CA1111575A - Apparatus and method for reconstructing data - Google Patents

Abstract

PATENT APPLICATION
by JOHN M. PAVKOVICH
for APPARATUS AND METHOD FOR RECONSTRUCTING DATA
Abstract of Disclosure In apparatus and method for reconstructing data, a fan beam of radiation is passed through an object, which beam lies in the same quasi-plane as the object slice to be examined. Radiation not absorbed in or scattered by the object slice is recorded on oppositely situated detectors aligned with the source of radiation. Relative rotation is provided between the source-detector configuration and the object.
Reconstruction means are coupled to the detector means, and may comprise a general purpose computer, a special purpose computer, and control logic for interfacing between said computers and controlling the respective functioning thereof for performing a convolution and back projection based upon non-absorbed and non-scattered radiation detected by said detector means whereby the reconstruction means converts values of the non-absorbed and non-scattered radiation into values of absorbed radiation at each of an arbitrarily large number of points selected within the object slice. Display means are coupled to the reconstruction means for providing a visual or other display or representation of the quantities of radiation absorbed at the points considered in the object.

Description

- Backqround of the In~ention ... , . _ . . _ .
This application is a continuation-in-part of my copending application, Serial No. 64~,896, filed on December 23, l975 and entitled "Apparatus and Method for Reconstructing Data", now United States Patent Number 4,149,24&.
Field of the Invention This invention relates to a method and apparatus for constructing a two-dimensional picture of an object slice from linear projections of radia~ion not absorbed or scattered by the object, useful in the fields of medical radiology, microscopy, and non-destructive testing. The branch of the invention employing x-rays for medical radiology is sometimes referred to as Computerized Tomography.
Description of the Prior Art It is useful in many technologies to construct a two-dimensional pictorial representation from a series of linear data resulting from sensory projections taken through - the quasi-plane within which lies the two-dimensional planar-slice of the ob~ect that one wishes to reconstruct. For example, in the case o~ utilizing x-rays to provide a pictorial representation of the inside of the human body it is known to pass X or- gamma radiation through the tissues of the body and measure the absorption of this radiation by the various tissues. The nature of the tissues may then be determined by the percentage extent ehb2013377 - 2 - ~6-~0 llllS75 of absorption in each tissue of the radiation, since different tissues are ~nown to absorb differinq amounts of radiation.
Passin~ a ~all of radiation through an ohiect and detecting the amount of absorption within the object by means of complementar~-sDaced detectors results in a three-dimensional object being projected onto a two-dimensional picture. This can result in the superim~osition of information and resulting lcss of sai~ infor~atior.. More ~ophisticated techni~ues must be devised if one wishes to examine a ~0 ~ody with gre2ter sensitivity to spatial variations in radiation absorption and fewer superimposition effects.
In a method kno~n as general to~osraphy a source of radiation and a photographic film are re~olved along an elliptical or other path near the body in such a way that . elements in one plane of the body remain substantially stationary. This techniaue is utilized to obtain relevant information along a two-dimensional ~lanar slice of the body.
This method has a disadvantage in that sh~do~s of bodily tiss~es on plane.s of the body other than the desired planar slice appear as background information partially obscuring .he information desired to be obtained frcm the cognizant slice.
In an attempt to obtain more accurate information, methods have be~n proposed whereby the radiation and detection of same all lie within the planaL- slice of the ob~ect to be examined. A t~o-di~ensional recon~tr~ction o~
the thin slice of the object is then perorMed, and r~p~ated for each slice desired tc b~ portray2~ or diagno~ed.
. In ~. M. Cormac~, "Representations of a Function by Line Inte~rals with Some Radiological ,~p?1ications", Journal of Applied ?hysics, Vol. 34, No. , pp. 2722-2727, 'Septemb~r 19~3), (reference 1), the a~thor used a colli~;~te~ 7 ~illimet~r lBl _ 3 _ 1111~75 diameter beam of cobalt 60 gamma rays and a collimated Geiger counter. About 20,000 counts were integrated for each 5 mill meter latcral displaccment of ~e beam which passed through a phantom 5 centimeters thick, and 20 centi-meters in diameter comprising concentric cylinders of alumin-um, aluminum alloy and wood. Because of symmetry of the phantom, measurements were made at only one angle. The re-sulting calculated absorption coefficients were accurate to plus or minus 1.5 percent.
io In October, 1964, the same author in "Representations of a Function by Line Integrals with 5Ome Radiological Appli-cations. II", Journal of Applied Physics, Vol. 35, No.10,pp 2908-2913, (2), separated the two-dimensional problem into a set of one-dimensional integral equations of a function with solely radial variation. The measurements were expanded in a sine series with coefficients identical to those of the , radial density function when expanded in a limited series ofZernicke polynomials. This method is mathematically equiva-lent to a Fourier transform technique but differs in prac-tical application, such as significance of artifacts intro-~ duced by interpolation. Cormack used a collimated 5 x 5 ¦ millimeter beam of cobalt 60 gamma rays and a collimated Geiger counter. About 20,000 counts were integrated for each beam position. The beam was displaced laterally by 5 millimeter intervals to form a parallel set of 19 lines and the set was repeated at 7.5 degree intervals for 25 separate angles. The phantom was 2.5 centimeters thick, 20 centi-meters in diameter, comprising an aluminum disc at the cen-ter, an aluminum ring at the periphery, an aluminum disc off axis, and the remainder LUCITE*. From 475 independent measurements, 243 constants were determined and used to synthesiæe the ~1115~
absorption distribution~ Tl-e resultinq accuracy of c~lculated absorption values was good on average but rin~ing w~s introduce~ b~ the sharp changes in density. Cormack's method is capable in theory of yielding a uni~ue Principal solution, but is nevertheless co~plicated, has li~ited practical application and is liable to error in its practically feasible forms.
D. J. DeRosier and A. Klug in "Reconstructicn of Three-Dimensio~al Structures from Electron Micro~raphs", ~ature, Vol. 217, pp. 130-134 ~January 13, 1968), (3), used Fourier transforma.tion of two-dimensional electron transmission images (electron micro~r2phs) at a number of angles ~30 for nonsymmetric objects) to produce a series.
-of sections representing the object in three dimensions.
L Resolution of the final three-dimensionai Fourier density map was 30 Angstroms, for a 250 Angstrom T4 bacteriophage tail.
~. G. Hart, "Electron MicroscGpy of Unstaine~
Biological Material: The Polytro~ic rlontage", Science, Vol. 159, pp. 1~64-1467 (March, 1968), (4), used 12 electron micrographs taken at different angles, a flying spot scanner, cathode ray tube and a CDC-3600 com~uter (Control ~t~
Corporation, Minneapoiis, Minnesot~) with 48 bit, 32 ~ word core to pro~uce a section disDlay by digital su~erposition.
Resolution approached 3 Anqstroms.
D. E. Kuhl, J. H~le and W. L. Eaton, "~ransmission Scanning: A Useful Adjunct to Conventional ~missi.on Scannina for Accurately Keyin~ Isotope DepositiGn to Radio~raphic Anato~v", Radioloav, Vol. 87, pp. 278-284, in Au~ust, 19~6, (5), (see FIG. 1~) installed a collimated radioactive source (100 ~illicuries of ~0 keV ~mericium-24l) 1~1 ' i217~ - 5 _ l~liS7~
OppOsite 011e detector of a scanner which had two opposed detectors. (Kuhl also su~ested that a 1 millicurie 30 keV Iodine-125 source could be installed opDosite each of the detectors of a two detector sytem.) A 6.3 millimeter hole was drilled in the collimator of the o-?posina detector.
The opposed detectors were translated together to scan th patient at each of a number of angles usually 15 degrees apart ~see Kuhl and Edwarcls, "Cylindrical and Section Radioisotope Scanning of Liver and Brain", Radiology, Vol. ~3, 926, Novemher, 1564, at paqe 932) t6). A CRT (cathode ray tube) beam was swept to form a narrow illuminated line corresponding to the orientation and position of the 6.3 millimeter gamma beam through the patient and as the scan proceeded the brightness of the line or. the CRT was varied according to the count rate in the detectcr; a transverse section image was thus built up on a film viewing the CRT. ~uhl found the transverse section transmission scan to be especially useful for an anatomic orientation of a simultaneous transverse section emission scan of the human thorax and mediastinum.
At the June, 1966 meeting of the Society of Nuclear Medicine in Philadelphia, Dr. Kuhl (D.`E. Kuhl and R. Q. Edwards, Abstract A-5 "Reorganizing Transverse Section Scan Data as a Rectilinear Matrix Using Digital Processing", Journal cf Nuclear Medicine, Vol. 7, P.
332, (June, 1966), ~7~, described the use of di~ital processing of his transverse section scan data to produce a rectilinear matri~ mage s~perior to the images obtained ~ith the above me~hod of ~ilm exposure sum~ation of count rate modulated CRT lines. The scan dat2 from each detector was stored on ~agnetic tape, com~rising a series 1~]

111157~
of scans at 2~ diffcrent ~ngles 7.5 de~rees 2part around the patient. One hundr~d eighty-one thousand operations were perEormed in 12 minutec. on this data to produce a tr~nsverse section image ~atrix of 10,000 elements.
The ~rocess is descri~ed in more detail in D. ~. I;uhl and R. Q. Edwards, "Reorganizing Data from Transverse Section Scans of the Brain Usina Diyital Processing", Radiology, Vol. 91, p. 975 (November, 1968 (8). The matrix comprised a 100 by 100 array of 2.5 milli~eter by 2.5 millimeter elements. For e2ch picture element the counts recorded on the sc~n line through the element at each of the 24 scan an~les were extracted by progra~ned search from drum storage, summe~, divided by 2 and stored on tape, after which they could be called ~- sequentially to produce a C~T raster ccan.
R. A. Crowther, D. J. DeRosier, end ~. Klu~, in UThe Reconstruction of a Three Dimensional Structure from Projections ar.d Its Application to Electron ~licroscopy", Proceedings of the Royal Society of ~ondon, 317A, 319 Z~ (1970), (g), developed a for~al solution of the problem of reconstructin~ three-dimensional absorption distributi~ns fro~ two-dimensional electron micro~raph projections, usina Fourier transformation. They considered a series of 5 degree t;lts from ~45 degrees to -45 de~rees and foun~ that at least ~ D/~ views are required to reconstruct a ~ody of diameter D to a resolution of d. p. 332.
M. Go tein, in "Three Di~.ensional Density Reconstruction from a Series of Two-Di~ensional Pro~ections", ~ucle~.r Instruments and Methods 101, 509 ~1972), (10), shows t~.at standard ~,atrix inversi.on techni~ues for two dimensional reconstructions re~uire too m~c~ stor~e s~acc.

lBl _ 7 _ lill575 ~e st~tes that a 501; word memory is re~uired for an inversion of a 225 x 22S matrix for a 15 x 15 element obiect grid and that ~ith use of overflo,~ memory the execution time increases as the sixth power of the number of cells alon~ the edge of the object grid, ~.511. ~e proposes an iter~tive relaxation procedure since an "exact solution" is not computationally accessable for a ty~ical object grid such as 100 x lQ0 elements. This technigue involves ad~usting the densit~
of any cell to fit all me~surements which involve that cell, "fit" being on the basis of least-s~uares minimization ~e used the Cormac~ (1964) phantom design as a model, simulated. i~ on a computer, "~easured" absor~tion w;th a scan of 51 transversely separated lines repeated at 40 uniformly spaced angles, introduced 1% ran~om error in the measurements znd computed the absorption distribution in a transverse section view on a 30 x 30 grid using 15 iterations. He also comp~ted absorption distributions in transverse section view using the or.iginal absorption data recorded by Cormack (1964) as well as data furnished by others from âlpha beam and X-ray beam transmission measurements.
D. Kuhl, R. Q. Edwards, A. R. Ricci and M. Reivich, in "Quantitative Secti.on Scanning Usin3 ~rthogonal Tangent Correction", Abstract, Journal of Nuclear Medicine Vol. 13, p.447 (June, 1972), (11), describe an iterative co~putation method combining the d~ta from a scan at one angle with the data from ~ scan at 90 degrees to this angle, an~ repeating this computation ~rocess for â multitude of angles. An iterative correction is cont;nued throuqh 311 anoles, re~uiring 10 minutes with a ~arian 16 ~it 8K word core compu~er ~arian Data ~1ach,nes, Irvine, Californiâ)~

lBl - 8 -~ 11 of these methods suf~er from certain de~iciencies.
The ~rrors inherent in such l~rior art t~chninues are not easily ascertainable. The time to gather the data is slo~;;
in the case of X-ray dia~nosis, this increases the ti~e the patient must be strapped in an uncom~-ortable position and limits the throughput, i.e., total pati~nt handlin~ -capacity, of the machine. It also means that for slices of bod~ regions such as the abdominal cavity, the ~atient's normal breathing produces motion in the object durin~
the taking of measurements and conseouent blurring of the output picture, which can mas~, for example, the presence of tumors. The time recuired to reduce the data to PiCture form is lengthy, typically on the order of a ouarter of an hour. Spatial resolution of the output pictu~e is often relatively poor.
D. Boyd, J. Coonrod, J. Dehnert, D. Chu, C. Lim, B. MacDonald, and V. Perez-Mendez, "A High Pressure Xenon Proportional Chamber for X-Ray Lamino~raphic Reconstruction Using Fan Beam Geometry," IEEE Transactions on Muclear Science, Vol. NS-21, No. 1 (Feb. 1974) (12), describe, at ~.
185, a reconstruction method for a fan beam source which employs a convolution methcd of data reconstr~ction. This use of a fan beam can result in a reduction in data-gatherinq time, and a more efficient utilization of radiation ~lux.
~owever, the fan bea~ rays âre first reordered into paral~el-beam rays~ then a known par~llel ray convolution method is employed. This ste~ of ~irst reordering the data introduces a delay. An additional problem with this method is that normal optir,lization of desi~n crit~ria in most aDplic~tions ~G reauir~s that the an~le bet~leen individual rays of the fan bea~ be less than the angl~ of arc between ~uls~ of th~

1~31 ' 121i7~ - 9 -llli~75 source. Thus, there is no one-to-one ma~chu? between fan beam rays and ~arallel beam rays. As a result, approximations ~ust be m~de durin~ the reordering step, causin~ a-diminution in resolution in the cut~ut ?icture.
Even in the case where there is a one-to-one relationship between fan beam rays and parallel bea~ ra~s, the distances between-the resultant parallel beam rays will be uneoual.
Therefore, ~nother set of resolution-di~inishin~ approximations must be made. Another problem with reorderin~ is that reorderina forces one to fix irrevocably the number of pulses per revolution of the source. This results in a loss of flexibility because, for exa~ple, the wider the object beinq pictured, the smaller the arcvate angle between pulsing re~uired for the ~ame resolution. If one does not reorder, one can design into the machine convenient me~ns whereby the operator may adjust the arcuate angle between pulsing depending upon the object size.

No prior art method combines the use of a fan beam source and the application of a convolu~ion me'hod of data reconstruction with no intervening reordering of the detected projection profiles o~rer each other. No prior art method is capable of providing an exact reconstruction of a two-~imensional picture from a series of one-dimensional projections when the superior fan bea~ source is emplo~ed.

lBl _ ~o -12~775 0l.3~CT~ 0&` Tll~. INVE~TI0~

It is therefore a primary object of the instant invention to provide an improved method and ap~aratus for reconstructing two-dimensional pictures from a succession of linear projections for use in computerized tomoqra~hy, electron microscopy/ and other fields of technology, and for creating three-dimensional pictures by piecing together a series of t~o-dimensional pictures.
It is a further object of the present invention to provide means for obtainina a two-dimensional Y.-ray pictorial representation of a slice of a human or other boay in which the body is exposed to radiation for a shorter time than in the prior art.
It is another object of the instant invention ~- to provide for more accurate two-dimensional reconstruction of an image based upon a succession of one-dimensional data obtained by projecting radiation through the object slice in its ~uasi-plane.
It is still another obiect of the instant invention to provide an improved means for reconstructing a two-dim~nsional pictorial representation of a planar slice of an object by passin~ radiation throu~h the object, in which the errors inherent in the pictorial reconstruction are readily ascertainable.
It is another object or the instant inVentiGr to provide an improved apparatus and method for reconstructing 2 two-dimensional Picture from a series of one-dimensional projection profiles in which the data ac~uisition an~ data r~construction times are ~oth substantially reduced over the prior art for the sa~e degree of resclution.

lBl lill~5 It is another object of the instant invention to project a fail beam of radiation through an objec- slice in its auasi-plal-e and reconstruct ~ picture of the object by means of a convolution method of data r~construction which operates on the detected projection profiles ~roduc-d by the fan bea~s with no intervening reordering of fan beam rays into a different set of rays.
It is y~t another object of the ins~ant in~ention to provide an exact reconstruction replica of a t~70 dimensional 0 . object slice when a f~n beam of radiation is passed through the object at different angles creating ~ set of one dimensional measured ~rojections.
It is another object of the instant invention to ~raphically picture a reconstruction of a two dimensional obiect based on a set of one-dimensional projection values corresponding to a~oun~s of detected radiation non-absorbed and non-scattered by the object, in which the resuolution is lmproved over the prior art for the same data acquisition and reconstruction times.

lBl - 12 -L~
` ~illS~5 ~SUM1~ Y OF INVJ::NTION
.
Brie~ly and in accordance with the ~bove ob~ects the present invention is concerned with a method and apparatus for constructing a two-dimensional picture of an ~bject slice from linear projections obtained by ~assing radiation in ~uasi-planar form through the quasi-plane of the object slice at various angles and ~easuring a~ounts of radiation not absorbed or scattered by said ob~ect slice. By quasi-plane is meant plane-like with small but finite thickness. This 1~ method and apparat~s can be used in the fields of me~ical radiology, microscopy, non-destructiYe testing, and other fields as well.
- A fan beam of radiation (which may be liqht, heat, sound, transmissive ultrasound, electro-magnetic radiation, X-rays, gamma rays, or sub-atomic particlec such as electrons, protons, neutrons, or heavy ions, or anv other form of tr~smissive radiation) is p2ssed through a~ object slice lying in a auasi-plane. The auasi-pl2ne has a small thickness, in the case of X-ray diagnosis, typically but not necessarily between about 1 millimeter and 15 millimet~rs. Tl~e entire three-dimensional object can be portrayed by picturing a series of- side-by-side slices each 1 tc 15 ~ thic~. The wnole series can be mapped 2nd pictured simult~neously.
The object absorbs so~e of the radiatio~, scatters some additional radiation,- and the rest is detected by an elong2te detector or detectors situated opPosite the source of fan radiation, and ~ligned therewith, lying in the sa~e pla~e as the source of radiation and the object slice. The an~ular distance be~ween each individual detection point is constant. Thus the detector ban~ is preferahlv arcuate in geometry, and may comprise an entire circle lBl - 13 -7~
or ellipse; OL' the detec~or bank lies in a strai~ht li~e with each individual detection point oriented tcward the sour~e.
A co~pens~tcr ~ay be pcsitioned around the object tc re~uc~
the variation in intensity of radiation reachins the detector(s) Data ~y h~ extracted from the detector(s) serially or in p~rallel, continuously or in pulses. In the case of ~a~2 or X-radiation, ezch individual detector ele~ent ~ay comprise 2 scintillator of gas, liauid, or solid e~ploying a crystalline substance such as sodium iodide and a photomultiplier or photodiode.
Alternatively, it m~y compris~ an ioniza~ion chamber filled with a high atomic numbered element such as xenon in ~as, liauid, or solid phase, with or without a lower atomic nu~bered element such as argon in similar form as the xenon to capture ~-emission X-ra~s. Alternatively, the detector may be a semiconductor SUC;l as hi~h purity germanium or cadmium te~luride or ~ercuric iodide, or it ma~ be an image intensifier. The detector may operate in current integration ~ode or it may count in~ividual gamma-ray or X-ray photons. The detector may comprise a scintillation screen-fil~ co~,bination moved perpendicular to the fan beam to record successive projection profiles at successive source angles, with a flyi~ spot scanner extractinq the dat~ fro~ the dev~lo~ed film.
The radiation beam and the detectors may be continuous but are usually discrete. In either case, the resultant ~etected radi~tion ~ay be fed into a com~uter for conversion into a two-di~ensicnal pictorial representation on a ~raphical displa~ ~evi.e such 2S a cathode ray tube (CRT~ cr a ~rinted she~t of paper capable of illustratin~ ~ensities or co~tours.
If a di~ital com~uter is enplcyed with a detector provi~
analo~ output, t~e infor~ation is first processed by an 1~1 lili~7~

analo~-to-di~ital converter. If an analo~ computer is - employed with a detector providing di~ital output, the information is irst processed by a diqital-to-analog converter; In either case, the com~uter calcula.es the degree of absorption for each cell in a mesh or grid superim?osed upon the object slice portrayed, and this dat~ is then processed and converted into an analo~ or digital t~:o-dimensional pictorial form.
In the case of using radiation to diagnose human and other animal bodies, it is possible to disting~ish and vividly portray aneurysms, hemorrhages, tu~ors, ~bnor~al c~vities, blood clots, enlarged organs, and ~bnormalities in ventricles (for example) since it is known that different tissues of the body absorb differing amounts of rzdiation, The instant invention is the first method and apparatus which uses a convolution reconstruction method on fan beam r~ys with no prior reordering of the fan beam rays into a new set of rays.

A "convolution process" of x, c(x), according to the mathematical literature, is any integral or summaticn function of the form:
c(x) = S f (x - x')g(x')dx' or c(x) - ~ f (x - Xn )g(xn ) By "convolution method" is meant any method which e~Dloys such a convolution process.

The convolution method is ~uch faster and provi~es enual or better resolution for the same amount of ~adiation than iterative metho~s used in the prior art. See "The Fourier ~econstruction of a Head Section", b~ r, A. Shepp and B. F. Logan, Eell Laboratories "~urray Hill, ~ew Jersey, 2Cl - 15 -July 1974, (13), ~p. 5,7, io ~hic~l a convolution J~ethod for parallel geometry (14), is favorab.y co~pared with an iterative method of s~lccessive a?proximations. ~he instant i~vention accentua,es this favorability because it uses a direct convolution method based u?on polar geometry.
No ~;~steful and error producing prior reordering of the rays into parall~l rays is required to take adv2nt~ge of the superior properties of fan beams.
The instant invention is also cap~ble of showing the relationship between the measurements taken and the errors inherent in them, since the discrete embodiment is re~lly a special case of the continuous embodiment, and one c~n ~auge the effect of each simplifying approximation in turn The advantzge to the fan beam is th~t it permits a faster d~ta gathering than with a parzl~el bea~l source produced by translating a source ~nd detector, and one can obtain a lzrge nu~ber of m2asurements without moving the source, thus reducing the effects of ~.echanical vibrations which can impair accuracy. In the case of X-radiation applied ~o the body of a human or other ani~21, the patient is forced to lié still for a much shorter period of time and more patients can be processed b~ the machine in ~ ~iven amount of time. Signific2ntly, since all the data re~uired for a cross-sectional scan can be acquired in the order of one second (one to two orders of m~qnituce faster than existing systems~ it is now possible for the first time to obtain accurate pictures of areas of the bcdy such as the abdominal cavity, without re~uirirg extensive periods of bre~th-holding by the patient and with lecs ~otion artifact due to peristalsis and other orsan motion ~Cl~
121~75 ~ e sourc~-detector array i5 typically rotaLed about the object ~lice in a circular path comprisin~ 360 degrees.
Alternativel~, the object may rotate ~ithin ~ station2ry source-detector assembly. Other configùrations are ~l~c possible and are described below in the Detailed Description of the Preferred Embodiment. The source may be rotated centinuously or step-wise in small anular steps. In either case, the radiation may emanate fro~ the source c~ntinuously (for exam~le, in the case where the source is radioactive), or in the form of periodic pulses or bursts. The detected cata is convolved and back-projected without the necessity for first reordering the data into a nPw set of rays, for example, a set of parallel rays, as in prior art fan beam systems.
By "back-projection" is meant the process of converting the convoived projection profile data 2sscciatea ~ith the detectors into values of absorbed aensity at an arbitrary number of points P preselected throu~hout the object slice under examination.
The instant invention enco~passes the first exact reconstruction of a two-dim~nsional picture of an object slice from a series of one-di~ensional projections of radiation non-absorbed by the slice when the superior fan be~. source is employed. This means that th~ accuracy ~nd resolu~ion of the output picture ~re good ~ven when d~ta gathering and data reconstructing times are small. Thus, the fan beam approach is stren~thened as a viable tool of scientific in~uiry.
In addition, since an exact reconstruction is achieved, (limited only b~r i~perfections in the inst-umentation e~ployed and intentionallv introduced apprcximating vari~tions to the ~eneral case!~ it is possi~ le to .~ore directly Perceive the ~elation~hi~? b~tween an introduced a~proximation and 2Cl 121~75 - 17 -1157~i the ~uality and s~eed of the out~ut respoinse. Th-~s, a more precise control over the resolution/speed tradeoff is obtained compared ~ith the known prior art.

BRIEF DESC~IPTIO~ OF T~IE DF~A~`7II~GS

These and other ~.ore detailed and specific objects and features of the instant invention are more fully disclosed in the following specification, reference bein~ haa to the accompanying drawings in which:

FIG. 1 is a ~artially schematic, partially block diagram of the embodiment of the system of the instant invention in which the data ac~uisition phase is continuous.

FIG. 2 is a partially schematic, ~artially block diagra~. of the embodiment of the instant invention in which the data acauisition phase is discrete.

FIG. 3 is a geometrical representation of a preferred method of data acaui~ition and~ reconstruction in the instant invention.

FIG. 4 is a flow diaqram which illustrates ~
typical embodiment of data acquisition ar.d reconctruction in the discrete em~odi~ent of the ir.C~ant invention.

2Cl 121~75 - ~ -FIG. 5 is a schematic representation of the rotatable gantry portion of apparatus of the type appearing in FIG. 2, and sets forth schematic information useful to analysis of certain aspects of the invention.
FIG. 6 is a schematic block diagram illustrating operation in a back projection mode for a preferred form of data reconstruction system utilizable with the present invention; and FIG. 7 is a schematic block diagram illustrating operation of the system of FIG. 6, in a convolution mode.

~hh7 nl l ~77 _ lQ~ L~

` ` lil~575 Detcliled Dr~cri~ticn of the Preferred ~mbcdiment .
FIG. 1 shows a schematic znd bloc~ diagram of the exact embodiment of the instant inventiGn, i.e., when the ~ata gatherina is ~erformed continuously. A source S of radiation, object slic~ 50, an~ a continuous detector 60 are lying in the same ~uasi-plane, which has a finite but small thickness, tvpically on the order of a few millimeters in the case of computerized tomography.
Source S and continuous detector 60 are aligned and are ~referably constructe~ so as to be always opposite each other; for example, they are each fixedly ~ounted on gantry 10 which rotates in a circular path around object slice 50. Alternatively, the object may rotate within a motionless source-detector assembly. Alternatively, one 360 degree continuous detector could be motionlessly mounted with just the source rotating. Or, a ~lurality of sources may be employed each over a portion of the circle; or else, one 360 degree continuous sourc~ could be employed with energization of only one point of said source at any given time, said point traversing the entire 360 degree arc over time.
The rotational force may be provided by a motor 13 which transmits energy to gantry gears 11 by means of drive aear 12. Continuous detector 60 preferably follows source S opposite thereto, ana is preferably arcuate in shz~e. When arcuate, its geometry is ~re~erably such that each ~oint on the detector is e~uidistant from the source S.
Source S may be any type of radiation such as an 3~ electron beam ir. the case of electron ~icrosco?y or X or ga~ma radiation for examining a human, or other body.

~Cl In the case where e~act data reconstruction is desired (see equation 33, in~ra.) the source is enerqized continuously throughout a co~plete 360 degree traversal of its circular path. OtXerwise the source may pulse. In the case of X-radiation, detector 60 is typically a scintillatcr fabricated of a crystalline material such as sodium iodide plus a photo~.ultiplier or photodiode; or it may comprise an ionization chamber filled with a substance such as xenon, or a mixture of substances such as xenon and ar~on, in gas, . liauid, or solid ~hase; or it may comprise an emulsive film.
Collimators 30 shape the beam of radiation emanating from source S into the shaPe of a fan, at least as ~7ide as object 50. Collimators 31 (parallel to the plane of the paper in Figure 1) are spaced one beside the other to ~ shape the fan into a thin quasi-~lanar bea~., which does not necessarily have to be of uniform thickness; for exa~ple, if a point source of radiation is used, the beam will fan in a vertical as well as a horlzontal direction. Detector collimators 61 serve to minimize the ef~ects of Compton

2~ scatter from planes other than the imaging quasi-plane.
Collimators 30, 31 and 61 are typically fabricated of lead, but may be made from any ma.terial which absorbs the radiation in unwanted directions. ln the case of X-ray dia5nosis the thickness of the fan, as defined by collimators 31, is typically between lmm ano 15~m at the ~.iddle of the object. The arc that is cut by the fan is sufficiently lar~e to cover the entire cbject slice.
. ComPensator 3~, which may be a ba~ filled with water or plastïc, ~ay optionall~ be positiored enshroudinq object 50 for the purposes oE attenuatin~ certain Ean beam r~-intensities and therebY reducin~ the ran~e cf intensitics over ~hich detect~r 6~ nust be respGnsive. The compensator ~ay 2Cl be ~ixedlv mounted on qantry 10 so as to rotate there~ith, or it may b~ mountcd fixe~ly with respect to object 50.
As the sourcc-detector array undergces relative rotation with respect to the object (continuously where exact reconstruction is desired) over a time o~ approxi~ately one to 15 seconds, r~adi~gs of non-absorbed and non-scattered radiation are time-continuously measure along detector 60. The data -acquisition is preferably co~leted during one relative revolution ~i.e., 360 degrees) of the system. Data ~rorn the ~etector may ~irst be smoothed, is convolved with other ~ata in a way which ~ill be described below, ~ay be smoothed again, and is then stored in computer 70 which, if analog, may comprise an analog store such as an acoustic ~Jave or video disc. If di~ital, the computer is preferably a hi~h-speed computer. The d~ta is later back-Drojected with other data to produce an output picture 8G which is a replica of object 50.
The output picture is portrayed on a visual display device 90 such as a CP~T (cathode ray tube) or an electrostatic output terminal ~7hich is capzble of showing density of the object being portrayed as depth, contour, shadings, or color. ~ photo~raph o. other hard copy of the CRT image may then be ta~en.
A series of two-dirensional pictures may be obtained by either takins a succession o pictures as above, or ~lse bv fabricating an array com~rising a plurality of source-detector configurations spacea beside each othcr e.~., mounted side-by-side cn gantry 10. ~n either case, the output may be ~ortrayed as a three-dimension~l picture, ~n for example, b~- portrayin~ each cutput element as ~ shaded 2Cl llilS7~
or color~d ~ranslucent b211 or cube. ~lternatively, a series of transparent light panels ~ay be used ~or three-dimensional display.
FIG. 2 is similar to ~IG. l; the only difference is that continuous aetector 60 has been replaced b~ an array or bank of discrete detectors 6~, and grid 66 has been added. In cases where the two figures are identical, the âescription employed above in connection with ~ig. 1 a~plie~
with eaual force to Fia. 2, which illustrates the discrete embodi~ent of the invention, a special case of the continuous e~bodiment. The radi2tion eman~ting from source S may be a continuous fan or 2 discrete set of pencil beams tformea, e.~., by a set of collimators) ~ith at least one bea~ ~er detector. Discrete detectors 65 typically number 300, although other values may be chosen.
The detector bank is ~ositioned in such 2 way that the angular aistance between detector elements is const2nt.
For example, the bank is arcuate in geometry, or the bank is in a straight line (because e~sier to b~ild) with ecch indiviaual detector ele~ent aligne~ with a straight line drawn from the detector element to the source. A ~rid 66 fabricated of an element such as lead ma~ be associated with each detector ele~ent 65 and ali~ned therewit~ to ~.ini~ize the effects of C¢mpton scatter lying in the same ~u2si-plane as the object slice. This 9rid is virtually essenti21 when the ra~iation emplGyea is X-radiation. This grid may o~tionally be used in the continuous case as well, i.e., where it is anticipated that Compton scatter in the same auasi-?lane as the object slice will be a problem. In that a~plication the qrid may be made to oscillate Ol- otherwise continucu~ly ~ove with resPect to detector 60 so that gr d lines do not aPpear on the output p~cture.

2Cl ~l~lS75 In the preferred ~mbodiment, a source-detector ~rray is rotatecl with ~antry 10 in a circular p~th. Pericdic~lly (t~7pically, durinq 3~0 short nloments in ti~e ~er rotati~r.,on the order of two ~,illiseco~ds each), radiation is pulsed from the source, absorption vzl~es ~re measllred bv the detectors 65, are digitalized, smoothed, and fed into a workin~
store within co~puter 70. Contrcls are built into the ~achine so that pulse duration and arcuate angle between pulsin~ c~n be ~uickly adjusted b~ the operator. These c2n ~lso bs employed in the ccntinuous e~bodiment where exact data reconstruction is not re~uired.
The data is then processed to yield absor~tion densities for a preselecteo plurality of points within obiect 50 and this reconstructed set of densities is portrayed as output picture 80. The computer ~ay be either hardwired, firm~2red (microproqrammed or PRO~-fused), or software pro~rammed ~or any combination of the ~bove) to perform the reauisite function~, ~Yhich is true in the continuous embodiment as well.
In one embodiment for use with patients in medical r~diology, the apparatus ~arameters could be as follows:

X-r~y tube volta~e 120 ~ V d.c.
X-ray tube average current 250 m~
X-ray tube avera~e power 3~ kW

X-ray exPosure pericd per 4 seco~ds object slice ~ntry rotatioll speed 0.25 rps ~umber of X-r~y pulses ~er objec~ 360 slic~

Exposure to s~lrface of oL)iect 8 raàs X-ray tube pulse current 1000 ~A
X-ray pu3se durztion 2.3 ~illisec 2~1 12~875 - 23 -~lli575 Interpulse duration 8.3 millisec Distance from axis to source 80 cm Distance from source to detector 160 cm Maximum ohject slice dimension 40 cm Fan beam angular spread 29 degrees Fan beam thic~ness at middle 8 mm of object Fan ray interval at mid~le of 1.5mm object ~umber of fan rays across 267 maximum object slice .
Nominal nu~ber of detector 300 elements Fan ray angular interval 0.109 degrees Source rotation interval 1 degree per pulse Interval between source 3.5 mm pulses at 40 cm di~meter object peri.phery X-ray photons per pulse 2.2 x 10 per detector element without object Primary photon transmission lt2000 through 40 cm water .
X-ray photons per pulse per 1.1 x 10 detector element through object Quantum statistical fluctuation 0.33~ rms per measurement Statistical error in total of 360 0.6~ rms measurements thro~gh one 1.5 ~m x 1.5 mm cell of object slice Number of reconstruction points 40,000 in 40 cm diameter image Spacin~ be~ween reconstruction 1.8 mm , points in 40 cm diameter image 2Cl - 24 -11:11575 If the fan rays were reordered into a r.ew set of parallel rays, bundles of 9 rays extenclin~ over successive 1 degree intervals of the fan woula be rec,rdered to successive source angul2r positions 1 degree zpart to obtain ~seuao-parallel raYs. The central rays of these '~-raY hundles would be parallel but their spacing would vary from 1.5 mm to 1.45 mm, an error 3~, depen~in~ on whether they came from the center or the edSe of the fan beam, due to the source moving on a circle rather than a straight line.

The spacing of individual rays of the fan beam to individual detector elements is 1.5 mm at the middle of the object. The spacing of the central ~xes of successively pulsed fan bea~s is 3.5 mm at the periphery of a 40 cm diameter object slice. Better resolution would be obtained for this `~ relatively lar~e size object if 720 pulses at 0.5 degree interv~ls of gantry rotation were employed, making the spacing of central rays at the object periphery comparable to the sDacing of rays within the fan, thereby oktaining more uniform resolution in all directions. The pulse duration would then be 1.4 milli-second and the interpulse duration ~lould be 4.1 millisecond for 4 seconds X-ray ex~osure per object slice, re~uirin~ faster d~ta extraction from the detector elements and twice the number of profile convolution and back-pro~ection com~u~ations.
Thus, the selection o~ 36~ Pulses at 1 degree intervals with a detector o~ nominal 300 elements represents a ~rzctical choice ~or objects ranging in size from a few cm to 40 c~ in diameter.

2Cl . .~ lii~S75 Let us no~l ex2~ine the ~ethod of data reconstruction for both the continuous and c~iscrete cases. Radon's formula for the density at a point P is .i c~
~ ( P ) - - ~T S d r f d r where r is measured from the point P and ~r) is the average of all line integrals o, the density over lines p2ssin~ a di~tance r fro~ the poir.t P. J. Radon, Ueber die Besti~.mun~
von ~unktionen durch ihre integralwerte laengs gewi ser ~annigfaltigkeiten (on-the determination of functions from their integrals alonq certain manifolds), Berichte Saechsische Akadamie der Wissenschaften (Leipzig), - Mathematische-Physische Xlasse 69, 262-277 (Germany 1517).
In this use D(P) represents the extent or density of radiatior absorbed at the point P.

Consider the diagram shown in FIG. 3. Define a ~easurement ~3p ( e,~ ) as the integral (or measure~.ent) of absorbed radiation 21cng a line defined by the angles e and startin~ at the source point S. In other wor~s, H=~ ~dx where dx iis the incre~ental distance along the co~nizant lir.e.
The subscript P denotes the fact that ~ is measured from the line dra~n fro~ the source S to the Point of interest P.
If we define I to be the measurement of resultin~ radiation reaching detector(s) 6~ or 65, and lo to be the radi2~ion which would reach the detector~s) in t~le ahsence of any object, cuch as object 50, which would attenu2te 2ny of the radiation 2s it leaves the source, then ~e kno-.~ fro~ ~asic -S~dX -H
physic~ s thet I=TOe =lo~ .
In other words, ~{=ln Io - ln I --ln(I~I

2Cl 121~75 - 26 -~ 7~

When the machine is initially calibrated, Io is chosen so as to be big enouqh to provide statistically adequate information (e.g., at least 103 X-rav or ga~.~a-ray photons per pulse at each detector element) but not so big as to harm the patient by means of an overdose of radiation in the case where the apparatus is e~ployed for X-ray diagnosis of a patient's body (less than 50 rads of X-ray or ~am~a-ray dose total for all pulses).

Using P~adon's formula we can write 10 ~P)=- ~ S dr~--ddr ~ 21~ S dY,Hp(e,~)} (3) 121~75 - 27 - .

~e must now change the variables of inte~ration from drl d~YI to de d~3. Now drl d~cl =JI d~3dQ

~;here Jl , the Jacobian, is given by ~r~ r' J~ ~ _ (4) ~- ~
~ ~e The coordinates defining the transformation are as follows:

r~=~stn~ ~5) , I ~ 2 ~ ~ (6) where RcOSe-D O~e ~ (7) ' ' -I Rsine lT~a~2TT
t~n _ lTÇ ~Ç 21~ (8) when we assume the principal ranae of tan~l is n to ~ .

2Cl 121~75 - 2~ -s Evaluatinq the Jacobian, we find that a r' - z cos ~ .

,~r~ RD sine sin~3 (10) . ~
R2 R D c o s ~3 -- - =
~ ~e z2 Thus, R2cos~ - ~D cos ( ~
Jl ~ ~ ~13) We must also consider the term ~ r ~ ~p ~ ? ~ ~ ~14) r ~P(e~ts)3= ,~e ~r~ r~

The derivatives ~ r and ~ r can be bbtained by implicitly differentiating equations t5~ and (6). Consider e~uation (S~. Its derivative is R D si n 9 s i n Ç3 ~ e ~ z cOs ~ @ ( 1 S ) 2Cl`
121~7~ - ~9 -~111575 Si~il<lrly the d^rivat;ve oE enuation ~6) with respect to rl is O R2-RD cos e ~e ~ (16) z 2 ,~ r~

- ~e ~
Solving eauations (lS) and (16) for ~ and we obtain ~e= Z (17) rl RD cos (~3 ~ 2 cos ~, ~3 RDcos e-R2 r~ Z ~I?D cos (~-O-R~os~3} (18) Substituting all these results into equation (3) we finally obta in ~P) = - I ~ d~ S d~

~ ~Hp~e,~ R2-RDcos~ a p(e,~) }

(19) , ~ Hp (e,~
Consider the term containin~ ~ e . It can be integrated by parts wi'h respect .~ e to obta~n 2C~
1~.575 - 30 -lli3 ~ (20 ) S2n ~ ,~Hp 52d 9 F~D sin O ~p ( e,~3) Thus we obtain 2T~2 50 ~ 50 ~}
Sdne R2 ~cose Sd~ a~

(21) 1~11S75 I~ obtaining the a~ove ~uation it has becn assu~ed that the object does not extend outside the arc covered by th~ fan beam e~anating from the source S. Thus the line integrals of density of rays tangent to circles centere~ at P such that at least a portion of such circles lie outside the arc coverea by the fan beam emanating from the source S are assumed to be zero, where said point of tangency also lies outside said fan beam.
If we again return to FIG. 3 and rewrite Eouatio~ (3) using ~z and GC~ rather than r~ and o~ we can obtain a second eguation for ~ (P). Thas ~(P~=~ Tr 5d~2 r;Z ~r;~2Tr SO ~ P ' 1~ (2~) Again we wish to char.ge the variables of integration from dr2 d~2 to d~ de . Proceeding as before we have .
rz = - Z ~in~ (23) ~C2 - 2 ~ 24 ) Evaluatina the Jacobian we find R2COS ~3 - Rl~ COS (~
J2 = ~ (25) Evaluating - and - ~ , we obtain 2 ~ ~2 (26) ~)n _ _ , rZ R~CQS ~3- R{) COS l~
- ~ R2-RDc~s e ~2. Z ~RDCOS(a~ 2COS~} (27) ~liiS75 Substitutin~ these expressions into Eauation (22 ), we obtain ~S 21T2 5~ ~Tr Z s i n ~
¦ ~ ~p ( e,~) R2~ RD cos e ~ ~p (e ~ 28 ) ~ e z2 ~3 J

A~3ain inte~rating ~--e with respect to ~3 , 2TT2 So Z3 S~rd~ sin 13 2Tr2 Sde z359 Sd~ I ~H

E~uations ~21) and (29) can now be added to obtain p ~ - IS d e R D si n e ~ l lp 1 9, ~ ) O ~ 30 ) 21T ~IT
S de R2-RDCc~s~ Sd~ P(~

O -lT

2Cl The followiny chanq~s can now b~ made to Eau~tion (30):
(1) Chanqe the v~riable of int~ration from to ~ where ~ 0 ~2) Change Hp (~ to Ho (~ ~ ) where the subscript O now reflects that S
is measured from the line connecting the point S ~nd 0, the center of rot~tion.

~3) Note that RD sin e R sln ~0 (31) z3 z2 (4) Note that R2- RD cos ~3R cos ~o (3 2 z3 zZ

Equation (30) then becomes r21T Rsin~50 lT .~Jo (~3~) ~3(P) = a,~ ~ z2 ~ sin (~-~0) ~?dtT~ F~Co5~osd,~ _ I Ho(~
4~2 z2 -lr sin ~ O) ~
~33, 12)h75 ~ 3~ -S

Eau2tion ~33) is the desired rcsult dn~ is the exat solution.
It covers the continuous case of data g~therin~. Although there appears to be a sin~ul2rity ~t S = SO, we ~re interested in the principal value cf the integral. ~ote that the inte~ral over S is in the form of a cor.volution.
Furthermor~, if R -i~ ~ , Eauation (33) reduces to the si~pler parallel geometry case.
Although one could evaluate EQu~tion (33) using an~lo~
methods, digital tdiscrete) techniaues are usually em?loye~
instead for the following reasons:
1) With par211el data extraction for ast data acquisition, it is convenient to use a num~r of discrete detector elements cou~led to an e~ual`nu~ber of discrete electronic amplifiers.

.- 2) Because of statistical variations in detected radiation values with finite total radiation exposure and hence finite number of radiation auanta, a point of diminishing returns is reached where dividing the discrete array detector into ~ l~r~er number of ~iner elements does not materially improve the auality oE the reconstructed image; therefore continuation of this division process to the limit of a continuous detector`
is not justified.

3) With continuous rctation of source angul~r position, the finite ~an beam thic~r.ess spreads over an e~uivalent a.ngular s~read cf the detecte~
dat~ and since there are limits to the accuracy with which this ancular spread car. be deconvolved, little inla~e auality is lost by exercisin~ the convcnience o~ usinq a firite number o source position angles.

~ ) The ~-csenc~ o th~ sin~ul~rity at S =
is no~ e~sily handled b~ analog t~chni~u~s.
5) Tlle accurac~ reouire~ is hi~her thzn that nor~lall~ obtaine~ with anala~ co~ut~tion metho~s.

Equ~ion t33) c~n be reduced to discrete form as fo]lows. The inte~rals over ~ cover the full ran~e from fro~ O to 2~ . Thus we zre free to begi~ and end at any point. Therefore E~u~tion (33) can be ~ritten as ~(P)= 4~2 5 d~ Sz2~ S~s i 0 - I Sd~3 Rcos~OSdlTS I ~Ho(C~

~low let a be the ~ngular distance ~etween me~sure.~.ents ana ~- f~r~her let ~= 4N . Th~n H~ 0~ ~) can be exp~nded in a finite Fourier series as follows:

Ho (~32~o'~~) ~ 2 ~ ~ ~n (9,~0) cos(n~) n-~ -bn(~ 0)sin(n~ ) cos(2N~) ~3~) n=J

Since S Sin n x C~ y _~2TT n ODD

s i n x O OTHERWISE ~~~

.

. ~37) ~lS~5 and (38 ) C cos n~ d O
~) si n x Ec~uation ~35) can be written as:

P)-- 2~ Sd z2 b (39) Sd~ Rcos~S ~e ~ ) -21~ 0 where Sa (e,~O) = ~ nan (e, ~io) 14~) n=l 8. OD~

2N-1 ~41 sb(e,~c,3= ~ bn(e7~o~
n-l & O~D

~ow ~iliS75 -(e~ 2~N ~ ~'O(~,~O~n~a) cOS ~mna) (~2 m O

2~ ~OHO(~ 0~m~) sin(mn~) Ther ef o re Sa t ~ 0) = 2 N ~ HO (~, ~O~ m ~} ~ n cos ~ n m Q ) m=O n= I
f3 O~D

4N-1 21~1-1 Sb~ O)=2lN ~ H~ ~mf~) ~sin(nml~ 5) m=O tl=l ~ ODD

The summ,ations over n can be evaluated 2N-1 r sin2~Nm~) ~ncos~nm~
~ d(mal l sin (ml~) J
n~ J
8~ ODD
r N2 m=O
cos~mh) m ODD ~, 2T~ (4~) sin2~ m~) 4N
m EVEN

- 3~ -and ~iil . sin2( Nml~) sl n ( n m l~) - s in ( m ~ ) n=l ODD
(47) ~ O m EVEN
=
sin(m~) a= ~2 If we now repl~ce 1~ by (2a) t then we can write .

Sa (e, ~0) = (l4Ta) H~ (e~O) ) a) ~ coS(m~) ~O(e,~
Tr si n21m a) m ODD
., and .

sb ~e ~ O ~e,~O~ma) m ODû

If we now substitute Eouations (4~) and (49) into Eauation (39) and repl~ce the integration over e by a cummation, ~re o~tain:

~ 3~ -.~

111157~

2 [ ~ T~2)~0 si~ 1~ 4~ ho(~O+

R~ 02'sO[ 81~ ~J(~o)' '' (~0) (~2)n ~nDsln'(ro 4) ~ }

This 2~ain can be simplified to (51) 1'0 g~(P)=~d zR2 ~ (~ cosS.

_ _ ~ c~s ~ ~ m ~ O~ m ~) ¦
2~T mODD - sin2 ~m/~) J
.
In both Equations (50) and (51), the limits on the summa-tion over nn have not been written_ m is summation-is taken over all detectors; ~ can be botll positive or-negative and is simply the number of detectors away from the detector at ,, ~ . , ~ m e expression within brackets in Equation (51), which ¦ must be evaluated first, represents a convolution and the ! remaining portion of Equation (51) represents a back-pro-jection. The slow way to evaluation Equation t51) would be to calculate the absorption density at each point P for each of the values detected; but there are faster ways of solving Equation (51) for many values of P at once. Typ-ically P's are about 40,000 in num~er, representing a 200 x 200 grid superimposed over object 50. The points P may be uniformly spaced or non-uniformly spaced. When 360 values were chosen for ~ and 300 detectors were selected, the data collection was performed in about 6 seconds.
_ ~n _ lili~7~

This is between one and two ord~rs of magnitude faster than the existing prior art for the same nuality picture. One sees that as ~ decreases and the numbers of measurements, ~ '5, and P's increase, a more accurate ~icture may be obtained but at a cost of ~reater data colleciion and reduction t;mes.
As stated before, the ~O (e,~ m~) s are obtained zs a result of measurements t~ken at detector elements65. The index m is measured from ~O , i.e., the location of the line through the point of interest P running from source S to the detector elements. In other words HO(e,~O) represents ~ th2t detector ele~.ent along the straight line running from S through P, Ho(9~o~ a), Ho(e~O~ 2 ~), etc. represent the detector ele~ents running sequentially in one direction from ~O ; and Ho(~)~o~ a3, Ho(~ ~ 2 ~ ), , etc., the detector elements running se~uentially in the opposite direction. The data from the detectors may be extracted serially or in parallel.

Por'each value of e and for each value of ~O , a single convolution profile value is calculated and stored in a storage device or arr2y C ~ e ,~0) . This calculati~n for all ~OSfor eache m~y'be perfor~ed as soon as the data-~atherin~ phase for that particular ~ is complete, i.e., while the source continues to rotate about its path. The outer loop (the back-projecticn portion) of Eauation (51) may also be complete~ for each e as soon as all measurements for that particular ~ have been read into stor~e and the inner loop tconvolution) is complete. Thus, measurements an~
calculation's are performed simultaneously; this is one, but by no means the most important, way the techni~ue oL the present inven~ion saves time.

~ t the time oF the back-pro~ection looP, interpolations are performe~ to take into account the fact that most of the P's do not lie along a line running from the source S to the mid-~oint of a detector element.
It is sufficient but not necessary that the interpolations be linear. The convoluticn profile values utilized in the linear interpolation are those zssociated with the midpoints (or other norm~l detection points) of those detectcr elements ~djacent to the point alonq the detector array cut by the straight line running from the source S through the point P in cuestion. This inter~olation could also be perfor~ed during the convolution steP.
After all calculations have been performed, the values of absorption densities at each point P may then be portrayed in graphic form as output picture 8~.
A greater insight into how an out~ut picture is pr~uced may be obtained by studyinq FIG. 4. The index for e is ~ and is initialized to zero. At ei r~aiation passes through the object and is read by the detector ele~ents.
as values of I. At this point th~ source is free to rotate to its next value of e; this in f~ct ~ould be done if the main criterion were to minimize the dat~ gatherin~ ti~e, or if two processors existed ~7ithin the computer, one for data gathering and one for data reduction. In the latter case, much of the data reduction could be performe~ simult~neousiv with the data collection. However (for purposes of discussion but withou, intending to limit in any way the invention), the f'ow chart shows a data-reconstruction e~bodi~ent in ~hich calculations are perfor~ed at this time, before th~ source rotates to its next value of ~ . In the case of administering X-radiation to humans, this does nct result in extra r~diaticn enterinq the body, bec3use the radiation is normally pulsed for just a shor~ time fcr e~ch value of ~ .
Next, H is calculated at each detector element and stored in the storage area or array R (~O)~ At th;s point, the source m2y rotate to its next value of e , and the same considerations ~overn as to the desirability of so doing.
For ei and each value of ~othe convolution profile value is o~tained and stored in a second stora~e area or array C ( e~ ~0 ) .
Again at this point the source m~y rotate to its next value Of e ~ however, the flow diagram illu5trates the case where .an additicn~l step is performed at this time.
It will readily be seen th~t many permut~tions of the steps are possible. The important point is that ~or each set of measurements taken for a particular ~ , either the convolution step or the convolution step plus the back projection step may be p~rformed ~t that time, with or withcut subseauent rotation and measurement gathering for ~dditional v21ucs 0 Unless the processor is extremely fast; if the same ~rocessor is both reading the dz.ta and performinq the convolution and bac~-projecting steps, then norm211y all of the data will first be read so as to minimize the data collection time.
If, on the other harJd, an ~ddition~l processor is e~loyed for just the data gathering step, then much time can b~ ~aved by providin~ for simultanecus perfor~.ance of the con~olution and back-projection stepS.

Notice that for each P, Z is uniaue ~nd may be pr~obtained; Z may also be thought of as a ~unction of e ar.d or ~ and m. Durinq the outer loo~ (the b~ck ~ro~ectior portion), a correction is performed by means of lir.ear or other interpolation to take into account the fact that the line running from the source ~ throu~h the point P will not normally strike a ~etector element at its midpoir.t ~or other voint ; in the detector element where the measurement is normally taken).
In other words, if the cognizant line strikes the de-tector bank 1/10 of the distance between the detection points of detector elements ml and m 2 then it is ass-umed for the purposes of the calculation that the equiva-lent value of C for this line is 9/10 the C based at ml+ 1/10 the C based at m 2 ~
Returning to FIG. 4, the index i is then incremented.
The question is asked, "Does i equal the maximum value pre-selected?" (A typical value for i~axis 360). If not, then the value of ~ is incremented accordingly and a new series of measurements or measurements plus calculations is performed. If i equals imaX, then we know that we are done with the data collection and pre-calculation portions I of the process and all that remains is to complete the cal-! culations and convert the ~ (P~s into picture form. In ¦ the case illustrated by FIG. 4, all that remains is, for each value of P, to convert the ~(P)'s into picture form for visual inspection by the observer.
Much of the above discussion, which pertains to the discrete embodiment, also applies to the continuous embod-iment, i.e., the graphic portrayal of equation (33).
The fol~owing listing of a computer program, written in the FORTRAN language for an IBM 3~0 computer, imple-ments a typical embodiment of the invention that was described above.

.

Ç~ .

111157~

MAY 72) OS/360 FORTRAN H

PILER OPTIONS - NAME = MAIN, OPT=02, LINECNT = 58, SIZE = OOOOK, SOURCE, EE~CDIC, NOLIST, DECK, LOAI~, MAP, NOEDIT,ID, NOXREF
C RECONSTRUCTION PROGRAM - FAN BEAM GEOMETRY 1.
C 2.
COMMON / CTITLE / TITLE(23) 3.
INTEGER*4 TITLE 4.
C 5.
COMMON /CMESH/ NX, NY, NTOT, Xl, X2, XINCR, Yl, Y2, YINCR 6.
C 7.
COMMON /COSSIN/ WCOS(1000), WSIN(1000) 8.
C 9.
DIMENSION RECON(2000) 10.
DIMENSION PJDATA(1000), PJCONV(1000) 11.
DIMENSION CCFNTN(1000), SCFNTN(1000) 12.
C 13.
DATA LBLNK/4H / 14.
DATA LMESH, LPROJ/'MESH', 'PROJ'/ 15.
C 16.
IQUIT = O 17.
READ (5, 25, END = 9999) (TITLE (1), 1 = 1, 20, 1) 18.
25 FORMAT (20A4) 19.
TITLE (21) = LBLNK 20.
CALL DATE (TITLE (22)) 21.
WRITE (6, 35) TITLE 22.
FORMAT (IHI, 'RECONSTRUClION TESr, 5X, 23A4) 23.

C 24.
C 25.
3 C READ DATA DEFINING PHANTOM 26.

llllS75 100 CALL ELDATA 27.
C 28.
C READ DATA DESCRIBING RECONSTRUCTION 29.
C 30.
200 READ (5, 215) NTEST 31.
215 FORMAT (A4) 32.
IF (NTEST .NE. LMESH) STOP 33.
READ (5, 255) NX, NY, XORG, YORG, XINCR, YINCR, PJNORM, XNORM 34.
225 FORMAT (2110, 6F10.5) 35.
XRANG = XINCR*FLOAT (NX - 1) 36.
YRANG = YINCR*FLOAT (NY -1) 37.
Xl = XORG - 0.5*XRANG 38.
X2 = XORG + 0.5*XRANG 39.
Yl = YORG - 0.5*YRANG 40.
Y2 = YORG + 0.5*YRANG 41.
WRITE (6, 235) NX, XORG, XINCR, Xl, X2, NY, YORG, YINCR, Yl, Y2, 42.
2350 FORMAT (lHO, lOX, 'MESH DATA',~lHO, 15X, 43.
'NX =', 14, 8X, 'XORG =', F10.5, 8X, 'XINCR =', F10.5, 8X, 44.
2 'Xl =', F10.5, 8X, 'X2 =', F10.5/lH, 15X 45.
3 'NY =', 14, 8X, 'YORG =', F10.5, 8X, 'YINCR =', F 10.5, 8X, 46.

4 'Yl =', F10.5, 8X, 'Y2 =', F10.5) 4~.
IF (P~NORM .NE. 0.0) WRITE (6, 236) PJNORM 48.
2360 FORMAT (lHO, 15X, 'RESULTS NORMALIZED TO
DENSITY*LENGTH', 49.
'FOR MAXlMUM MEASUREMENT OF', F11.5) 50.

IF (XNORM .NE. 0.0) WRITE (6, 238) XNORM 51.
238 FORMAT (lHO, 15X, 'RESULTS NORMALIZED TO ', F10.5) 52.
C 53.
-45(a)-~illS75 C READ PROJECTION DATA 54.
C 55.
READ (5, 245) NTEST 56.
245 FORMAT (A4) 57.
IF (NTEST .NE. LPROJ) STOP 58.
C 59.
0 READ (5, 255) NPROJ, NRAYS, TOTANG, WIDRAT, RADIUS, THETA 60.
OFFSET, NETYPE 61.
255 FORMAT (2110, 5F10.5, 110) 62.
WRITE (6, 265) NPROJ, TOTANG, RADIUS, OFFSET, NRAYS, WIDRAT, THETA 63.
2650 FORMAT (lHO, 10X, 'PROJECTION DATA'/lHO, 15X, 64.
'NPROJ =', 14, 5X, 'TOTAL ANGLE =', F10.5, 5X, 65.
2 'SOURCE RADIUS =', F10.5, 5X, 'OFFSET =', F10.5/lH, 15X, 66.
3 'NRAYS =', 14, 5X, 'WIDTH RATIO =', F10.5, 5X, 67.
4 'FIRST ANGLE =', F10.5) 68.
C 69.
IF (NETYPE .NE. 0) CALL ERDATE (NETYPE) 70.
C 71.
300 RAYSPG = TOTANG/(57.29578*FLOAT (NRAYS- 1)) 72.
RAYWID = WIDRAT*RAYSPG 73.
C 74.
CENTER = 0.5*FLOAT (NRAYS + 1) 75.
DO 380 N = 1, NRAYS, 1 76.
BETA = RAYSPG*(FLOAT (N) - CENTER - OFFSET) 77.
WSIN (N) = SIN (BETA) 78.
WCOS (N) = COS (I~ETA) 79.

380 CONTINUE 80.
C 81.

lill575 C CLEAR RECONSTRUCTION AREA 82.
C 83.
600 NTOT = NX*NY 84.
IF (NTOT .LE. 2000) GO TO 640 85.
IQUIT = 1 86.
WRITE (6, 615) 87.
615 FORMAT (lHO, 5X, 'TOO MANY MESH POINTY) 88.
GO TO 700 89.
640 DO 660 N = 1, NTOT, 1 90.
RECON (N) = 0.0 91.
1 0 660 CONTINUE 92.
C 93.
C GET CONVOLUTION FUNCTION 94.
C 95.
700 CALL CNVLFN (NRAYS, TOTANG, CCFNTN, SCFNTN, INCR) 96.
IF (IQUIT .NE. 0) GO TO 20 97.
C 98.
C PERFORM RECONSTRUCTION 99.
C 100.
1000 ALPHO = THETA/57.29578 101.
DALPH = 6.283185/FLOAT(NPROJ) 102.
AZERO = 0.5*FLOAT(NRAYS + 1) + OFFSET 103.
CO = 0.999992/RAYSPG 104.
C2 = - 0.332127/RAYSPG 105.
C4 = 0.173750/RAYSPG 106.
C 107.
NPLUS = NE~AYS/2 108.
XNl = FLOAT (NX - 1) 109.

YNl = FLOAT (NY - 1) 110.

3 -46(a)-S~S

0 RCALC = RADIUS*AMINI (ABS(WSINtl)), ABS(WSIN(NRAYS)) 111.
- 0.5*RADIUS*RAYSPG 112.
C 113.
PJMAX = 0.0 114.
DO 1400 NP = l,NPROJ, 1 115.
ALPHA = ALPHO + DALPH*FLOAT (NP- 1) 116.
0 IF (NETYPE .GE. 0) CALL FANGEN(ALPHA, 117.
PJDATA, RADIUS, NRAYS, RAYSPG, RAYWID) 118.
0 IF (NETYPE .NE. 0) CALL FANERR(ALPHA, 119.
1 PJDATA, RADIUS, NRAYS, RAYSPG, RAYWID) 120.
DO 1020 NR = 1, NRAYS, 1 121.
IF (PJDATA(NR) .GT. PJMAX), PJMAX = PJDATA(NR) 122.
1020 CONTINUE 123.
C 124.
C INITIALIZE CONTROL PARAMETERS 125.
C 126.
SINT = SIN(ALPHA) 127.
COST = COS(ALPHA) 128.
DRDX =- XINCR*COST 129.
DRDY =- YINCR*SINT 130.
2 0 DZDX = + XINCR*SINT 131.

DZDY =- YINCR*COST 132.
RORG = RADIUS- Xl*COST- Yl*SINT 133.
ZORG = Xl*SINT - Yl*COST 134.
C 135.
TMIN = ZORG/RORG 136.
TMAX = TMIN 137.
C 138.

RTEMP = RORG + XNI*DRDX 139.
ZTEMP = ZORG + XNI*DZDX 14û.
TEMP = ZTEMP/RTEMP 141.
IF (TEMP .LT. TMIN) TMIN = TEMP 142.
IF (TEMP .GT. TMAX) TMAX = TEMP 143.
C 144.
RTEMP = RORG + YNl*DRDY 145.
ZTEMP = ZORG + YNl*DZDY 146.
TEMP = ZTEMP/RTEMP 147.
IF (TEMP .LT. TMIN) TMIN = TEMP 148.
IF (TEMP .GT. TMAX) TMAX = TEMP 149.
C 150.
RTEMP = RORG + XNl*DRDX + YNl*DRDY 151.
ZTEMP = ZORG + XNl*DZDX + YNl*DZDY 152.
TEMP = ZTEMP/RTEMP 153.
IF (TEMP .LT. TMIN) TMIN = TEMP 154.
IF (TEMP .GT. TMAX) TMAX = TEMP 155.
C 156.
AA = TMIN*TMIN 157.
ALPHA = TMIN*(CO ~ AA*(C2 + AA*C4)) + AZERO 158.
NRl = IFIX(ALPHA) 159.
IF (NRl .LE. 0) NRl = 1 160.
C 161.
AA = TMAX*TMAX 162.
ALPHA = TMAX*(CC + AA*(C2 + AA*C4)) + AZERO 163.
NR2 = IFIX(ALPHA) + 1 164.
IF (NR2 .GT. NRAYS) NR2 = NRAYS 165.

C 166.

-47(a)-~ i~L1575 r C CONVOLVE PROJECTION DATA WITH CONVOLUTION
FUNCTION 167.
C 168.
DO 1200 NR = NRl, NR2, 1 169.
CSUM = 0.0 170.
SSUM = 0.0 171.
JSUM = MINO(NR - 1, NRAYS- NR) 172.
JMAX = MAXO(NR - 1, NRAYS- NR) 173.
IF (JSUM .EQ. JMAX) GO TO 1140 174.
1 0 JMIN = INCR*((JSUM + lNCR - l)/INCR) + 1 175.
IF (JMIN .GT. JMAX) GO TO 1140 176.
IF (NR .LE. NPLUS) GO TO 1080 177.
JADD = - INCR 178.
JJ = NR- JMIN- JADD 179.
GO TO 1100 180.
1080 JADD = INCR 181.
JJ = NR + JMIN - JADD 182.
1100 DO 1120 J = JMIN, JMAX, INCR 183.
JJ = JJ + JADD 184.
CSUM = CSUM + CCFNTN(J + l)*
PJDATA (JJ) 185.
SSUM = SSUM + SCFNTN(J + l)*
PJDATA (JJ) 186.
1120 CONTINUE 187.
IF (JADE~ .LT. 0) SSUM = - SSUM 188.
1140 IF (JSUI~I .EQ. 0) GO TO 1180 189.
DO 1160 J = 1, JSUM, INCR 190.
0 CSUM = CSUM + CCFNTN(J + l)* 191.
(PJDATA(NR+J) + PJDATA(NR-J)) 192.

0 SSUM = SSUM + SCFNTN(J + 1)* 193.

1~1575 (PJDATA(NR+J) - PJDATA(NR-J)) 194.
1160 CONTINUE 195.
C 196.
1180 CSUM = CSUM + CCFNTN(l)*PJDATA(NR) 197.
PJCONV(NR) = WCOS(NR)*CSUM - WSIN
(NR)*SSUM 198.
1200 CONTINUE 199.
C 200.
C PERFORM BACK PROJECTION OPERATION 201.
1 0 C 202.
YN =- 1.0 203.
DO 1380 IY = 1, NTOT, NX 204.
YN = YN + 1.0 205.
YTEST = Yl + YN*YINCR 206.
TEMP = RCALC**2- YTEST**2 207.
IF (TEMP .LE. 0) GO TO 1380 208.
XTEST = SQRT(TEMP) 209.
NXl = IFIX((- XTEST- Xl)/XINCR) 210.
IF (NXl .LT. 0) NXl = 0 211.
NX2 = IFIX((XTEST- Xl)/XINCR) 212.
IF (NX2 .GE. NX) NX2 = NX - 1 213.
IF (NXl .GT. NX2) GO TO 1380 214.
XN = FLOAT(NX1 - 1) 215.
Rl = RORG + YN*DRDY 216.
Z1 = ZORG + YN*DZDY 217.
NXl = NXl + lY 218.
NX2 = NX2 + lY 219.
DO 1360 I = NX1, NX2, 1 220.
XN = XN + 1.0 221.
R = Rl + XN*DRDX 222.

-48(a)-ll`~l.S75 Z = Zl + XN*DZDX 223.
A = Z/R 224.
AA = A*A 225.
ALPHA = A*(CO + AA*(C2 +
AA*C4)) + AZERO 226.
IDX = IFIX(ALPHA) 227.
T2 = ALPHA - FLOAl (IDX) 228.
Tl = 1.0- T2 229.
DD = R**2 ~ Z**2 230.
0 RECON(I) = RECON(I) ~ (T l* PJCONV 231.

(IDX) + T2*PJCONY(IDX+l))/DD 232.
1360 CONTINUE 233.
1380 CONTINUE 234.
1400 CONTINUE 235.
WRITE (6, 1415) PJMAX 236.
1415 FORMAT (lHO, 10X, 'MAXIMUM MEASUREMENT =', lPE12.5) 237.
C 238.
C NORMALIZE RESULT 239.
C 240.
TMAX = RECON(l) 241.
DO 1500 N = 2, NTOT, 1 242.
IF (RECON(N) .GT. TMAX) TMAX = RECON(N) 243.
1500 CONTINUE 244.
FACT = RADIUS /(RAYSPG*FLOAT(NPROJ)) 245.
IF (P~NORM .NE. 0.0) FACT = PJNORM*FACT/PJMAX 246.
TE~flP ~ FACT*TMAX 247.
WRITE (6, 1525) TEMP 248.

1525 FO~MAT (lHO, 10X, 'MAXIMUM VALUE =', lPE12.5) 249.

C 250.
IF (IXDRM .NE. 0) FACT = XNORM/TMAX 251.
DO 1600 N = 1, NTOT, 1 252.
RECON(N) = FACT*RECON(N) 253.
1600 CONTINUE 254.
C 255.
C PRINT RESULT 256.
C 257.
CALL PTDENS(RECON) 258.
GO TO 20 259.
C 260.
9999 STOP 261.
C 262.
END 263.

where ELDATA reads data describing an object if simulated data is to be generated;
CNVLFN calculates the function to be convolved with the measured or calculated data;
FANGEN reads measured data from tape or calculates data in the case of a simulated object;
FANERR introduces errors into the data if desired;
PTDENS prints the results.

-49(a)-liiiS75 In a preferred data reconstruction system in accordance with the present invention the computer 70 appearing in FIG. 1 ~ay be embodied in the form of a general purpose computer or "reconstructor". The latter is specifically adapted for use with the present invention, and so designed as in conjunction with the ~eneral purpose computer, to enable extremely rapid manipulative processing of data obtained pursuant to the invention. Nore specifically, the reconstructor may receive its input data and instructions from the aforementioned general purpose or host computer.
After performing a specified sequence of computations, the reconstructor transfers the results back to the general purpose computer, which in turn functions during display, recording and/or for further processing.
- The reconstructor which will be more fully described in connection with FIGS. 5 through 7, basically constitutes a pipeline processor. Control functions for the pipeline processor are provided by control lo~ic which interfaces with the general purpose host computer -- which may comprise for example a device such as the Varian Data Machines Model V76. The pipeline may thus be characterized as a highspeed special purpose computing device, which in fact performs the ~reat bulk of reconstruction computations re~uired by the present invention. In a general sense the reconstructor pipeline processor, as will become evident, can function in either a back projection or in a convolution mode of operation~ In the convolution mode of operation the pipeline computes convolved data functions. In the back projection mode of operation the pipeline computes mass densities at each picture element (or "pixel") for display 90~ The pipeline may al50 operate in a contro~ mode of operation, during which phase the aforementioned control loqic communicates skO1~377 -50- 7~-60 ~ ~illS75 with the general purpose computer or ~ith the pipeline.
In order to more fully appreciate the manner in which the special purpose computer or reconstructor of FIGS. 6 and 7 functions, reference may initially be had to FIG~ 5 herein, setting forth a schematic representation of the rotatable gantry portion 10 of the present apparatus, and including various schematically portrayed information useful to analysis of the operation now to be described.
In this schematic depiction it is seen that the penetrating radiation source, in this instance representatively shown as an x-ray source 102, is positioned at one side of the circle 101, representing the rotatable gantry 10. The radius of the said circle 101 ~s indicated as R. Directly opposed to x-ray source 102 at the opposite side of circle 101 there is schematically shown the array 103 of detectors.
These are shown to be arranged along a straight line 105, which is a preferable arrangement for these devices; (as disclosed e.g. in the issued United States Patent Number 4,190,772 (February 26, 1980), . ~ ~~
~nd ;. entitled "Tomographic ~canning Apparatus") bu~
they can also be arrayed along a curvelinear arc, as is known in the art, and illustrated in FIGS. 1 through 3.
When the straight line locus is used, the plates constitutin~
~rid 66 are inclined, as may be the axis of each detector cell 65, so that each such cell and associated pair of collimator plates are oriented along a radius to source lQ2.
At the center of circle 101, a plurality of picture elements or "pixels" 107 are schematically illustrated.
The actual number of such pixels in the reconstructed image which is eventually derived in accordance with the invention, as has previously been discussed, is actually much larger than indicated in the schematic showing; thus skO11377 -51 - 76-60 ~11575 the total number of pixels 107 is representatively 65,536 -- since there are typically 256 rows and 256 columns of same in the reconstructed image. X and Y coordinate axes are defined in the Figure, so that the location of individual pixels may be readily specified: Thus the uppermost pixel at the lefthand corner of the array is indicated as having coordinates XOand Yo; the pixel immediately therebelow has coordinates XO and Yl and so forth~
Line 108 extending from x-ray source 102 throu~h - 10 the center of rotation 109 to detector array 103, serves with the X axis to define the projection angle ~ , i.e.
the angle at which data is being generated at the detector array in consequence of passage of radiation through the body portion being analyzed. A coordinate axis Z is also defined, which coincides with whichever of the X or Y
axes is more nearly perpendicular to line 108. A distance specified as ~ is also identified, representing the distance from x ray source 102 to a particular pixel of interest (e.g. at XO, YO). The distance S is also defined, representing the distance in the direction perpendicular to the Z-axis between source 102 and a pixel of interest. This distance will have further significance in the discussion set forth below. Finally, and of considerable siqnificance for present purposes, is the back projection angle (BPA~, which is defined as the angle between the x-ray beam center line 10~ and the line 111 from the source 1~2 to the pixel of interest. This back proiection angle BPA also, of course, correspo~ds to the angle ~in Figure 3 herein;
~or purposes of the present analysis, however, and for fur'her clarity, the said angle will be consistently referred to as "BPA".

In considering Figure 5 it may initially be noted skO11377 - 52- 76-60 llli57~

that for any particular pixel located at coordinates XO, Y~ the line from the x-ray source passing through such pixel and incident on detector array intersects the Z-axis at a location given by the expression:
oS ~ S
CoS ~ ~ ~o ~/~C05~ o= R ~`1 s~-~
"= tQ~ (~;~) The expression (54) for angle BPA wlll shortly be useful, as will hereinbelow be set forth.
Referring now to the schematic showings of FIGS. 6 and 7, the special purpose computer, in the form of a pipeline 113 is set forth, which consists of six segments or slices, i.e. slices 1 through 6. Each slice performs one part of the total computation, and the output of each slice is utilized as the input to the next successive slice.
The pipeline 113 is synchronized by a master clock -- not shown in the present schematic depiction. Each of the s'ices begins its operation on the falling edge of the clock pulse.
The clock period is rendered sufficiently long that all slices 2~ have ample time to finish their operation before the next falling edge of the clock. Each set of data originates in slice 1 during a given clock cycle (cycle N), passes to slice 2 in cycle N+l etc., and reaches slice 6 in cycle N+5.
Considering the back projection operation in general terms, the task is one of computing from the convolved data functions a cross-sectional picture of the subject being examined. This is effected by segmenting the said picture into the aforementioned large number of pixels, and computing the mass density within each such pixel.

In a typical such scheme, as already mentioned, the picture is divided into a 256 x 256 array of rows and columns, skO11377 -53 - 76-60 S7~

for a total of 65,536 pixels. Each pixel density is typically derived in this arrangement from 360 scanner projection angles, each represented by 12QO convolved data (CD) values. In overview the back projection procedure is as follows:
~ or each of the scanner projection positions ~ the back projection angle for each pixel is computed~ From that angle the convolved data functions CD are evaluated.
More specifically such CD functions are evaluated by interpolation from a more limited number of values, i.e.
typically from 1200 such values which are stored in a suitable memory, i.e. as a table of such values. It should be appreciated that the convolved data function CD is a function of the BPA. The said C~ functlon is actually continuous but arbitrarily is calculated at a limited number of points, i.e. in this instance at 1200 equally spaced locations along the coordinate with interpolation being utilized to calculate the intermediate values of same.
Having evaluated the convolved data functions for the BPA of each pixel, the resultant value is scaled by the factor (R /Q) to yield the density contribution to the ?articular pixel at the projection angle ~ then under consideration.
This value is added to a suitable memory, which accumulates the density contributions to the individual pixels for the successive projection an~les at which the foregoing operations are repeated to derive other partial ~ensity contributions for the individual pixe]s.
The manner in which the aforementioned o~erations are brought about during back projection may be ~est understood skO11377 - 54 - 76-6~

`` iiiiS75 by reference to FIG. 6. For each of the pixels 107, slice 1 firs,t computes the variable Z, which measures the pixel , position ~ithin the row or column. Sincè t~e pixels are esually spaced, Z(N+l) = Z(N) +~Z, where 4 Z is the incremental distance between pixels on the Z axis of FIG. 5. This latter value is provided by general purpose computer 115 through control logic 117, linell8 and memory 119, and thus to adder 121 -- which also receives the Z value of the prior pixel from memory 123 via line 125. The variable ~ proceeding from memory 123 is cumulative by virtue of the loop 127 returning to memory 123 from adder 121. Thus the varia~le Z proceeds via line 129 during each cycle of the pipeline --this result being loaded into the Z memory 131 on the n2xt clock pulse.
The act~lal scheme utilized is seen to be one wherein the integral portion N of Z is provided to Z memory 131, with` the fractional portion K bypassing same and being provided to mu~tiplier 133. Z memory 131 has previously been loaded with the values of BPA for a limited number of Z locations -- which BPA values have been calculated by application of equation (54)~ The objective at this point is to calculate the BPA for the particular Z value pertinént to the pixel of interest from these tabulated values, using a linear interpolation scheme. Thus the integral portion of Z provided via line 135 is utilized to read out from Z memory 131 at -output line 137 the BPA
corresponding to the said integral portion -- indicated in the Figure as BPA (~l). At the same time the increment in the BPA between BPA for the value N and the next higher ~PA, i.e. the quantity BPA (N*l) - BP" (N), is read out from the Z memory via line 139. The fractional portion K
of ~ proceeding via line 141 is then multiplied with the skO1~377 -5~ - 76-~0 ~ill~

output in line 139 at multiplier 133. m us in line 143 from multiplier 133 the quantity R x ~BPA (N+l)- BPA(N) appears. This quantity is added to the output in line 137 at adder 145, so that the resulting quantity emerging from adder 145 at line 147 is BPA (Z), and thus:
(55) BPA(Z) = BPA(N) + K x [BPA(N~ BPA(N)]
During the back projection mode of operation illustrated in Figure 5, logic 117 acting through line 197 sets the data selector 199 to pass the data on line 147 on to the slice 1 output on line 198, thus yielding at such output the back projection angle BPA corresponding to the pixel of interest.
It may parenthetically be noted in connection with the prior several paragraphs that a "line" such as at 129 proceeding from memory 123, may actually consist of a plurality of separate electrically distinct signal carriers, i.e. such "line" may be in the nature of a data bus. Accordingly the splitting off of plural lines such as at 135 and 141 (which carry distinct signals) merely represents in the schematic Figure the divergence of signal carriers which were in fact distinct from one another when incorporated into the common bus.
The function performed in the next successive slice, i.e. slice 2, is one of providing the convolved data CD
at a given scanner position as a function of the developed back pro3ection angle BPA. In order to simplify certain calculations herein, consideration should momentarily be given to the fact previously elucidated that the density of a.particular pixel is given by the expression:
Density of Pixel = ~ CD (~PA) x (R/Q) where Q = ~
~CoS ~

skO11377 -~6 - 76-60 ~i~1575 We may arbitrarily define an auxiliary function CD*
which is also a geometrically scaled function of the back projection angle, as follows:
c~ c~ C~s~ sl-~@~ PP~-~3ence, the density of the pixel may also be given by the expression:

where S has already been identified by reference to FIG. 5.
Since S is constant for each column of pixels and each B, one only is required to compute S for each 256 pixels --where the auxiliary function CD* is utilized. T~is simplifies the requirements of computation and eases the burden upon the computer 115, and also enables the calculations to be effected at differing times, thus making better use of computer 115. Returning to slice 2 in FIG. 6 it will thus be seen that the F memory 147 actually tabulates the auxiliary function CD* (BPA) -- which has previously been provided to F memory 147 via line 151 fro~ computer 115 ~ia logic 117. Computer 115, in turn, calculates CD
values from CD values which are determined during operation of pipeline 113 in its convolution mode -- which will be discussed hereinbelow.
The same type of linear interpolation scheme that has previously been discu~sed in connection with slice 1 is utilized for slice 2. In particular it is seen that the back projection angle of the pixel of interest, i~e~ proceeding via line 198, is split, with only the integral portion thereof, being furnishea to F memory 130- The fractional portion thereof, L, proceeds via the bypassing line 153, and is furnished to a multiplier 155. There is thus read out from F memory 130 at line 157 a value CD*(M) corresponding to the address M, and at line 159 the incremental value skO11377 - 57 - 76-60 llllS75 between CD*(M) and the next higher value of CD~ -- i.e.
the quantity CD*(M+l) - CD*(~). This last quantity i5 furnished to the multiplier 155. Thus the output from the multip~ier 155 is the quantity which is then added to the output in line 157 at adder 161 to finally yield an output for slice 3, i.e. at line 163 the value CD*(BPA) =
CD (M) + L x [CD~(M~ CD~(M)].
At the next successive slice of pipeline 113, i~e. at slice 4, the convolved data CD* proceeding from slice 3 via line 163, is multiplied by a geometrical scaling factor. This factor, i.e. (R / S)2is read out from the K memory 165, which is provided with these scaling guantities from computer 115 and logic 117 via line 167. The two factors proceeding to multiplier 169 at slice 4 tnus yield at the output line 171 from such multiplier the quantity CD*~BPA) x (R /S) = CD (BPA) x (R / Q~ . Thus it is seen that at this point the auxiliary function CD* is returned to the desired form for contribution to the total density of the pixel then being considered.
Finally, and continuing in the analysis of the back projection operation, at slices 5 and 6 of pipeline 113, the contributions of density to each pixel are accumulated.
Specifically it is seen that the output from slice 4 via line 171 is provided to adder 173 at slice 5, and the output of the adder is furnished via line 175 to the M
memory 177 which holds the picture density for each pixel.
It is noted that adder 173 is in a loop with an output from M memory 177 at line 179. Hence a cummulative action is effected, i.e. M memory 177 holds the cummulative density for the individual pixels as a result of contributions effected during operations at each projection angle Q .
The M memory communicates via an output line 181 with skO11377 - 58 - 76-60 loqic 117 and general purpose computer 11~, to enable display or the like of the desired viewing image.
In FIG. 7 herein, the operation of pipeline 113 is shown during the convolution operation. In the course of such operation, as has been discussed in early portions of the specification, the convolved data functions CD
are derived from the scanner projection data, i.e. from the raw data provided from the detector array, and from a specified convolution function. In the Figure, the operations occurring at the various slices discussed in connection with back projection operation are again set forth.
In considering the convolution operation it may be initially noted that the general mathematical operation involved in computing the convolved data functions ~or integral values of BPA's, i.e. CD(M) is given by the expression CD (M) = ~ 9~A(~) X C~(~ ~), ( ) where K is the particular detector number in the detector element array, the summation being effected over the variable K for the products of the DATA (K) with the convolution function CF for (M-K).
Luring convolution operation the only portion of slice 1 which actively participates is the F memory address counter 183, which is in communication with logic 117 through line 185 as indicated. The addresses N are successively read-out through line 187. Logic 117 acting throuqh line 197 sets data selector 199 to pass the data on line 137 onto the slice 1 output on line 198 and into the F memory 130 of slice 2. Since the portions of slice 1 in Fig 6 which are connected to line 147 are not used where selector 199 is in the position shown in Fig. 7, such further elements are omitted from Fig. 7 for c~arity.
F-Memory 130, which has previously been discussed in skO11377 - 59 - 76-60 connection with Figure 6, is provided with the convolution functions through logic 117 and line 189, with such functions being generated at computer 115. The convolution functions for successive integral values N are read out at line 137 from slice 2. The other element in the memory read-out, which provides the incremental difference between convolution functions for M and for ~ = 1 via line 139 to multiplier 133, is in this instance nullified by providing a zero read-in throu~h line 141 to multiplier 133. Thus the output from the multiplier line 134 is in turn zero. Hence, the output from adder 145 at line 147 is the function CF(N), this function entering slice 4 of the pipeline 113.
K memory 165, previously discussed in connection with FIG. 6, is provided during the convolution operation ` with the raw data from the detector element array, and the output from K memory 165, which receives its control and also raw data input via line 191 from logic 117 and computer 115, provides an output of DATA (K) via line 170 to multiplier 169. Multiplier 169 then forms the multiplications CF(N) x DAT~(K) to yield the various terms for the convolution data functions forming part of equation~) above. These functions are provided at slice S to the adder 173, with the output of same being ~urnished to M memory 177 in slice 6 which stores the final results CD(M).
The M memory 177 is seen to be in a loop with the adder 173 via the additional line 179, whereby the M memory serves again to accumulate the plurality of terms, which in accordance with e~uation (55) above constitutes the convolved da~a functions CD(~I). This convolved data is then furnished to the computer 115, and as was previous~y discussed, utili~ed in connection with the back projection operations heretofore analy~ed.

ehb2011376 - 60- 76-60 ~1~1575 While the principles of the invention have now been made clear in the illustrated embodiment sho~n above, there will be obvious to those reasonably skilled in the art many modifications in arrangement of components and choices of variables used in the practice of the invention without departing from the above enunciated principles.
For example, other convolution functions than the one detailed herein may be employed. Further, it must be remembered that the technique of the invention can be employed over a wide range of applications, such as transmissive ultrasonics, electron microscopy, and others, as long as radiation in the shape of a fan beam can be caused to pass through an object at a plurality of angles - and then detected.
The appended claims are intended to cover and embrace any such modification within the limits only of the true spirit of the invention.

ehb2011377 - 61 - 76-60

Claims (12)

I CLAIM:

1. Apparatus for constructing a two-dimensional representation of an object lying in a quasi-plane comprising:
a radiation source for providing radiation in the form of a fan beam positioned so that at least some of said radiation passes through said object;
detector means positioned opposite said source and aligned therewith and lying in said quasi-plane for detecting radiation in said quasi-plane not absorbed by said object;
means for effecting relative rotation between said object and said source-detector means combination about an axis of rotation such that said source and detector means remain in said quasi-plane;
reconstruction means coupled to said detector means and comprising a general purpose computer, a special purpose computer, and control logic for interfacing between said computers and controlling the respective functioning thereof, for performing a convolution and back projection based upon said non-absorbed radiation detected by said detector means without first reordering said fan beam rays into a different set of rays, wherein said reconstruction means converts values of said non-absorbed and non-scattered rad-iation into values of absorbed radiation at each of an arbitrarily large number of points selected within said object; and readout means coupled to said reconstruction means for providing a display of said amounts of absorbed radiation.

2. Apparatus in accordance with claim 1 wherein said special purpose computer comprises a pipeline processor.

3. Apparatus in accordance with claim 2 wherein said pipeline processor is operable in at least (a) a convolution mode of functioning wherein convolved data functions associated with said selected points of said object are computed and (b) a back projection mode of operation wherein mass densities are computed for said points;
and wherein said convolution or back projection mode of operation is enabled by said control logic.

4. Apparatus in accordance with claim 3, wherein said pipeline processor includes first means for determining the back projection angle for said points selected within said object; second means downstream of said first means for providing a geometrically scaled function indicative of the convolved data function correlated with each said back projection angle determined at said first means;
third means downstream of said second means, for scaling the functions from said second means to reflect the distance from said radiation source to said points, to thereby provide partial density functions for said points; and fourth means downstream of said third means for accumulating the said partial densities applicable for each of said points, to thereby yield the total appropriate densities for said points.

5. Apparatus in accordance with claim 4, wherein said second means includes memory means for storing said function indicative of said convolved data from prior calculations thereof incident to operation of said pipeline in said convolution mode.

6. Apparatus in accordance with claim 5, wherein said second means includes means for linearly interpolating between values stored in said memory means, to yield values for said functions in addition to those stored.

7. Apparatus in accordance with claim 5, wherein said memory means during operation of said pipeline processor in said convolution mode, tabulates convolution functions provided from said general purpose computer, said pipeline processor including second memory means for storing data from said detector means; means for reading out said stored data and for operating thereupon by said tabulated convolution functions; and means for storing the resultant convolved data functions at the output of said pipeline for use in back projection.

8. Apparatus in accordance with claim 2, wherein said detector means comprises an elongate detector capable of measuring amounts of radiation at each point along said detector and converting said measurements into measurement signals.

9. Apparatus in accordance with claim 2 wherein said detector means comprises a plurality of detector elements.

10. A system for reconstructing into intelligible form data generated by fan beam radiation transmitted from a plurality of positions through an object to be inspected and onto detector means whereat a multiplicity of signals are received and detected; said system comprising:
means to perform a non-reordering convolution operation on manipulations of the signals received by said detector means at each position of the fan beam relative to said object to arrange said manipulations in convolved form representative of radiation transmitted through said object at each of said positions, and means for back projecting said convolved manipulations to form a two-dimensional reconstruction of the object;
said means being coupled to said detector means and including a general purpose computer, a special purpose computer, and control logic for interfacing between said computers and controlling the respective functioning thereof.

11. A system in accordance with claim 10 wherein said special purpose computer comprises a pipeline processor.

12. Apparatus in accordance with claim 11 wherein said pipeline processor is operable in at least (a) a convolution mode of functioning wherein convolved data functions associated with selected points of said object are computed and (b) a back projection mode of operation wherein mass densities are computed for said points and wherein said convolution or back projection mode of operation is enabled by said control logic.