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Algorithms for two-dimensional reconstruction
This content has been downloaded from IOPscience. Please scroll down to see the full text. 1977 Phys. Med. Biol. 22 994 (http://iopscience.iop.org/0031-9155/22/5/020) View the table of contents for this issue, or go to the journal homepage for more
Download details: IP Address: 130.63.180.147 This content was downloaded on 02/10/2015 at 20:22
Please note that terms and conditions apply.
PHYS. MED. BIOL.,
1977, VOL. 22, NO. 5 , 994-997.
@ 1977
Scientific Note
Algorithms for Two-dimensional Reconstruction A. M. CORMACK and BRIAN J. DOYLE Physics Department, Tufts University, Medford, M a 02155, U.S.A.
Received 9 August 1976, in$nal f o r m 18 October 1976
1. Introduction
The two-dimensional reconstruction problem which occurs in radiography, radioastronomy, electron microscopy and many other fields is as follows. A two-dimensional object which can be characterized by a variable density is scanned by some means in such a way as to provide a series of measurements of the line integrals of the density along say N . straight lines which intersect the object. The problem is to calculate the density a t points in the object from the measurements of the line integrals. A numberof algorithms isnow available for performing this calculation. When these algorithms are used in practice most seem to use all of the hTmeasures of the line integrals to calculate the density a t each point in the object. It is the purpose of this note to point out that in algorithmsof this kindthe use of all ATpoints for each density calculation is unnecessary. Use of the unnecessary data, points, while it may contribute to noise reduction, generates anoise-like error in the reconstruction a t all but a few points a t which reconstruction of the density is made. Argument I n fig. 1 let A be the object which is to be reconstructed, and let us suppose that the density in the object, g, is variable and is non-zero only inside the boundary of A. Iff is the line integral of the density along the straight line L, then f can be considered to be a function of the polar coordinates ( p ,4) which 2.
Fig. 1 . A n object A of variable density is intersected by a straight line L defined by ( p , 4).
Algorithms for Reconstructions
995
specify the line L relative to some origin 0 chosen in the object. Now a finite object like A can be enclosed in a circle of radius R as shown in fig. 2(a). If 0 is chosen to be the centre of this circle then f, the Radon transform of the object, will be zero outside a circle of radius R as indicated by the hatched area in fig. 2(b).
[a)
lbl
IC)
id1
Fig. 2. ( a ) An object; ( b ) its Radon transform; (c) the same object with a holein i t ; ( d ) Radon transform of (c).
Now suppose that a hole of radius r , centred on 0, is cut' out of the object as shown in fig. 2(c). Then the Radon transform of the object with the hole in it will be identical to thetransform of the original objectin the annulusbetween the radii r and R in the domain off as shown in fig. 2(d). Inside the radius r the transform willof course be different. Now suppose that it is desired to reconstruct the density of the object a t a point Q using some precise mathematical formula which permits one to transform f to the density g . Then if Q lies between the radii r and R, the formula must yield the same value for g regardless of which of the Radon transforms offig. 2(b) or fig. 2(d) is used. Furthermore, r can be chosen so that' Q is just outside the circle of radius r . What has been shown is that the values off in a circle of radius less than the distance of Q from 0 cannot affect the value obtained for the density g a t Q. The values off in such a circle will be referred to as 'the interior values off '. This result by itself is not new. Explicit formulae for finding g without the use of the interior values offwere given by one of the aut'hors (Cormack 1963), the general result was proved in a different way by Ein-Gal (1974))and the result for circularly symmetrical objects is well known. What is interesting are the implications of the result. For thosealgorithms which are based on closed-form solutions tothe 1917, Bracewell 1956, Cormack reconstruction problem (for example: Radon 1963, 1964, Crowther, De Rosier and Klug 1972)' and which use all values of f to calculate g a t a point, theimplication is that thecontribution of the interior values off to t'he value of g must vanish identically. However, if all we have to work with is a sampling o f f a t a finite number of discrete interior points then it will be only by accident that the contributions of the interior values off to the value of g will vanish identically, so, only by accident will the contributions of the interior values off not contribute a noise-like error to the value of g. I n addition to eliminating a source of noise by not using the interior values o f f there should be a saving in computer time by not using them, Let f be
996
A . M . Cormack and B r i a n J. Doyle
determined a t N points approximately uniformly distributed over a circular region, where N is determined by thehighest spatial frequency sought and the resulting necessary number of projections. Then if it is desired to calculate g a t M points, M N calculations will be necessary if all N points are used in each calculation of g. By excluding the interior points offit can be shown by asimple integration that only M N j 2 calculationsare necessary. It is assumed that N and M are large. This fact'or of one-half appears to be related to that found by Klug and Crowther (1972), who point out that from M projections the maximum number of independent samples of the density which can be obtained is M 2 / 2 .
3. An illustration
As an illustration we state how excluding the interior values off effects one formula for the inversion of the Radon transform, namely Radon's (1917 ) formula; the proof will be published elsewhere. To find the density g(r, e) at a point defined by polar coordinates ( r ,0) referred to the origin in fig. 1 the formula is
++
I n this formula f ( p , 7 ~ =) f( p ,5). The contribution of the interior values of f can be proved to vanish identically, and eqn (1) reduces to
Having the partial derivative off under the integral may be undesirable when the integral has to be evaluated numerically from experimental data, but, as with the circularly symmetrical case, the differentiation can be taken outside of the integration sign and eqn (2) becomes
This form may be more suitable for computation. The p-integration will be replaced by a weighted sum of values off rather than af/ap, the +-integration represents an averaging which is characteristic of the Radon solution, and the differentiation is now of a function which will be fairly smooth as a result of the two prior summations. I n this example the exclusion of the interior values off is achieved by the straightforward alteration of the p-integration in eqn(1) to exclude the region ( - v , T ) , as in eqn (2). Likewise, the exclusion of the interior values o f f from any algorithm based on eqn ( 1 ) can be achieved by simply omitting themfrom the algorithm. This is of course, not generally true. The effect of excluding the interior values off on algorithms not based on eqn (1) must be examined for each of those algorithms separately. The saving in computer time may then
Algorithms for Reconstructions
997
be more or less than thefactor of one-half mentioned above since the computer operations required for the modified algorithm will not necessarily be the same as for the original algorithm. 4.
Noisereduction
Several people who read this note in its original form pointed out that the use of the interior points, while introducing its own ‘noise’, might be useful in reducing the effects of noise in the data; one could, for example, use them to average out the noise in the remaining points. This is certainly true and we can imagine circumstances in which the reduction of the effects of noise in the data might outweigh the ‘noise’ generated by use of the interior points. We are presently unableto give criteria as towhich of these effects will be the larger or which will be of sufficient generality t o cover the wide spectrum of applications of two-dimensional reconstruction. On the one hand, one has to deal with relatively small variations of density about some mean value. On the other hand, one has to deal with the case where there is no data at all for values p less than some number, and inwhich case some form of ‘onion-peeling’ must be used. (A primary problem in interferometry a t wind-tunnels is that there is an opaque object at thecentre of the tunnel.) Between these extremes is an intermediate case, for example seeking densityvariationsinthesoft tissue near an approximately circular bone. Here the exclusion of the interior sufficiently points might reducethe effects of the sharp discontinuity in density to offset any reduction in the effects of noisy data achieved by their inclusion. Clearly, results here will depend very much on the natureof the problem being studied. We are happy to acknowledge the comments provided by R. N. Bracewell,
R.A. Crowther, A. Klug, A. H. Uhlir, Jr., C. M. Vest and the referees, all of whom read this note in its original form.
REFERENCES BRACEWELL, R. N., 1956, Aust. J . Phys., 9, 198. R.A., DE ROSIER,D. J.,and &U@, A., Proc. R.Soc. A, 317, 319. CROWTRER, A. M., 1963, J . Appl. Phys., 34, 2722. CORMACK, A. M., 1964, J . Appl. Phys., 35, 2908. CORMACK, EIN-GAL,M., 1974, Stanford University Information Systems Laboratory, Technical Report 6851-1. Kma, A., and CROWTHER, R.A., 1972, Nature, Lond., 238, 435. RADON,J., 1917, Ber. J’erh. Sachs. Akad. Wiss., 67, 262.
Search
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About
Contact us
My IOPscience
Algorithms for two-dimensional reconstruction
This content has been downloaded from IOPscience. Please scroll down to see the full text. 1977 Phys. Med. Biol. 22 994 (http://iopscience.iop.org/0031-9155/22/5/020) View the table of contents for this issue, or go to the journal homepage for more
Download details: IP Address: 130.63.180.147 This content was downloaded on 02/10/2015 at 20:22
Please note that terms and conditions apply.
PHYS. MED. BIOL.,
1977, VOL. 22, NO. 5 , 994-997.
@ 1977
Scientific Note
Algorithms for Two-dimensional Reconstruction A. M. CORMACK and BRIAN J. DOYLE Physics Department, Tufts University, Medford, M a 02155, U.S.A.
Received 9 August 1976, in$nal f o r m 18 October 1976
1. Introduction
The two-dimensional reconstruction problem which occurs in radiography, radioastronomy, electron microscopy and many other fields is as follows. A two-dimensional object which can be characterized by a variable density is scanned by some means in such a way as to provide a series of measurements of the line integrals of the density along say N . straight lines which intersect the object. The problem is to calculate the density a t points in the object from the measurements of the line integrals. A numberof algorithms isnow available for performing this calculation. When these algorithms are used in practice most seem to use all of the hTmeasures of the line integrals to calculate the density a t each point in the object. It is the purpose of this note to point out that in algorithmsof this kindthe use of all ATpoints for each density calculation is unnecessary. Use of the unnecessary data, points, while it may contribute to noise reduction, generates anoise-like error in the reconstruction a t all but a few points a t which reconstruction of the density is made. Argument I n fig. 1 let A be the object which is to be reconstructed, and let us suppose that the density in the object, g, is variable and is non-zero only inside the boundary of A. Iff is the line integral of the density along the straight line L, then f can be considered to be a function of the polar coordinates ( p ,4) which 2.
Fig. 1 . A n object A of variable density is intersected by a straight line L defined by ( p , 4).
Algorithms for Reconstructions
995
specify the line L relative to some origin 0 chosen in the object. Now a finite object like A can be enclosed in a circle of radius R as shown in fig. 2(a). If 0 is chosen to be the centre of this circle then f, the Radon transform of the object, will be zero outside a circle of radius R as indicated by the hatched area in fig. 2(b).
[a)
lbl
IC)
id1
Fig. 2. ( a ) An object; ( b ) its Radon transform; (c) the same object with a holein i t ; ( d ) Radon transform of (c).
Now suppose that a hole of radius r , centred on 0, is cut' out of the object as shown in fig. 2(c). Then the Radon transform of the object with the hole in it will be identical to thetransform of the original objectin the annulusbetween the radii r and R in the domain off as shown in fig. 2(d). Inside the radius r the transform willof course be different. Now suppose that it is desired to reconstruct the density of the object a t a point Q using some precise mathematical formula which permits one to transform f to the density g . Then if Q lies between the radii r and R, the formula must yield the same value for g regardless of which of the Radon transforms offig. 2(b) or fig. 2(d) is used. Furthermore, r can be chosen so that' Q is just outside the circle of radius r . What has been shown is that the values off in a circle of radius less than the distance of Q from 0 cannot affect the value obtained for the density g a t Q. The values off in such a circle will be referred to as 'the interior values off '. This result by itself is not new. Explicit formulae for finding g without the use of the interior values offwere given by one of the aut'hors (Cormack 1963), the general result was proved in a different way by Ein-Gal (1974))and the result for circularly symmetrical objects is well known. What is interesting are the implications of the result. For thosealgorithms which are based on closed-form solutions tothe 1917, Bracewell 1956, Cormack reconstruction problem (for example: Radon 1963, 1964, Crowther, De Rosier and Klug 1972)' and which use all values of f to calculate g a t a point, theimplication is that thecontribution of the interior values off to t'he value of g must vanish identically. However, if all we have to work with is a sampling o f f a t a finite number of discrete interior points then it will be only by accident that the contributions of the interior values off to the value of g will vanish identically, so, only by accident will the contributions of the interior values off not contribute a noise-like error to the value of g. I n addition to eliminating a source of noise by not using the interior values o f f there should be a saving in computer time by not using them, Let f be
996
A . M . Cormack and B r i a n J. Doyle
determined a t N points approximately uniformly distributed over a circular region, where N is determined by thehighest spatial frequency sought and the resulting necessary number of projections. Then if it is desired to calculate g a t M points, M N calculations will be necessary if all N points are used in each calculation of g. By excluding the interior points offit can be shown by asimple integration that only M N j 2 calculationsare necessary. It is assumed that N and M are large. This fact'or of one-half appears to be related to that found by Klug and Crowther (1972), who point out that from M projections the maximum number of independent samples of the density which can be obtained is M 2 / 2 .
3. An illustration
As an illustration we state how excluding the interior values off effects one formula for the inversion of the Radon transform, namely Radon's (1917 ) formula; the proof will be published elsewhere. To find the density g(r, e) at a point defined by polar coordinates ( r ,0) referred to the origin in fig. 1 the formula is
++
I n this formula f ( p , 7 ~ =) f( p ,5). The contribution of the interior values of f can be proved to vanish identically, and eqn (1) reduces to
Having the partial derivative off under the integral may be undesirable when the integral has to be evaluated numerically from experimental data, but, as with the circularly symmetrical case, the differentiation can be taken outside of the integration sign and eqn (2) becomes
This form may be more suitable for computation. The p-integration will be replaced by a weighted sum of values off rather than af/ap, the +-integration represents an averaging which is characteristic of the Radon solution, and the differentiation is now of a function which will be fairly smooth as a result of the two prior summations. I n this example the exclusion of the interior values off is achieved by the straightforward alteration of the p-integration in eqn(1) to exclude the region ( - v , T ) , as in eqn (2). Likewise, the exclusion of the interior values o f f from any algorithm based on eqn ( 1 ) can be achieved by simply omitting themfrom the algorithm. This is of course, not generally true. The effect of excluding the interior values off on algorithms not based on eqn (1) must be examined for each of those algorithms separately. The saving in computer time may then
Algorithms for Reconstructions
997
be more or less than thefactor of one-half mentioned above since the computer operations required for the modified algorithm will not necessarily be the same as for the original algorithm. 4.
Noisereduction
Several people who read this note in its original form pointed out that the use of the interior points, while introducing its own ‘noise’, might be useful in reducing the effects of noise in the data; one could, for example, use them to average out the noise in the remaining points. This is certainly true and we can imagine circumstances in which the reduction of the effects of noise in the data might outweigh the ‘noise’ generated by use of the interior points. We are presently unableto give criteria as towhich of these effects will be the larger or which will be of sufficient generality t o cover the wide spectrum of applications of two-dimensional reconstruction. On the one hand, one has to deal with relatively small variations of density about some mean value. On the other hand, one has to deal with the case where there is no data at all for values p less than some number, and inwhich case some form of ‘onion-peeling’ must be used. (A primary problem in interferometry a t wind-tunnels is that there is an opaque object at thecentre of the tunnel.) Between these extremes is an intermediate case, for example seeking densityvariationsinthesoft tissue near an approximately circular bone. Here the exclusion of the interior sufficiently points might reducethe effects of the sharp discontinuity in density to offset any reduction in the effects of noisy data achieved by their inclusion. Clearly, results here will depend very much on the natureof the problem being studied. We are happy to acknowledge the comments provided by R. N. Bracewell,
R.A. Crowther, A. Klug, A. H. Uhlir, Jr., C. M. Vest and the referees, all of whom read this note in its original form.
REFERENCES BRACEWELL, R. N., 1956, Aust. J . Phys., 9, 198. R.A., DE ROSIER,D. J.,and &U@, A., Proc. R.Soc. A, 317, 319. CROWTRER, A. M., 1963, J . Appl. Phys., 34, 2722. CORMACK, A. M., 1964, J . Appl. Phys., 35, 2908. CORMACK, EIN-GAL,M., 1974, Stanford University Information Systems Laboratory, Technical Report 6851-1. Kma, A., and CROWTHER, R.A., 1972, Nature, Lond., 238, 435. RADON,J., 1917, Ber. J’erh. Sachs. Akad. Wiss., 67, 262.
