past and future directions in x-ray computed … and future directions in x-ray computed tomography...

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Past and Future Directions in X-Ray Computed Tomography (CT) Frank Natterer 1 , Erik L. Ritman 2 1 Institut fu ¨ r Numerische und Instrumentelle Mathematik, Westfalische Wilhelms-Universitat, Einsteinstrasse 62, D-48149 Munster, Germany 2 Department of Physiology and Biophysics, Mayo Clinic, Rochester, MN 55905 Received 28 February 2002; Accepted 25 June 2002 ABSTRACT: We give a short account of the history of CT from motion tomography in the early 1930’s to sprial CT. We discuss the physical and the mathematical background. Finally we give an outlook on possible future developments of CT and related techniques. © 2002 Wiley Periodicals, Inc. Int J Imaging Syst Technol, 12, 175–187, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/ima. 10021 I. INTRODUCTION In 1895, Ro ¨ntgen first indicated that X-rays could be used to image internal structures such as bones (Ro ¨ntgen, 1895), and the method was commercialized almost immediately afterwards (Blume, 1992). Computed tomography (CT), as we know the imaging modality today, was first commercially introduced for clinical use in 1971 (Ambrose and Hounsfield, 1972) and rapidly led to many companies building scanners, each with their slight improvements. Many books have been published about the history and recent developments of X-ray CT imaging (Dolby and Alker, 1997; Holtz- mann Kevles, 1997; Webb, 1990). The story is interesting, but like so many histories of important technological discoveries and/or developments, it is difficult to extract a constructive lesson about what factors favored the development and implementation of such discoveries. It is tempting to see the evolution of CT, indeed, of technological developments in general, as an analog of Darwinian evolution, but this has turned out to be a good metaphor yet not a useful model (Ziman, 2000). Hence, we appear to be no closer to understanding the basic factors in the evolution of CT. The story does tell us that serendipity in the conjunction between a new idea (the “seed”), the environment of technological capabilities and prac- tical needs (the “soil”), and the inventor and/or protagonist (the “sower”) are essential components of the evolution of the new imaging modality. Although outside of the scope of this article, it is of interest to see how the development of X-ray CT greatly accel- erated, and perhaps gave direction to, and acceptance of, other CT imaging modalities such as magnetic resonance imaging (MRI) (Mattson and Simon, 1996). II. TECHNICAL DEVELOPMENT Tomographic imaging (i.e., imaging of a slice through, rather than a projection of, a three-dimensional (3D) structure) was recognized early on in the development of X-ray imaging as being a way to overcome the limitations of projection imaging such as superposi- tion and foreshortening of structures. This approach, based on ana- log mechanisms for generating the tomographic images, evolved until the scanner mechanism and the image generation process in many ways were very similar to current CT systems (Andrews and Stava, 1937; Vallebona, 1931), especially as described by Frank in 1940 (Figure 1). In addition, those systems provided image infor- mation (Figure 2) very reminiscent of that made by early CT scanners (Ledley et al., 1974), even to the extent of multislice gated imaging of the heart (Carey et al., 1971). These developments, however, apparently had little direct impact on the development of computed tomography. Indeed, the latter rapidly cast the sophisti- cated analog tomographic systems (Coulam et al., 1981; Littleton, 1976) into virtual oblivion. As is often the case with new developments and discoveries, the development of CT occurred in several places over the decade from the late 1950s on. Some developments were poorly disseminated for a number of reasons, the main ones being lack of communication for understandable sociopolitical reasons, one being the Pole Wloka (1953; Figure 3), who may well have been the first to actually use the Radon Transform for generating tomograms, and another being the Russian Korenblyum (Figure 4; Korenblyum et al., 1958). The motivations and approaches used to develop tomographic imaging scanner systems involved three people who had quite different motivations and apparently little awareness of the tomographic imaging field at that time. Cormack (1963), the South African physicist who needed to make accurate estimates of the spatial distribution of therapeutic radiation absorption distribution within the body, developed an inversion formula of his own (Figure 5). Oldendorf, an American neurosurgeon interested in accurately esti- mating local concentrations of radiolabeled indicators within the brain, needed to overcome the problem of the superimposed radio- activity. This he partially achieved with development of an unfil- tered backprojection method (1963). The other major development was by the Englishman Hounsfield (1968 –1972), whose approach solved the problem of discriminating white from grey matter in the X-ray images of the brain (Figure 6). He used an algebraic type of reconstruction algorithm [see Eq. (9)]. It was his scanner, built by the EMI Company (his employer), that first launched X-ray CT on © 2002 Wiley Periodicals, Inc.

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Past and Future Directions in X-Ray Computed Tomography (CT)

Frank Natterer1, Erik L. Ritman2

1Institut fur Numerische und Instrumentelle Mathematik, Westfalische Wilhelms-Universitat, Einsteinstrasse 62,D-48149 Munster, Germany

2Department of Physiology and Biophysics, Mayo Clinic, Rochester, MN 55905

Received 28 February 2002; Accepted 25 June 2002

ABSTRACT: We give a short account of the history of CT from motiontomography in the early 1930’s to sprial CT. We discuss the physicaland the mathematical background. Finally we give an outlook onpossible future developments of CT and related techniques. © 2002Wiley Periodicals, Inc. Int J Imaging Syst Technol, 12, 175–187, 2002; Publishedonline in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/ima.10021

I. INTRODUCTIONIn 1895, Rontgen first indicated that X-rays could be used to imageinternal structures such as bones (Rontgen, 1895), and the methodwas commercialized almost immediately afterwards (Blume, 1992).Computed tomography (CT), as we know the imaging modalitytoday, was first commercially introduced for clinical use in 1971(Ambrose and Hounsfield, 1972) and rapidly led to many companiesbuilding scanners, each with their slight improvements.

Many books have been published about the history and recentdevelopments of X-ray CT imaging (Dolby and Alker, 1997; Holtz-mann Kevles, 1997; Webb, 1990). The story is interesting, but likeso many histories of important technological discoveries and/ordevelopments, it is difficult to extract a constructive lesson aboutwhat factors favored the development and implementation of suchdiscoveries. It is tempting to see the evolution of CT, indeed, oftechnological developments in general, as an analog of Darwinianevolution, but this has turned out to be a good metaphor yet not auseful model (Ziman, 2000). Hence, we appear to be no closer tounderstanding the basic factors in the evolution of CT. The storydoes tell us that serendipity in the conjunction between a new idea(the “seed”), the environment of technological capabilities and prac-tical needs (the “soil”), and the inventor and/or protagonist (the“sower”) are essential components of the evolution of the newimaging modality. Although outside of the scope of this article, it isof interest to see how the development of X-ray CT greatly accel-erated, and perhaps gave direction to, and acceptance of, other CTimaging modalities such as magnetic resonance imaging (MRI)(Mattson and Simon, 1996).

II. TECHNICAL DEVELOPMENTTomographic imaging (i.e., imaging of a slice through, rather than aprojection of, a three-dimensional (3D) structure) was recognized

early on in the development of X-ray imaging as being a way toovercome the limitations of projection imaging such as superposi-tion and foreshortening of structures. This approach, based on ana-log mechanisms for generating the tomographic images, evolveduntil the scanner mechanism and the image generation process inmany ways were very similar to current CT systems (Andrews andStava, 1937; Vallebona, 1931), especially as described by Frank in1940 (Figure 1). In addition, those systems provided image infor-mation (Figure 2) very reminiscent of that made by early CTscanners (Ledley et al., 1974), even to the extent of multislice gatedimaging of the heart (Carey et al., 1971). These developments,however, apparently had little direct impact on the development ofcomputed tomography. Indeed, the latter rapidly cast the sophisti-cated analog tomographic systems (Coulam et al., 1981; Littleton,1976) into virtual oblivion.

As is often the case with new developments and discoveries, thedevelopment of CT occurred in several places over the decade fromthe late 1950s on. Some developments were poorly disseminated fora number of reasons, the main ones being lack of communication forunderstandable sociopolitical reasons, one being the Pole Wloka(1953; Figure 3), who may well have been the first to actually usethe Radon Transform for generating tomograms, and another beingthe Russian Korenblyum (Figure 4; Korenblyum et al., 1958). Themotivations and approaches used to develop tomographic imagingscanner systems involved three people who had quite differentmotivations and apparently little awareness of the tomographicimaging field at that time. Cormack (1963), the South Africanphysicist who needed to make accurate estimates of the spatialdistribution of therapeutic radiation absorption distribution withinthe body, developed an inversion formula of his own (Figure 5).Oldendorf, an American neurosurgeon interested in accurately esti-mating local concentrations of radiolabeled indicators within thebrain, needed to overcome the problem of the superimposed radio-activity. This he partially achieved with development of an unfil-tered backprojection method (1963). The other major developmentwas by the Englishman Hounsfield (1968–1972), whose approachsolved the problem of discriminating white from grey matter in theX-ray images of the brain (Figure 6). He used an algebraic type ofreconstruction algorithm [see Eq. (9)]. It was his scanner, built bythe EMI Company (his employer), that first launched X-ray CT on

© 2002 Wiley Periodicals, Inc.

the commercial radiological scene. The developments up to thatpoint in time can be separated quite distinctly from the subsequentdevelopments, which were mostly (with a few exceptions) incre-mental refinements and extensions of the CT imaging technique.Since the latter half of the 1990s, developments in fast, multislice,helical scanning CT appears to have rejuvenated the whole-body CTfield.

X-ray CT involves several disciplines such as mathematics (Ra-don, 1917), physics (Rontgen, 1895), computer science (Goldstine,1988; Queisser, 1988), technology (Coolidge, 1913; Hine, 1977),and medicine (Baker, 1975), as well as the spectrum of personalitiesinvolved in those disciplines. Each of these disciplines evolved in itsown right, but only occasionally was the connection between thedevelopments in the different disciplines realized or acknowledged.The mathematical basis is crucial, but the effective mathematicalmethods turn out to be only practically useable with a programmabledigital computer. Such machines, suitable for the large-scale com-putations involved in CT, did not become generally available untilthe 1960s, when the computer chip was developed (Queisser, 1988).Thus, as indicated to us by H. Barrett, the Dutchman Bockwinkel(1906) developed a mathematical approach to solve a crystalo-graphic problem that could well have been used as the basis of CTin the late 1800s. The Austrian Radon (1917) had little direct impacton the initial development of X-ray CT imaging, even thoughRadon’s Transform turned out to be a more direct match to the needsof CT. Only after aspects of that Transform were developed inde-pendently some 60 years later was it realized that Radon had alreadypublished this approach. Analog ray-tracing backprojection methods(Figure 7) of generating a tomographic image were limited by theinability to subtract light (including X-ray). This had two conse-

quences. There was bias on the image (Barrett and Swindell, 1977)(which would today be removed by a suitable digital filter) and thedynamic range of the analog approach was limited to around 100:1(Coulam et al., 1981). These two limitations essentially reduced theimage contrast resolution, the very attribute of CT that made itattractive to the neuroradiologists in the first place. Another aspectof digital CT is that the reconstruction algorithm could be morereadily adjusted to accommodate the practical constraints of thescanner itself, whereas analog tomography involved complex me-chanical movement of the X-ray source and imaging device, withresulting bulky, inconvenient equipment that would be prone to lossof adjustment. The use of analog electrical signals proportional to alight signal (e.g., generated by an X-ray to light-converting screen)made subtraction feasible, but this was still limited by poor dynamicrange. In the 1950s, development of X-ray scintilators coupled tophotoelectric cells, primarily for radionuclide imaging (Hine, 1977),made this possible. It was the development of the analog-to-digitalconverter and the means to record the digital values representingtransmitted X-ray intensity that made Hounsfield’s approach feasi-ble. Thus, digital memory and the associated digital computationaltechnology now provided the subtraction capability, high dynamicrange, and accommodation of scanner geometry.

The evolution of the CT scanner and its applications after theinitial introduction of the (EMI) head scanner (Ambrose andHounsfield, 1972) is an interesting and instructive process in its ownright. This evolution can be seen as involving “perfection” of thescanner’s original purpose; that is, it was made faster [for improvedlogistics and reduced motion artifacts (Figure 8) and higher spatial

Figure 1. Example of analog tomography processes that closelyresembles several of the computed tomography processes.

Figure 2. Transverse layergraph developed by Kishi et al. (1969).

Figure 3. Page from Wloka’s 1953 paper clearly referencing Ra-don’s Transform as being a control feature of computed tomography.

176 Vol. 12, 175–187 (2002)

and contrast resolution], as well as an “extension” of the conceptfrom a head scanner to whole-body transaxial scanning (Figure 9),3D imaging (Figure 10) (Herman and Liu, 1977), rapid repetitivescanning of dynamic structures (Figure 11) (Bove et al., 1985; Robbet al., 1975), and quantitation of pathophysiologic processes (Figure12) (Ritman, 1987). This evolution of commercially available scan-

ners (Figure 13) (Hemmy et al., 1994) involved further refinementsin mathematics, engineering, computer technology, and science, butmost importantly, an increasing appreciation of the spectrum ofpotential clinical applications of this imaging modality.

The first substantive increase in speed of the scanning procedureresulted from the implementation of a fan-beam geometry. This wasmade possible by the use of a linear array of detectors, therebyremoving the need for the time-consuming transaxial linear scan ofa single detector at each angle of view. The rebinning of the rayswould then be used to generate a nonuniformly sampled parallel raygeometry reconstruction problem. Several, more direct mathemati-cal solutions rapidly followed (Gordon et al., 1970; Ramachandranand Lakshminarayanan, 1971). This approach did introduce theproblem of increased scatter due to reduced X-ray collimation(Johns and Yaffe, 1982) and of ring artifacts (the so-called third-generation artifact) (Shepp and Stein, 1977), due to incompletelycalibrated X-ray detectors.

The next step in speeding up the scanning procedure wasachieved primarily by more powerful X-ray sources and fastertransmission and recording of the digital data. The limit appeared tobe about 1 s, primarily due to the mechanical problems introducedby rotating the X-ray source around the subject at greater speeds.This limitation has been reduced by a factor of 2 in the late 1990s,but is likely to be reduced by a factor of 3, so that a 180° plus

Figure 4. The apparatus of Korenblyum et al. for CT. The upperfigure is a profile and the lower, a plan view (from Korenblyum et al.,1958).

Figure 5. The reconstruction of some original data of Cormack. Theshaded areas are aluminum; the unshaded interior area is Lucite (fromCormack, 1964).

Figure 6. Polaroid picture of the CT scan of the bottled specimen ofa brain, showing a tumor in the third ventricle (Ambrose, 1996). Theinterdigitating pattern of the white matter can be seen but not thecortex. The white spots correspond to hemorrhages in the tumor.

Vol. 12, 175–187 (2002) 177

half-fan angle scan would be acquired in �200 ms. An obvioustechnological solution involved the use of multiple X-ray sources, asolution proposed by a number of people (Franke, 1977–1979; Lill,1977–1978; Sturm et al., 1975; Wagner, 1975–1978) (Figures 14,15, 16), but was only implemented for biomedical purposes in theDynamic Spatial Reconstructor (Ritman et al., 1985). This scannerhad the added feature that it used cone-beam geometry. Although itused cone-beam backprojection it also used a modified Feldkampalgorithm (Feldkamp et al., 1984) to generate a stack-of-fan recon-structions rather than true cone-beam geometry. It is of interest tonote that the American Feldkamp developed his early practicalsolution to the cone-beam reconstruction problem because he usedcone-beam geometry to provide the magnification needed for themicro-CT application to inert, isolated bone specimens, not thecurrent need for high speed, multislice data acquisition. In themid-1980s, the American Boyd, developed a scanning electron

beam, fixed anode, X-ray source that completed scans in 50 ms andrepeated this over a period of about 0.224 s to scan 8 parallel slices(Boyd and Lipton, 1983) (Figure 17). This scanner, while effectivefor imaging the moving heart (or children who cannot stop wrig-gling), has not been attractive for most abdominal scans, especiallyin heavy patients. An economically viable 3D imaging methodturned out to be helical scanning CT, first introduced in the late1980s (Mori, 1986; Nishimura and Miyazaki, 1988).

Developments in these fields were greatly facilitated by theenormous power of Moore’s law (Reed, 1984) which resulted incomputers that could handle highly compute-intensive algorithms,which allowed for extension into 3D imaging by way of cone-beamand helical scanning, as well as increasingly incomplete data setsresulting from technical and/or cost limitations on packing of X-raysources and detector arrays. Similarly, the development of X-raypower supplies that could be integrated into the X-ray source nowfreed the scanner from the need to accommodate bulky, high-voltagepower cables and similarly, the recent development of high-resolu-tion imaging arrays (Street et al., 1990), which should permit inno-vative scanning arrangements to be implemented in the future.

Technical capability is apparently not always sufficient for aninnovation to “diffuse” into the user community. Several examplesare provided by the Dynamic Spatial Reconstructor (DSR) project inthe 1980s (Jorgensen et al., 1990). The primary purpose of thatscanner was to provide an image of a 3D volume (sufficiently largeto incorporate an entire organ such as the heart or lung, as illustrated

Figure 10. Surface rendering of a dog’s myocardium generatedfrom a DSR 3D image (Robb, 1982).

Figure 7. Takahashi’s apparatus for reconstructing the outline of anobject from a discontinuous sinogram (from Takahashi, 1957).

Figure 8. Chronological improvement in scan times of some CTscanners.

Figure 9. Thoracic cross section generated by Ledley et al. (1974).

178 Vol. 12, 175–187 (2002)

in Figure 18) within a sufficiently short period of time to reducemotion artifacts to within the limits of the image spatial resolution(�1 mm) and to repeat those scans sufficiently frequently (up to60/s) so as to capture the motion dynamics of the heart, lungs, gut(Seide and Ritman, 1984), and of contrast dilution process througha moving organ such as the heart (Wang et al., 1989), lungs, andkidney. It is of interest to see that there has been, and remains, aconsiderable effort to achieve comparable goals with tomosynthesis(Dobbins, 1990), multislice helical-scanning CT scanners (Mori,1986; Nishimura and Miyazaki, 1988), and rotating array detectors(Bidaut et al., 1998; Ning and Kruger, 1996), which now have thebenefit of superior X-ray generation and X-ray image capture tech-nology. Nonetheless, an example that technological capability itselfis not sufficient to lead to development of a useful technique is alsoillustrated by the DSR project. It had a patient table that moved thepatient through the scanner when an entire thorax or even the entiretorso needed to be imaged (Krayer et al., 1989). Yet at not time didthe investigators use this system to implement helical scan recon-struction, even though, in retrospect, that was an obvious possibility.Similarly, because of the limited dynamic range of the detector

systems and overexposure saturation consequences of the scanner’simaging system, two innovations were implemented. One innova-tion was designed to minimize the overexposure resulting wheneverthere was raw X-ray beam imaged alongside the body. This was

Figure 11. Coronary arteriographic views in adult human multipleorientation cine arteriograms versus single angiogram DSR scan(from Bove et al., 1985).

Figure 12. Physiologic parameters from dynamic CT (from Ritman,1987).

Figure 13. CT equipment survey in 1977 (from Norman et al., 1977).

Figure 14. Franke’s multiple X-ray CT scanner. The system neededto rotate only 120° to collect full projection data sets (from Franke,1977–1979).

Vol. 12, 175–187 (2002) 179

particularly bothersome in terms of the nonlinearity of the X-ray toelectrical signal conversion of the image isocon video camera used,and the “spillover” of signal to contiguous (but not overexposed)detectors (Ritman et al., 1973). This was largely eliminated by useof dynamic shutters, which “followed” the outline of the silhouettedbody, thereby largely eliminating the raw beam exposure (Jorgensen

et al., 1992). The other innovation (Ritman et al., 1985) was tocompensate for the fact that the exposure in the anteroposteriordirection of the thorax need only be approximately half that in the

Figure 15. Schematic of DSR multi-X-ray source CT scan (Sturm etal., 1990). The entire system rotated continuously at 15 rpm, whileX-ray sources were pulsed on sequentially over an 11-ms period, asequence that was repeated 60 times/s.

Figure 16. Wagner’s multiple x-ray source scanner. The whole setof sources and detector banks executes small angular rotations foreach sequential firing until each source has rotated completely tooccupy the position of its neighbor at the start of scanning. (FromWagner (1976–9))

Figure 17. Schematic of Boyd’s electron beam CT scanner (Boyd etal, Proc IEEE, 1983).

Figure 18. Noninvasive numerical “vivisection” (24 year old malescanned supine at total lung capacity) generated with the DSR scan-ner in 1983.

180 Vol. 12, 175–187 (2002)

lateral direction, in order to maintain the signal-to-noise similarly inall views of the body. The X-ray exposure was varied as a functionof the angle of view relative to the patient. It is only in recent years,with the need to keep X-ray exposure at a minimum, that thevariable exposure approach has been entertained (Kalender et al.,1999). This example illustrates that even the clear demonstration ofa capability is not necessarily implemented in other, existing, scan-ner systems—somehow the need for this capability needs to beapparent before such implementation proceeds.

III. THE MATHEMATICS BEHIND THE SCENECT relies on mathematics far beyond the level used in traditionalX-ray imaging. It is one of the ironies of the history of science thatthe basic result, Radon’s inversion formula, was known long beforethe advent of CT, but nobody in the field knew. Even worse, imagereconstruction algorithms were used in remote fields, such as radioastronomy and electron microscopy, long before CT but did not havethe slightest impact on the development in CT. Only recently hasthere been a general awareness that all the imaging techniques inradiology, science, and technology are based on the same mathe-matical principles and can be done by similar algorithms. A generalmathematical theory of imaging emerged and found much attention.As general references to the mathematics of CT see Herman (1980),Kak and Slaney (1987), Natterer and Wubbeling (1995).

Figure 19 demonstrates what the reconstruction problemamounts to. The abdominal cross section (top) has been computedfrom the “sinogram” (bottom). Each of the rows of the sinogramcontains the detector readings for one source position. In contrast totraditional X-rays, looking directly at the detector readings is use-less. A sophisticated mathematical machinery is required to produceuseful cross-sectional images. This machinery is the subject of thepresent section.

A. The 2D Case. In the simplest case, CT requires the recon-struction of a function in the plane from its line integrals. Let f besuch a function, and let Rf be its Radon transform (i.e., the set of allline integrals of f ). We represent each line in the plane in the formx � � � s, where � � �(�) � (sin �

cos �) is a unit vector and s a realnumber. Then, we can think of Rf as a function of �, s:

�Rf���, s� � �x���s

f�x� dx � ���

��

f�s� � t��� dt, (1)

where �� � (cos ��sin �) is perpendicular to �. Radon’s celebrated 1917

inversion formula reads:

f �1

4�R*H

�sg, g � Rf. (2)

Here, H is the Hilbert transform,

�Hg��s� �1

� ���

�� g�t�

s � tdt, (3)

and R* is the backprojection operator,

�R*g��x� � �0

2�

g��, x��) d�. (4)

Obviously R*g is the average over all lines through x. Radon’sinversion formula, Eq. (2) gave rise to the filtered backprojectionalgorithm (FBP), which is probably the most widely used algorithmin CT. In FBP, the operator (1/4�) H(�/�s) is approximated by aconvolution with a ramp filter w, resulting in the approximation

f � R*�w � g�, (5)

�w � g���, s� � ���

��

w�s � t�g��, t� dt. (6)

FBP is just a discretized version of Eqs. (5) and (6). In the medicalfield, FBP was introduced by Ramachandran and Lakshminarayananin 1971. However, Bracewell (Bracewell and Riddle, 1967) used thisalgorithm as early as 1956 in radio astronomy, and Wloka derived itfrom Radon’s inversion formula in his pioneering paper (1953) thathas been completely overlooked by the scientific community.

FBP is by far the most popular reconstruction algorithm in CT.The structure of the algorithm is very simple and supports the usualmanner of collecting the data: As soon as the data g for a direction� is measured, the filtering w � g can be carried out, and the filtereddata for different directions are simply superimposed. FBP is easilyextended to fan-beam scanning. Here, the X-ray tube is running ona circle of radius r, so that the measured line integrals are

g��, � � � f�r���� � s�� � ��� ds.

A completely different inversion formula was derived by Cor-mack in 1963. Using expansions of f, g � Rf in Fourier series,

f�r�� � �����

��

f��r�ei��,

Figure 19. Tomogram of abdomen (top) and data (bottom).

Vol. 12, 175–187 (2002) 181

g��, s)� �����

��

g�(s)ei��,

he showed that

f��r� � �1

� �r

�s2 � r2��1/2T����s

r�g��s� ds, (7)

with the Chebysheff polynomials T��� of the first kind. For practicalwork, this formula is useless, because T���(s/r) increases exponen-tially as a function of ��� for s r. Therefore, Eq. (7) is subject tounbearable numerical instabilities. In 1964, Cormack replaced Eq.(7) with the stable formula:

f��r� � �1

�r ��r

� �s2

r2 � 1��1/2�s

r� �s2

r2 � 1�1/2�����

g����s� ds

��0

r

U����1�1

r�g��s� ds�with the Chebysheff polynomials U��� of the second kind. Successfulreconstruction has been done with Eq. (7); see, for example, Hansen(1981). However, these algorithms never achieved the popularity ofFBP, because they are not adjusted as well as FBP to the way thedata are collected.

A completely different kind of reconstruction algorithms is basedon the relation

f��)�(2�)�1/2(Rf)�(�, ) (8)

between the 2D Fourier transform of f,

f��� �1

2� �R2

e�ix��f�x� dx

and the 1D Fourier transform of Rf,

�Rf����, � � �2���1/2 ���

��

e�is�Rf���, s� ds.

Eq. (8) is usually referred to as projection-slice or central sectiontheorem; in statistics it is called the Cramer–Wold theorem. At a firstglance, it looked as if it were the perfect solution to the reconstruc-tion problem: Taking 1D Fourier transforms of the data g � Rfyields the 2D of f. Hence, a 2D inverse Fourier transform yields f.If this is done by the fast Fourier transform (FFT), we obtain a veryefficient reconstruction algorithm, much more efficient than FBP. Infact, Fourier algorithms were used as early as 1970 by Crowther etal. (1970) in electron microscopy. First attempts in the medical fieldfailed because Fourier algorithms were much less accurate thanFBP. Only in 1985, O’Sullivan showed that Fourier reconstructioncan be made accurate by the gridding technique introduced in 1975by Brouw in radio astronomy. Gridding can be thought of as amethod for doing the FFT on a nonequispaced grid, a topic that has

found much attention recently; see Dutt and Rokhlin (1993), Beylkin(1995), Steidl (1998), Fourmont (1999). An alternative Fourierreconstruction method is the linogram method of Edholm et al.(1988). In the linogram method, the sampling of g is done so as tosupport the chirp-z-version of the FFT. Attempts have been made tospeed up the backprojection process to make FBP competitive withFourier reconstruction; see Gotz and Druckmuller (1996), Nilsson(1997). By and large, Fourier methods have not yet found much usein clinical CT. One should remark that in magnetic resonanceimaging (MRI), they are the methods of choice.

Often, explicit inversion procedures cannot be applied directly,because not all of the necessary data are available. An example isregion-of-interest tomography in which only a small part of the bodyis to be imaged, and, naturally, only rays hitting this small part aremeasured. To a certain extent, this can be done by local inversionformulae such as

�f � R*�2

�s2 g,

which can be evaluated locally (i.e., for the computation of �f( x),one needs only lines that pass a small neighbourhood of x). �f is notidentical to f, but it can be shown that f and �f have the samediscontinuities. This means that edges of f can be seen in �f(Faridani et al., 1990).

The reconstruction algorithm employed in the first commercialCT scanner (Hounsfield’s 1973 EMI scanner) was iterative. It waslater recognized as Kaczmarz’s method of 1937, which is simply amethod for solving linear systems of equations. Let �f � g be sucha system with an (m, n)-matrix A, the sought-for n-vector f, and thegiven m-vector g. Starting out from an initial approximation f 0

(usually 0), a sequence f i of approximations to f is computed by

f i � f i�1 � �ai

Tf i�1 � gi

ai2 , i � 1, . . . , m, (9)

where aiT is the i-th row of A, ai the Euclidean norm of ai, and �

a relaxation factor between 0 and 2. Equation (9) describes acomplete sweep of the Kaczmarz method. It is repeated until con-vergence. In the application to CT the matrix, � is a discreteapproximation to the Radon transform, either by pixelation (Her-man, 1980) or by the so-called blob functions (Lewitt, 1992). Be-cause no analytical tools, such as Fourier transforms or inversionformula, are used, the method is usually referred to as algebraicreconstruction technique (ART). Many versions of ART have be-come known in the literature (Herman, 1980). For fast convergence,the ordering of the equations in the system Af � g is decisive. Thisfact was recognized by Hamaker and Solmon (1993) as early as1978, was rediscovered over and over again (Herman and Meyer,1993), and has been extended to the EM algorithm by Hudson andLarkin (1994). The EM algorithm by Shepp and Vardi can beviewed as the multiplicative version of Kaczmarz’s method. If A isnormalized such that AT1m � 1n (1n is an n-vector with only 1’s asentries) it reads:

f i � f i�1ATg

Af i�1 ,

where multiplication and division are component wise. Iterativemethods, in particular the EM algorithm, gained some popularity in

182 Vol. 12, 175–187 (2002)

positron emission tomography and single photon emission computedtomography (PET and SPECT). One of the reasons is that no exactinversion formula for SPECT has been known until very recently,when Novikov (2000) derived such a formula for the attenuatedRadon transform.

B. The 3D Case. Interest in truly 3D reconstruction started in theradiological field in the early 1980s with the DSR (Jorgensen et al.,1990). Again, as so often in CT, the mathematical foundation werelaid much earlier by Kirillov (1961) and Orlov (1976).

In principle, 3D imaging in CT can be done by imaging 2Dslices. Slice-by-slice imaging leads to long scanning times. A firststep toward truly 3D imaging was helical scanning, introduced byKalender et al. in 1990. Here, the patient is moved continuously inthe axial direction of a CT scanner during the scanning process.Equivalently, we may assume that the X-ray source is running on ahelix around the patient. Assuming that the axis of the scanner is thex1 axis and that the radius of the helix is r, the data

g�x1, �, � � � f �x1, r���� � s�� � �� ds

are collected. Reconstruction is usually done slice by slice, thefan-beam data for a slice being computed by interpolation. Furtherspeed-up of the scanning process can be achieved by using morethan one linear detector array in a helical scanner.

The ultimate solution to truly 3D imaging is the cone-beamscanner. Here, the X-ray source is moving on a curve A, and thedetector is 2D. The data are the so-called cone-beam transform Df off, that is,

g�a, �� � �Df��a, �� � � f�a � t�� dt, a � A, � � S2.

A natural choice for the source curve A would be a circle, as in theDSR. Unfortunately, a circle does not permit stable reconstruction.Nevertheless, a stable approximate formula, the Feldkamp–Davis–Kress (FDK) formula (1984), is available for A with a circle ofradius r and a flat 2D detector. It reads:

f�x� �S1

r2

�r � x � ��2 � w�y2 � y2�g��, y2��

� y3e3�r dy2

�r2 � y22 � y3

2 d�,

y2 �r

r � x � �x � ��, y3 �

r

r � x � �x3, (10)

where g(�, y) is the integral of f along the line joining the sourceposition r� in the x1 � x2 plane. The detector sits at the point y �y2�� � y3e3 in the detector plane ��, which is spanned by the unitvector e3 in x3-direction and the unit vector �� perpendicular to �and e3. w is the familiar ramp filter from Eq. (6).

For stable reconstruction, the Kirillov–Tuy condition (Kirillov,1961; Tuy, 1983) has to be satisfied: Every plane hitting the objectintersects the source curve transversally. If this condition is met,exact inversion formulas are available. The derivation of these

formulas parallels the 2D case. We start out from Eqs. (5) and (6),that is,

W � f�x� � �S2

�R1

w�x � � � s��Rf���, s� ds d�,

where R is the 3D Radon transform and W � R*w, that is,

W�x� � �S2

w�x � �� d�.

Following Dietz (1999), we perform an integration by parts, obtain-ing

W � f�x� � �S2

�R1

v�x � � � s��

�s�Rf���, s� ds d�,

where v � w. Let a � a( ), � � � �1 be a parametricrepresentation of the source curve A. We want to substitute for sin the inner integral by putting s � a( ) � �. If the Kirillov–Tuycondition is met, then each s for which (Rf )(�, s) � 0 can bereached in this way. However there may be more than one inwhich s � a( ) � �; that is, there may be more than one source inthe plane x � � � s. Therefore, we choose a function M(�, ) suchthat

a� ����s

M��, � � 1

for each s. The simplest choice is M(�, ) � 1/n(�, a( ) � � )where n(�, s) is the number of ’s for which s � a( ) � �, that is,the number of source in the plane x � � � s. The result of thesubstitution s � a( ) � � is

W � f�x� � �S2

��

v��x � a� � ���

�s�Rf�

� ��, a � ��M��, ��a � �� d d�,

where a � a( ). The link between (�/�s) Rf and Df is provided byGrangeat’s celebrated formula

�s�Rf���, a � �� � G�a, ��,

G�a, �� � ����S2

��g�a, �� d�,

where �/�� stands for the directional derivative in direction � withrespect to the second argument of g. We obtain

W � f�x� � �S2

��

v��x � a� � ��G�a, ���a � ��M��, � d d�.

Vol. 12, 175–187 (2002) 183

This can be written as

W � f�x� � ��

Gv�a, x � a� d ,

Gv�a, x� � �S2

v�x � ��G�a, ��a � ��M��, � d�. (11)

This formula is the starting point of a reconstruction algorithm of thefiltered backprojection type. Further simplifications are possible. Forv � �(1/8�2)�, Gv becomes a function homogeneous of degree�2 in the second argument and W � �. Hence, we have

f�x� � ��

�x � a��2Gv�a,x � a

�x � a�� d ,

Gv�a, �� � �1

8�2 �S2

��� � ��G�a, ���a � ��M��, � d�, w � S2.

(12)

This formula was obtained by Defrise and Clack (1994), and Kudoand Saito (1994).

Contrary to Eq. (10), the Eqs. (11) and (12) are exact. However,they suffer from a serious drawback: They need the data functiong(a, � ) for all � � S2 and thus cannot handle truncated projections.The development of formulas for truncated projections (long-objectproblem) is still going on. Promising approaches include the PI-method of Danielsson et al. (1999) and algorithms of Tam et al.(1998) and Kudo et al. (1998).

IV. FUTURE DEVELOPMENTSIt is not clear what governs the formulation of an imaging need intoan expression of technological requirements. So a look into thefuture can at best be constrained to describing the technological andphysics limitations of the CT scanner components. Most wouldagree that, based on the sustained validity of Moore’s law, digitalcomputers are unlikely, in the near future, to represent a limitationon the design of scanners. The main limitation of X-ray CT appearsto be a radiation overexposure, so that efforts to reduce X-rayexposure will directly limit tomographic image quality (Brooks andDiChiro, 1976; Chesler et al., 1977; Motz and Danos, 1978). Con-sequently, every effort needs to be made to utilize all X-ray photonstransmitted through the body. The main physics limitation of X-rayCT appears to be the current dependence on X-ray attenuation by thebody. Implementation of mechanisms other than attenuation, such asphase delay, are plausible, but it is not clear whether the socioeco-nomic will and/or capability to proceed with such implementationswill be there.

At the present time, there are still several important, inherentlimitations of the CT scanner. The X-ray photon energy should bemonochromatic (Ross, 1928) and tunable (Grodzins, 1983) if theimages are to eliminate the beam-hardening artifact and for the greyscale of the CT image to be a truly quantitative representation of theattenuation coefficient of the body’s tissues. This can be achievedcurrently with synchrotron radiation (Dilmanian, 1992), but that is

an unrealistic source for routine clinical application. Current at-tempts involving filtration of the X-ray beam in order to “harden” itare not very effective and result in great loss of X-ray beam inten-sity, which increases the scan duration and places an inordinate heatburden on the X-ray tube, which in turn reduces the logistic through-put rate of the scanner. Hence, software methods for mitigating thebeam hardening are also a routine package in most scanners (Josephand Spital, 1978). While quite effective, these retrospective methodsresult in selective elimination of X-ray signal information and henceinvolve patient exposure with incomplete benefit to the patient(Motz and Danos, 1978). Another major limitation is X-ray scatter(Wagner et al., 1989). This has become an increasing issue with theuse of fan-beam and more recently with progression to cone-beamgeometry. The detection of the scatter can be minimized by suitablecollimation of the transmitted X-ray at the detector [as long as thedetector “pixel” is not used as a virtual focal spot for a fan or conebeam, such as is used in the Imatron Electron Beam CT (EBCT)scanner (Boyd and Lipton, 1983), but this generally results in loss ofsensitivity (i.e., some of the nonscattered beam is rejected also) andtherefore results in increased patient exposure to X-rays for a givensignal-to-noise of the CT image. Another approach is to limit thepatient exposure to just the region of interest, which results inmathematically incomplete data sets. If a priori knowledge is avail-able (Lewitt, 1979) or a “local” reconstruction algorithm (Faridani etal., 1990) is suitable, then this could also be utilized. In the future,if very brief (picoseconds) X-ray pulses can be generated, anddetected, in clinically practical scanners, scatter rejection by time-of-flight methods would not be handicapped by this “collateral” lossof signal photons. This approach would require very fast, high-resolution, detector systems similar, in principle, to those currentlyused in some PET scanners (Ter Pogossian et al., 1982), but capableof dealing with orders of magnitude higher event rates.

With the ever-increasing desire to increase contrast resolutionand contrast characteristics, several physics principles, beyond thecurrently used attenuation of the transmitted X-ray beam, could beinvoked. Dual-energy subtraction imaging is a modified, attenua-tion-based method that has already been explored (Alvarez andMacovski, 1976; Riederer et al., 1981), but is limited by the diffi-culty of obtaining high-flux, tunable, monochromatic X-ray sources.Nonetheless, with suitable selection of contrast agent (Zeman andSidons, 1990), low doses of contrast agent would still provide theneeded discrimination of blood vessels from the surrounding tissue.Another plausible method involves the use of phase-shift rather thanattenuation of X-ray as the signal of interest. This approach has beenused in micro-CT approaches (Beckman et al., 1997), but its use inhumans still appears to present a formidable technical challenge atthe X-ray generation, X-ray detection, and image processing levels.Nonetheless, this approach is attractive, because it would generate agrey scale closely proportional to true tissue density.

V. CONCLUSIONWithin the medical field, various aspects of the CT approach havebeen applied to ultrasound tomographic imaging (Greenleaf et al.,1975) and MRI (Lauterbur, 1973). However, due to the physicalproperties of the radiations involved, these methods have undergonedivergent evolution (Johnson et al., 1979; Kaveh et al., 1981; Mans-field and Maudsley, 1977; Natterer and Wubbeling, 1995).

This overview of the development and evolution of medical CTimaging does not include the development and evolution of themethodology in nonmedical arenas. Essentially independent devel-opment (primarily of the mathematical tools) has occurred in as-

184 Vol. 12, 175–187 (2002)

tronomy (Bracewell and Riddle, 1967), geological sciences (Sniederand Tramper, 2000), microscopic imaging (Crowther et al., 1970),crystalography (Bragg and Bragg, 1915), and nondestructive testing(Flannery et al., 1987), with little cross-fertilization among the fieldsin the early stages of their development. There is increasing evi-dence of mutual awareness in recent years, after the independentdevelopments were well established, and subsequent cross-fertiliza-tion has been accelerated.

REFERENCES

R.E. Alvarez and A. Macovski, Energy-selective reconstructions in x-raycomputerized tomography, Phys Med Biol 21 (1976), 733–744.

J. Ambrose and G.N. Hounsfield, Computerized transverse axial tomogra-phy, Br J Radiology 46 (1972), 148–149.

J. Ambrose, You never know what is around the corner, Rivista di Neuro-radiologia 9 (1996), 399–404.

J.R. Andrews and R.J. Stava, Planigraphy 2: Mathematical analyses of themethods, description of apparatus, and experimental proof, Am J Roentgen-ology 38 (1937), 145–151.

H.L. Baker, Jr., The impact of computed tomography on neuroradiologicpractice, Radiology 116 (1975), 637–640.

H.H. Barrett and W. Swindell, Analog reconstruction methods for transaxialtomography, Proc IEEE 65 (1977), 89–107.

F. Beckman, U. Bonse, F. Busch, and O. Gunnewig, X-ray microtomography(mCT) using phase contrast for the investigation of organic matter, JCAT 21(1997), 539–553.

G. Beylkin, On the fast Fourier transform of functions with singularities,Appl Comp Harm Anal 2 (1995), 363–381.

L.M. Bidaut, C. Laurent, M. Piotin, P. Gailloud, M. Muster, J.H.D. Fasel,D.A. Rufenacht, and F. Terrier, Second-generation three-dimensional recon-struction for rotational three-dimensional angiography, Acad Radiol 5(1998), 836–849.

S.S. Blume, Insight and industry: On the dynamics of technological changein medicine, MIT Press, Cambridge, MA, 1992.

H.B.A. Bockwinkel, Over de voortplanting van licht in een twee-assig kristalrondom een middelpunt van trilling, Natuurkunde 62 (1906), 636–651.

A.A. Bove, M. Block, H.C. Smith, and E.L. Ritman, Evaluation of coronaryanatomy using high speed volumetric CT scanning, Am J Cardiology 55(1985), 528–584.

D.B. Boyd and M.J. Lipton, Cardiac computed tomography, Proc IEEE 71(1983), 298–307.

R.N. Bracewell and A.C. Riddle, Inversion of fan beam scans in radio-astronomy, Astrophys J 150 (1967), 427–434.

W.H. Bragg and W.L. Bragg, X-rays and crystal structure, G. Bell & Sons,London, 1915. Chapter 2.

R.A. Brooks and G. DiChiro, Statistical limitations in x-ray reconstructivetomography, Med Physics 3 (1976), 237–240.

W.N. Brouw, Aperture synthesis, Methods in Comput Phys B 4 (1975),131–175.

L.S. Carey, G.L. Mansour, G.E. Ressmeyer, D.A. Svihel, V. Long, and E.J.Cooper, Synchronized pulsed cardiac planigraphy, Radiology 100 (1971),169–174.

D.A. Chesler, S.J. Riederer, and N.J. Pelc, Noise due to photon countingstatistics in computed xray tomography, J Comput Assist Tomogr 1 (1977),64–74.

W.D. Coolidge, Vacuum Tube, Application to the U.S. Patent Office, May13, 1913.

A.M. Cormack, Representation of a function by its line integrals, with someradiological applications I, J Appl Physics 34 (1963), 2722–2727.

A.M. Cormack, Representation of a function by its line integrals, with someradiological applications II, J Appl Physics 35 (1964), 195–207.

C.M. Coulam, J.J. Erickson, and S.J. Gibbs, “Image and equipment consid-erations in conventional tomography,” C.M. Coulam, J.J. Erickson, F.D.Rollo, and A.E. James, Jr. (Editors), The Physical Basis of Medical Imaging,Norwalk, CT, Appleton-Century-Crofts, pp. 123–140.

R.A. Crowther, D.J. DeRosier, and A. Klug, The reconstruction of a three-dimensional structure from its projections and its application to electronmicroscopy, Proc R. Soc London A317 (1970), 319–340.

P.E. Danielsson, P. Edholm, J. Eriksson, M. Magnusson Seger, and H.Turbell, The original PI-method for helical cone-beam CT, Proceedings ofthe 1999 International Meeting on Fully Three-Dimensional Image Recon-struction in Radiology and Nuclear Medicine, Egmond aan Zee, June 23–26,1999.

M. Defrise and R. Clack, A cone-beam reconstruction algorithm usingshift-variant filtering and cone-beam backprojection, IEEE Trans Med Imag13 (1994), 186–195.

R.L. Dietz, Die approximative Inverse als Rekonstruktions-methode in derRontgen-Computertomographie, Dissertation, Universitat Saarbrucken,Saarbrucken, Germany, 1999.

F.A. Dilmanian, Computed tomography with monochromatic x-rays. AMJ Physiol Imaging 7 (1992), 175–193.

J.T. Dobbins, Matrix inversion tomosynthesis improvements in longitudinalx-ray slice imaging, United States Patent 4,903,204, 1990.

T. Dolby and G. Alker, Origins and development of medical imaging,Carbondale and Edwardsville, Southern Illinois University Press, 1997.

A. Dutt and V. Rokhlin, Fast Fourier transforms for nonequispaced data,SIAM J Sci Comput 14 (1993), 1368–1393.

P. Edholm, G.T. Herman, and D.A. Roberts, Image reconstruction fromlinograms: Implementation and evaluation, IEEE Trans Med Imag 7 (1988),239–246.

A. Faridani, F. Keinert, F. Natterer, E.L. Ritman, and K.T. Smith, “Local andglobal tomography,” F.A. Grunbaum, J.W. Helton, and P. Khargonekar(Editors), Signal Processing, Part II (IMA Control Theory and Applications),Vol 23, Springer-Verlag, New York, 1990, pp. 241–255.

L.A. Feldkamp, L.L. Davis, and J.W. Kress, Practical cone-beam algorithm,J Opt Soc Am 1 (1984), 612–619.

B.P. Flannery, H.W. Deckman, W.G. Roberge, and K.L. D’Amico, Threedimensional x-ray microtomography, Science 237 (1987), 1439–1444.

K. Fourmont, Schnelle Fourier-Transformation bei nicht-aquidistanten Git-tern und tomographische Anwendungen, Dissertation, Fachbereich Math-ematik und Informatik der Universitat Munster, Munster, Germany, 1999.

G. Frank, Verfahren zur Herstellung von Korperschnittbildern mittels Ront-genstrahlen, German Patent 693374, 1940.

K. Franke, Tomographic apparatus for producing transverse layer images,U.S. Patent 4150293, 1977–1979.

H.H. Goldstine, The computer from Pascal von Neumann, Princeton Uni-versity Press, Princeton, NJ, 1988.

R. Gordon, R. Bender, and G.T. Herman, Algebraic reconstruction technique(ART) for three-dimensional electron microscopy and x-ray photography, JTheor Biol 29 (1970), 471–481.

W.A. Gotz and H.J. Druckmuller, A fast digital Radon transform: Anefficient means for evaluating the Hough transform, Pattern Recogn 29(1996), 711–718.

J.F. Greenleaf, S.A. Johnson, W.F. Samayoa, and F.A. Duck, “Algebraicreconstruction of spatial distributions of acoustic velocities in tissues fromtheir time-of-flight profiles,” Newell Booth (Editor), Acoustical holography,Vol. 6, 1975, Plenum Press, New York, pp. 71–90.

L. Grodzins, Optimal energies for x-ray transmission tomography of smallsamples, Nucl Instrum Methods 206 (1983), 541–545.

Vol. 12, 175–187 (2002) 185

C. Hamaker and D.C. Solmon, The angles between the null spaces of X-rays,J Math Anal Appl 62 (1978), 1–23.

E.W. Hansen, Circular harmonic image reconstruction, Appl Optics 20(1981), 2266–2274.

D.C. Hemmy, F.W. Zonneveld, S. Lobregt, and K. Fukuta, A decade ofclinical three-dimensional imaging: A review: Part I. Historical development,Invest Radiol 29 (1994), 489–496.

G.T. Herman and H.K. Liu, Display of three dimensional information incomputed tomography, J Comput Assist Tomogr 1 (1977), 155–160.

G.T. Herman, Image reconstruction from projection: The fundamentals ofcomputerized tomography. Academic Press, New York, 1980.

G.T. Herman and L. Meyer, Algebraic reconstruction techniques can bemade computationally efficient, IEEE Trans Med Imag 12 (1993), 600–609.

G.J. Hine, The inception of photoelectric scintilation counters, J Nucl Med18 (1977), 868.

B. Holtzmann Kevles, Naked to the bone: Medical imaging in the twentiethcentury, Rutgers University Press, New Brunswick, NJ, 1997.

G.N. Hounsfield, Computerized transverse axial scanning tomography: PartI: Description of the system. Br J Radiol 46 (1973), 1016–1022.

G.N. Hounsfield, A method of and apparatus for examination of a body byradiation such as X or gamma radiation, U.K. Patent 1283915, 1968–1972.

H.M. Hudson and R.S. Larkin, Accelerated EM reconstruction using orderedsubsets of projection data, IEEE Trans Med Imag 13 (1994), 601–609.

J. Mattson and M. Simon, “The pioneers of NMR and magnetic resonance inmedicine, The story of MRI, Bar-Ilan University Press, 1996, p 838.

P.D. Johns and M. Yaffe, Scattered radiation in fan beam imaging systems,Med Phys 9 (1982), 464–472.

S.A. Johnson, J.F. Greenleaf, B. Rajagopalan, and R.C. Bahn, “Ultrasoundimages corrected for refraction and attenuation: A comparison of new highresolution methods,” in J. Raviv, J.F. Greenleaf, and G.T. Herman (Editors),Computer aided tomography and ultrasonics in medicine, North Holland,Amsterdam, 1979, pp. 55–71.

S.M. Jorgensen, S.V. Whitlock, P.J. Thomas, R.W. Roessler, and E.L.Ritman, The dynamic spatial reconstructor: A high speed, stop action, 3-D,digital radiographic imager of moving internal organs and blood, Proc SPIE,Ultrahigh- and High-Speed Photography, Videography, Photonics, and Ve-locimetry 1346 (1990), 180–191.

S.M. Jorgensen, S.V. Whitlock, P.J. Thomas, and E.L. Ritman, Dynamicimage-adaptive x-ray beam limiters, Biomed Instrum Tech 26 (1992), 328–344.

P.M. Joseph and R. Spital, A method for correcting bone induced artifacts inCT scanners, J Comput Assist Tomogr 2 (1978), 100–108.

S. Kaczmarz, Angenaherte Auflosung von Systemen linearer Gleichungen,Bulletin de l’Academie Polonaise des Sciences et des Lettres A35 (1937),355–357.

A.C. Kak and M. Slaney, Principles of computerized tomography imaging,IEEE Press, New York, 1987. Reprinted as SIAM classics in applied math-ematics, SIAM, Philadelphia, 2001.

A. Kalender, W. Klotz, and E. Vock, Spiral volumetric CT with singlebreath-hold technique, continuous transport, and continuous scanner rotation,Radiology 176 (1990), 181–183.

W.A. Kalender, H. Wolf, C. Suess, M. Gies, H. Greess, and W.A. Bautz,Dose reduction in CT by on line tube current control: Principles and vali-dation on phantoms and cadavers, Eur Radiol 9 (1999), 323–328.

M. Kaveh, M. Soumekh, and R.K. Meuller, A comparison of Born and Rytovapproximations in acoustic tomography, Acoust Imaging 11 (1981), 325–335.

A.A. Kirillov, On a problem of I.M. Gelfand, Soviet Math 2 (1961), 268–269.

K. Kishi, S. Kurihara, M. Yuasa, H. Gouke, Radiotherapy treatment planningapparatus, Toshiba Rev vol. 43, 1969, 36–41.

B.I. Korenblyum, S.I. Tetel’baum, and A.A. Tyutin, About one scheme oftomography, Izv VUZ Radiofiz 1 (1958), 151–157.

S. Krayer, K. Rehder, J. Vettermann, E.P. Didier, and E.L. Ritman, Positionand motion of the human diaphragm during anesthesia-paralysis, J Anesthe-siol 70 (1989), 891–898.

H. Kudo and T. Saito, Derivation and implementation of a cone-beamreconstruction algorithm for nonplanar orbits, IEEE Trans Med Imag 13(1994), 196–211.

H. Kudo, F. Noo, and M. Defrise, Cone-beam filtered backprojection algo-rithm for truncated and helical data, Phys Med Biol 43 (1998), 2885–2909.

P.C. Lauterbur, Image formation by induced local interactions: Examplesemploying nuclear magnetis resonance, Nature 242 (1973), 190–191.

R.S. Ledley, G. DiChiro, A.J. Ruessentrop, and H.L. Twigg, Computerizedtransaxial x-ray tomography of the human body, Science 186 (1974), p. 207.

R.M. Lewitt, Processing of incomplete measurements data in computedtomography, Med Phys 6 (1979), 412–417.

R.M. Lewitt, Alternatives to voxels for image representations in iterativereconstruction algorithms, Phys Med Biol 37 (1992), 705–716.

B.H. Lill, Apparatus for computerized tomography having improved anti-scatter collimators, U.S. Patent 4101768, 1997–1998.

J.T. Littleton, “Tomography: Physical principles and clinical applications,”in L.L. Robins (Editor), Section 17 of Golden’s diagnostic radiology, Wil-liams & Wilkins, Baltimore, MD, 1976.

P. Mansfield and A.A. Maudsley, Medical imaging by NMR, Brit J Radiol 50(1977), 188–194.

I. Mori, Computerized tomographic apparatus utilizing a radiation source,U.S. Patent 4630202, 1986.

J.W. Motz and M. Danos, Image information content and patient exposure,Med Phys 5 (1978), 8–22.

F. Natterer and F. Wubbeling, A propagation–backpropagation method forultrasound tomography, Inverse Problems 11 (1995), 1225–1232.

F. Natterer and F. Wubbeling, Mathematical methods in image reconstruc-tion, SIAM, Philadelphia, 2001.

S. Nilsson, “Application of fast backprojection techniques for some inverseproblems of integral geometry,” Linkoping studies in science and technol-ogy, Dissertation No. 499, Department of Mathematics, Linkoping Univer-sity, Linkoping, Sweden, 1997.

R. Ning and R.A. Kruger, Image intensifier-based computed tomographyvolume scanner for angiography, Acad Radiol 3 (1996), 344–350.

H. Nishimura and O. Miyazaki, CT system for specially scanning subject ona moveable bed synchronized to x-ray tube revolution, U.S. Patent 4 789 929,1988.

R.G. Novikov, An inversion formula for the attenuated X-ray transform,Preprint, Departement de Mathematique, Universite de Nantes, 2000.

D. Norman, M. Korobkin, Th. Newton, Computed tomography, 1977,Mosby, St. Louis, 362 pg.

W.H. Oldendorf, Radiant energy apparatus for investigating selected areas ofthe interior of objects obscured by dense material, U.S. Patent 3106640,1963.

S.S. Orlov, Theory of three dimensional reconstruction: II. The recoveryoperator, Sov Phys Crystallogr 20 (1976), 429–433.

H. Queisser, The conquest of the microchip. Harvard University Press,Cambridge, MA, 1988.

J. Radon, Uber die Bestimmung von Funktionen durch ihre Integralwertelangs gewisser Mannigfaltigkeiten, Berichte Sachsische Akademie der Wis-senschaften, Leipzig 69 (1917), 262–277.

186 Vol. 12, 175–187 (2002)

G.N. Ramachandran and A.V. Lakshminarayanan, Three-dimensional recon-struction from radiographs and electron micrographs: Application of convo-lutions instead of Fourier transforms, Proc Nat Acad Sci USA 68 (1971),2236–2240.

S.J. Riederer, R.A. Kruger, and C.A. Mistretta, Limitations to iodine isola-tion using a dual beam non-K-edge approach, Med Phys 8 (1981), 54–61.

E.L. Ritman, S.A. Johnson, R.E. Sturm, and E.H. Wood, The televisioncamera in dynamic videoangiography, Radiology 107 (1973), 417–427.

E.L. Ritman, Three-dimensional anatomy and function of the pulmonaryarterial tree: Overview of results with the dynamic reconstructor, Innov TechBiol Med 8 (NE Special 1) (1987), 37–50.

E.L. Ritman, S.M. Jorgensen, M.H. Rhyner, R.W. Roessler, S.V. Whitlock,and P.E. Caskey, X-ray image chain for a computed tomography approach tointravenous coronary arteriography, Proc Int Symp CAR 85 (1985), 65–69.

E.L. Ritman, R.A. Robb, and L.D. Harris, Imaging physiological functions:Experience with the dynamic spatial reconstructor, Praeger, Philadelphia,PA, 1985.

R.A. Robb, X-ray computer tomography and Engineering Synthesis ofmultiscientific principals. Crit Revs in BioMed 7 (1982), 265–334.

R.A. Robb, J.F. Greenleaf, S.A. Johnson, E.L. Ritman, R.E. Sturm, B.K.Gilbert, P.A. Chevalier, and E.H. Wood, Three-dimensional reconstructionand display of the working canine heart and lungs by multiplanar x-rayscanning videodensitometry, Proc International Workshop, Brookhaven Na-tional Lab, Spec Publ BNL 20425 (1975), 99–106.

W.C. Roentgen, Uber eine neue Art van Strahlen, Wurzburg, Stahel, 1895.

P.A. Ross, A new method of spectroscopy for faint x-radiations, J Opt SocAm 16 (1928), 433–437.

K. Seide and E.L. Ritman, Three-dimensional dynamic x-ray computedtomography imaging of stomach motility, Am J Physiol 247 (Gastro IntestLiver Physiol 10) (1984), G574–G587.

L.A. Shepp and J.A. Stein, “Simulated reconstruction artifacts in computer-ized x-ray tomography,” M.M. Ter-Pogossian, M.E. Phelps, G.L. Brownell(Editors), Reconstruction tomography in diagnostic radiology and nuclearmedicine, University Park Press, Baltimore, 1977, pp. 33–48.

L.A. Shepp and Y. Vardi, Maximum likelihood for emission tomography,IEEE Trans Med Imag (1982), 113–122.

R. Snieder and J. Tramper, “Linear and nonlinear inverse problems,” A.Dermanis, A. Grun, and F. Sanso (Editors), Geomatic methods for theanalysis of data in the earth sciences, Springer, Berlin, 2000, pp. 93–164.

J.D. O’Sullivan, A fast sinc function gridding algorithm for Fourier inversionin computer tomography, IEEE Trans Med Imag 4 (1985), 200–207.

G. Steidl, A note on fast Fourier transforms for nonequispaced grids, AdvComput Math 9 (1998), 337–352.

R.A. Street, S. Nelson, L.E. Antonuk, and V. Perez Mendez, Amorphoussilicon arrays for radiation imaging, Mat Res Soc Proc 192 (1990), 441–452.

R.E. Sturm, E.L. Ritman, and E.H. Wood, Quantitative three-dimensionaldynamic imaging of structure and function of the cardiopulmonary andcirculatory systems in all regions of the body, D.C. Harrison, H. Sandler, andH.A. Miller (Editors), Cardiovascular Imaging and Image Processing, The-ory and Practice 72 (1975), 103–122.

S. Takahasi, Rotation radiography. Tokyo, Japan Society for the Promotionof Science.

K.C. Tam, S. Samasekara, and F. Sauer, Exact cone-beam CT with a spiralscan, Phys Med Biol 43 (1998), 1015–1071.

M.M. Ter Pogossian, D.C. Ficke, M. Yamamoto, and J.T. Hood, Sr., SuperPETT 1: A positron emission tomograph utilising time-of-flight information,IEEE Trans Med Imag MI-1 (1982), 179–187.

H.K. Tuy, An inversion formula for cone-beam reconstruction, SIAM J ApplMath 43 (1983), 546–552.

A. Vallebona, Una modalita di technica per la dissociazione radiografica,Radiologia Medica 17 (1931), 1090–1097.

W. Wagner, Device for measuring local radiation absorbtion in a body, U.S.Patent 4057725, 1975–1978.

F.C. Wagner, A. Macovski, and D.G. Nishimura, Two interpolating filters forscatter estimation, Med Phys 16 (1989), 747–757.

T. Wang, X. Wu, N. Chung, and E.L. Ritman, Myocardial blood flowestimated by synchronous, multislice, high speed computed tomography,IEEE Trans Med Imaging 8 (1989), 70–77.

S. Webb, From the watching of shadows—the origins of radiological tomog-raphy, Adam Hilger, Bristol and New York, 1990.

J. Wloka, Tomografia. Protokolle des Seminars Anwendungen der Math-ematik (Lukaszewicz, Perkal, Steinhaus) der Universitat Breslau, 1953.

H.D. Zeman and D.P. Sidons, Contrast agent choice for intravenous coronaryangiography, Nucl Instrument Methods Phys Res A 291 (1990), 67–73.

J. Ziman (ed.), Technological innovation as an evolutionary process, Cam-bridge University Press, Cambridge, UK, 2000.

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