Journal of Microscopy, Vol. 167, Pt 2, August 1992, p p . 169-179. Received 4 November 1991; revised 5 March 1992; accepted 16 March 1992

Three-dimensional analysis of cell nucleus structures visualized by confocal scanning laser microscopy by P. KETT*t, B. G E I G E R * t , v. E H E M A N Nand " D. K O M I T O W S K I*Institute *, of Experimental Pathology, German Cancer Research Center, Im Neuenheimer Feld 280,D-6900 Heidelberg, and +Institute of Applied Physics I , University of Heidelberg, Albert- Ueberle-Strasse 3-5, 0-6900 Heidelberg, Germany

w O R D S . Automatic three-dimensional image analysis, three-dimensional structure of cell nuclei, confocal scanning laser microscopy.

KEY

SUMMARY

Studies of the three-dimensional (3-D) organization of cell nuclei are becoming increasingly important for the understanding of basic cellular events such as growth and differentiation. Modern methods of molecular biology, including in situ hybridization and immunofluorescence, allow the visualization of specific nuclear structures and the study of spatial arrangements of chromosome domains in interphase nuclei. Specific methods for labelling nuclear structures are used to develop computerized techniques for the automated analysis of the 3-D organization of cell nuclei. For this purpose, a coordinate system suitable for the analysis of tri-axial ellipsoidal nuclei is determined. High-resolution 3-D images are obtained using confocal scanning laser microscopy. The results demonstrate that with these methods it is possible to recognize the distribution of visualized structures and to obtain useful information regarding the 3-D organization of the nuclear structure of different cell systems. INTRODUCTION

Nuclei are highly specialized cell organelles with a complex structure which specifically changes during the processes of cell growth and differentiation. The most prominent nuclear structures are the envelope, nucleolus and chromosomes. Chromosomes can be identified only during cell division. Different hypotheses have been proposed to explain the organization of the chromosomes, and thus of the cell genome in the interphase between two cell divisions (Rabl, 1885; Boveri, 1909; Bennett, 1982; Blobel, 1985). By using modern methods of molecular biology, it has recently become possible to prove these hypotheses and obtain reliable information regarding the compartmentalization of nuclear structures (Rieder, 1982; Manuelidis, 1985; Rattner, 1986). Methods such as in situ hybridization and immunofluorescence have allowed distinct chromosomal structures to be labelled and arrangements of individual chromosomes and chromosomal subregions in interphase nuclei to be identified (Moroi et al., 1981; Brakenhoff et al., 1985; Brinkley et al., 1986; Moyzis et al., 1987; Lichter et al., 1988). It seems that the interphase chromosomes, although not identifiable as separate structures, occupy well-defined functional domains within the nucleus

0 1992 The Royal Microscopical Society 169

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(Hadlaczky et al., 1986). Changes in the spatial distribution of these domains characterize different cellular events, including differentiation and neoplasias (Brakenhoff et al., 1985; Manuelidis, 1985). Sophisticated methods of image analysis and threedimensional (3-D) image reconstruction have been used to quantify the spatial organization of cell nuclei (Ringertz et al., 1986; Manuelidis & Borden, 1988; van Dekken et al., 1990; Geiger et al., 1991). In this paper we present a fully automated system for the description of the 3-D organization of cell nuclei, by means of a coordinate system adapted to the individual form of the nuclei. Confocal scanning laser microscopy was used to obtain improved resolution. The aim is to develop methods for the evaluation of changes in the spatial arrangement of nuclear structures indicating altered cell functions. MATERIALS A N D METHODS

Specimens In order to develop algorithms with more general applicability, different cell systems, such as normal human lymphocytes, lympho-blastoid cells and PTK (Potorous tridactylis kidney) epithelial cells were used. T o visualize individual nuclear structures, different methods were applied. For centromeric regions of chromosome 1, in situ hybridization followed by fluorescein isothiocyanate (FITC) labelling was performed (Lichter e t al., 1988). Briefly, the cells cultured in RPMI (Roswell Park Memorial Institute) 1640 medium containing 10% FCS (fetal calf serum) were fixed with 10% paraformaldehyde. Biotin-labelled DNA centromeric probes of chromosome 1 mixed with salmon testis DNA dissolved in buffer containing 50% formamide, 10% dextran sulphate and 40% SSC (standard saline citrate) were used for hybridization. The solution was dropped onto glass slides and incubated for 12 min at 75°C. The samples were then incubated at 37°C in a moist chamber overnight, rinsed in 50% formamide, and once in SSC, and then stained with FITC-labelled anti-biotin antibodies. The total DNA was counterstained with propidium iodide (1 pg/ml). The specific DNA probes were kindly provided by T. Cremer (Institute of Human Genetics, University of Heidelberg). The kinetochores were visualized by the CREST method, detailed descriptions of which are given in Moroi et al. (1980) and Ehemann et al. (1988). The cells cultured in RPMI medium as described above were fixed in 10% paraformaldehyde. The dried slides were rinsed in PBS (phosphate-balanced saline) and incubated with antibody-solution containing 1:200 CREST serum, 0.5% BSA (bovine serum albumin) and 0 5 % Tween 20 in PBS for 40 min at 37°C. The samples were then washed thoroughly in PBS and stained with FITC-labelled anti-human antibodies for 40 min at 37°C. The autoantibodies were a gift from H. Ponstingl (German Cancer Research Center, Heidelberg). Data acquisition Data were obtained by a confocal scanning laser microscope (CSLM; Leica LaserTechnik, Heidelberg). The confocal principle permits optical sectioning with a real resolution along the optical axis (Pawley, 1990). This ensures that the image planes are free from out-of-focus projections. For the fluorescence stimulation of the F I T C dye, a wavelength of 488 nm from an air-cooled argon-ion laser (Siemens) was used. The fluorescence light was filtered in the range from 530 to 580 nm by a T K 510 and a T K 580 beamsplitter and a cut-off filter O G 530 (Leitz). The light intensity was detected by a photomultiplier. Three-dimensional images were directly recorded with a x 63, NA = 1.4, plan-apo oil-immersion objective (Leitz) by scanning at ascending z-levels, using a computer-controlled variable step-size stage (with a sampling rate of 150nm per voxel). The image size was 256 x 256 x (50-80) with an 8-bit grey-level resolution (256 grey

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levels). Using the variable hardware zoom of the CSLM, the image size was individually adapted to the nuclear size (or at least sampled at the Nyquist distance, which was about x y = 100 nm; Oppenheim et al., 1983, chapter 8 ) . The actual voxel sizes were computed automatically by the system and stored together with the 3-D image. Every plane was recorded three times for image averaging. The recording time for one nucleus with sixty-four z-planes was about 5 min. Computer hardware For control of the CSLM, an ELTEC E5-VME-bus system with a Motorola 68020 CPU,12-Mbyte RAM, 2-Mbyte frame buffer, 60-Mbyte hard disk, 400-Mbyte optical disk (WORM) and an ethernet interface were used. The system ran under the real-time multi-user and multi-task operating system os9. The recorded 3-D images were stored on the optical disk and then transferred via ethernet to a VAX 11/750computer which also handled the O S ~ N E Tethernet protocol of the E5. From the VAX 11/750 the image files were transferred to a Convex C210 for further computations. The Convex C2 10 computer, running under the UNIX operating system, was equipped with a vector processor, 64-Mbyte RAM and 6-Gbyte disk space. The 3-D visualization was calculated on a Silicon Graphics 4D80GT. IMAGE A N A L Y S I S

Three-dimensional segmentation of the nucleus A cluster analytical procedure was applied for automatic segmentation of the cell nuclei (Coleman & Andrews, 1976). Considering the wide variations in microscopical appearance of the nuclei, this procedure enabled us to discriminate the region occupied by the nucleus without any prior information (Fig. 1). The grey values of the voxels were chosen as the only discrimination feature. T o initiate cluster centres, the 20th and 80th percentiles of the grey-level histograms were used. At the beginning of cluster generation each of the voxels in the space was assigned to one of the initial clusters; we computed the Euclidian distance of the voxel to the cluster centres for this purpose. The mean grey values of the clusters generate new cluster centres and induce reorganization of the initial clusters. This procedure was iteratively repeated until steady state (no further changes in cluster centres) was reached. Fewer than ten iterations was usually sufficient to reach stability. T o eliminate the influence of noise effects, the segmentation was post-processed. This was accomplished by applying a 3-D filter to the cluster-labelled image, in which the background had a grey value of 0 and the identified nuclear region a grey value of 1. For our grid topology, the voxel neighbourhood is 1 voxel in all directions (3 x 3 x 3 - 1 = 26 voxels). The filter tested the probability of the relation to the clusters for each voxel. The probability that a voxel belongs to a cluster i is proportional to the number n; of voxels surrounding the tested voxel, which belongs to cluster i . The tested voxel is assigned to the cluster with the highest probability. The 3-D filter was also applied to the original image for further processing by the hierarchical object segmentation. Low grey-level regions within the nucleus are classified as belonging to the background after the grey-value-orientated cluster-segmentation. These regions were detected in the cluster-labelled image and were filled with the label of nucleus cluster. The result was a homogeneous 3-D nuclear region. Form-speciJc coordinate system As each nucleus has an individual 3-D geometry, to determine the position of the nuclear structures a coordinate system independent from that of the 3-D image matrix is required. The nuclear surface was approximated by an ellipsoid to introduce such a coordinate system. The three half-axes of this ellipsoid define the Euclidean space for

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the Cartesian coordinate system (Fig. 2). In order to measure how the ellipsoid fitted to the surface of the nucleus, a quality factor (QF), the root-mean-square deviation of the number of boundary voxels to the approximated ellipsoid, was used. A typical value for QFin our investigation was 0.2, i.e. on average, the actual boundary was 0.2 voxels away from the fitted ellipsoid. The ellipsoid-fitting method in our studies actually combines the methods of multiple non-linear regression analysis (Sachs, 1978, pp. 344-356) with a main axis transformation (Bronstein 8z Semendjajew, 1981, pp. 232-233; see Appendix). The half-axes are given in the implicit order of a > b > c, which permits direct comparison of nuclear eccentricities. Each object within the nucleus (see below) can be localized with the coordinate system introduced above. Hierarchical object segmentation algorithm A special local adaptive thresholding, which segments hierarchically ordered objects, was developed to detect the labelled structures within the segmented nucleus. Starting at the local area of the basic object (the nucleus background), the grey-value threshold was progressively increased. This was performed until changes in the object, such as splitting or a noticeable decrease in size, occurred. The regions found by this threshold were treated as distinct objects within the nuclear background. This procedure was then applied recursively to every object to search for new objects. Each of these quantitative parameters (i.e. volume, position, range of grey values and the mean of the grey values) was determined. The result, after all iterations, was a tree graph of subobjects within the nucleus. The flow chart of the parameterized algorithm is shown in Fig. 3. RESULTS

The two successive steps of analysis-grey-value cluster image binarization and local recursive object detection-were applied to identify both nucleus and nuclear structure. With the ellipsoid approximation and introduction of a nucleus-specific coordinate system the topography of the labelled objects within the nucleus was specified. Figure 4(a) shows the segmentation of typical nuclei with regular shape. Figure 4 (b,c) displays the labels for the nucleus and the two chromosome regions. The computed ellipsoid is overlain on the original nuclear shape. The whole 3-D data set is used for the fitting. Even in the z-direction the nuclear region detected appears to be homogeneous. Although the form of the nucleus is not exactly ellipsoidal, the extracted individual nuclear coordinate system ensures an adequate specification of the shape. The geometrical proportions are well preserved. The hierarchical object segmentation makes it possible to detect the two objects as different regions, although they are situated close together. Figure 5 demonstrates the reconstructed nucleus in a pseudo 3-D visualization. The upper part of the nuclear boundary has been removed using an appropriate setting of a virtual cutting plane. This provides a view inside the nucleus. DISCUSSION

It is generally accepted that there is a close association between spatial organization of nuclear structures and nuclear functions. Quantitative data defining the topology of nuclear structures are important for the recognition of functional compartmentalizaFig. 1. (a) Equatorial slice (focal plane) of the original image at z = 28. (b) Vertical slice of the original image cut at x = 128. (c) Vertical slice (focal plane) of the original image cut at y = 128. (d) Segmentation of the nucleus by grey-value cluster analysis ( z = 28).

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X

I Fig. 2. Scheme of theellipsoid model.

Start of recursion Start threshold = lowest grey value of base object + 1 Threshold =start threshold )eat until maximum grey value of base object is reached 'hreshold base object (IIvoxels obtained may form daughter obiects If no voxels obtained, base object is last object and indivisible. Store in resun list. Leave recursion level. Untainea voxels form only one connected component (i.e. one object). Is the volume of the new object less/equal a certain

Increment threshold by certain amount.

Base object is independent object Store in resuil list. Daughter object is new base object for next recursion level.

Divide base object into daughter objects Daughter objects will be base objects in next recursion level.

Ease object is independentobjecl Store in resun list. Daughter objects are new base objectsfor next recursion level.

Fig. 3. Schematic flow chart of the recursive object segmentation algorithm.

tion in the nuclei. Such data could be provided by the techniques for computerized analysis of cell nucleus images discussed in this paper. Our studies have focused on 3-D analyses of images obtained by confocal scanning laser microscopy. For the purpose of this analysis we have defined a coordinate system that considers individual variability in the shape of nuclei. Programs for the localization of labelled objects within the nuclei Fig. 4. (a) Segmented and labelled image ( z = 28); the smaller spots located at the broader plateau are assumed to be the two chromosomes (lower part of the nucleus). (b) Ellipsoid approximation to the nucleus and the two chromosomes ( z = 28; the other labels were discarded). (c) Ellipsoid approximation to the nucleus and the two chromosomes (vertical slice, y = 128; the other labels were discarded).

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Fig. 5. Surface visualization of the nucleus and two segmented chromosomes. The upper part of the nucleus has been removed using a virtual cutting frame (white line) from the rendering program.

were also developed. In this respect, the specificity of the programs for 3-D analysis should be emphasized. Although occasionally the 3-D analysis is considered to be only a result of the 2-D analyses carried out on serial sections through the object, there are major differences between both types of analyses. Algorithms predominantly applied to 2-D analyses, such as those given by Serra (1982, pp. 414-478, 1988, pp. 181-202) could be adapted for 3-D analyses; however, they are extremely time-consuming. Thus, the development of effective algorithms specifically designed for 3-D analyses was an important part of our studies. The cluster algorithm for binarization of CSLM images should also be mentioned. The algorithm clearly distinguishes the nuclear region, visualized with ethidium bromide as a counterstaining to the FITC-labelled structures, and background. Problems related to the size of the objects arise when attempting to use cluster algorithms for segmentation of the structures included in the nuclear region. This is because small objects do not create distinguishable modes in the histogram. Application of the cluster algorithm is also limited in the segmentation of objects that belong to a complex structure. Binary images obtained with cluster analytical techniques and improved by means of image equalization are useful in analysis of nuclei with approximately regular shape. Analysing topologically more complex nuclei shapes, such as those typical for neoplasias, requires other methods (VOSS,1988, pp. 73-101). The final goal of the 3-D analysis of cell nucleus images is the recognition of spatial relationships between specifically labelled structures. T o establish adequate models capable of describing positional relationships between the nuclear structures we have used shape approximation by fitting an ellipsoid. Prior knowledge indicates that most nuclei, especially those of cultured cells, tend to have a boundary which is, piece-bypiece, convex. The nuclei, although oval, are more or less irregular objects, and the fitting of an ellipsoid is the best compromise amongst different non-ideal solutions for

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the approximation of cell nuclear shape. In addition, a measure of the quality of ellipsoid-fitting may be obtained by the root mean square deviation. With near spherical nuclei, the coordinate system is indeterminate, and the voxel system can be applied. The origin of the system, however, should be shifted to the centre of the sphere. Although the analysis is confined to the positioning of singular specifically labelled nuclear structures, the object detection is a critical step in the 3-D analysis. T h e reason is the hierarchical organization of nuclear structure in which large objects and the small subobjects within them must be distinguished. Examples are chromosomal domains and subchromosomal structures, beginning from centromeres and kinetochores and going down to DNA segments of only a few kilobases. Consequently, we developed a global model for nuclear structure, in which nuclear space, together with included objects, is presented as a complex of stacked 3-D plateaus with defined grey levels. The segmentation algorithm identifies the objects from the bottom to the top (recursive approach) by different parameters, such as changes in the size or the difference between the maximal grey level in the object and the threshold value. Through appropriately chosen parameters even small voxel deviations, which could be considered as noise, were suppressed. This is a heuristic approach that relies on subjectively defined parameters. T o select these parameters prior knowledge, gathered from observations by conventional microscopy, is important. As an alternative to heuristic methods, theoretically founded methods for 2-D segmentation of objects with similar characteristics should be taken into consideration. The disadvantage of these 2-D methods adapted for 3-D segmentation is the extremely long time for computation. The anisotropy of the voxels is another problem which should be considered by evaluating the capabilities of the methods used for 3-D object detections. The anisotropy relates to the sampling range in the z-direction and to the 26-connectivity imposed by our algorithm. There are major difficulties involved in eliminating the anisotropy due to z-direction sampling. Every interpolation made for this purpose is associated with a blurring of the recorded intensities. In regard to 26-connectivityY several approaches were proposed to overcome the anisotropic character of the augmented neighbourhood of the cubic lattice. There are some special detector systems which deliver a hexagonal grid of pixels. However, to keep this isotropy in the third dimension, the images must be shifted from plane to plane. Because most detector systems deliver a quadratic grid, different software approaches to overcome the anisotropy have been reported (Acharya & Laurette, 1987; Acharya e t al., 1990; Carvajal-Gonzalez e t al., 1990; Meyer, 1990, 1992; Conan et al., 1990, 1992). In our method the anisotropy occurs where neighbourhoods are involved, i.e. in filter operations. T o eliminate this anisotropy related to 26-connectivityYa distance weight for the corner voxels has been used. Despite the restrictions discussed above, the advantages of the algorithms presented are obvious. The fully automated segmentation of the nuclei from different cell systems and the ability to provide quantitative assessments of the topology of specifically labelled nuclear structures should be mentioned in particular. This might be accomplished by investigating topographically relevant parameters (Konig et al., 1991) which could be used for statistical analysis. REFERENCES Acharya, R.S., Cheng, P.C., Samarabandu, J., Chen, L.H. & Summers, R.G. (1990) Multidimensional image analysis for confocal microscopy. Proc. Roy. Microsc. SOC.25, S48. Acharya, R.S. & Laurette, R. (1987) Cuboctahedric sampling mesh for mathematical morphology in three dimensions. Appl. Control, Filtering and Signal Processing, Proc. IASTED, pp. 191-193. Academic Press, London. Bennett, M.D. (1982) Nucleotypic basis of the spatial ordering of chromosomes in eukaryotes and the

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implicationsof the order for genome evolution and phenotypic variation. Genome Evolution (ed. by G. A. Dover and R. B. Flavell), pp. 239-260. Academic Press, London. Blobel, G. (1985) Gene gating: a hypothesis. Proc. Natl. Acad. Sci. U S A , 82, 8527-8529. Boveri, T. (1909) Die Blastomerenkernevon Ascaris megalocephala und die Theorie der Chromosomenindividualitt. Arch. exp. Zellforschung, 3, 181-268. Brakenhoff, G.J., van der Voort, H.T.M., van Spronsen, E.A., Linnemans, W.A. & Nanninga, N. (1985) Three dimensional chromatin distribution in neuroblastoma nuclei shown by confocal scanning microscopy. Nature, 317,748-749. Brinkley, R.R., Brenner, S.L., Hall, J.M., Tusson, A., Balczon, R.D. & Valdivia, M. (1986) Arrangement of kinetochores in mouse cells during meiosis and spermiogenesis. Chromosoma, 94, 309-3 17. Bronstein, I.N. & Semendjajew, K.A. (1981) Taschenbuch der Mathematik. Verlag Harri Deutsch, Thun. Carvajal-Gonzalez, S., Rigaut, J.P., Vassy, J. & Konig, D. (1990) Three-dimensional architecture analysis in cellular pathology by confocal microscopyand point process modelling. Proc. Roy. Microsc. SOL.25, S49. Coleman, G.B. & Andrews, H.C. (1976) Image segmentation and clustering. Proc. IEEE, 67,773-785. Conan, V., Gesbert, S., Howard, C.V., Jeulin, D., Meyer, F. & Renard, D. (1992) Geostatistical and morphological methods applied to 3-D microscopy. J. Microsc. 166, 169-184. Conan, V., Howard, V., Jeulin, D., Renard, D. & Curnmins, P. (1990) Improvement of 3D confocal microscope images by geostatistical filters. Proc. Roy. Microsc. SOL.25, S48. van Dekken, H., van Rotterdam, A., Jonker, R., van der Voort, H.T.M., Brakenhoff, G.J. & Baumann, J.G.J. (1990) Confocal microscopy as a tool for the study of the intranuclear topography of chromosomes. 3. Microsc. 158, 207-214. Ehemann, V., Walz, C.H., Kett, P. & Komitowski, D. (1988) Immunocytochemical labelling of the kinetochores in several cell-lines and analysis by a confocal laser microscope. Eur. J. Cell. Biol. 46, Suppl. 22, 18. Geiger, B., Komitowski, D., Jauch, A., Hausmann, M. & Cremer, C. (1991) Optional sectioning and 3Dimage reconstruction to determine the volume of specific chromosome regions in human interphase cell nuclei. Optik, 86, 113-1 19. Hadlaczky, G.Y., Went, M. & Ringertz, N.R. (1986) Direct evidence for the non-random localization of mammalian chromosomes in the interphase nucleus. Exp. Cell. Res. 167, 1-15. Konig, D., Carvajal-Gonzalez, S., Downs, A.M., Vassy, J. & Rigaut, J.P. (1991) Modelling and analysis of 3-D arrangements of particles by point processes with examples of application to biological data obtained by confocal scanning light microscopy. J. Microsc. 161,405-433. Lichter, P., Cremer, T., Borden, H., Manuelidis, L. &Ward, D.C. (1988) Delineation of individual human chromosomes in metaphase and interphase cells by in situ suppression hybridization using recombinant DNA libraries. Hum. Genet. 80, 224-234. Manuelidis, L. (1985) Individual interphase domains revealed by in situ hybridization. Hum. Genet. 71, 288-293.

Manuelidis, L. & Borden, J. (1988) Reproducible compartmentalization of individual chromosome domains in human CNS cells revealed by in situ hybridization and three dimensional reconstruction. Chromosoma, 96,397-410. Meyer, F. (1990) 3D mathematical morphology. Proc. Roy. Microsc. SOL.25, S47. Meyer, F. (1992) Mathematical morphology: from two dimensions to three dimensions. J. Microsc. 165, 5-28.

Moroi, Y., Harunann, A,, Nakane, P.K. & Tan, E.M. (1981) Distribution of kinetochore (centromere) antigen in mammalian cell nuclei. 3. Cell Biol. 90, 254-259. Moroi, Y., Peebels, C., Fityer, M., Steigerwald, J. & Tan, E.M. (1980) Antibody to centromere (kinetochores) in scleroderma sera. Proc. Natl. Acad. Sci. U S A , 77, 1627-1631. Moyzis, R.K., Albright, K.L. & Batholdi, M.F. (1987) Human chromosome-specific repetitive DNA sequences: novel markers for genetic analysis. Chromosoma, 95, 375-386. Oppenheim, A.V., Willsky, A.S. &Young, I.T. (1983) Signals and Systems. Prentice Hall, Englewood Cliffs. Pawley, J. (1990) Fundamental limits in confocal microscopy. Handbook of Biological Confocal Microscopy (ed. by J. B. Pawley), pp. 15-26. Plenum Press, New York. Rabl, C. (1885) ober Zellteilung. Morph. Jb. 10, 214-330. Rattner, J.B. (1986) Organization within the mammalian kinetochore. Chromosoma, 93, 515-520. Rieder, C.L. (1982) The formation structure and composition of the mammalian kinetochore and kinetochore fiber. Int. Rev. Cytol. 79, 1-59. Ringertz, G., Hadlaczky, G., Hallman, H., Nyman, U., Petterson, I. & Sharp, G.C. (1986) Computer analysis of the distribution of nuclear antigens: studies on the spatial and functional organization of the interphase nucleus. J. Cell. Sci. Suppl. 4, 11-28. Sachs, L. (1978) Angewandte Statistik. Springer-Verlag, Berlin. Serra, J. (1982) Image Analysis and Marhematical Morphology. Academic Press, London. Serra, J. (1988) Image Analysis and Mathematical Morphology, Theoretical Advances. Academic Press, London. Voss, K. (1988) Theoretische Grundlagen der digitalen Bildverarbeitung. Akademie-Verlag, Berlin.

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APPENDIX

Using the general ellipsoid equation with the coefficients (a0 . . . as): aOx2 a l y 2

+ a2z2+ a3xy + a4xz + a s y z + a& + a7y +

= 1,

and all surface voxel coordinates, a regression analysis (least-squares method) was made. As a result we obtained a linear equation system with nine equations. With the coefficients (ao. . . as), the linear translation vector a, could be extracted by solving

x,

(

as12

a0

a1

4 2 a512

4 2

a2

4 2

1

= -(a69

a7,

ad.

Transforming the surface coordinates by vector x,,

x’= x - x,, the ellipsoid equation was reduced to box”

+ bly” + b&’ + b 3 ~ ‘ y+‘ b4X’Z’ + b5Y’Z’ = 1.

A second regression analysis, analogous to the first, was performed to compute the coefficients (bo . . . b5).Written in the matrix form:

X’BX‘‘= 1 ,

This matrix was diagonalized by a main-axis transformation to

with

x”cx‘ - 1. f

With the eigenvalues (Ao, A1,

A’) I$’

t

in the ellipsoid equation,

+ Ai’2 + 1’;”

the half-axes (a, b, c) were determined as a = A-0 l / z, b = 2 1-

1/2 ,

,

=1

=