European Journal of Radiology. 12 (1991) I l-16 Elsevier
EURRAD
11
00117
An alternative method of three-dimensional reconstruction from two-dimensional CT and MR data sets W. Wrazidlo,
H.J. Brambs, W. Lederer, S. Schneider,
B. Geiger and Ch. Fischer
Department of Diagnostic Radiology, University of Heidelberg, Heidelberg, F. R. G. (Received 21 April 1990; accepted
after revision 20 August 1990)
~__Key words: Hip, MRI; Liver, MRI; Magnetic resonance,
image processing
Abstract
Cross-sectional images for medical diagnosis and therapy are obtained by sonography, CT or MRI. We propose an alternative solution to the problem of constructing a set of cross-sectional contours from two-dimensional (2D) CT or MR data sets. The method reduces the problem of constructing a shape over the cross-sections to one of constructing a sequence of partial shapes, each of them connecting two cross-sections lying on adjacent planes. The solution makes use of a spatial mathematical formalism (Delaunay triangulation). MR investigations were carried out on different objects (hips, livers) to illustrate the two-dimensional MR data sets by the reconstruction method. The resulting images represent the original image data in a way that is more suitable for observation of 3D relationships than the conventional cross-sectional viewing model.
Introduction Modern medical imaging modalities, such as CT and MRI produce image data in slices and these are conventionally viewed as an array of two-dimensional grayscale slice images. The advantages of comprehension in three dimensions have spurred the development of methods of 3D display of this data. Several methods have been proposed in the literature [9, 12, 14-171. Two approaches are possible. In the first the data constitute a 3D image consisting of voxels. It is problematic to find and follow the interfaces of the voxels, which are on the boundary of the objects. However, frequently we do not have any 3D objects consisting only of planar contours extracted from the crosssectional images. An entirely new approach to reconstruct a volumetric model of an object represented by a finite number of planar MRI cross-sections is presented. This method can handle cases where the objects contain multiple contours and where the number of contours varies from one cross-section to the other. Address for reprints: Wiesloch-2, F.R.G. 0720-048X/91/$03.50
W. Wrazidlo.
M.D., Griinlingweg
0 1991 Elsevier Science Publishers
2, 6908
Unlike other methods [ 1,4-8, lo], our method does not construct the surface of the object, but its volume. At the end of the procedure, it will be easy to produce the surface of the object by looking at the boundary of the volume obtained. We describe the method and present a series of images from CT and MRI cross-sections to illustrate the potential uses of this method and to show the quality of images that can be produced. Materials and Methods Hip joints of healthy test persons as well as patients with liver hemangiomas were examined. Data were recorded with a Picker magnetic resonance tomograph (Vista MR-2055 HP) with a field strength of 1.0 Tesla. The MR examinations of the hip joints were made in coronal and sag&al slice orientation with 3-mm slice thickness without a gap. We used a gradient echo sequence with fat-water phase coherence (TE 14/TR 400) to display the hip joint cartilage with high signal strength (acquisition time about 8 min). For liver examinations, we used a spin-echo-multi-echo sequence with TE times up to 16 ms for diagnosis of liver hemangio-
B.V. (Biomedical
Division)
12
mas. The slice thickness was 8 mm without any gap and with axial slices (acquisition time about 12 min). The data stored on magnetic tape were then fed into a mainframe computer (IBM 4831) for decoding. The data were then transferred through a network (ether net) to the disk of a workstation (HP 9000 835 turbo SRX or SUN-3) and converted from the Picker format into a corresponding gray-scale image (Gray scale matrix). Finally, the desired MR image appeared within a few minutes on a high-resolution monitor (1024 x 1024) of the workstation. The contours of interest of the organ to be reconstructed (hip-joint cartilage, liver hemangiomas) were then recorded slice for slice by tracing with an electronic pen, since it was possible to correct the inputs as desired. A specially developed REPROS programming system then performed the reconstruction to form the wire-mesh model. The triangles of the wire-mesh model, which correspond to the surface of the organ, were coded with different colors with a computer graphics program developed by us (shading method), so that an enclosed organ surface could be displayed. The CPU times in seconds have been measured on a SUN-3 workstation. The number of tetrahedra as well as the CPU times vary almost linearly with the number of points. The CPU time was always less than 65 ms per point on a SUN-3 workstation. Total CPU times (2D-triangulation, 3Dtriangulation and reconstruction) were about 80 s for a wire-mesh model of the liver. The reconstruction times for a 3D display of the hips or of the livers were about 15 min on average, including the manual tracing of the organs. We have used an algorithm published for the first time in 1984 by Boissonnat [2, 31 for the 3D reconstruction of the CT or MRI images. In contrast to other previously known methods this algorithm reconstructs not only the outer skin but the entire volume of the body by triangulation of a sequence of sections which are prescribed in planes parallel to one another. The result is a polyhedron consisting of tetrahedra which satisfy the Delaunay condition [ 11, 181. The definition of the Delaunay triangulation is based on the Voronoi diagram. For a D-dimensional Euclidean space E and a set M of points M, . . . . . M,, the associated Voronoi diagram is a sequence (Vi . . . . . V,) of convex polyhedra covering E, where Vi consists of all the points of E that have Mi as a nearest points in the set M. Thus: vi = {P&EVj.l
Fig. 1. Voronoi diagram and its dual in a 2D case.
called the Delaunay triangulation of M. Fig. 1 shows an example of a Voronoi diagram and its dual in a 2D simple case. In the 3D case, it is possible to show that two elements are disjoint or have one vertex in common, or that they have two vertices and consequently the entire interface joining them. Moreover, the union of the elements of the Delaunay triangulation is equal to the interior of the convex hull of M. Because of the definition, when no five points are co-spherical, the circumspheres do not contain any point of M in their interior. In the case of more than five co-spherical points the elements can be broken down into several tetrahedra so that in every case the Delaunay triangulation is composed of tetrahedra. Moreover, the least three neighbors Mj are roughly the nearest neighbors of Mi in the different directions. Indeed, an inversion is considered
~j~N.d(P.M,)~d(P.M,)}
where d denotes the Euclidean distance. The geometrical dual of the Voronoi diagram obtained by linking the points M of which Voronoi polyhedra are adjacent, is
Fig. 2. CT image of the thoracic aorta. The contours of interest are traced with an electronic pen (arrows).
with Mi as center, which associates with a point Mi the point M;, on the line MiMj, whose distance from Mi satisfies the equation M,M x M&i = k x k. The reciprocal images of the half-spaces limited by the interfaces of the convex hull of the images M; . . . . . A4,‘,of M, . . . . . M, are the interiors of the spheres passing through Mi and three other points of M. Because such half-spaces are empty, the interiors of the corresponding spheres are also empty; moreover, these half-spaces are the only empty half-spaces passing through three points ofM, so the spheres are the Delaunay spheres passing through Mi. Thus, the neighbors of Mi are the points Mj whose images by an inversion with Mi as center are vertices of the convex hull of the images of Mi . . . . . A4,. Thus the Delaunay triangulation is a 3D-connected graph on M-embedded R3 which defines symmetrical and isotropic neighborhood relationships between the points. The Delaunay triangulation contains polyhedra and satisfies the first and second conditions of the definition of a polyhedron which is a decided advantage. Another advantage of the Delaunay triangulation is that it can be computed efficiently. Reconstruction
technique and results
Triangulation of a set of points in a plane is a Delaunay triangulation if the circumscribing circle of each of its triangles contains no further points. Accordingly, the circumscribing spheres in a 3D Delaunay tetrahedral triangulation are computed between two consecutive planes. The result is discoid polyhedra, the convex envelopes of the body slices sought for. All superfluous tetrahedra must then be removed in a second step the actual reconstruction step. In collaboration with the INRIA in France, we have developed a programming system REPROS (reconstruction from planar cross-sections) which implements the principle of Boissonnat’s reconstruction algorithm. REPROS is written in the programming languages C and Fortran. The Fortran modules handle the 2D Delaunay triangulation, the C module computes the 3D Delaunay triangulation and the subsequent reconstruction [ 131. The practical course of the reconstruction is demonstrated by taking the CT display of the thoracic aorta (Fig. 2) which slice for slice is traced with an electronic pen. The slices created with the electronic pen, are displayed in Plate I which lie plane-parallel to one another. Subsequently, the three-dimensional tetrahedral triangulation is computed and the organ is reconstructed to form a wire mesh model. Plate IIa shows the wire-mesh model with the triangles of the tetrahedra which form the organ surface. An enclosed surface is obtained by coloring the triangles, the so-called shading
Fig. 3. Coronal MR image of the right hip of an 35year-old volunteer. Hyaline cartilage of the femoral head shows a high signal intensity by using a gradient field echo sequence (arrow).
method, which is performed with a special computer graphics program (Plate IIb). One of 14 coronal MR tomographs of a test person’s hip is shown in Fig. 3. The hyaline joint cartilage is displayed with high signal strength by selection of a suitable gradient echo sequence (TE 14/TR 400). Plate IIIa shows the reconstructed three-dimensional wire-mesh model of the same hip joint cartilage. In comparison, Plate IIIb shows the same model after color-coding of all triangles. The head of the femur cartilage surface can be displayed less well in comparison to the wire-mesh model. However, compared to the wire-mesh model, the shape and position of the neck of the femur in relation to the acetabulum can be assessed better in the shading model. Apart from hip joints, we also reconstructed changes in the liver. Fig. 4a and b show MR tomograms of the liver with hemangiomas in the right lobe of the liver. The hemangiomas can be clearly displayed in shape and number and separated spatially in the wire-mesh model (Fig. 5a and b). Discussion Various organs can be correctly reconstructed threedimensionally by using a suitable reconstruction algorithm and developing corresponding programming systems. The method we have presented reconstructs not only the outer integument but the entire volume of an organ in comparison to other surface reconstruction methods from planar contours [ 1,4,5, 181. In this way,
Plate I. Slices of the thoracic aorta created with an electronic pen. Plate II. (a) Wire-mesh model ofthe thoracic aorta with the triangles of the tetrahedra. (b) Shading model with the surface obtained by coloring the triangles from (a). Vena cava (arrow).
Plate III. (a) Wire-mesh surface approximation of the same hip reconstructed from 14 coronal MR slices. The number of surface triangles are about 1000. The hyaline cartilage of the femoral head is well visualized. (b) Shaded model of the same hip. The shape of the hip is well visualized.
Fig. 4. Typical hemangiomas. (a) An SE (2000/160) image at 1.0 T demonstrates two hyperintensive lesions to the right lobe ofthe liver. (b) An SE (2000/160) image of the same patient shows a large hyperintensive lesion to the caudal part of the right lobe of the liver adjacent to the gallbladder (arrow).
even hip joints can be reconstructed three-dimensionally with a quality which enables an optimally matched artificial counterpart to be produced. The wire-mesh model is shaped as if the surface and the boundaries of the organ are reconstructed by tetrahedra. Depending on the clinical problem, either the wire-mesh model or the shading model is suitable for displaying the organ structure of interest, and the triangles can be coded with different colors. Relevant anatomical or pathological details can be emphasized by partial color-coding as shown in the examples. As shown in the example of the liver, the programming system reconstructs even complex surface structures correctly. Here, several contours in a plane are recognized clearly as such. Another advantage is that the time for reconstruction
is about 15 min, which is very reasonable with regard to its potential usefulness. The method is also suitable for reconstruction of CT images. The prerequisites for correct organ reconstruction are high-resolution MR images with slices as thin as possible. The advantages compared with the voxel technique are a clearly shorter computing time and considerably reduced storage capacity. This enables all models to be rotated in real time on the monitor along three axis. Of course it would be better to use the electronic pen only to roughly outline the object and set limits for density like the Ray-trace method, but a reconstructed Ray-trace model can not rotate in real time on the monitor. Future examples of application of the method pres-
Fig. 5. (a) Wire-mesh model ofthe same liver reconstructed from 14 transaxial MR slices. Orientation ofthe liver from caudal to cranial, where all hemangiomas can be displayed. The number of surface triangles are about 2500. (b) Rotation of the same model about 180” around the horizontal axis. Note the better spatial orientation with the large hemangioma in the caudal part of the right lobe of the liver.
16
ented by us are the development of simulation programs before corrective osteotomy in cases of injury to the femur head cartilage, reconstruction of the vascular tree of the liver for the assignment to segments of pathological liver processes before possible partial resection, as well as imaging of joints and of the heart in the tine mode. Besides the demonstrated examples, the method is also very useful for craniofacial reconstructive surgery [ 19,201. References Artzy E, Frieder G, Hermann G. The theory, design, implementation and evaluation of a three-dimensional surface detection algorithm. Comput. Graphics Image Process 1981; 15: i-24. Boissonnat J. Shape reconstruction from planar cross-sections. Comput Vision, Graphics Image Processing 1988; 44: l-29. Boissonnat J. Geometric surfaces for three-dimensional shape reconstruction. ACM Tram Graphics 1984; 3: 266-286. Chen L, Herman G, Reynolds R, Udupa J. Surface shading in the Cuberille environment. Comput Graph Appl 1985; 5: 33-43. Christiansen H, Sederberg T. Conversion of complex contour line definitions into polygonal element mosaics. Comput Graphics 1978; 13: 187-192. Fuchs H, Kedem Z, Uselton S. Optilam surface reconstruction from planar contours. Commun ACM 1977; 20: 693-702. Ganapathy S, Dennehy T. A new triangulation method for planar contours. Comput Graphics 1982; 16: 69-78. Gordan D, Reynolds R. Image space shading of threedimensional objects. Comput Vis 1985; 29: 361. 9 Herman G. Three-dimensional imaging on a CT or MR scanner. J Comput Assist Tomogr 1988; 12: 450-458.
10 Keppel E. Approximating
complex surfaces by triangulation of contour lines. IBM J Res Dev 1975; 19: 2-11. 11 Lee D, Schachter B. Two algorithsm for constructing a Delaunay triangulation. Intern J Comput Inform Sci 1980; 9: 219-242. 12 Levin D, Hu X, Tan K, Galhotra S. Surface of the brain: three-dimensional MR images created with volume rendering. Radiology 1989; 171: 277-280. 13 Mtiller H, Geiger B. Rekonstruktion komplexer K&per aus ebenen Schnitten und deren hochqualitative graphische Darstellung. In 17. GI-Jahrestagung, Proceedings, Informatik Fachberichte 156. Heidelberg: Springer Verlag, 1987; 571-583. 14 Pate D, Resnick D, Andre M. Perspective: three-dimensional imaging ofthe musculoskeletal system. AJR 1986; 147: 545-551. 15 Rusinek H, Mourino M, Firooznia H, Weinreb J, Chase N. Volumetric rendering of MR images. Radiology 1989; 171: 269-272. 16 Stimac G, Sundsten J, Prothero JS, Prothero JW, Gerlach R, Sorbonne R. Three-dimensional contour surfacing of the skull, face and brain from CT and MR images and from anatomic sections. AJR 1988; 151: 807-810. 17 Vannier M, Marsh J, Warren J. Three-dimensional reconstruction images for craniofacial surgical planning and evaluation. Radiology 1984; 150: 179-184. 18 Watson D. Computing the n-dimensional Delaunay tessellation with application to Voronoi polytopes. Comput J 1981; 24: 167-172. 19 Wrazidlo W, Schneider S, Brambs HJ, Richter GM, Kauffmann GW, Geiger B, Fischer Ch. New method of three-dimensional reconstruction from two-dimensional MR data jets. RSNA, November 1989, Works in progress-physics, Chicago. 20 Wrazidlo W, Lederer W, Kauffmann GW, Richter GH, Schneider S, Geiger B, Fischer Ch. 3D-MR-Rekonstruktionen und deren hochqualitative graphische Darstellung. 6th Grazer Radiologisches Symposium, October 1989.
EURRAD
11
00117
An alternative method of three-dimensional reconstruction from two-dimensional CT and MR data sets W. Wrazidlo,
H.J. Brambs, W. Lederer, S. Schneider,
B. Geiger and Ch. Fischer
Department of Diagnostic Radiology, University of Heidelberg, Heidelberg, F. R. G. (Received 21 April 1990; accepted
after revision 20 August 1990)
~__Key words: Hip, MRI; Liver, MRI; Magnetic resonance,
image processing
Abstract
Cross-sectional images for medical diagnosis and therapy are obtained by sonography, CT or MRI. We propose an alternative solution to the problem of constructing a set of cross-sectional contours from two-dimensional (2D) CT or MR data sets. The method reduces the problem of constructing a shape over the cross-sections to one of constructing a sequence of partial shapes, each of them connecting two cross-sections lying on adjacent planes. The solution makes use of a spatial mathematical formalism (Delaunay triangulation). MR investigations were carried out on different objects (hips, livers) to illustrate the two-dimensional MR data sets by the reconstruction method. The resulting images represent the original image data in a way that is more suitable for observation of 3D relationships than the conventional cross-sectional viewing model.
Introduction Modern medical imaging modalities, such as CT and MRI produce image data in slices and these are conventionally viewed as an array of two-dimensional grayscale slice images. The advantages of comprehension in three dimensions have spurred the development of methods of 3D display of this data. Several methods have been proposed in the literature [9, 12, 14-171. Two approaches are possible. In the first the data constitute a 3D image consisting of voxels. It is problematic to find and follow the interfaces of the voxels, which are on the boundary of the objects. However, frequently we do not have any 3D objects consisting only of planar contours extracted from the crosssectional images. An entirely new approach to reconstruct a volumetric model of an object represented by a finite number of planar MRI cross-sections is presented. This method can handle cases where the objects contain multiple contours and where the number of contours varies from one cross-section to the other. Address for reprints: Wiesloch-2, F.R.G. 0720-048X/91/$03.50
W. Wrazidlo.
M.D., Griinlingweg
0 1991 Elsevier Science Publishers
2, 6908
Unlike other methods [ 1,4-8, lo], our method does not construct the surface of the object, but its volume. At the end of the procedure, it will be easy to produce the surface of the object by looking at the boundary of the volume obtained. We describe the method and present a series of images from CT and MRI cross-sections to illustrate the potential uses of this method and to show the quality of images that can be produced. Materials and Methods Hip joints of healthy test persons as well as patients with liver hemangiomas were examined. Data were recorded with a Picker magnetic resonance tomograph (Vista MR-2055 HP) with a field strength of 1.0 Tesla. The MR examinations of the hip joints were made in coronal and sag&al slice orientation with 3-mm slice thickness without a gap. We used a gradient echo sequence with fat-water phase coherence (TE 14/TR 400) to display the hip joint cartilage with high signal strength (acquisition time about 8 min). For liver examinations, we used a spin-echo-multi-echo sequence with TE times up to 16 ms for diagnosis of liver hemangio-
B.V. (Biomedical
Division)
12
mas. The slice thickness was 8 mm without any gap and with axial slices (acquisition time about 12 min). The data stored on magnetic tape were then fed into a mainframe computer (IBM 4831) for decoding. The data were then transferred through a network (ether net) to the disk of a workstation (HP 9000 835 turbo SRX or SUN-3) and converted from the Picker format into a corresponding gray-scale image (Gray scale matrix). Finally, the desired MR image appeared within a few minutes on a high-resolution monitor (1024 x 1024) of the workstation. The contours of interest of the organ to be reconstructed (hip-joint cartilage, liver hemangiomas) were then recorded slice for slice by tracing with an electronic pen, since it was possible to correct the inputs as desired. A specially developed REPROS programming system then performed the reconstruction to form the wire-mesh model. The triangles of the wire-mesh model, which correspond to the surface of the organ, were coded with different colors with a computer graphics program developed by us (shading method), so that an enclosed organ surface could be displayed. The CPU times in seconds have been measured on a SUN-3 workstation. The number of tetrahedra as well as the CPU times vary almost linearly with the number of points. The CPU time was always less than 65 ms per point on a SUN-3 workstation. Total CPU times (2D-triangulation, 3Dtriangulation and reconstruction) were about 80 s for a wire-mesh model of the liver. The reconstruction times for a 3D display of the hips or of the livers were about 15 min on average, including the manual tracing of the organs. We have used an algorithm published for the first time in 1984 by Boissonnat [2, 31 for the 3D reconstruction of the CT or MRI images. In contrast to other previously known methods this algorithm reconstructs not only the outer skin but the entire volume of the body by triangulation of a sequence of sections which are prescribed in planes parallel to one another. The result is a polyhedron consisting of tetrahedra which satisfy the Delaunay condition [ 11, 181. The definition of the Delaunay triangulation is based on the Voronoi diagram. For a D-dimensional Euclidean space E and a set M of points M, . . . . . M,, the associated Voronoi diagram is a sequence (Vi . . . . . V,) of convex polyhedra covering E, where Vi consists of all the points of E that have Mi as a nearest points in the set M. Thus: vi = {P&EVj.l
Fig. 1. Voronoi diagram and its dual in a 2D case.
called the Delaunay triangulation of M. Fig. 1 shows an example of a Voronoi diagram and its dual in a 2D simple case. In the 3D case, it is possible to show that two elements are disjoint or have one vertex in common, or that they have two vertices and consequently the entire interface joining them. Moreover, the union of the elements of the Delaunay triangulation is equal to the interior of the convex hull of M. Because of the definition, when no five points are co-spherical, the circumspheres do not contain any point of M in their interior. In the case of more than five co-spherical points the elements can be broken down into several tetrahedra so that in every case the Delaunay triangulation is composed of tetrahedra. Moreover, the least three neighbors Mj are roughly the nearest neighbors of Mi in the different directions. Indeed, an inversion is considered
~j~N.d(P.M,)~d(P.M,)}
where d denotes the Euclidean distance. The geometrical dual of the Voronoi diagram obtained by linking the points M of which Voronoi polyhedra are adjacent, is
Fig. 2. CT image of the thoracic aorta. The contours of interest are traced with an electronic pen (arrows).
with Mi as center, which associates with a point Mi the point M;, on the line MiMj, whose distance from Mi satisfies the equation M,M x M&i = k x k. The reciprocal images of the half-spaces limited by the interfaces of the convex hull of the images M; . . . . . A4,‘,of M, . . . . . M, are the interiors of the spheres passing through Mi and three other points of M. Because such half-spaces are empty, the interiors of the corresponding spheres are also empty; moreover, these half-spaces are the only empty half-spaces passing through three points ofM, so the spheres are the Delaunay spheres passing through Mi. Thus, the neighbors of Mi are the points Mj whose images by an inversion with Mi as center are vertices of the convex hull of the images of Mi . . . . . A4,. Thus the Delaunay triangulation is a 3D-connected graph on M-embedded R3 which defines symmetrical and isotropic neighborhood relationships between the points. The Delaunay triangulation contains polyhedra and satisfies the first and second conditions of the definition of a polyhedron which is a decided advantage. Another advantage of the Delaunay triangulation is that it can be computed efficiently. Reconstruction
technique and results
Triangulation of a set of points in a plane is a Delaunay triangulation if the circumscribing circle of each of its triangles contains no further points. Accordingly, the circumscribing spheres in a 3D Delaunay tetrahedral triangulation are computed between two consecutive planes. The result is discoid polyhedra, the convex envelopes of the body slices sought for. All superfluous tetrahedra must then be removed in a second step the actual reconstruction step. In collaboration with the INRIA in France, we have developed a programming system REPROS (reconstruction from planar cross-sections) which implements the principle of Boissonnat’s reconstruction algorithm. REPROS is written in the programming languages C and Fortran. The Fortran modules handle the 2D Delaunay triangulation, the C module computes the 3D Delaunay triangulation and the subsequent reconstruction [ 131. The practical course of the reconstruction is demonstrated by taking the CT display of the thoracic aorta (Fig. 2) which slice for slice is traced with an electronic pen. The slices created with the electronic pen, are displayed in Plate I which lie plane-parallel to one another. Subsequently, the three-dimensional tetrahedral triangulation is computed and the organ is reconstructed to form a wire mesh model. Plate IIa shows the wire-mesh model with the triangles of the tetrahedra which form the organ surface. An enclosed surface is obtained by coloring the triangles, the so-called shading
Fig. 3. Coronal MR image of the right hip of an 35year-old volunteer. Hyaline cartilage of the femoral head shows a high signal intensity by using a gradient field echo sequence (arrow).
method, which is performed with a special computer graphics program (Plate IIb). One of 14 coronal MR tomographs of a test person’s hip is shown in Fig. 3. The hyaline joint cartilage is displayed with high signal strength by selection of a suitable gradient echo sequence (TE 14/TR 400). Plate IIIa shows the reconstructed three-dimensional wire-mesh model of the same hip joint cartilage. In comparison, Plate IIIb shows the same model after color-coding of all triangles. The head of the femur cartilage surface can be displayed less well in comparison to the wire-mesh model. However, compared to the wire-mesh model, the shape and position of the neck of the femur in relation to the acetabulum can be assessed better in the shading model. Apart from hip joints, we also reconstructed changes in the liver. Fig. 4a and b show MR tomograms of the liver with hemangiomas in the right lobe of the liver. The hemangiomas can be clearly displayed in shape and number and separated spatially in the wire-mesh model (Fig. 5a and b). Discussion Various organs can be correctly reconstructed threedimensionally by using a suitable reconstruction algorithm and developing corresponding programming systems. The method we have presented reconstructs not only the outer integument but the entire volume of an organ in comparison to other surface reconstruction methods from planar contours [ 1,4,5, 181. In this way,
Plate I. Slices of the thoracic aorta created with an electronic pen. Plate II. (a) Wire-mesh model ofthe thoracic aorta with the triangles of the tetrahedra. (b) Shading model with the surface obtained by coloring the triangles from (a). Vena cava (arrow).
Plate III. (a) Wire-mesh surface approximation of the same hip reconstructed from 14 coronal MR slices. The number of surface triangles are about 1000. The hyaline cartilage of the femoral head is well visualized. (b) Shaded model of the same hip. The shape of the hip is well visualized.
Fig. 4. Typical hemangiomas. (a) An SE (2000/160) image at 1.0 T demonstrates two hyperintensive lesions to the right lobe ofthe liver. (b) An SE (2000/160) image of the same patient shows a large hyperintensive lesion to the caudal part of the right lobe of the liver adjacent to the gallbladder (arrow).
even hip joints can be reconstructed three-dimensionally with a quality which enables an optimally matched artificial counterpart to be produced. The wire-mesh model is shaped as if the surface and the boundaries of the organ are reconstructed by tetrahedra. Depending on the clinical problem, either the wire-mesh model or the shading model is suitable for displaying the organ structure of interest, and the triangles can be coded with different colors. Relevant anatomical or pathological details can be emphasized by partial color-coding as shown in the examples. As shown in the example of the liver, the programming system reconstructs even complex surface structures correctly. Here, several contours in a plane are recognized clearly as such. Another advantage is that the time for reconstruction
is about 15 min, which is very reasonable with regard to its potential usefulness. The method is also suitable for reconstruction of CT images. The prerequisites for correct organ reconstruction are high-resolution MR images with slices as thin as possible. The advantages compared with the voxel technique are a clearly shorter computing time and considerably reduced storage capacity. This enables all models to be rotated in real time on the monitor along three axis. Of course it would be better to use the electronic pen only to roughly outline the object and set limits for density like the Ray-trace method, but a reconstructed Ray-trace model can not rotate in real time on the monitor. Future examples of application of the method pres-
Fig. 5. (a) Wire-mesh model ofthe same liver reconstructed from 14 transaxial MR slices. Orientation ofthe liver from caudal to cranial, where all hemangiomas can be displayed. The number of surface triangles are about 2500. (b) Rotation of the same model about 180” around the horizontal axis. Note the better spatial orientation with the large hemangioma in the caudal part of the right lobe of the liver.
16
ented by us are the development of simulation programs before corrective osteotomy in cases of injury to the femur head cartilage, reconstruction of the vascular tree of the liver for the assignment to segments of pathological liver processes before possible partial resection, as well as imaging of joints and of the heart in the tine mode. Besides the demonstrated examples, the method is also very useful for craniofacial reconstructive surgery [ 19,201. References Artzy E, Frieder G, Hermann G. The theory, design, implementation and evaluation of a three-dimensional surface detection algorithm. Comput. Graphics Image Process 1981; 15: i-24. Boissonnat J. Shape reconstruction from planar cross-sections. Comput Vision, Graphics Image Processing 1988; 44: l-29. Boissonnat J. Geometric surfaces for three-dimensional shape reconstruction. ACM Tram Graphics 1984; 3: 266-286. Chen L, Herman G, Reynolds R, Udupa J. Surface shading in the Cuberille environment. Comput Graph Appl 1985; 5: 33-43. Christiansen H, Sederberg T. Conversion of complex contour line definitions into polygonal element mosaics. Comput Graphics 1978; 13: 187-192. Fuchs H, Kedem Z, Uselton S. Optilam surface reconstruction from planar contours. Commun ACM 1977; 20: 693-702. Ganapathy S, Dennehy T. A new triangulation method for planar contours. Comput Graphics 1982; 16: 69-78. Gordan D, Reynolds R. Image space shading of threedimensional objects. Comput Vis 1985; 29: 361. 9 Herman G. Three-dimensional imaging on a CT or MR scanner. J Comput Assist Tomogr 1988; 12: 450-458.
10 Keppel E. Approximating
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