In: Proc. Intern. Conf. on Biomedical Engineering (Biomed) 2006, Kuala Lumpur, Malaysia, 11-14 December 2006. - 1 -

Warping a Neuro-Anatomy Atlas on 3D MRI Data with Radial Basis Functions

H.E. Bennink, J.M. Korbeeck, B.J. Janssen, B.M. ter Haar Romeny

Eindhoven University of Technology - Department of Biomedical Engineering Group Biomedical Image Analysis

Eindhoven, the Netherlands

Abstract— Navigation for neurosurgical procedures must be highly accurate. Often small structures are hardly seen on pre-operative scans. Fitting a 3D electronic neuro-anatomical atlas on the data assists with the localization of small structures and dim outlines. During surgery also brainshifts occurs. With intra-operative MRI the pre-operative MRI can be warped to the real 3D situation. The paper describes a general 3D landmark-based warping method, based on radial basis functions (thin plate splines) for data of any number of dimensions, including all code in Mathematica.

Keywords— Brain atlas, image warping, radial ba-sis functions, neurosurgery, computer vision

I. INTRODUCTION

Neurosurgical operations require very precise naviga-tion. Not only during resection surgery of brain tumors, also for deep brain stimulation (DBS), i.e. the placement of electrodes for the electrical stimulation of some spe-cific deep-brain structures. In particular, the placement of electrodes for stimulation of the Nucleus Subthalamicus (STN) in the deep brain may enable Parkinson patients to reduce their spontaneous tremor, which is often invalidat-ing them.

Today, with Magnetic Resonance Imaging and Com-puted Tomography scanners it is possible to image the patient in three dimensions at high resolution, typically 0.5-1 mm in each direction.

Fig. 1. Open MRI scanner on the neurosurgical theatre enables imag-ing during surgery (Medtronic - Odin N20, 0.15 Tesla, Maastricht

University Hospital, the Netherlands).

However, two major problems exist: - it is often difficult to find the many individual brain

nuclei on these scans due to the low contrast and the noise. The mapping of an electronic brain atlas on the scans substantially helps the navigation (fig. 2) [3]. The atlas is based on a mean patient, and will not fit the data. So a 3D warping method is necessary.

- When the skull is opened, brain shift occurs due to pressure changes and loss of liquids. The pre-operative data do no longer correspond to the real situation, making precise navigation, e.g. with avoidance of bloodvessels, difficult. Recently, an open mobile low-field MRI scan-ner (fig. 1) has been installed in the Maastricht University Hospital (the first in the Netherlands), to image the pa-tient after the trepanation. Due to the lower image quality of the intra-operative scans, registration and warping of the pre-operative scans is necessary.

Fig. 2. Electronic brain atlas mapping on Magnetic Resonance slices.

Fully automatic registration and 3D warping on the grayscale images is difficult, if not impossible. In this paper we describe a warping based on registration of manually selected anatomical landmarks in 3D. Neuro-surgeons are very familiar with such landmarks, and good descriptions [6][7] and neuro-anatomical atlases [8] exist. The 3D deformation field is constructed by means of an interpolation based on thin plate splines [1],[2], a special form of the radial basis functions. In Mathematica [16] the code is very short, readable and efficient.

In: Proc. Intern. Conf. on Biomedical Engineering (Biomed) 2006, Kuala Lumpur, Malaysia, 11-14 December 2006. - 2 -

II. THIN PLATE SPLINES

In order to warp a brain atlas onto an MRI scan of the brain we identify so called landmarks of which the corre-spondence between the atlas and the scan is known. Be-cause of this correspondence we can find the vector field that can be used for the warping procedure. The vector field is found by interpolating the know vectors at the location of the landmarks. Hence we first introduce inter-polation of arbitrarily spaced points in multiple dimen-sions.

Imagine a thin metal plate of infinite extent that is

fixed at certain points },{ iii yxx

at the heights

ifi , (and neglect gravity). The metal plate has a

shape such that its surface is minimally bent. The bending energy of such a plate ),( yxs equals

dxdyy

yxs

yx

yxs

x

yxssEbend 2

22

2

2 ),(),(2

),()(

2

This is an instance of a semi-norm proposed by Jean Duchon. In order to find the shape of a plate that is fixed at the points mentioned above we will have to find the minimizer of this equation such that the constraints,

ifxs ii ,)(

are met. The shape can be approxi-

mated by finding that s that minimizes

ibendii sEfxssE )(|)(|)( 2

The first part of this convex energy functional makes sure the constraints are met and the second part smooths the result.

is a parameter that controlls the quality (deviation from the constraints) of the approximation. A smaller value of

results in a better approximation, ultimately achieving interpolation when

tends to 0. Duchon was one of the first who recognised that the solu-tion of the variational problem can be written in the form

)(||)(||)( xpxxwxsi

ii

(1)

Here odd

even

dkif

dkif

r

rrr

dk

dk

k 2

2)log()(

2

2

2

is a so called radial basis function. We neglect k, which is a setting for the order of the norm that will be minimized

)2(k . d sets the number of dimensions, for our fidu-

cial metal plate 2d . )(xp is a polynomial that lies in

the null space of the differential operator that appears in the bending energy. In our example it thus takes the form

yaxaaxp 321)( . In order to find the interpolating

(or approximating) function of the form of equation (3) we can simply solve a linear system of equations. We are searching for the solution of

0

f

a

w

OQ

QIBT

(2)

where ||)(||, jiji xxB , yxQ iji ,,1,, I is

the identity matrix and O =

000

000

000. Those w and

a that solve the linear system of equation (2) can be plugged into equation (1) in order to find the expression for the shape we are looking for.

III. THE LANDMARKS

Well-known neuro-surgical atlases exist, such as by Talairach-Tournoux [11][12]. Recently, a family of elec-tronic atlases has been brought to market, such as the Cerify series of atlases [8] by Nowinski et al. In these atlases landmark coordinates can conveniently be found [6],[7]. In this study we selected 148 well-defined land-marks in coronal, sagittal and transversal slices in the atlas and in 3D in the patient’s MRI dataset. Some exam-ples are shown in fig. 3.

Fig. 3. Examples of landmark points in coronal slice TT88c / +24mm (top) and in sagittal slice TT88s / R-4mm (bottom). In total 148

landmarks were set.

IV. IMPLEMENTATION

We implemented the method that is outlined above (for a 2D image/metal plate) for N dimensions in Mathe-matica [16] (code: see Appendix). The module takes input similar to the built-in Interpolation function and returns a compiled function.

There is a great advantage to design this method in the high level software system Mathematica. The code is amazingly compact, and can be easily transferred and made public.

D30L D32L

D28L

D31L

D29L

D30L D32L

D28L

D31L

D29L

TT88s / L+5mm

D30LD30L D32LD32L

D28LD28L

D31LD31L

D29LD29L

D30LD30L D32LD32L

D28LD28L

D31LD31L

D29LD29L

TT88s / L+5mm

In: Proc. Intern. Conf. on Biomedical Engineering (Biomed) 2006, Kuala Lumpur, Malaysia, 11-14 December 2006. - 3 -

A 2D EXAMPLE

The Mathematica code for the radial basis function in-terpolation is given in the Appendix. Below, the example code (fig. 4) shows the local warping on a 2D image by randomly moving a small set of 7 randomly chosen landmark points. The radial basis function is a Bessel function in this case (to show the versatility).

source Table Random Real, 10, 90 , 7 , 2 ;

target source Table Random Real, 5, 5 , 7 , 2 ;

grid Table x, y , x, 1, 100, 2.5 , y, 1, 100, 2.5 ;

Show Graphics Gray, Line grid, Line grid , Blue,

PointSize 0.02 , Point source, Red,

Circle #1, 2 & target , AspectRatio 1,

Frame True ;

rbn v_ : If # 0, 0.5,BesselJ 1, 1

5#

1

5#

& v.v ;

rbi RadialBasisInterpolation source, target ,

RadialBasisNorm rbn ;

newgrid Map rbi #1 &, grid, 2 ;

Show Graphics Gray, Line newgrid, Line newgrid ,

PointSize 0.02 , Blue, Point rbi #1 & source,

Red, Circle #, 2 & target , AspectRatio 1,

Frame True ;

Fig. 4. Mathematica code to warp a grid based on 7 random location shifts. The main code, which is suitable for data of any dimensionality,

is given in the Appendix.

After the warping the full new image is interpolated towards the new coordinates. The new points are placed exactly at the location of the old points, the environment is smoothly deformed by radial basis interpolation. The deformation of the coordinate grid can be clearly appreci-ated (fig. 5).

0 20 40 60 80 100

0

20

40

60

80

100

0 20 40 60 80 100

0

20

40

60

80

Fig. 5 Radial basis interpolation of a regular grid, based on the random motion of 7 landmarks.

V. RESULTS

In total 148 landmarks were set in both the 3D MRI data and the electronic atlas, in coronal, sagittal and tranversal slices. After 3D warping by thin plate splines, the result was good around the set landmarks, but unsatisfactory in regions far away from the landmarks. New points were added in those regions (typically 3-10) for a second warp-

ing iteration. After three iterations the result was satisfac-tory. Some examples of the warped atlas on the data are shown in fig. 6.

Fig. 6. Left: integration of the warping result in the Cerefy students atlas. Right: one of the transversal slices of the final result.

VI. CONCLUSIONS

An elegant and efficient method is described to warp data in any dimensions based on sets of corresponding landmarks. Automated landmark extraction should be rthe next step. Fast implementations of this method are appearing, based on graphics card hardware, and splines interpolation. Code in Mathematica [16] is short, easily transferable, educational and applicable to data of any dimension.

ACKNOWLEDGEMENT

The authors like to thank the A*Star Bio-Informatics Institute in Singapore for making available the Cerify Student brain atlas, and invaluable help in the landmark allocation. We thank Markus van Almsick for the Math-VisionTools library (www.mathvisiontools.net).

VII. REFERENCES

[1] Bookstein, F. (1989) Principal warps: thin-plate splines and the decomposition of deformations. IEEE Trans. Pattern Analysis Ma-chine Intelligence 11(6): 567-585.

[2] Carr J.C., Fright W.R., Beatson R.K. (1997) Surface interpolation with radial basis functions for medical imaging. IEEE transactions on medical imaging, Vol. 16, No. 1.

[3] Free S.L., O’Higgins P, Maudgil D.D., Dryden I.L., (2001) Land-mark-Based morphometrics of the normal adult brain using MRI. NeuroImage 13 801-813.

[4] Grachev I.D., Berdichevsky D., Rauch S.L. (1999) A method for assessing the accuracy of intersubject registration of the human brain using anatomic landmarks. NeuroImage 9, 250-268.

[5] Levin D., Dey D., Slomka P.J. (2004) Acceleration of 3D, nonlin-ear warping using standard video graphics hardware: implementa-tion and initial validation. Computerized Medical Imaging and Graphics 28: 471-483.

[6] Nowinski W.L., Thirunavuukarasuu A. (2001) Atlas-assisted localization analysis of functional images. Medical Image Analy-sis 5, 207-220.

[7] Nowinski W.L., (2001) Technical Report: Modified Talairach Landmarks. Acta Neurochir 143: 1045- 1057 Wien.

In: Proc. Intern. Conf. on Biomedical Engineering (Biomed) 2006, Kuala Lumpur, Malaysia, 11-14 December 2006. - 4 -

[8] Nowinski W.L., Belov D. (2003) The Cerefy Neuroradiology Atlas: a Talairach-Tournoux atlas-based tool for analysis of neuro-images available over the internet. NeuroImage 20, 50-57.

[9] Rizzo G., Scifo P., Gilardi M.C. (1997) Matching a computerized brain atlas to multimodal medical images, NeuroImage 6, 59-69.

[10] Talairach J., Tournoux P. (1988) Co-planar stereotactic atlas of the human brain. Georg Thiem Verlag/Thieme Medical Publishers, Stuttgart New York.

[11] Talairach J., Tournoux P. (1993) Referentially oriented cerebral MRI anatomy. Atlas of stereotaxic anatomical correlations for gray and white matter. GeorgThieme Verlag/Thieme Medical Pub-lishers, Stuttgart New York.

[12] Verard L., Allain P., Travere J.M. (1997) Fully automatic identifi-cation of AC and PC landmarks on brain MRI using scene analy-sis. IEEE transactions on medical imaging, Vol. 16, No. 5.

[13] M.Arigovindan,M.Sühling,P.Hunziker,and M.Unser. Variational image reconstruction from arbitrarily spaced samples:A fast mul-tiresolution spline solution. IEEE-TIP,14(4):450-460,April 2005.

[14] J. Duchon, in Constructive theory of functions of several vari-ables,W. Schempp and K. Zeller, editors. Splines minimizing rota-tion-invariant semi-norms in Sobolev spaces. Springer Verlag, 1976.

[15] J. Kybic, T. Blu, and M. Unser. Generalized sampling: a varia-tional approach, part I: Theory. IEEE Transactions on Signal Proc-essing, 50:1965, 1976, August 2002.

[16] Mathematica 5.2, URL: www.wolfram.com.

VIII. APPENDIX: MATHEMATICA CODE

RadialBasisInterpolation[{x1,y1},{x2,y2},…,{xn,yn}]] con-structs a compiled approximate function that interpolates the y-value at any given x-value. xi and yi can be tensors in any number of dimensions. The interpolation function

is a function of the form: n

i ii xxwaxxy1

)()( .

Here a is the linear part of the interpolation function, iw is

the weight of basis ix and

is a function that calculates

the vector norm of the vector )( ixx . The implemented

radial basis norm functions are:

ThinPlateSplineNorm[p] (fig. 7) with dkp 2 and GaussianNorm[ ]. Thin plate splines are fundamental solutions of 0k .

In d dimensions these solutions are given by rrr dkd log)( 2)( for dk and d even and

dkd rr 2)( )(

for dk and d odd.

0.5 1 1.5 2

-0.2

0.2

0.4

0.6

0.8

1

1.2

Fig. 7. ThinPlateSplineNorm[p] functions for p = 2 (red), p = 3 (green) and p = 4 (blue).

The Gaussian norm function is implemented as a com-piled function calculating a Gaussian weighted vector

norm 2

2

2)(r

er .

Mathematica code:

Options RadialBasisInterpolation

RadialBasisNorm Automatic, Smoothness 0. ;

ThinPlateSplineNorm 1

Compile #1, _Real, 1 , #1.#1 ;

ThinPlateSplineNorm 2

Compile #1, _Real, 1 ,

If #1 0., #1 Log #1 , 0. & #1.#1 ;

ThinPlateSplineNorm p_?OddQ

Compile #1, _Real, 1 , #1.#1 p 2, p, _Integer ;

ThinPlateSplineNorm p_?EvenQ

Compile #1, _Real, 1 ,

If #1 0., #1p 2 Log #1 , 0. & #1.#1 ,

p, _Integer ;

GaussianNorm _ Compile #1, _Real, 1 ,

If #1 0.,#12 2 , 1. & #1.#1 , , _Real ;

RadialBasisInterpolation data_List, opts___?OptionQ :

Module points, values, n, pointDim, valueDims,

valueRank, , B, Q, O, A, , x, w, a, slots ,

, RadialBasisNorm, Smoothness . opts .

Options RadialBasisInterpolation ;

points, values N Transpose data ;

If Depth points 2, points List points ;

n, pointDim Dimensions points ;

valueDims Rest Dimensions values ;

valueRank Length valueDims ;

If Automatic,

ThinPlateSplineNorm Max 2, pointDim ;

O Table 0., pointDim 1 , pointDim 1 ;

B Outer #1 #2 &, points, points, 1 ;

Q Prepend #1, 1 & points;

A MapThread Join,

MapThread Join,B IdentityMatrix n Q

Q O;

b PadRight Transpose values,

RotateRight Range valueRank 1 ,

Append valueDims, n pointDim 1 , 0 ;

x Map LinearSolve A, #1 &, b, valueRank ;

w Map Take #1, n &, x, valueRank ;

a Map Take #1, pointDim 1 &, x, valueRank ;

slots Array Slot, pointDim ;

Compile #1, #3.#4 Transpose #5 #6 #2,

#4 _ , _Real, 0 & #1, _Real & slots,

a.Prepend slots, 1 , w, , slots, Transpose points

Address of the corresponding author:

Author: Bart M. ter Haar Romeny Institute: Eindhoven University of Technology Street: Den Dolech 2 – WH2.108 City: NL-5600 MB Eindhoven Country: the Netherlands Email: B.M.terHaarRomeny@tue.nl