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An attempt to reconstruct the cerebral bloodvessels from a lateral and a frontal angiogram
Pattern Retognition Vol. 16, No. 5, pp. 517 524, 1983. printed in Great Britain.
0031 3203/83 $3.00+ .00 Pergamon Press Ltd. © 1983 Pattern Recognition Society
AN ATTEMPT TO RECONSTRUCT THE CEREBRAL BLOODVESSELS FROM A LATERAL AND A FRONTAL ANGIOGRAM P. SUETEN~1 A. HAEGEMANS~2 A. OOSTERLINCK3 and J. GYBELS1 1Department of Neurology and Neurosurgery, A.Z. St. Rafaeq, Kapucijnenvoer 35, B-3000 Leuven, 2Department of Computer Sciences, Celestijnenlaan 200A, B-3030 Heverlee, and 3Center for Human Genetics, A.Z. St. Rafael, Minderbroedersstraat 12, B-3000 Leuven, Belgium (Received 14 June 1982; received for publication 21 March 1983)
Abstract--We prove that, contrary to expectation, no part of any bloodvessel of the brain can be reconstructed if only two orthogonal subtraction angiograms are given and a priori knowledge is not used. We further discuss other possible approaches to reconstruct the cerebral vessels. Reconstruction from projections Computerized tomography Cerebral angiography Stereotactic neurosurgery
Binary patterns
True three-dimensional computed tomography would be a possibility for fast scanning and the idea is Considering the capabilities of conventional X-ray not new. t2 1ol However, the complexity and cost of a methods, two main limitations become obvious. First, large CT-scanner that can take a sequence of pictures it is impossible to display within the framework of a in a fraction of a second, as is being constructed at the two-dimensional X-ray picture all the information Mayo Clinic, Rochester, Minnesota, 111-18) may rule contained in the three-dimensional scene under view. out the use of such a machine worldwide. Because of the obstacles in conventional X-ray Objects situated in depth are superimposed, causing confusion to the viewer. Second, conventional X-rays methods and 2D-computerized tomography, we examined the possibilities of two orthogonal subcannot distinguish between soft tissues. These limitations,m however, can be overcome traction angiograms of the brain, obtained using when an image of the bloodvessels of the brain is teleradiography. In this way problems due to sensirequired. Bloodvessels can be made visible by using tivity, spatial resolution and scantime disappear. radio-opaque dyes and a stereoscopic view can give a When one divides the 3D-objects into thin slices three-dimensional impression. Nevertheless it can perpendicular to the two projection planes, the probsometimes be desirable to compute the positions lem is reduced to the reconstruction of a twoof arteries and veins, e.g. during brain stereotactic dimensional object from two orthogonal oneinterventions. dimensional projections. The possibility of obtaining a two-dimensional Computerized tomography, on the other hand, yields virtually total three-dimensional information. reconstruction from a series of one-dimensional proPictures are based on the separate examination of a jections is shown by the so-called central-slice or series of contiguous cross sections, as if we looked at projection-slice theorem. 19) In its usual form, this the body separated into a series of thin slices. More- theorem states that the Fourier transform of a oneover, this technique is enormously more sensitive dimensional projection of a two-dimensional object is than conventional X-ray methods. This sensitivity, a radial slice through the two-dimensional Fourier however, has not proven to be sufficient to detect the transform of the object itself. Thus, a large number smaller bloodvessels of the brain without contrast of one-dimensional projections under different injection. Even when using a radio-opaque dye, some angle~theoretically an infinite number--is necessary difficulties remain. On the one hand there is the limited to reconstruct an arbitrary two-dimensional object. In the case of subtraction angiograms, however, the spatial resolution--it is likely that machines in the future will be designed to provide considerably higher object can be considered as binary; since radio-opaque resolution--and on the other hand present-day CT- material can be assumed to be homogeneous and all scanners have a too long scantime to gather the other material may be assumed to be cancelled out in information for a three-dimensional dynamic image of the subtraction image, the pattern of interest may be the cerebral vessels, because the contrast dye passes assumed to be a binary pattern with 1 representing the presence and 0 the absence of the radio-opaque through the brain in a few seconds. 517 I. INTRODUCTION
518
P. SUET[NS,
A. HAEGEMANS, A. OOSTERLINCKand J. GYBELS
material. Binary two-dimensional objects have the property that some of them can be partially or even completely reconstructed from two orthogonal onedimensional projections. A number of computer programs have been proposed to solve this problemJ 19-22~ The technique described in Chang and Chow O9) can be used effectively to reconstruct any convex symmetric binary object with a piecewise linear boundary. For practical applications, the images are always quantized. Therefore, the objects can be assumed to have piecewise linear boundaries. The authors discuss the applicability of their method to cardiac cineangiography. In Chang, t2°) an A L G O L program for the reconstruction of an arbitrary binary pattern from its horizontal and vertical projections is given. If more than one binary pattern with the same orthogonal projections exists, only one of them is reconstructed. In Wee, {21~ a number of algorithms are developed related to the reconstruction of ambiguous patterns, i.e. different patterns with identical orthogonal projections. A first algorithm identifies the unambiguous subpatterns of a given pattern, a second algorithm detects and reconstructs the unambiguous subpatterns from its projections and a third algorithm computes the number of possible patterns if two orthogonal projections are given. In Wang, ~22~ two characterization questions of binary patterns and their projections are solved. First, the necessary and sufficient condition for a binary twodimensional object to be ambiguous or unambiguous is discussed. Second, given two orthogonal projections, the necessary and sufficient condition that this projection set yields none, one or more than one solution for the reconstruction problem, is derived. We have applied these theories O9-22) in an attempt to reconstruct the cerebral bloodvessels from a lateral and a frontal subtraction angiogram. I1. A S S U M P T I O N S
Suppose that a three-dimensional object is given, represented by a function of three variables that only takes on the value 0 or 1. Imagine that a set of parallel straight lines .is drawn through the three-dimensional object. A projection of the object perpendicular to these lines is a two-dimensional picture which is obtained by integrating the function along these parallel lines. The goal is to reconstruct an object from two orthogonal projections. When dividing the object into thin slices orthogonal to the two projection planes, the problem is reduced to the reconstruction of a two-dimensional object from two orthogonal onedimensional projections. In discrete form, it is reduced to the reconstruction of a binary matrix from its row and column projections. With certain assumptions, subtraction angiograms can be considered as the projections of a threedimensional binary pattern with 1 representing the presence and 0 the absence of radio-opaque material.
First, the sources are assumed to emit parallel X-rays through the object. Second, the radio-opaque material is assumed to be homogeneous. Third, all other material is assumed to be cancelled out in the subtraction image. Fourth, the response of the X-ray image system has to be linear. The first condition Can be well approximated using teleradiography and the second assumption is generally well-fulfilled. Because of the dynamic behavior of the contrast dye, a short interval between the two orthogonal exposures is necessary. To fulfil the third condition, the system parameters must be controlled accurately; the emitted energy spectrum before and after contrast injection has to be identical and no head motion may occur. Noise, however, can never be completely avoided and using film radiographs still other inaccuracies occur; slight changes in the developing circumstances, film inhomogeneities and non-linearity of the photographical subtraction process may give a subtraction image in which bone and brain tissue are not completely cancelled out. Moreover, due to the nonlinearity of the sensitometric curve, the fourth condition--linear response of the Xray imaging system---cannot be fulfilled. These last disadvantages, however, can largely be removed by using a filmless digital X-ray imaging system. Even if one assumes that the original object to be reconstructed is binary, there still remains a difficulty. Indeed, the original obRct will not remain binary after digitization, since a picture element could be only partially occupied by a bloodvessel. Nevertheless, let us assume that the pixels are so small that round-off errors after reconstruction are insignificant. III. T H E O R Y O F BINARY O B J E C T R E C O N S T R U C T I O N FROM ORTHOGONAL PROJECTIONS
We will discuss only those theoretical aspects that are necessary to understand the further discussion in this paper. In Chang and Chow, "9~ Chang, ~2°) Wee 121) and Wang, ~22) a variety of definitions is given and sometimes the same terminology is used with different meanings. Therefore, we will define the terminology we USe.
A. Notations F and G are binary m x n matrices. The i-th row projection Fx(i) is the number of l's in the i-th row ofF. Thej-th column projection Fy(j) is the number of l's in thej-th column of F. The row projection vector F x and the column projection vector Fy are the vectors with elements Fx(i), i = l(1)m and Fr(j), j = l(1)n, respectively. M is the set {1, 2,..., m} and N the set {1, 2, .... n}. G x and Gy are defined analogously. Without being restrictive, we will assume that the elements of Fx and Fy are arranged in order of size. Remark that Fx(m ) <<,n and F~(n) <~ m and that m
n
~, Fx(i ) = ~ Fy(j). i =1
j -1
Cerebral bloodvessels
i 0
i
I
n' •
1"11 I
I
N'
-
'
N'
-
N
=-'I ,LI
Fig. 1. Representation of a not-completely ambiguous matrix when row and column projections are arranged in order of size.
B. Definitions 1. F and G are similar matrices ifF x = Gx and Fy = Gy. 2. F is said to be ambiguous if a similar matrix G exists and F(i, j) ~ G(i, j) for at least one (i, j). 3. F is completely ambiguous if for each (i,j) a similar matrix G exists such that F(i, j) ~ G(i, j). 4. A submatrix of F is unambiguous if there exists no matrix G, similar to F, that differs from F in one of the elements of this submatrix.
C. Properties 1. Obviously a necessary condition for F to be completely ambiguous is 1 ~< Fx(1), Fx(m ) <~ n - 1 and 1 ~< F~,(1), Fr(n) ~< m - 1.
(1)
2. Suppose F is not completely ambiguous and (1)is satisfied. Then it is always possible to find an unambiguous zero submatrix and an unambiguous unity submatrix (i.e. all elements 1) in F corresponding to the
____0_
Fig. 2. Schematical representation of a not-completely ambiguous matrix F, satisfyingcondition (l). The zero and unity submatrices are unambiguous, the shaded submatrices are completely ambiguous. PR 16:5-F
519
sets of matrixpoints M" × N' and M' × N", respectively, such that M' ~ M" = ~ , N' c~ N" = ~ , M' w M" = M and N' u N" = N. The proof of property 2 can implicitly be found in Wee.~2~)It is also proven in the Appendix. Because F x and Fy are arranged in order of size, F looks as shown in Fig. 1. If the submatrix corresponding to M' x N' (or M" x N") is not completely ambiguous, property 2 can be applied to this submatrix. By repeatedly applying this property, a not-completely ambiguous matrix F, satisfying condition (1), can be subdivided into a number of unambiguous zero and unity submatrices and a number of completely ambiguous submatrices, as is schematically represented in Fig. 2. Consider now an arbitrary unambiguous unity submatrix in F. Obviously, this submatrix is identical to or belongs to an unambiguous unity submatrix corresponding to M~ x N1, such that the submatrix corresponding to ( M \ M O x (NkN1) is an unambiguous zero submatrix. This important property will be used in the next paragraph.
IV. APPLICATIONTO THE BLOODVESSELS OF THE BRAIN We can now investigate the possibility of reconstructing the cerebral bloodvessels from two orthogonal subtraction angiograms without the use of a priori knowledge. Consider a lateral and a frontal angiogram of the brain (Fig. 3). After subtraction, and assuming optimal conditions, only the injected contrast dye remains visible. Dividing the object into thin slices orthogonal to the two projection planes, the problem is reduced to the reconstruction of a twodimensional binary object from two orthogonal onedimensional projections, i.e. two corresponding horizontal thin strips on the lateral and the frontal subtraction angiogram, respectively. We now suppose that at least a part of one of the contrast dyed bloodvessels in the considered slice can be reconstructed unambiguously. This part can be represented by a unity submatrix corresponding to M 0 x No, as schematically represented in Fig. 4. If we exclude the situation in which a whole row or column consists of l's (contrast dye), we have seen that the only possibility for unambiguously reconstructing a unity submatrix occurs when it is identical to or belongs to an unambiguous unity submatrix corresponding to M1 × N1, such that the submatrix corresponding to (M~.MI) x (N'~NI) is an unambiguous zero submatrix. The situation M o = MI, N O = N x is schematically represented in Fig. 5a and the situation M o c Mx, N O c N~ in Fig. 5b. Both cases, however, assume that each bloodvessel is overlapped by at least one other bloodvessel in one ot the two orthogonal projections of the considered slice. Figure 3 shows that this situation is quite impossible ; when one selects an arbitrary slice (i.e. two cor-
520
P. SUI!TENS,A. HAEGEMANS,A. OOSTERLIN('Kand J. GYBELS
Fig. 3. Lateral and frontal angiogram of the brain. responding horizontal thin strips on the lateral and frontal angiogram), it is mostly not very complicated to find one vessel that is not overlapped by another vessel in both projection strips of that particular slice. Consequently, we may safely assert that it is impossible to reconstruct any part of a bloodvessel if only two orthogonal subtraction angiograms of the brain are given and no other a priori knowledge is used. V. DISCUSSION
We have proven that no part of any bloodvessel of the brain can be reconstructed from only two orthogonal subtraction angiograms, even in optimal conditions when several assumptions can be made. A priori information about the object and/or more projections could carry us a step forward. In Chang and Wang, (23) the general three-dimensional object reconstruction problem from three orthogonal projections is considered and an attempt is made to solve it in an heuristic way; the authors state, however, that an efficient reconstruction algorithm is yet to be devised. In Shliferstein and Chien, (24) the concepts of switching components are generalized to systems with more than two projections and they are incorporated into a partial reconstruction technique. The algorithm of Hsia et al325) uses four X-ray scans spaced 45 ° apart. The reconstruction procedure is iterative and image matrix elements are corrected by alternately matching the two sets of orthogonal scan data. In Tsikos and
Bajcsy {26) two orthogonal conventional X-ray projections, CT scans perpendicular to the X-ray planes and a priori knowledge, derived from an anatomy atlas, are used to reconstruct three-dimensional objects. These methods, however, all seem to be impractical to us, because of the high architectural complexity involved when more than two angiograms taken from a large distance are needed and because a priori information derived from an anatomy atlas clearly does not suffice without the enormous store of practical knowledge of a radiologist regarding anatomy, pathology and patient-specific data.
I
.....
_
i
I :
MoE-
l.,,-.,d
No Fig. 4. Schematical representation of a part of one of the contrast dyed bloodvessels in the slice being reconstructed.
Cerebral bloodvessels
°
i
N\No ,
521
..-N\N 1
- -~
I
0
~
ItO
i
_
--
0
No
b Fig. 5. Schematical representation of partially and unambiguously reconstructable matrices. We are investigating two other approaches to reconstruct the cerebral bloodvessels. The first approach uses a stereo-pair of subtraction angiograms. Our aim is to match corresponding bloodvessels on both images. To do this, the bloodvessels are seen in a first step as strings of micropatterns, characterized by an orientation, width and intensity, and similar micropatterns on the left and the right image of the stereo-pair are selected. In a second step, the previous result is iteratively improved using the consistency property of the bloodvessels as a priori knowledge. The second approach--we term it computerized conventional tomography--uses the geometry of conventional circular or linear radiographic tomography. Instead of radiographic film, a digital radiography device has to be used to collect the projections. To overcome the problem of missing projection data for a significant range of angles needed for a true threedimensional reconstruction, we use the property that the object to be reconstructed has a compact support and its Fourier transform is analytical In Lent and Tuy t27) an appropriate iterative method for analytic continuation is given. Since the subspace of the object that is filled with contrast dye (bloodvessels) is relatively small as compared with the complete support, many projection values will be zero or, in the presence of noise, very small. This strongly limits the set of possible solutions and it can be expected that this strong limitation will yield a fast convergence, such that noise does not cause the abruption of the iteration process before a satisfying result is obtained. SUMMARY Cerebral bloodvessels may be made visible using radio-opaque dyes and a stereoscopic view can generate a three-dimensional impression. Nevertheless, it can sometimes be desirable to measure the positions of arteries and veins in a quantitative way, e.g. during
brain streotactic interventions. Computerized tomography, as it is currently used, has too long a scantime to gather the information for a three-dimensional reconstruction of large dynamic objects such as the contrast dye filled bloodvessels. Because of the obstacles of conventional X-ray methods, the idea arose to examine the possibilities of two orthogonal subtraction angiograms obtained using teleradiography. Because most of the bloodvessels are clearly visible on both angiograms, it seemed to be a reasonable attempt. With several assumptions, the object (bloodvessels) can be regarded as a binary pattern with 1 representing the presence and 0 the absence of radio-opaque material. In this way the problem is reduced to the reconstruction of a binary pattern from two orthogonal projections. We have applied the theory of binary pattern reconstruction in an attempt to reconstruct the cerebral bloodvessels from a lateral and a frontal subtraction angiogram and we have proven that, contrary to our expectations, no part of any bloodvessel can be reconstructed when a priori knowledge is not used. A priori information about the object and/or more projections could carry us a step further. Several methods, described in the literature, are discussed, but none of them seems to be of practical value to us. Two new approaches, which are currently under investigation, are briefly discussed. The first approach tries to match corresponding bloodvessels on a stereo-pair of subtracted angiograms. The second approach--we term it computerized conventional tomography---can be considered as a true three-dimensional computerized tomography method. Acknowledgments--The authors would like to thank Dr, G. T. H f~RMANof the University of Pennsylvania at Philadelphia for his remarks and Miss R. NARTUSfor typing the manuscript. This research is supported by the F.G.W.O. (Belgium) under grant number 3.0021.81.
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A.
HAEGEMANS,
REFERENCES 1. G. N. Hounsfield, Computed medical imaging, Science, N.Y. 210, 22-28 (1980). 2. G. Kowalski, Multislice reconstruction from twin-cone beam scanning, IEEE Trans. Nucl. Sci. NS-26, 2895-2903 (1979). 3. E. Levitan, On true 3-D object reconstruction from line integrals, Proc. IEEE 67, 1679-1680 (1979). 4. O. Nalcioglu and Z. H. Cho, Reconstruction of 3-D objects from cone beam projections, Proc. IEEE 66, 1584-1585 (1978). 5. R.V. Denton, B. Friedlander and A. J. Rockmore, Direct three-dimensional image reconstruction from divergent rays, IEEE Trans. Nucl. Sci. NS-26, 4695-4703 (1979). 6. M.D. Altschuler, G. T. Herman and A. Lent, Fully threedimensional image reconstruction from cone-beam sources. Proceedings of the IEEE Computer Society Conference on Pattern Recognition and Image Processing, 31 May-2 June, Chicago, IL, pp. 194-199 (1978). 7. G. Minerbo, Maximum entropy reconstruction from cone-beam projection data, Comput. Biol. Med. 9, 29 37 (1979). 8. G. Minerbo, Convolutional reconstruction from conebeam projection data, IEEE Trans. Nucl. Sci. NS-26, 2682-2684 (1979). 9. M.Y. Chiu, H. H. Barret, R. E. Simpson, C. Chou, J. W. Arendt and E. R. Gindi, Three-dimensional radiographic imaging with a restricted view angle, J. opt. Soc. Am. 69, 1323-1333 (1979). 10. G. T. Herman, Image Reconstruction from Projections-- The Fundamentals of Computerized Tomography, chapter 14, Truly three-dimensional reconstruction, pp. 237-259. Academic Press, New York (1980). 11. R.A. Robb, E. L. Ritman, J. F. Greenleaf, R. E. Sturm, G. T. Herman, P. A. Chevalier, H. K. Liu and E. H. Wood, Quantitative imaging of dynamic structure and function of the heart, lungs and circulation by computerized construction and subtraction techniques, Comput. Graphics 10, 246-256 (1976). 12. R.A. Robb, E. L. Ritman, L. D. Harris and E. H. Wood, Dynamic three-dimensional X-ray computed tomography of the heart, lungs and circulation, IEEE Trans. Nucl. Sci. NS-26, 1646-1660 (1979). 13. R.A. Robb, E. L. Ritman, B. K. Gilbert, J. H. Kinsey, L. D. Harris and E. H. Wood, The DSR : a high-speed threedimensional X-ray computed tomography system for dynamic spatial reconstruction of the heart and circulation, IEEE Trans. Nucl. Sci. NS-26, 2713-2717 (1979). 14. L. D. Harris, R. A. Robb, T. S. Yuen and E. L. Ritman, Display and visualisation of three-dimensional reconstructed anatomic morphology: experience with the thorax, heart, and coronary vasculature of dogs, J. Comput. assisted Tomography 3, 439-446 (1979). 15. R. A. Robb and E. L. Ritman, High speed synchronous volume computed tomography of the heart, Radiology 133, 655-661 (1979). 16. R. A. Robb, Three-dimensional dynamic imaging of the heart, lungs and circulation by r6ntgen-video computed tomography, Medical Imaging Techniques. A Comparison, pp. 185-198. Plenum, New York (1979). 17. E. L. Ritman, J. H. Kinsey, R. A. Robb, L. D. Harris and B. K. Gilbert, Physics and technical considerations in the design of the DSR : a high temporal resolution volume scanner, Am. J. Roentctenol. 134, 369-374 (1980). 18. E. L. Ritman, J. H. Kinsey, R. A. Robb, B. K. Gilbert, L. D. Harris and E. H. Wood, Three-dimensional imaging of heart, lungs and circulation, Science, N.Y. 210, 273-280 19.--"""~.-~l~U.)'Changand C. K. Chow, The reconstruction of three-dimensional objects from two orthogonal projections and its application to cardiac cineangiography, IEEE Trans. Comput. C-22, 18-28 (1973).
A. OOSTERLINCK and J. GYBELS 20. S.-K. Chang, Binary pattern recognition from projections, Communs. Ass. comput. Much. 16, 185-186 (1973). 21. W. G. Wee, On reconstruction of ambiguous patterns, Proceedings of the First International Joint Conference on Pattern Recognition, Washington D.C., 30 Oct.-I Nov. 1973, pp. 293-301. IEEE Publ. 73-CHO-821-9C (1973). 22. Y. R. Wang, Characterization of binary patterns and their projections, IEEE Trans. Comput. C-24, 1032-1033 (1975). 23. S.-K. Chang and Y. R. Wang, Three-dimensional object reconstruction from orthogonal projections, Pattern Recognition 7, 167-176 (1975). 24. A. Shliferstein and Y. T. Chien, Switching components and the ambiguity problem in the reconstruction of pictures from their projections, Pattern Recognition 10, 327-340 (1978). 25. T.C. Hsia, S. C. Smith and B. M. T. Lantz, Computerized tomography using a modified orthogonal tangent correlation algorithm, Comput. Programs Biomed. 6, 136-141 (1976). 26. C. J. Tsikos and R. K. Bajcsy, 3-D reconstruction of objects from incomplete data and a priori knowledge, Proc. 5th Int. Conf. on Pattern Recognition, pp. 393-395 (1980). 27. A. Lent and H. Tuy, An iterative method for the extrapolation of band-limited functions, J. math. analysis Applic. 83, 554-565 (1981).
APPENDIX
A
Definition A switching component is a set of matrixpoints {(i', j'), (i',j"), (i",f), (i",j")} satisfying F(i',j') = F(i",j") = 1 - F(i',j") = 1 - F ( i " , j ' ) = O o r 1. Proof of property 2 Because F is not completely ambiguous, there is at least one (io, Jo) for which no similar matrix G with G(io,Jo) ~ F (io,Jo) can be found. This means that (i0, Jo) does not belong to a switching component. Let us define M' = {i': F(i',jo ) = 1},M" = {i":F(i",jo)= 0}, N' = {j': F(io, j' ) = 0} and S " = {j": F(io, j" ) = 1}. Because (1) is satisfied, none of these subsets is empty. Assume F(io, Jo) = 1 (if F(io, Jo) = 0 an analogous deduction can be made). For F(io, Jo) not to belong to a switching component, the matrix corresponding to M" x N' obviously has to be a zero matrix. By proper row and column permutations, F looks as shown in Fig. A1. Let Z be the set of matrixpoints M" x N' and let us expand this set as follows:
i !
'I
0
0•
iI I
i
1
I I I
j ~
rl
...
-
Il v" N'
~
jo N"
Ill I
"1
Fig. A1. Representation ofmatrix F in the proof ofproperty 2 after a first series of row and column permutations.
Cerebral bloodvessels
523
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!
0
....
- -~- /
-
I
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,I
n,N' o - - ~
I.I -
N'
=~-
0
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0
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I
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_i
,_)......]_L i-
------ !()
7.
n,
p,
ml
t
I
I
I
i
I
N"
.I I
I ,
Fig. A2. Representation of matrix F in the proof of property 2 after a second series of row and column permutations.
i 1,
N'= Nk÷I k
d
N"
-'-
N
4 :
(1) for all j2 of N" for which F(i",j2 ) = 0 for all i" of M", the Fig. A3. Representation of matrix F in the proof ofproperty 2 after a third series of row and column permutations. elements (i", J2) are added to Z ; (2) for all i~ of M' for which F(il, j') = 0 for all j' of N', the elements (il, j') are added to Z. (M',,M~+I) x (N.Nk_l). If all its elements are one, a zero By proper row and column permutations, F now looks as submatrix M~+ 1 x N k ~and a unity submatrix (M".MI+ 1) × shown in Fig. A2. (N'\Nk_~) in F are found and consequently property two is By further row and column permutations, it can easily be proven. seen that F can be arranged as shown in Fig. A3. Z is now Otherwise, at least one zero element in (M~".M~+I) x redefined as (N'..Nk_l) can be found. Let (i**,j**) be the zero element or (M 0 x Nk+l) W (M l x Nk) L; ... k~ (Mk+ l × No). All one of the zero elements belonging to the column with highest elements of Z are zero and all elements of the matrices number. This element belongs to the submatrix (M{..Mt+ t ) × (Mo\M1) x (Nk~Nk+l), (MI\M2} x (Nk_l\Nk), ..., (N\Nk_l) or to the submatrix (Mp\MI) x (N\Nk_l). If it (Mk\Mk+x) × (No\N1)are one. belongs to (M{.,M~+I) x (NINk_~), the set {(i*,j*), (i*,j**), Consider the submatrix (M",Mo) × (N\Nk+I). If all its (i**,j*), (i**,j**)} is a switching component in F. If it belongs elements are one, a zero submatrix Mo x Nk+l and a unity to (Mp'.Ml) x (N',Nk_~) , the element (i**,]*) can be a zero submatrix (M\M0) x ( N W k + I ) in F are found and conelement. In this case, the set {(i**, j*), (i**,j'), (i', j*), (i',j')} sequently property two is proven. where (i',j*) is an element of (Ml',,Ml+l) x (Nk_{'.Nk_l+l) , In the other case, at least one zero element in (M\Mo) x (i**,j') is an element of (Mp'\Mp + 1) × (Nk- p'Nk- p + 1) •... w ( N W k + ~) can be found. Let (i*,j*) be the zero element or one (Mt_{.,MI) x (Nk_l+ l',.Nk_~+ 2)and(i',j')isanelement ofZ, is of the zero elements belonging to the column with highest a switching component in F. It is clear that, in both cases, a n u m b e r and let us define p = 0. Further, let (M\MI) x matrix similar to F can be deduced by proper switching, (N\JVk_l+l) be the submatrix with highest l that (i*, j*) where (i*, j**) is a zero element. This means that the zero belongs to. Two possibilities can be distinguished. element (i*, j*) is shifted to the right. Remark that this shift (a) l = k + 1. In this case (i*,j*) belongs to ( M \ M o ) × never can cause this zero element to enter the submatrix (N\No). It is clear that F(i*,j') is one for at least onej' of N' and ( M \ M o ) x (N~..No),because in that case (io,Jo) could be made F(i',j*) is one for at least one i" of M". The set {(i",j'), (i",j*), zero by proper switching, as explained in (a), and this would (i*, j'), (i*, j*)} is a switching component in F. From F, a be in conflict with the premises. similar matrix G can be deduced by switching (i",j') with (i", From (b) we conclude that either (M'..M~+ 1) × (N'.Nk_~) is j*) and (i*,j') with (i*,j*). Thus, G(i",j') = G(io, Jo ) = 1, G(i", a unity submatrix, in which case property two is proven, or at Jo) = G(io,j') = 0 and the set {(io,J0), (io,j'), (i",Jo), (i",j')} is a least one submatrix (M'.,Mr+t) x ( N \ N k _ r ) with 1' < I and switching component in G. where the zero element (i*,j**) does not belong, still exists. In From G, a new matrix similar to G, and thus similar to F, this last case, consider that submatrix with the lowest value of can be deduced by switching (io,Jo) with (io,j') and (i",jo) with l'; redefining p .-- I + 1, l ,--- l' and j* ~ ] * * and repeating the (i",j'). The value in point (io,Jo) is now zero. However, this is in procedure described in (b), it is clear that after one or more conflict with the premises. Consequently, the case ! = k + 1 is steps, a unity submatrix (M'\M~+ t) x (N\Nk i) will be found excluded. and consequently property two is proven. (b) / < k + 1. In this case we consider the submatrix
A b o u t the Author--PAuL SUETENSgraduated in 1977 as an Engineer of Computer Sciences at the University
of Leuven, Belgium. Since 1978 he has worked as a Research Assistant at the Department of Neurology and Neurosurgery, University of Leuven, Belgium, where he is preparing his Ph.D. thesis entitled 'A threedimensional image of the cerebral bloodvessels for use in stereotactic neurosurgery'. A b o u t the A u t h o ~ A N:~ H AE(iEMANSgraduated in 1973 as an Electronical Engineer and received a Ph.D. in
Computer Science in 1975 at the University of Leuven, Belgium. From 1973 till 1978, she was an aspirant of the N.F.W.O. (National Science Foundation Belgium). Since 1978 she has been Professor in Computer Science. Her main interests are numerical linear algebra and multiple integration.
524
P. S/;ITENS, A. HAIiGI!MAN%,A, OOSTI RI INCK and J. G'~ Bl!I_S
About the Author--ANDRI ~,OC~STERLINCKreceived a degree in Electrical Engineering from the University of Leuven in 1972. He gained his Ph.D. in Electrical Engineering (also at Leuven) in 1977. His postgraduate education continued in Leuven, Pasadena, Irvine and West Lafayette. As a fellow of the National Foundation of Scientific Research Belgium (NFWO), he is now head of the Image Processing Group of the Department of Human Genetics at the University of Leuven (Belgium). About the Author J ANG VaELSreceived a Medical Degree in 1963 and a Ph.D. in 1968, from the University of Leuven. He is currently Professor of Neurology and Neurosurgery at the same university.
0031 3203/83 $3.00+ .00 Pergamon Press Ltd. © 1983 Pattern Recognition Society
AN ATTEMPT TO RECONSTRUCT THE CEREBRAL BLOODVESSELS FROM A LATERAL AND A FRONTAL ANGIOGRAM P. SUETEN~1 A. HAEGEMANS~2 A. OOSTERLINCK3 and J. GYBELS1 1Department of Neurology and Neurosurgery, A.Z. St. Rafaeq, Kapucijnenvoer 35, B-3000 Leuven, 2Department of Computer Sciences, Celestijnenlaan 200A, B-3030 Heverlee, and 3Center for Human Genetics, A.Z. St. Rafael, Minderbroedersstraat 12, B-3000 Leuven, Belgium (Received 14 June 1982; received for publication 21 March 1983)
Abstract--We prove that, contrary to expectation, no part of any bloodvessel of the brain can be reconstructed if only two orthogonal subtraction angiograms are given and a priori knowledge is not used. We further discuss other possible approaches to reconstruct the cerebral vessels. Reconstruction from projections Computerized tomography Cerebral angiography Stereotactic neurosurgery
Binary patterns
True three-dimensional computed tomography would be a possibility for fast scanning and the idea is Considering the capabilities of conventional X-ray not new. t2 1ol However, the complexity and cost of a methods, two main limitations become obvious. First, large CT-scanner that can take a sequence of pictures it is impossible to display within the framework of a in a fraction of a second, as is being constructed at the two-dimensional X-ray picture all the information Mayo Clinic, Rochester, Minnesota, 111-18) may rule contained in the three-dimensional scene under view. out the use of such a machine worldwide. Because of the obstacles in conventional X-ray Objects situated in depth are superimposed, causing confusion to the viewer. Second, conventional X-rays methods and 2D-computerized tomography, we examined the possibilities of two orthogonal subcannot distinguish between soft tissues. These limitations,m however, can be overcome traction angiograms of the brain, obtained using when an image of the bloodvessels of the brain is teleradiography. In this way problems due to sensirequired. Bloodvessels can be made visible by using tivity, spatial resolution and scantime disappear. radio-opaque dyes and a stereoscopic view can give a When one divides the 3D-objects into thin slices three-dimensional impression. Nevertheless it can perpendicular to the two projection planes, the probsometimes be desirable to compute the positions lem is reduced to the reconstruction of a twoof arteries and veins, e.g. during brain stereotactic dimensional object from two orthogonal oneinterventions. dimensional projections. The possibility of obtaining a two-dimensional Computerized tomography, on the other hand, yields virtually total three-dimensional information. reconstruction from a series of one-dimensional proPictures are based on the separate examination of a jections is shown by the so-called central-slice or series of contiguous cross sections, as if we looked at projection-slice theorem. 19) In its usual form, this the body separated into a series of thin slices. More- theorem states that the Fourier transform of a oneover, this technique is enormously more sensitive dimensional projection of a two-dimensional object is than conventional X-ray methods. This sensitivity, a radial slice through the two-dimensional Fourier however, has not proven to be sufficient to detect the transform of the object itself. Thus, a large number smaller bloodvessels of the brain without contrast of one-dimensional projections under different injection. Even when using a radio-opaque dye, some angle~theoretically an infinite number--is necessary difficulties remain. On the one hand there is the limited to reconstruct an arbitrary two-dimensional object. In the case of subtraction angiograms, however, the spatial resolution--it is likely that machines in the future will be designed to provide considerably higher object can be considered as binary; since radio-opaque resolution--and on the other hand present-day CT- material can be assumed to be homogeneous and all scanners have a too long scantime to gather the other material may be assumed to be cancelled out in information for a three-dimensional dynamic image of the subtraction image, the pattern of interest may be the cerebral vessels, because the contrast dye passes assumed to be a binary pattern with 1 representing the presence and 0 the absence of the radio-opaque through the brain in a few seconds. 517 I. INTRODUCTION
518
P. SUET[NS,
A. HAEGEMANS, A. OOSTERLINCKand J. GYBELS
material. Binary two-dimensional objects have the property that some of them can be partially or even completely reconstructed from two orthogonal onedimensional projections. A number of computer programs have been proposed to solve this problemJ 19-22~ The technique described in Chang and Chow O9) can be used effectively to reconstruct any convex symmetric binary object with a piecewise linear boundary. For practical applications, the images are always quantized. Therefore, the objects can be assumed to have piecewise linear boundaries. The authors discuss the applicability of their method to cardiac cineangiography. In Chang, t2°) an A L G O L program for the reconstruction of an arbitrary binary pattern from its horizontal and vertical projections is given. If more than one binary pattern with the same orthogonal projections exists, only one of them is reconstructed. In Wee, {21~ a number of algorithms are developed related to the reconstruction of ambiguous patterns, i.e. different patterns with identical orthogonal projections. A first algorithm identifies the unambiguous subpatterns of a given pattern, a second algorithm detects and reconstructs the unambiguous subpatterns from its projections and a third algorithm computes the number of possible patterns if two orthogonal projections are given. In Wang, ~22~ two characterization questions of binary patterns and their projections are solved. First, the necessary and sufficient condition for a binary twodimensional object to be ambiguous or unambiguous is discussed. Second, given two orthogonal projections, the necessary and sufficient condition that this projection set yields none, one or more than one solution for the reconstruction problem, is derived. We have applied these theories O9-22) in an attempt to reconstruct the cerebral bloodvessels from a lateral and a frontal subtraction angiogram. I1. A S S U M P T I O N S
Suppose that a three-dimensional object is given, represented by a function of three variables that only takes on the value 0 or 1. Imagine that a set of parallel straight lines .is drawn through the three-dimensional object. A projection of the object perpendicular to these lines is a two-dimensional picture which is obtained by integrating the function along these parallel lines. The goal is to reconstruct an object from two orthogonal projections. When dividing the object into thin slices orthogonal to the two projection planes, the problem is reduced to the reconstruction of a two-dimensional object from two orthogonal onedimensional projections. In discrete form, it is reduced to the reconstruction of a binary matrix from its row and column projections. With certain assumptions, subtraction angiograms can be considered as the projections of a threedimensional binary pattern with 1 representing the presence and 0 the absence of radio-opaque material.
First, the sources are assumed to emit parallel X-rays through the object. Second, the radio-opaque material is assumed to be homogeneous. Third, all other material is assumed to be cancelled out in the subtraction image. Fourth, the response of the X-ray image system has to be linear. The first condition Can be well approximated using teleradiography and the second assumption is generally well-fulfilled. Because of the dynamic behavior of the contrast dye, a short interval between the two orthogonal exposures is necessary. To fulfil the third condition, the system parameters must be controlled accurately; the emitted energy spectrum before and after contrast injection has to be identical and no head motion may occur. Noise, however, can never be completely avoided and using film radiographs still other inaccuracies occur; slight changes in the developing circumstances, film inhomogeneities and non-linearity of the photographical subtraction process may give a subtraction image in which bone and brain tissue are not completely cancelled out. Moreover, due to the nonlinearity of the sensitometric curve, the fourth condition--linear response of the Xray imaging system---cannot be fulfilled. These last disadvantages, however, can largely be removed by using a filmless digital X-ray imaging system. Even if one assumes that the original object to be reconstructed is binary, there still remains a difficulty. Indeed, the original obRct will not remain binary after digitization, since a picture element could be only partially occupied by a bloodvessel. Nevertheless, let us assume that the pixels are so small that round-off errors after reconstruction are insignificant. III. T H E O R Y O F BINARY O B J E C T R E C O N S T R U C T I O N FROM ORTHOGONAL PROJECTIONS
We will discuss only those theoretical aspects that are necessary to understand the further discussion in this paper. In Chang and Chow, "9~ Chang, ~2°) Wee 121) and Wang, ~22) a variety of definitions is given and sometimes the same terminology is used with different meanings. Therefore, we will define the terminology we USe.
A. Notations F and G are binary m x n matrices. The i-th row projection Fx(i) is the number of l's in the i-th row ofF. Thej-th column projection Fy(j) is the number of l's in thej-th column of F. The row projection vector F x and the column projection vector Fy are the vectors with elements Fx(i), i = l(1)m and Fr(j), j = l(1)n, respectively. M is the set {1, 2,..., m} and N the set {1, 2, .... n}. G x and Gy are defined analogously. Without being restrictive, we will assume that the elements of Fx and Fy are arranged in order of size. Remark that Fx(m ) <<,n and F~(n) <~ m and that m
n
~, Fx(i ) = ~ Fy(j). i =1
j -1
Cerebral bloodvessels
i 0
i
I
n' •
1"11 I
I
N'
-
'
N'
-
N
=-'I ,LI
Fig. 1. Representation of a not-completely ambiguous matrix when row and column projections are arranged in order of size.
B. Definitions 1. F and G are similar matrices ifF x = Gx and Fy = Gy. 2. F is said to be ambiguous if a similar matrix G exists and F(i, j) ~ G(i, j) for at least one (i, j). 3. F is completely ambiguous if for each (i,j) a similar matrix G exists such that F(i, j) ~ G(i, j). 4. A submatrix of F is unambiguous if there exists no matrix G, similar to F, that differs from F in one of the elements of this submatrix.
C. Properties 1. Obviously a necessary condition for F to be completely ambiguous is 1 ~< Fx(1), Fx(m ) <~ n - 1 and 1 ~< F~,(1), Fr(n) ~< m - 1.
(1)
2. Suppose F is not completely ambiguous and (1)is satisfied. Then it is always possible to find an unambiguous zero submatrix and an unambiguous unity submatrix (i.e. all elements 1) in F corresponding to the
____0_
Fig. 2. Schematical representation of a not-completely ambiguous matrix F, satisfyingcondition (l). The zero and unity submatrices are unambiguous, the shaded submatrices are completely ambiguous. PR 16:5-F
519
sets of matrixpoints M" × N' and M' × N", respectively, such that M' ~ M" = ~ , N' c~ N" = ~ , M' w M" = M and N' u N" = N. The proof of property 2 can implicitly be found in Wee.~2~)It is also proven in the Appendix. Because F x and Fy are arranged in order of size, F looks as shown in Fig. 1. If the submatrix corresponding to M' x N' (or M" x N") is not completely ambiguous, property 2 can be applied to this submatrix. By repeatedly applying this property, a not-completely ambiguous matrix F, satisfying condition (1), can be subdivided into a number of unambiguous zero and unity submatrices and a number of completely ambiguous submatrices, as is schematically represented in Fig. 2. Consider now an arbitrary unambiguous unity submatrix in F. Obviously, this submatrix is identical to or belongs to an unambiguous unity submatrix corresponding to M~ x N1, such that the submatrix corresponding to ( M \ M O x (NkN1) is an unambiguous zero submatrix. This important property will be used in the next paragraph.
IV. APPLICATIONTO THE BLOODVESSELS OF THE BRAIN We can now investigate the possibility of reconstructing the cerebral bloodvessels from two orthogonal subtraction angiograms without the use of a priori knowledge. Consider a lateral and a frontal angiogram of the brain (Fig. 3). After subtraction, and assuming optimal conditions, only the injected contrast dye remains visible. Dividing the object into thin slices orthogonal to the two projection planes, the problem is reduced to the reconstruction of a twodimensional binary object from two orthogonal onedimensional projections, i.e. two corresponding horizontal thin strips on the lateral and the frontal subtraction angiogram, respectively. We now suppose that at least a part of one of the contrast dyed bloodvessels in the considered slice can be reconstructed unambiguously. This part can be represented by a unity submatrix corresponding to M 0 x No, as schematically represented in Fig. 4. If we exclude the situation in which a whole row or column consists of l's (contrast dye), we have seen that the only possibility for unambiguously reconstructing a unity submatrix occurs when it is identical to or belongs to an unambiguous unity submatrix corresponding to M1 × N1, such that the submatrix corresponding to (M~.MI) x (N'~NI) is an unambiguous zero submatrix. The situation M o = MI, N O = N x is schematically represented in Fig. 5a and the situation M o c Mx, N O c N~ in Fig. 5b. Both cases, however, assume that each bloodvessel is overlapped by at least one other bloodvessel in one ot the two orthogonal projections of the considered slice. Figure 3 shows that this situation is quite impossible ; when one selects an arbitrary slice (i.e. two cor-
520
P. SUI!TENS,A. HAEGEMANS,A. OOSTERLIN('Kand J. GYBELS
Fig. 3. Lateral and frontal angiogram of the brain. responding horizontal thin strips on the lateral and frontal angiogram), it is mostly not very complicated to find one vessel that is not overlapped by another vessel in both projection strips of that particular slice. Consequently, we may safely assert that it is impossible to reconstruct any part of a bloodvessel if only two orthogonal subtraction angiograms of the brain are given and no other a priori knowledge is used. V. DISCUSSION
We have proven that no part of any bloodvessel of the brain can be reconstructed from only two orthogonal subtraction angiograms, even in optimal conditions when several assumptions can be made. A priori information about the object and/or more projections could carry us a step forward. In Chang and Wang, (23) the general three-dimensional object reconstruction problem from three orthogonal projections is considered and an attempt is made to solve it in an heuristic way; the authors state, however, that an efficient reconstruction algorithm is yet to be devised. In Shliferstein and Chien, (24) the concepts of switching components are generalized to systems with more than two projections and they are incorporated into a partial reconstruction technique. The algorithm of Hsia et al325) uses four X-ray scans spaced 45 ° apart. The reconstruction procedure is iterative and image matrix elements are corrected by alternately matching the two sets of orthogonal scan data. In Tsikos and
Bajcsy {26) two orthogonal conventional X-ray projections, CT scans perpendicular to the X-ray planes and a priori knowledge, derived from an anatomy atlas, are used to reconstruct three-dimensional objects. These methods, however, all seem to be impractical to us, because of the high architectural complexity involved when more than two angiograms taken from a large distance are needed and because a priori information derived from an anatomy atlas clearly does not suffice without the enormous store of practical knowledge of a radiologist regarding anatomy, pathology and patient-specific data.
I
.....
_
i
I :
MoE-
l.,,-.,d
No Fig. 4. Schematical representation of a part of one of the contrast dyed bloodvessels in the slice being reconstructed.
Cerebral bloodvessels
°
i
N\No ,
521
..-N\N 1
- -~
I
0
~
ItO
i
_
--
0
No
b Fig. 5. Schematical representation of partially and unambiguously reconstructable matrices. We are investigating two other approaches to reconstruct the cerebral bloodvessels. The first approach uses a stereo-pair of subtraction angiograms. Our aim is to match corresponding bloodvessels on both images. To do this, the bloodvessels are seen in a first step as strings of micropatterns, characterized by an orientation, width and intensity, and similar micropatterns on the left and the right image of the stereo-pair are selected. In a second step, the previous result is iteratively improved using the consistency property of the bloodvessels as a priori knowledge. The second approach--we term it computerized conventional tomography--uses the geometry of conventional circular or linear radiographic tomography. Instead of radiographic film, a digital radiography device has to be used to collect the projections. To overcome the problem of missing projection data for a significant range of angles needed for a true threedimensional reconstruction, we use the property that the object to be reconstructed has a compact support and its Fourier transform is analytical In Lent and Tuy t27) an appropriate iterative method for analytic continuation is given. Since the subspace of the object that is filled with contrast dye (bloodvessels) is relatively small as compared with the complete support, many projection values will be zero or, in the presence of noise, very small. This strongly limits the set of possible solutions and it can be expected that this strong limitation will yield a fast convergence, such that noise does not cause the abruption of the iteration process before a satisfying result is obtained. SUMMARY Cerebral bloodvessels may be made visible using radio-opaque dyes and a stereoscopic view can generate a three-dimensional impression. Nevertheless, it can sometimes be desirable to measure the positions of arteries and veins in a quantitative way, e.g. during
brain streotactic interventions. Computerized tomography, as it is currently used, has too long a scantime to gather the information for a three-dimensional reconstruction of large dynamic objects such as the contrast dye filled bloodvessels. Because of the obstacles of conventional X-ray methods, the idea arose to examine the possibilities of two orthogonal subtraction angiograms obtained using teleradiography. Because most of the bloodvessels are clearly visible on both angiograms, it seemed to be a reasonable attempt. With several assumptions, the object (bloodvessels) can be regarded as a binary pattern with 1 representing the presence and 0 the absence of radio-opaque material. In this way the problem is reduced to the reconstruction of a binary pattern from two orthogonal projections. We have applied the theory of binary pattern reconstruction in an attempt to reconstruct the cerebral bloodvessels from a lateral and a frontal subtraction angiogram and we have proven that, contrary to our expectations, no part of any bloodvessel can be reconstructed when a priori knowledge is not used. A priori information about the object and/or more projections could carry us a step further. Several methods, described in the literature, are discussed, but none of them seems to be of practical value to us. Two new approaches, which are currently under investigation, are briefly discussed. The first approach tries to match corresponding bloodvessels on a stereo-pair of subtracted angiograms. The second approach--we term it computerized conventional tomography---can be considered as a true three-dimensional computerized tomography method. Acknowledgments--The authors would like to thank Dr, G. T. H f~RMANof the University of Pennsylvania at Philadelphia for his remarks and Miss R. NARTUSfor typing the manuscript. This research is supported by the F.G.W.O. (Belgium) under grant number 3.0021.81.
522
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A.
HAEGEMANS,
REFERENCES 1. G. N. Hounsfield, Computed medical imaging, Science, N.Y. 210, 22-28 (1980). 2. G. Kowalski, Multislice reconstruction from twin-cone beam scanning, IEEE Trans. Nucl. Sci. NS-26, 2895-2903 (1979). 3. E. Levitan, On true 3-D object reconstruction from line integrals, Proc. IEEE 67, 1679-1680 (1979). 4. O. Nalcioglu and Z. H. Cho, Reconstruction of 3-D objects from cone beam projections, Proc. IEEE 66, 1584-1585 (1978). 5. R.V. Denton, B. Friedlander and A. J. Rockmore, Direct three-dimensional image reconstruction from divergent rays, IEEE Trans. Nucl. Sci. NS-26, 4695-4703 (1979). 6. M.D. Altschuler, G. T. Herman and A. Lent, Fully threedimensional image reconstruction from cone-beam sources. Proceedings of the IEEE Computer Society Conference on Pattern Recognition and Image Processing, 31 May-2 June, Chicago, IL, pp. 194-199 (1978). 7. G. Minerbo, Maximum entropy reconstruction from cone-beam projection data, Comput. Biol. Med. 9, 29 37 (1979). 8. G. Minerbo, Convolutional reconstruction from conebeam projection data, IEEE Trans. Nucl. Sci. NS-26, 2682-2684 (1979). 9. M.Y. Chiu, H. H. Barret, R. E. Simpson, C. Chou, J. W. Arendt and E. R. Gindi, Three-dimensional radiographic imaging with a restricted view angle, J. opt. Soc. Am. 69, 1323-1333 (1979). 10. G. T. Herman, Image Reconstruction from Projections-- The Fundamentals of Computerized Tomography, chapter 14, Truly three-dimensional reconstruction, pp. 237-259. Academic Press, New York (1980). 11. R.A. Robb, E. L. Ritman, J. F. Greenleaf, R. E. Sturm, G. T. Herman, P. A. Chevalier, H. K. Liu and E. H. Wood, Quantitative imaging of dynamic structure and function of the heart, lungs and circulation by computerized construction and subtraction techniques, Comput. Graphics 10, 246-256 (1976). 12. R.A. Robb, E. L. Ritman, L. D. Harris and E. H. Wood, Dynamic three-dimensional X-ray computed tomography of the heart, lungs and circulation, IEEE Trans. Nucl. Sci. NS-26, 1646-1660 (1979). 13. R.A. Robb, E. L. Ritman, B. K. Gilbert, J. H. Kinsey, L. D. Harris and E. H. Wood, The DSR : a high-speed threedimensional X-ray computed tomography system for dynamic spatial reconstruction of the heart and circulation, IEEE Trans. Nucl. Sci. NS-26, 2713-2717 (1979). 14. L. D. Harris, R. A. Robb, T. S. Yuen and E. L. Ritman, Display and visualisation of three-dimensional reconstructed anatomic morphology: experience with the thorax, heart, and coronary vasculature of dogs, J. Comput. assisted Tomography 3, 439-446 (1979). 15. R. A. Robb and E. L. Ritman, High speed synchronous volume computed tomography of the heart, Radiology 133, 655-661 (1979). 16. R. A. Robb, Three-dimensional dynamic imaging of the heart, lungs and circulation by r6ntgen-video computed tomography, Medical Imaging Techniques. A Comparison, pp. 185-198. Plenum, New York (1979). 17. E. L. Ritman, J. H. Kinsey, R. A. Robb, L. D. Harris and B. K. Gilbert, Physics and technical considerations in the design of the DSR : a high temporal resolution volume scanner, Am. J. Roentctenol. 134, 369-374 (1980). 18. E. L. Ritman, J. H. Kinsey, R. A. Robb, B. K. Gilbert, L. D. Harris and E. H. Wood, Three-dimensional imaging of heart, lungs and circulation, Science, N.Y. 210, 273-280 19.--"""~.-~l~U.)'Changand C. K. Chow, The reconstruction of three-dimensional objects from two orthogonal projections and its application to cardiac cineangiography, IEEE Trans. Comput. C-22, 18-28 (1973).
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APPENDIX
A
Definition A switching component is a set of matrixpoints {(i', j'), (i',j"), (i",f), (i",j")} satisfying F(i',j') = F(i",j") = 1 - F(i',j") = 1 - F ( i " , j ' ) = O o r 1. Proof of property 2 Because F is not completely ambiguous, there is at least one (io, Jo) for which no similar matrix G with G(io,Jo) ~ F (io,Jo) can be found. This means that (i0, Jo) does not belong to a switching component. Let us define M' = {i': F(i',jo ) = 1},M" = {i":F(i",jo)= 0}, N' = {j': F(io, j' ) = 0} and S " = {j": F(io, j" ) = 1}. Because (1) is satisfied, none of these subsets is empty. Assume F(io, Jo) = 1 (if F(io, Jo) = 0 an analogous deduction can be made). For F(io, Jo) not to belong to a switching component, the matrix corresponding to M" x N' obviously has to be a zero matrix. By proper row and column permutations, F looks as shown in Fig. A1. Let Z be the set of matrixpoints M" x N' and let us expand this set as follows:
i !
'I
0
0•
iI I
i
1
I I I
j ~
rl
...
-
Il v" N'
~
jo N"
Ill I
"1
Fig. A1. Representation ofmatrix F in the proof ofproperty 2 after a first series of row and column permutations.
Cerebral bloodvessels
523
_
!
0
....
- -~- /
-
I
j-:1-,
,I
n,N' o - - ~
I.I -
N'
=~-
0
0
0
" i
JoL
n I
I
N"
_i
,_)......]_L i-
------ !()
7.
n,
p,
ml
t
I
I
I
i
I
N"
.I I
I ,
Fig. A2. Representation of matrix F in the proof of property 2 after a second series of row and column permutations.
i 1,
N'= Nk÷I k
d
N"
-'-
N
4 :
(1) for all j2 of N" for which F(i",j2 ) = 0 for all i" of M", the Fig. A3. Representation of matrix F in the proof ofproperty 2 after a third series of row and column permutations. elements (i", J2) are added to Z ; (2) for all i~ of M' for which F(il, j') = 0 for all j' of N', the elements (il, j') are added to Z. (M',,M~+I) x (N.Nk_l). If all its elements are one, a zero By proper row and column permutations, F now looks as submatrix M~+ 1 x N k ~and a unity submatrix (M".MI+ 1) × shown in Fig. A2. (N'\Nk_~) in F are found and consequently property two is By further row and column permutations, it can easily be proven. seen that F can be arranged as shown in Fig. A3. Z is now Otherwise, at least one zero element in (M~".M~+I) x redefined as (N'..Nk_l) can be found. Let (i**,j**) be the zero element or (M 0 x Nk+l) W (M l x Nk) L; ... k~ (Mk+ l × No). All one of the zero elements belonging to the column with highest elements of Z are zero and all elements of the matrices number. This element belongs to the submatrix (M{..Mt+ t ) × (Mo\M1) x (Nk~Nk+l), (MI\M2} x (Nk_l\Nk), ..., (N\Nk_l) or to the submatrix (Mp\MI) x (N\Nk_l). If it (Mk\Mk+x) × (No\N1)are one. belongs to (M{.,M~+I) x (NINk_~), the set {(i*,j*), (i*,j**), Consider the submatrix (M",Mo) × (N\Nk+I). If all its (i**,j*), (i**,j**)} is a switching component in F. If it belongs elements are one, a zero submatrix Mo x Nk+l and a unity to (Mp'.Ml) x (N',Nk_~) , the element (i**,]*) can be a zero submatrix (M\M0) x ( N W k + I ) in F are found and conelement. In this case, the set {(i**, j*), (i**,j'), (i', j*), (i',j')} sequently property two is proven. where (i',j*) is an element of (Ml',,Ml+l) x (Nk_{'.Nk_l+l) , In the other case, at least one zero element in (M\Mo) x (i**,j') is an element of (Mp'\Mp + 1) × (Nk- p'Nk- p + 1) •... w ( N W k + ~) can be found. Let (i*,j*) be the zero element or one (Mt_{.,MI) x (Nk_l+ l',.Nk_~+ 2)and(i',j')isanelement ofZ, is of the zero elements belonging to the column with highest a switching component in F. It is clear that, in both cases, a n u m b e r and let us define p = 0. Further, let (M\MI) x matrix similar to F can be deduced by proper switching, (N\JVk_l+l) be the submatrix with highest l that (i*, j*) where (i*, j**) is a zero element. This means that the zero belongs to. Two possibilities can be distinguished. element (i*, j*) is shifted to the right. Remark that this shift (a) l = k + 1. In this case (i*,j*) belongs to ( M \ M o ) × never can cause this zero element to enter the submatrix (N\No). It is clear that F(i*,j') is one for at least onej' of N' and ( M \ M o ) x (N~..No),because in that case (io,Jo) could be made F(i',j*) is one for at least one i" of M". The set {(i",j'), (i",j*), zero by proper switching, as explained in (a), and this would (i*, j'), (i*, j*)} is a switching component in F. From F, a be in conflict with the premises. similar matrix G can be deduced by switching (i",j') with (i", From (b) we conclude that either (M'..M~+ 1) × (N'.Nk_~) is j*) and (i*,j') with (i*,j*). Thus, G(i",j') = G(io, Jo ) = 1, G(i", a unity submatrix, in which case property two is proven, or at Jo) = G(io,j') = 0 and the set {(io,J0), (io,j'), (i",Jo), (i",j')} is a least one submatrix (M'.,Mr+t) x ( N \ N k _ r ) with 1' < I and switching component in G. where the zero element (i*,j**) does not belong, still exists. In From G, a new matrix similar to G, and thus similar to F, this last case, consider that submatrix with the lowest value of can be deduced by switching (io,Jo) with (io,j') and (i",jo) with l'; redefining p .-- I + 1, l ,--- l' and j* ~ ] * * and repeating the (i",j'). The value in point (io,Jo) is now zero. However, this is in procedure described in (b), it is clear that after one or more conflict with the premises. Consequently, the case ! = k + 1 is steps, a unity submatrix (M'\M~+ t) x (N\Nk i) will be found excluded. and consequently property two is proven. (b) / < k + 1. In this case we consider the submatrix
A b o u t the Author--PAuL SUETENSgraduated in 1977 as an Engineer of Computer Sciences at the University
of Leuven, Belgium. Since 1978 he has worked as a Research Assistant at the Department of Neurology and Neurosurgery, University of Leuven, Belgium, where he is preparing his Ph.D. thesis entitled 'A threedimensional image of the cerebral bloodvessels for use in stereotactic neurosurgery'. A b o u t the A u t h o ~ A N:~ H AE(iEMANSgraduated in 1973 as an Electronical Engineer and received a Ph.D. in
Computer Science in 1975 at the University of Leuven, Belgium. From 1973 till 1978, she was an aspirant of the N.F.W.O. (National Science Foundation Belgium). Since 1978 she has been Professor in Computer Science. Her main interests are numerical linear algebra and multiple integration.
524
P. S/;ITENS, A. HAIiGI!MAN%,A, OOSTI RI INCK and J. G'~ Bl!I_S
About the Author--ANDRI ~,OC~STERLINCKreceived a degree in Electrical Engineering from the University of Leuven in 1972. He gained his Ph.D. in Electrical Engineering (also at Leuven) in 1977. His postgraduate education continued in Leuven, Pasadena, Irvine and West Lafayette. As a fellow of the National Foundation of Scientific Research Belgium (NFWO), he is now head of the Image Processing Group of the Department of Human Genetics at the University of Leuven (Belgium). About the Author J ANG VaELSreceived a Medical Degree in 1963 and a Ph.D. in 1968, from the University of Leuven. He is currently Professor of Neurology and Neurosurgery at the same university.