-ness of a slice and scaling parameters. In order to take into accountthese parameters, the imaging model will combine an affine trans-formation accounting for intrinsic scaling parameters, the Euclid-ean transformation between the scanner frame and thestereotactic frame, and an orthographic projection of a thin sliceonto the CT-image.

Let F0 the reference frame attached to the fiducials, Fct asthe frame attached to the scanner, and FI as a scaled frameattached to the CT-image with pixel units. The origin of theseframes will be noted, respectively, O0, Oct , and OI . The imagingmodel relates the coordinates of a 3D point, ctPj, expressed inFct and the coordinates of the corresponding point IQ j in theCT-image expressed as

ctPj ¼xj

yj

zj

264375¼ ctPp

pSIIQ j ¼

1 00 10 0

264375

|fflfflfflffl{zfflfflfflffl}orthographic

sx g

0 sy

� �|fflfflfflffl{zfflfflfflffl}

intrinsic parameters

uj

v j

� �|ffl{zffl}

pixel

ð1Þ

if IQ j ¼ uj v j½ �T , ctPp is a ð3� 2Þ matrix accounting for the ortho-graphic projection onto the cutting plane, p, and the non-nullentries of matrix pSI are the intrinsic parameters of the scanner.They consist of two scaling factors sx and sy and a shearing param-eter g accounting for a gantry tilt angle error or table bending dur-ing the scan. Usually, this parameter is very small and it is oftenneglected. However, it may be identified, since it may decreasethe registration accuracy in some circumstances, like for a heavypatient [38] or for MRI [39]. Using rigid-body transformations, Pj

can be expressed in F0 as 0Pj ¼ RctPj þ t, where R is a rotationmatrix (RT R ¼ I3, det(R) = 1) and t is a translation vector. Then,

0Pj ¼ r1 r2½ �pSIIQ j þ t; ð2Þ

where rk is the kth column of R, so that (2) is a compact represen-tation for the transformation FI !F0 including the orthographicprojection. Let 0LI be the following ð3� 2Þ real matrix:

0LI ¼ r1 r2 r3½ �1 00 10 0

264375

p

SI ¼ r1 r2½ �pSI ¼ l1 l2½ �; ð3Þ

then, from the orthonormality of any rotation matrix, it comes

lT1l1 ¼ s2

x ; lT2l2 ¼ s2

y þ g2 and lT1l2 ¼ sxg: ð4Þ

Usually, the shearing parameter, g, is neglected. Therefore, in theremaining, the following equation will be used:

lT1l1 ¼ s2

x ; lT2l2 ¼ s2

y and lT1l2 ¼ 0: ð5Þ

Finally, if homogenous coordinates are used for IQ j, likeIQ j ¼ ðuj;v j;1ÞT , then, (2) can be rewritten as

0Pj ¼ 0LI t� �

IQ j: ð6Þ

2.2. The fiducials model

Fiducials used in stereotactic regsitration are usually made ofrods (see Fig. 3a) and can be represented with straight lines[40,23,25]. Let Dj be the jth line fiducial. This line may be repre-sented by its origin Oj and a unit vector yj. Its intersection withthe scanner plane p is (generally) a point Pj ¼ Dj \ p (seeFig. 3b). From (6), it comes that

0Pj ¼ 0Oj þ kj0yj ¼ 0LI t

� �IQ j; kj 2 R; ð7Þ

where Oj is selected as the orthogonal projection of the origin of theframe F0 onto Dj, thus satisfying 0OT

j � 0yj ¼ 0. Therefore, to achievethe registration, one must solve (7) for 0LI , t and the fkjg’s, that is forð9þ nÞ unknowns where n is the number of line fiducials. Conse-quently, the number of equations to solve increases with the num-ber of rods, leading to a large number numerical computations [41].At this point, we introduce the Plückerian representation [33] oflines, in order to separate the identification of 0LI from the fkjg’s.Using yj and a new vector wj, defined by wj ¼ yj � Oj, the Plückeriancoordinates of a 3D line, Dj, are the pair of orthogonal vectorsðyj;wjÞ (see Fig. 4). Using this definition and the expression of0OPj (see Eq. (7)), it follows that

½0yj�� 0LI t� �

IQ j ¼ 0wj; ð8Þ

where ½yj�� is the ð3� 3Þ skew-symmetric (singular) matrix associ-ated to yj. Eq. (8) will be used to identify 0LI and t. Then, the fkjg aregiven by

kj ¼ 0yTj

0PTj ¼ 0yT

j0LI t� �

IQ j: ð9Þ

Generally, intersections of straight lines with the cutting planeshould provide as many spots as there are lines (see Fig. 3b). In

Fig. 4. A 3D line Di crossing the cutting plane p. The pair of vectors ðyj;wjÞ is thePlückerian representation of the line.

B. Maurin et al. / Computer Vision and Image Understanding 113 (2009) 878–890 881

practice, several spots may be missing in the image or, in contrary,some artifacts may appear [25]. Usually, line fiducials are bounded,i.e., kmin

j 6 kj 6 kmaxj , with known bounds (the fiducial geometry is

known a priori). Then, using (9), it is possible to check for any valueof the registration parameters (R,t) the relevance of the correspond-ing spot.

3. Scanner parameters calibration

In this section, two new algorithms are presented for thedetermination of the geometric transformation 0LI t

� �with an

uncalibrated scanner. The first algorithm provides an analytical(closed-form) solution. The second algorithm uses a non-linearprogramming method. Starting from Eq. (8), one must solve

IQ Tj � ½0yj��

h i l1

l2

t

264375¼0wj ð10Þ

subject to lT1l2 ¼ 0, where � is the kronecker matrix product. Given

n lines/points correspondences, (10) can be expressed as a minimi-zation problem with equality constraints:

minxkAx� bk2 subject to xT Cx ¼ 0; ð11Þ

where x ¼ lT1 lT

2 tTh iT

, C is a ð9� 9Þ symmetrical matrix with nullentries except for C14 ¼ C25 ¼ C36 ¼ C41 ¼ C52 ¼ C63, A is a ð3n� 9Þmatrix, and b is a ð3n� 1Þ, respectively, defined as

A ¼

IQ T1 � ½0y1��

..

.

IQ Tn � ½0yn��

26643775; b ¼

0wT1

..

.

0wTn

26643775: ð12Þ

3.1. Solution with a linear algorithm

The linear system (10) has nine unknowns and each fiducialbrings two independent linear equations. If n is the number of linefiducials, one must have 2n P 9, that is n P 5, leading to a full rankmatrix A, in order to solve this linear system (without the orthog-onality constraint lT

1l2 ¼ 0). Thus, vector x can be computed as thesolution minimizing kAx� bk2 with the least-squares (LS) method.Then, AT Ax ¼ AT b, and x can be computed provided AT A is of fullrank, i.e., if the following conditions are satisfied:

Feature geometrical constraints:

(1) three lines should not lie in the same plane, otherwise thethree corresponding image points are collinear,

(2) three image points should not be collinear. This may happeneven if the previous condition is verified,

(3) at least three lines should have distinct directions, so thatthe matrices ½0yi�� are linearly independent in order to com-pute a consistent position vector t.

Once 0LI and t have been estimated, the scaling factors sx and sy inpSI can be computed with (5). Usually, sx and sy are assumed to bepositive. As a matter of fact, the sign of these parameters dependson the definition of the scanner frame, Fct , with respect to theimage frame, FI . The first two columns of the rotation matrix Rare recovered by dividing the first and second columns of 0LI bysx and sy, respectively, and the third column of R is the cross-prod-uct of the first two. The well-known drawback with the LS algo-rithm is that if an outlier is introduced in the data, a bias isdistributed on all the estimated parameters. In Section 5, we willshow how to deal with the presence of outliers among the set ofimage points. Note that this first approach does not guarantee thatlT1l2 ¼ 0. Therefore, R is not exactly a rotation matrix.

3.2. Error propagation with the least-squares

Once the minimization problem has been solved with the least-squares algorithm, a numerical approach can be used to computeerror bounds on vector x. This is useful to guarantee a minimumaccuracy. To do so, a first-order error model is applied toAT Ax ¼ AT b such that A0 ¼ ðAþ dAÞ gives a resulting x0 ¼ xþ dxwhere dx is the estimation error vector. Then, the sensitivity ofdx can be derived from (12):

dA ¼ A0 � A ¼

du1½0y1�� dv1½0y1�� 0

..

. ... ..

.

dun½0yn�� dvn½0yn�� 0

0BB@1CCA; ð13Þ

where the non-zero components of the sparse matrix dA are comingfrom feature extraction errors and are related to the variationsðdui; dv iÞ of image point, IQ i, coordinates. On the opposite, it isassumed that the vector b that contains the fiducials descriptionis accurate, so that AT Ax ¼ AT b always holds. Therefore, the least-squares solution with matrix Aþ dA is satisfying ðAþ dAÞT

ðAþ dAÞx0 ¼ ðAþ dAÞT b. After computations and keeping first-orderterms, we get

AT Adx ¼ dAT b� ðATdAþ dAT AÞx: ð14Þ

Using a computer-assisted symbolic software, a linear representa-tion in dui and dv i can be easily derived from (14)

AT Adx ¼ Mðui;v i;0yi;

0wi;R; tÞdE ð15Þ

with dE ¼ du1; . . . ; dun dv1; . . . ; dvn½ �T and assuming that thematrix M is properly defined. Finally, dx can be computed withdx ¼ ðAT AÞ�1MdE. Therefore, by knowing bounds on image pointsdetection error, i.e., dEmin 6 dE 6 dEmax, a linear programming tech-nique can be used to get the min/max values of the error compo-nent dxi:

mindE

dTi ðA

T A�1MdE 6 dxi 6maxdE

dTi ðA

T AÞ�1MdE ð16Þ

with di ¼ ½0; . . . ;0;1;0; . . . ; 0�T , i.e., 1 at the ith position. Using otherorthogonal directions for given image point detection error bounds,a smaller polytope can be built around ðAT AÞ�1AT b, where the truevalue of x should lie.

3.3. Numerical solution with equality constraint

In this section, (11) is considered with the orthogonalityconstraint lT

1l2 ¼ 0. To do so, we propose to use the Newton–Raphson

882 B. Maurin et al. / Computer Vision and Image Understanding 113 (2009) 878–890

(N–R) numerical method with equality constraints. The Lagrangianfunction corresponding to this problem can be expressed as

L ¼ f ðxÞ þ lhðxÞ; ð17Þ

where l is the Lagrangian multiplier, f ðxÞ ¼ kAx� bk2 andhðxÞ ¼ xT Cx.

The necessary Kuhn–Tucker conditions for the existence of aminimum are

ðrL ¼ rf þ lrh ¼ 0hðxÞ ¼ 0

�with

rf ¼ 2ATðAx� bÞ;rh ¼ 2Cx:

(ð18Þ

The solution of (18) based on an iterative method consists in findingthe best direction of descent. To this end, let

F ¼rLh

� �; Y ¼

xl

� �such that FðYÞ ¼ 0: ð19Þ

According to the N–R method, the solution can be iteratively foundas

Ykþ1 ¼ Yk þ DYk with ½rF�TkDYk ¼ �FðYkÞ: ð20Þ

In our case, we have

½rF�k ¼2AT Aþ 2lkC 2Cxk

2ðCxkÞT 0

" #¼ ½rF�Tk ; ð21Þ

DYk ¼DxDl

� �k

¼ �ð½rF�kÞ�1FðYkÞ ð22Þ

or equivalently,

D~xlkþ1

" #¼ �ð½rF�kÞ

�1 2ATðAxk � bÞxT

k Cxk

" #: ð23Þ

The initialization of this iteration process may be set to the solutionof the linear problem in order to avoid local minima far away fromthe true solution.

4. Registration as a pose estimation problem

Assuming that a calibrated scanner is available, i.e., the intrin-sics parameters are known, this section aims at designing fast algo-rithms for estimating the rigid registration parameters.

First, the registration problem is solved with the minimumnumber of features required for a linear algorithm with rank char-acteristics. Then, an algorithm is presented which solves the prob-lem with any number, n, of features ðn 6 4Þ.

4.1. A linear four lines/points algorithm

We tackle the problem of R and t recovery with the minimumnumber of fiducials needed by means of a linear algorithm.Remember that five features were necessary for the calibrationproblem in Section 3.1. Given a single image, and considering theunknown vector n ¼ rT

1 rT2 tT

� �T , Eq. (10) becomes

IQ Tj � ½0yj��

h iS9n ¼ 0wj; ð24Þ

where

S9 ¼pST

I � I3 00 I3

" #: ð25Þ

With exactly four lines/points correspondences, (24) is a deficient-rank system which can be solved (up to two solutions) thanks to therotation properties. Indeed, except for some arrangements of thefiducials enumerated in Section 3.1, the matrix A in (12) has a rankequal to 8 when components contain uncorrupted data. However,

with noisy data, the rank may be equal to 9). Hence, in order to en-force the rank value, (24) is solved thanks to the Singular ValueDecomposition (SVD), leading to a one-parameter family of solu-tions. Let summarize the algorithm as follows:

(1) Find the SVD of A ¼ UDVT , with the diagonal entries di of D indescending numerical order.

(2) Let b0 ¼ b01 b02 . . . b09� �T ¼ UT b.

(3) Build the vector z defined by zi ¼ b0i=di, for i ¼ 1; . . . ;8 andz9 ¼ 0.

(4) The general solution is n ¼ S�19 ðVzþ cv9Þ, where v9 is the

last column of V and c is a free parameter.(5) Compute c (see Appendix A) using the relationship between

r1 ¼ ½ n1 n2 n3 �T and r2 ¼ ½ n4 n5 n6 �T .

Eq. (24) can be solved provided that all the features verify the threeconditions mentioned in the previous section. Because of the pres-ence of noise, R is not exactly a rotation matrix. One may enforce Rto be a rotation matrix by computing the SVD, R ¼ URVT , and bysetting singular values to 1. Then, R0, the true rotation matrix, isgiven by R0 ¼ U diagð1;1;detðUVÞÞVT (cf. [42]).

4.2. A fast linear n-lines/points algorithm

One can notice that for determining the rigid-body transforma-tion in the previous algorithm, up to three SVD should be com-puted. Since this algorithm may be called many times during anautomatic registration procedure, it is an important issue to speedup the computations. Furthermore, the rank of matrix A is notequal to 8 with more than four correspondences, while the previ-ous algorithm which solves the deficient-rank linear system pro-vides an accurate solution only for n ¼ 4. Here, a simplemodification is presented in order to handle more correspondencesand to avoid some numerical computations. From Eq. (24), theexpression for eight unknowns can be derived as a one-parameterfamily of solutions of a linear system with nine unknowns. If theone-dimensional solution space is spanned by a component ofthe position vector t ¼ ½ tx ty tz �T , say tz, we obtain

½IQ Tj � ½0yj���S

09

r1

r2

tx

ty

2666437775 ¼ 0wj � tz½0yj��

001

264375 ð26Þ

with S09 ¼ S9I8

0T

� �. Then, similarly to the previous Section 4.1, the

Moore–Penrose inversion of AS09 is applied to (26) and the orthogo-nality constraint, rT

1r2 ¼ 0, is used to compute tz from a quadraticequation (cf. Appendix A).

5. Automatic registration

In this section, an automatic registration algorithm based onthe robust estimation theory is presented. The purpose is todetermine the best Euclidean transformation, given a set of no

line fiducials and a set of nI image points. Usually, no and nI

are not equal for numerous reasons. For example, a line mayhave no intersection with the cutting plane, or artifacts may ap-pear in the image, because of other parts inside the scanningarea. These artifacts act as outliers and they must be discarded.Furthermore, even if nI ¼ no, the fiducial-based pose algorithmneeds a one-to-one correspondence between a subset of linefiducials and a subset of points. Moreover, the matching processmust be fast and with low algorithmic complexity. Therefore, itis important to design a pose estimation algorithm with a fewfeatures even if the final registration computation takes advan-tage of all the available inliers.

B. Maurin et al. / Computer Vision and Image Understanding 113 (2009) 878–890 883

There exist many robust estimation algorithms and we havechosen the classical hypothesize-and-test approach where a smallset of correspondences are first hypothesized, then, the corre-sponding transformation is computed. A well-known example ofthis approach is the RANdom SAmple Consensus (RANSAC) algo-rithm of Fischler and Bolles [43] which is able to cope with a largeamount of outliers. Applied to the current registration problem,this algorithm randomly selects a set of four image points for everyarrangement of four lines and four points, since 4 is the number offeatures needed in order to compute the rigid-body transformationwith a linear method. For that candidate transformation, a supportvalue is measured by the number T of (line,point) couples that areconsistent. This random selection is repeated N times, where thenumber N of samples is also computed by the algorithm and de-pends on the fraction � of outliers, the sample size, s ¼ 4, and theprobability, p, that guarantees at least one sample is free from out-liers. Thus, w ¼ 1� � is the probability that any selected sample isan inlier and N, ðN � nIÞ, is such that [33]

ð1�wsÞN ¼ 1� p: ð27Þ

If no contextual information is available to constrain the search, ahigh probability of success can be achieved with the RANSAC algo-rithm in Oðn4

onIÞ operations and in Oðn4oNÞ operations in average.

However, with our registration algorithm, a geometric constraint isintroduced to speed up the rejection of some most probably unconsis-tent samples. It consists in looking for triplets of collinear imagepoints within a sample. As mentionned in Section 3, such a configura-tion leads to a lower rank of the matrix A and should be discarded. Werefer to this as the collinearity constraint Ccol that is computed, giventhree points Q1; Q2 and Q3, as the minimum sinus angle among thethree angles of the triangle (Q1Q2Q3). A threshold value, sc , is usedto define a minimum acceptable value for this criterion. Now, let sd

as a distance threshold, such that an image point is an inlier with alarge probability (that is the line/point correspondence is consistentwith the putative transformation). Assuming a Gaussian measure-ment error with zero mean and standard deviation r, Hartley andZisserman [33] have provided a method for computing such kind ofthreshold for several geometric transformations. In accordance withthis, the threshold distance value is set to sd ¼

ffiffiffi6p

r. Then, looking forall object samples, Nno transformations are computed, and for each ofthem a consensus set of correspondences is built and a standard devi-ation is derived. If the size T of that consensus set is strictly greaterthan the sample size s, the associated transformation is labelled asqualified. Then, the selected transformation is the one with the small-est standard deviation among qualified ones (see Fig. 5).

6. Results

This section presents simulation and experimental results. First,the calibration algorithms of Section 3.1 are compared through sim-ulations. Then, experiments are conducted with rods embedded inplastic cubes in order to assess the relative accuracy of the differentregistration algorithms. Finally, the automatic line to point matchingand registration procedure is validated through simulations.

For simulation purposes, images data have been synthesized asfollows. Given a rigid-body transformation, the image point, IQ j, ofthe jth line fiducial represented by its Plückerian coordinatesðoyj;

owjÞ, is computed according to the formula

uj

v j

� �¼ pS�1

I ðFTj FjÞ�1FT

j ðowj � ½oyj��tÞ; ð28Þ

with Fj ¼ ½oyj��R1 00 10 0

24 35, which is derived from (8). These points

have been included in the set of image points and displayed only

if kj is in the range kminj ; kmax

j

h i, in accordance with (9). To evaluate

the noise sensitivity of the proposed models and methods, a uni-form noise distribution has been added to the pixel coordinates ofeach spot centroid. For the robustness w.r.t. outliers, random pointshave been included. Performances of the proposed algorithms arecharacterized by the following criteria:

� the 3D rotation and position errors,� the error on scale parameters sx; sy for the calibration,� the error on the estimated image points location.

The rotation error is computed in the following manner: when aknown rotation matrix R is used to compute n simulated image

points IQ j, then the estimated rotation bR allows to compute the

rotation error, EðbRÞ ¼ k 180p mk. k � k is the Euclidean norm and m

is the absolute instantaneous rotation vector obtained from the

skew-symmetric matrix ðR� bRÞRT . The position error is defined

as EðbtÞ ¼ kt� btk, similarly to the scaling factors error computed

with the matrix pSI , EðpbSIÞ ¼ kpSI � pbSIk. Finally, the error on thelocation of the overall image points is defined by assuming the

estimated position bQ j ¼ uj v j� �T of each rod has been computed

using Eq. (28), with the transformation ðbR;btÞ, so that

Eð bQ Þ ¼ 1n

Pnj¼1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðuj � ujÞ2 þ ðv j � v jÞ2

q.

6.1. Calibration

A calibration procedure has been carried with both the LS (III-A)and the N–R minimization (III-C). For simulation purposes, thefiducial model with an arrangement of eight lines has been consid-ered, such that conditions listed in Section 3 hold and the true scal-ing factors are set to sx ¼ sy ¼ 0:5 mm/pixel. Simulation results arepresented in Figs. 6 and 7 and have been drawn for various levels ofnoise, with a set of 100 random rotation angles in the range of ±20�around each axis and with a set of 100 random positions in therange of [±50, ±50, 30 ± 10] in millimetres (for each noise level)so as to ensure that most of fiducials are crossing the CT-plane.These figures show that all parameters are well estimated, espe-cially the scaling factors. Both algorithms perform well. However,the iterative N–R algorithm which is able to handle the orthogonal-ity constraint provides the best estimation.

6.2. Sensitivity of the rigid-body registration algorithms

In this part, we compare performances of the proposed algo-rithms for the rigid-body registration problem, that is when thescalings are known. Four algorithms are compared:

� Least-squares SVD: This is the algorithm described in Section 4.1which uses the orthogonality constraint (31) in the appendix forthe computation of parameter c.

� Least-squares SVD and data redundancy: This is the algorithmdescribed in IV-A which uses the three quadratic constraints(29)–(31) for the computation of parameter c.

� Least-squares Tz: This is the fast algorithm which solves Eq. (26)in IV-B with the third component of the position vector t as thepivoting value. Eq. (31) is used for the computation of tz (withtz ¼ c).

� Newton–Raphson minimization: It is described in Section 3.3, andused for both calibration and registration.

The noise sensitivity has been studied for n ¼ 4 and n ¼ 8 by add-ing a uniform noise distribution to the ideal locations of imagepoints. Errors for n = 4 are reported in Fig. 8. With a noise level

Fig. 5. The algorithm for the determination of the stereotactic registration simultaneously with the line/point correspondence problem.

884 B. Maurin et al. / Computer Vision and Image Understanding 113 (2009) 878–890

of 0.5 pixel, the 3D position/orientation errors are found to bearound 1 mm/3� with closed-form algorithms and 0.5 mm/1.5�with numerical approach.

The same analysis with eight fiducials (see Fig. 9) provides er-rors around 0.35 mm/0.8� with closed-form algorithms and0.2 mm/0.3� with the N–R method. The accuracy factor when dou-

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.5

1

1.5

2

Erro

r on

Rot

atio

n (d

eg) Least Squares

Newton−Raphson

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

1

2

3

4

5

Erro

r on

Tran

slat

ion

(mm

)

Uniform Noise Amplitude (in pixels)

Least SquaresNewton−Raphson

Fig. 6. Errors on the estimation of 3D parameters with eight features: angles (top) and positions (bottom), w.r.t. uniform noise levels.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

1

2

3

4x 10−3

Erro

r on

Scal

e (m

m/p

ix) Least Squares

Newton−Raphson

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.5

1

1.5

2

2.5

Erro

r on

Rec

onst

ruct

ed P

ixel

(pix

)

Uniform Noise Amplitude (in pixels)

Least SquaresNewton−Raphson

Fig. 7. Errors on scaling parameters (top) and reconstructed spot locations (bottom) w.r.t. uniform noise levels.

B. Maurin et al. / Computer Vision and Image Understanding 113 (2009) 878–890 885

bling the features number is significant, and in all cases, thenumerical approach provides accurate results.

6.3. Experimental validation of calibration and 3D pose

Experiments were conducted with a Siemens Somatom Plus CTscanner and with fiducials composed of block cubes with six rodseach (see Fig. 3) of 30 mm length and 1 mm diameter. They canbe easily placed on a robotic assistant (see Fig. 1). The rods aresmaller to those used in [23], but they are sufficiently large so as

to prevent the partial volume effect [44]. Each cube of fiducialshas been calibrated with a Mitutoyo measuring machine, a preci-sion of better than 10 lm is achieved with this material. Each spotcentroid was located by means of an image thresholding techniquewith a constant value of 2000 in Hounsfield Unit [44], followed bythe computation of each centroid [40]. As the DICOM formatincludes the scalings (assuming their values are the ground truth),the calibration as well as the rigid registration can be validated. InFig. 10, we have reported the scaling values while parameters areestimated with both LS and N–R methods with 43 CT-images.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

1

2

3

4

5

6

Erro

r on

posi

tion

(mm

) LS−SVDLS−SVD (DR)LS−TZN−R

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

2

4

6

8

10

12

14

Erro

r on

rota

tion

(deg

) LS−SVDLS−SVD (DR)LS−TZN−R

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.5

1

1.5

2

2.5

Erro

r on

spot

cen

troid

s

l

ocat

ion

(pix

)

Uniform noise level (pixel)

LS−SVDLS−SVD (DR)LS−TZN−R

Fig. 8. The mean 3D position error (top), the mean angular error (middle) and the reconstructed spot location errors (bottom) for the stereotactic registration with n ¼ 4 andw.r.t. varying uniform noise levels computed with the orthogonality constraint (except for case (ii)) and for: (i) SVD and orthogonality constraint, (ii) SVD and the threequadratic constraints, (iii) least-squares with pivot c ¼ tz , and (iv) Newton–Raphson.

886 B. Maurin et al. / Computer Vision and Image Understanding 113 (2009) 878–890

These methods provide very similar results and the mean errors(w.r.t. DICOM data: 0.5 mm/pix) are less than 5%. Note also thatstandard deviations all along the sequence are very smallðrðsxÞ 6 4� 10�3; rðsyÞ 6 2� 10�3Þ.

To assess the accuracy of the pose recovery, two block cubeshave been used. Their relative position is constrained by a guiderail on which they are screwed on (see Fig. 11). Then, it is pos-sible to verify the coherency of the pose estimation with a singleimage by registering only one fiducial cube at a time. To thisend, a helicoidal sequence has been captured while a constanttranslation of the table is performed. In Fig. 13, we present theestimated relative position while the orientation is approxi-mately the identity matrix (it differs to the identity matrix byless than 10�5 on each component). The registration has beenexecuted for each cube with the LS method and the N–R minimi-zation. The position T ¼ ½0� 118:290�T mm was measured duringa calibration procedure. This value (dotted line in Fig. 13) can becompared to the estimates. Results in this figure and in Table 1show that a mean position error always less than 3.6 mm isachieved with only five line fiducials, and less than 0.6 mm(and always <1.3 mm) with six fiducials, both with the N–Rnumerical approach. As shown in Fig. 12, the 3D pose algorithmworks very well on experimental data, as the registration of eachcube allows to predict the position of rods of the other cube. Theestimated error bounding-boxes with a supposed spot locationerror of 0.25 pixel are displayed. All spots are inside a box withboundaries corresponding to 3D errors less than 2 mm.

6.4. Automatic registration

First of all, the automatic registration algorithm has been carriedout with simulated data. With several rigid-body transformations,outliers have been artificially incorporated (as random points) tothe set of image points. Then, varying levels of noise have been addedto the overall image points. For a value of 0.5 pixel error in the cen-troid locations, we have reported in Table 2 the measured computing(min–max) times with a Pentium IV 2.8 GHz, with six line fiducials,and with a set of 15 registrations for the three following cases:

� the collinearity constraint, Ccol, is not taken into account,� Ccol is included in the registration process, with two following

threshold values sc ¼ sinð10Þ or sc ¼ sinð20Þ.

The computing time is less than 1 s with at most one outlier in theimage data and less than 2 s in average with four outliers. We havenoticed that it is very robust since a solution is still found with 24outliers. We have also applied the registration algorithm to the CTimages of the two cubes (see Fig. 11b), one making 6 inliers andone 6 outliers. Finally, Fig. 14 shows a CT-image with one cubeand the needle-holder. In this image, nine spots were found butonly six of them are corresponding to images of the fiducials andmatched in less than 2 s. As typically no more than such two reg-istrations (including features matching) are done in clinical situa-tions, we consider these time results compatible with surgeonsand physicists’ interventions.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.5

1

1.5

2

Erro

r on

posi

tion

(mm

)

LS−SVDLS−SVD (DR)LS−TZN−R

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

1

2

3

4

5

6

7

Erro

r on

rota

tion

(deg

)

LS−SVDLS−SVD (DR)LS−TZN−R

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.5

1

1.5

2

Erro

r on

spot

cen

troid

s

lo

catio

n (p

ix)

Uniform noise level (pixel)

LS−SVDLS−SVD (DR)LS−TZN−R

Fig. 9. The mean 3D position error (top), the mean angular error (middle) and the reconstructed spot location errors (bottom) for the stereotactic registration with n ¼ 8 andw.r.t. varying uniform noise levels computed with the orthogonality constraint (except for case (ii)) and for: (i) SVD and orthogonality constraint, (ii) SVD and the threequadratic constraints, (iii) least-squares with pivot c ¼ tz , and (iv) Newton–Raphson.

0 5 10 15 20 25 30 35 40 454.7

4.75

4.8x 10−1

Sx(m

m/p

ix)

0 5 10 15 20 25 30 35 40 454.73

4.74

4.75

4.76x 10−1

Sy(m

m/p

ix)

Image #

Least SquaresNewton−Raphson

Fig. 10. Estimation of the scalings sx and sy with the least squares and the Newton–Raphson methods. The mean value is corresponding to the red line.

B. Maurin et al. / Computer Vision and Image Understanding 113 (2009) 878–890 887

7. Discussion

Clearly, the most accurate algorithm for registration with noisydata is the N–R algorithm that iteratively computes an estimate ofthe registration with the scanner calibration parameters. Onepossible explanation for this better performance is that the N–Ralgorithm solves the LS minimization problem under the orthogo-nality constraint, while the other algorithms solve a LS problem

and then proceed to enforce the orthogonality conditions a poste-riori using some kind of projection.

The automatic line-to-point matching is realized using theRANSAC statistical method with a linear algorithm for registration.From the experiments, it can be seen that the simple LS algorithmwith a pivot c ¼ tz and the use of the orthogonality constraint (31)is accurate enough to solve the registration problem with a limitednumber of features ðn ¼ 4Þ.

Fig. 11. (a) The two cubes used for the experiments. (b) image picked up from a CT helicoidal sequence with image size of ð512� 512Þ pixel.

0 5 10 15 20 25 30−123

−122

−121

−120

−119

−118

−117

−116

Erro

r on

Posi

tion

Ty (m

m)

Frame #

LS−SVDN−R

0 5 10 15 20 25 30−121

−120

−119

−118

−117

−116

Erro

r on

Posi

tion

Ty (m

m)

Frame #

LS−SVDN−R

Fig. 13. Relative positions between the cubes during the acquisition (slice thickness is 0.5 mm) while translating the table. The first plot (up) is with five fiducials for theestimation while the second (down) is with six fiducials.

Fig. 12. Images of the cubes. Each yellow box in one cube is corresponding to 2 mm of error bounds computed with the registration of the other cube. (For interpretation ofthe references to color in this figure legend, the reader is referred to the web version of this paper.)

888 B. Maurin et al. / Computer Vision and Image Understanding 113 (2009) 878–890

The automatic registration algorithm is fast enough for applica-tions with real-time requirements where a robotic assistant wouldbe directly controlled by the scanner image while tracking a mov-ing target. Up to now, we propose to follow the ‘‘look and move”

principle. In the future, we plan to tackle the compensation ofsmall motions for which the performances are depending on thelocation of the stereotactic markers w.r.t. the patient. For the nee-dle positioning, we chose to place them on the robot basis which in

Table 2Computing times (min–max in seconds) for the automatic registration with no ¼ 6line fiducials without (col. 2) or with the collinearity constraint (col. 3 and 4).

% of outliers ðnIÞ sc ¼ 0 sc ¼ sinð10Þ sc ¼ sinð20Þ

0 (6) 0.38 0.17– 0.33 0.05–0.1620 (7) 0.85 0.29–0.68 0.12–0.4240 (10) 5.1 1.3–3.2 0.5–2.460 (14) 24.2 5.7–13.2 3.7–9.580 (30) 645 365 171

Fig. 14. A DICOM image including several spots. A small part of them arecorresponding to the image of the fiducials.

Table 1Mean 3D position error and standard deviation (in mm) between the two cubes offiducials with least-squares (LS) and Newton–Raphson (N–R), for five and six lines.

n �tx rðtxÞ �ty � Ty rðtyÞ �tz rðtzÞ

5 0.77 1.03 0.75 0.93 4.86 3.03 LS6 0.88 0.99 1.04 0.68 0.98 2.81 LS5 0.26 0.97 1.11 0.52 2.18 4.64 N–R6 0.15 0.91 0.79 0.35 0.74 2.32 N–R

B. Maurin et al. / Computer Vision and Image Understanding 113 (2009) 878–890 889

turn is fixed on the abdomen: this situation allows the robot to fol-low some regular motions as the respiratory motion, and with arespiratory control device based on precise volumetric control ofthe airway, a generalized predictive control can quickly learn thefrequency of such motion. However, the performances w.r.t. someabrupt and unpredictable motions are also depending on the char-acteristics of the fluoroscopic mode of the imaging device like theresolution in that mode and the video rate.

8. Conclusion

A fast and automatic stereotactic registration algorithm forintra-operative CT imaging has been presented. This algorithmdoes not require a specific configuration of the fiducials. A geomet-ric imaging model with 8 degrees of freedom has been proposed toachieve the registration. For this purpose, we have presented atechnique which simultaneously solves the correspondence andregistration problems with line fiducials and which needs onlyone image and few fiducials (4). We show it can be solved usingboth closed-form solution and numerical approaches in less than1/3 s with a standard PC. Moreover, a major benefit of this new for-mulation is to point out the fiducials arrangements inside the ste-reotactic cube which lead to inconsistent registrations. As many

artifacts may occur, a RANSAC-like robust registration has beendesigned to deal with missing data and outliers and to automatethe registration. Simulations and experiments with CT-imagesand two cubes of six straight rods have been carried out. Experi-mental results have shown that the 3D position accuracy is betterthan 1.2 mm in average with six fiducials for that 2D/3D registra-tion computed in less than 2 s with 50% of outliers in the data.Future work could include the study of optimal configurations ofthe line fiducials.

Appendix A. Derivation of the scalar c

When n ¼ 4, a general solution for n ¼ rT1 rT

2 tT� �T is given in

Section 4.1, as a one-parameter family of solutions n ¼ Vzþ cv9.Since the first 6 components of n are the first two columns of therotation R, they may be expressed as affine functions of c:r1 ¼ a1 þ cb1 and r2 ¼ a2 þ cb2, with kr1k ¼ 1; kr2k ¼ 1; rT

1r2 ¼ 0.By substitution, we get

kb1k2c2 þ 2aT1b1cþ ka1k2 ¼ 1; ð29Þ

kb2k2c2 þ 2aT2b2cþ ka2k2 ¼ 1; ð30Þ

bT1b2c2 þ ðaT

1b2 þ aT2b1Þcþ aT

1a2 ¼ 0: ð31Þ

Eq. (31) is the orthogonality constraint and may be solved in orderto determine two solutions for c. Since the scalar c is also used todetermine the position vector t, a more robust estimation of c canbe found. The goal is to directly get the solution from a system ofredundant polynomials. The three second degree polynomials(29)–(31) are equivalent to

g1 g2 g3

g01 g02 g03g001 g002 g003

264375 c0

cc2

264375 ¼ GC ¼ 0: ð32Þ

These equations are linearly, but not algebraically, independent. Letthe SVD of G be URVT , the vector ½c0; c; c2�T is the right singular vec-tor associated to the smallest singular value of G. Then, c can becomputed using C2=C1, C3=C2 or the average of these two values.Note that n ¼ 3 would be the absolute minimum number of fea-tures necessary to recover R and t. However, it would lead to theresolution of three quadratic equations with three unknowns thathas up to eight solutions.

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