Method and Apparatus for X-Ray or .Gamma.-Ray 3-D Tomography Using a Fan Beam

A multiwire radiation detector, of the type wherein the wires of the detector are parallel to the divergent rays of penetrating radiation, forms the subject matter of and is claimed in copending U.S. application Ser. No. 528,025 filed Nov. 29, 1974 and assigned to the same assignee as the present invention. The method and apparatus for 3-D X- or .gamma.-ray tomography employing a fan-shaped beam forms the subject matter of and is claimed in copending U.S. application Ser. No. 528,026 filed Nov. 29, 1974 and assigned to the same assignee as the present invention.

The present invention relates in general to fan beam X- or .gamma.-ray 3-D tomography and more particularly to such tomography utilizing a position sensitive detector.

Heretofore, it has been proposed to employ collimated beams of penetrating radiation to derive a set of angularly displaced shadowgraph data from which to reconstruct a 3-D tomograph of a slice of the body. The 3-D tomograph was reconstructed by a method of computing the absorption or transmission coefficients for a matrix of elements of cross sectional area intersected by the angularly displaced sets of parallel rays. The coefficients were refined by a process of successive approximations to derive the final 3-D tomograph. Such a method is proposed in U.S. Pat. No. 3,778,614 issued Dec. 11, 1973.

In this prior patent, the shadowgraph data is derived by either of two methods. In a first method, a collimated source of penetrating radiation passes through the body to a detector in alignment with the beam path. The detector and source are then rectilinearly translated laterally of the body to derive a given set of shadowgraph data. The source and detector are then angularly rotated to a second position and again laterally translated relative to the body to obtain a second set of shadowgraph data, and so forth.

In the second method, a fan-shaped array of collimated beams of penetrating radiation, each beam having a detector in alignment therewith, is caused to be laterally rectilinearly translated relative to the body and then rotated to a second position with lateral translation at the second position, and so forth and so on, to derive angularly displaced sets of shadowgraph data.

The advantage of the second scheme relative to the first scheme, is that the lateral translation can be cut by a factor of 1/N where N is the number of detectors, such as 6 or 7. However, this prior art patent discloses that the paths of penetrating radiation through the body should all have a constant width and that this is an essential requirement for accurate computer calculations which are to follow for reconstruction of the 3-D tomograph. Also, the algorithms presented therein for reconstruction of the 3-D tomograph are based upon sets of parallel rays. However, in the case of the collimated divergent beams, there is no disclosure of how one obtains shadowgraph data based upon sets of parallel rays. Furthermore, there is no teaching nor suggestion of how the divergent rays passing through the body could be made to traverse paths of constant width. Thus, there is no teaching in the subject patent of a method for reconstruction of 3-D tomographs from sets of divergent rays of penetration as would be obtained from a divergent fan beam.

It has also been proposed in the prior art relating to 3-D X-ray or .gamma.-ray tomographic reconstructions to reorder divergent fan beam shadowgraphic data into parallel ray shadowgraphic data from which to compute the 3-D reconstruction. Such a proposal is found in an article titled, "Reconstruction of Substance From Shadow" appearing in the Proceedings of the Indian Academy of Sciences, Vol. LXXIV, No. 1, Sec. A (1971) pages 14-24.

The problem with this reconstruction proposal is that it provides only a rather abstract algorithm for transforming a continuous distribtuion of divergent X-ray or .gamma.-ray shadowgraphic data into a parallel ray continuous distribution of X-ray or .gamma.-ray shadowgraphic data. However, in a practical system, the data is acquired not as a continuous distribution but as incremental data not only as a function of distance X perpendicular to the central ray of the fan beam but also as a function of .theta., the angle of rotation of the source about the axis of revolution. There is no teaching in this article of a way to transform such incremented divergent fan beam shadowgraphic data into corresponding sets of incremented parallel ray shadowgraphic data. There is, however, a mention of a required interpolation which is not defined in the article (see page 22 therein).

The principal object of the present invention is the provision of an improved method and apparatus for reconstruction 3-D X- or .gamma.-ray tomographs derived from sets of shadowgraphic data derived employing a fan-shaped beam of penetrating radiation.

In one feature of the present invention, a fan-shaped beam divergent ray of penetrating radiation is angularly moved relative to the body to obtain angularly displaced shadowgraph data and wherein the shadowgraph data is reordered to derive shadowgram data based upon angularly displaced sets of parallel rays, whereby reconstruction of a 3-D tomograph is facilitated.

In another feature of the present invention, the 3-D X- or .gamma.-ray tomograph is reconstructed from sets of shadowgraphic data derived by use of a fan-shaped beam of penetrating radiation, such reconstruction being by a method of convolutions.

In another feature of the present invention, the divergent penetrating radiation detected is detected at angular spacing less than and preferably one half the angular spacing between adjacent shadowgrams.

In another feature of the present invention, the divergent ray data is converted into parallel ray data of equal lateral spacing.

Other features and advantages of the present invention will become apparent upon a perusal of the following specification taken in connection with the accompanying drawings wherein:

FIG. 1 is a schematic transverse sectional diagram of a penetrating ray 3-D tomography apparatus of the present invention and including a shadowgraph produced by the apparatus,

FIG. 2 is an enlarged sectional view of a portion of the structure of FIG. 1 delineated by line 2--2,

FIG. 3 is an enlarged detail view of a portion of the structure of FIG. 1 delineated by line 3--3,

FIG. 4 is a view of the structure of FIG. 3 taken along line 4--4 in the direction of the arrows,

FIG. 5 is an enlarged detailed view of a portion of the structure of FIG. 1 taken along line 5--5 in the direction of the arrows,

FIG. 6 is a schematic diagram of a fan beam X-ray 3-D tomographic apparatus incorporating features of the present invention,

FIG. 7 is a longitudinal sectional view of a position sensitive X-ray detector employed in the apparatus of the present invention,

FIG. 8 is a view of the structure of FIG. 7 taken along line 8--8 in the direction of the arrow and including associated circuitry in block diagram form,

FIG. 9 is a schematic line diagram, partly in block diagram form, of a data processing portion of the apparatus of the present invention,

FIG. 10 is a schematic line diagram depicting how a rotating fan beam produces sets of parallel rays,

FIG. 11 is a view similar to that of FIG. 10 depicting extrapolation of the arrangement of FIG. 10 to twice as many detectors and to approximations of parallelism for the added intermediate rays,

FIG. 12 is a schematic diagram depicting the process for correction of the set of detected parallel rays to sets of parallel ray data of equal lateral spacing,

FIG. 13 is a schematic diagram depicting the process for compensating for non-equal spacing between the detected parallel rays,

FIG. 14 is a shadowgraph converted to 1n of the ratio of detected intensity as a function of lateral position I'(y) divided by the beam intensity Io measured without absorption,

FIG. 15 is a plot of a function utilized in the 3-D reconstruction method,

FIG. 16 is a plot for the convolution of the function of FIG. 15 with a single point on the function of FIG. 14,

FIG. 17 is a plot of the convolution of the function of FIG. 15 with the shadowgraph function of FIG. 14,

FIG. 18 is a schematic line diagram depicting the process for back projecting and adding the contributions of the convoluted shadowgraphic data,

FIG. 19 is a schematic line diagram representing the positional uncertainty when a fan beam is detected by a rectilinear detecting array,

FIG. 20 is a longitudinal sectional view of a preferred multiwire radiation detector,

FIG. 21 is an enlarged sectional view of the structure of FIG. 20 taken along line 21-21 in the direction of the arrows, and

FIG. 22 is a flow chart for a computerized method for reconstruction of the 3-D tomographs.

Referring now to FIG. 1, there is shown an apparatus for deriving penetrating radiation shadowgraphs of a body to be examined. More particularly, the patient 11 to be examined is supported on a couch 12, as of a suitable plastic material. A suitable point source of penetrating radiation 13, such as X-rays or .gamma.-rays is disposed above the body for projecting a fan-shaped beam of divergent penetrating radiation through a narrow elongated slot 14 in a collimator 15, as of lead. The fan-shaped beam is relatively thin and comprises divergent rays of penetrating radiation which are directed onto the body 11 to be examined.

The penetrating radiation is partially absorbed in the body 11 in accordance with the density of the various portions of the body penetrated by the radiation. In a typical example of a torso tomograph, the lungs would have relatively low density, whereas the spinal column would have relatively high density. The penetrating radiation emerging from the body is passed through a second fan beam collimator 20 and thence through a focussed grid collimator 16 which is shown in greater detail in FIGS. 3 and 4. The second collimator 20 is similar to the fan source collimator 15 and the focussed grid collimator 16 comprises an array of lead vanes 17 embedded in a plastic filler material 18, as of polyethylene. The vanes 17 have a thickness of, for example, 0.5 millimeters and the plane of the vanes is directed parallel to the divergent rays emanating from the source 13. In a typical example, the collimating vanes 17 are spaced apart by approximately 5.0 millimeters for blocking scattered radiation emerging from the body 11 from passing into a position sensitive radiation detector 21. Less than 1% of the scattered radiation reaches the detector 21.

In a preferred embodiment, the position sensitive detector 21 includes an array of closely spaced detecting wires as more fully disclosed below with regard to FIGS. 7, 8, 9, and 20 and 21. Generally, there is one detecting element of the array in alignment with the center of each of the bins of collimated divergent rays passing through the collimator 16. In a typical example, the position sensitive detector 21 would have a length of approximately 50 centimeters and would include 150 individual detecting elements at 1/2.degree. intervals. The fan beam typically subtends an arc .psi. of 75.degree..

The penetrating radiation such as X-rays or .gamma.-rays, in passing through the body 11, are variously attenuated or absorbed by the different portions within the body such as the lungs, spinal column, etc. to produce a shadowgraph of detected intensity versus distance as shown in FIG. 1 by curve 22.

In a typical example, the X-ray or .gamma.-ray source 13, which is shown in greater detail in FIG. 2 comprises a cylindrical body 23 of high atomic number Z material, such as lead or tantalum, and includes a central re-entrant bore 24 containing a cylindrical insert 25. The insert typically comprises a plastic body 26 having a pellet 27 of radioactive material embedded in the outer end thereof. A shutter 28 of high Z material is pivotably mounted to the body 23 at 29 and is pivoted for closing off the source of radiation and held in the closed position via a spring clasp 31. Typical source materials for the pellet 27 include materials that will provide X-ray or .gamma.-ray radiation having intensities falling within the range of 50-100 keV. The source radiation is preferably monochromatic. Materials of this type include Gadolinium.sup.153 having a half life of approximately 242 days and having a very stable predictable decay rate. However, other types of sources 13 could be employed such as an X-ray tube utilizing various different types of secondary .gamma.- or X-ray emitting materials.

Referring now to FIG. 6 there is shown the apparatus of FIG. 1 mounted for rotation about an axis of rotation 33 disposed centrally of the body 11. The source 13, detector 21, and collimator 16 are mounted to a ring 34 for rotation about the axis of rotation 33. The ring 34 is driven from a friction drive wheel 35 which is connected to a drive motor 36 via a suitable drive means, such as a drive belt 37. The ring 34 is supported via the drive wheel and an idler wheel 38 rotationally mounted to a base support structure 39. The ring 34 includes an array of axially directed pins 41 disposed at one degree intervals around the periphery of the ring 34. A photocell detector 42 is mounted in fixed relation relative to the ring 34 and pins 41 so that as the ring 34 is rotated successive pins 41 come into registration with the optical path of reflected light from the respective pin 42 to the photocell 42 for giving an output signal indicative of the angular position of the ring 34 and, thus, the source and detector relative to the body 11.

This electrical signal, representative of the position of the ring 34, is fed to one input of a sequencer 40. The output of the sequencer 40 is fed to the motor 36 for driving the ring 34 around the body 11. For each 1.degree. of angular position .theta., a 151 point shadowgraph is derived so that a set of shadowgraphs is obtained there being one shadowgraph for each degree of rotation of the source around the patient. In a typical example employing a 75.degree. fan beam of radiation, the sequencer 40 is set to rotate the source continuously around the patient for a total of 255.degree. to obtain 255 sets of shadowgraph data. The reason for 255 sets of data is explained below.

Referring now to FIGS. 7 and 8 there is shown a position sensitive detector 21. The detector includes an elongated channel member 45, as of G-10 fiberglass, having a base portion 46 and two upstanding side wall portions 47 and 48. The channel 45 is closed at its ends via transverse walls 49 and 51. In a typical example, the detector 21 has a length of 50 centimeters. A penetrating ray transparent gas-tight window 52, as of Mylar, is sealed across the open side of the channel 45. An array of transversely directed anode wires 53 extend for the length of the detector 21. Two arrays of longitudinally directed cathode wires 54 and 55 are disposed on opposite sides of the anode array 53.

In a typical example, the anode wires 53 are spaced apart by 2.5 millimeters and the wires have a diameter of 0.025 millimeters (25 microns). The cathode wires 54 and 55 are of tungsten having a diameter of 0.1 millimeters and are spaced apart by approximately 2.5 millimeters. The cathode wires are operated at ground potential, whereas the anode wires 53 are operated at +3kV. The chamber defined by the interior of the closed channel 45 is filled with an ionizable gaseous medium such as xenon at atmospheric pressure. The cathode wires are spaced above and below the anode wires by approximately 3 millimeters.

The anode wires 53 pass through the side wall 48 of the channel 45 in gas tight sealed relation therewith and are affixed at equally spaced intervals to and along a helical delay line 56 operating at anode potential. Opposite ends of the delay line 56 are connected to respective pulse discriminators 57 via the intermediary of pulse amplifiers 58. The outputs of the discriminators 57 are fed to a time-to-amplitude converter 59 which converts the timing between successive pulses, as derived from the discriminators 57, to a potential proportional to the timing between such pulses. The potential output of the time-to-amplitude converter 59 is fed to one input of an A-to-D converter 61 for converting the amplitude information to a digital output which is thence fed to a computer 62 which will be more fully described below.

In operation, a quantum of ionizing radiation passing through the body 11 passes through the window 52 and into the ionizable gas filled chamber 45. Due to the high electrical field region surrounding the individual anode wires 53, when a quantum of ionizing radiation is absorbed in the ionizable gas, ionization occurs which triggers an avalanche of current flow between the anode and cathode resulting in a current pulse on the respective anode wire 53 which is closest to the ionizing event. That pulse of avalanche current is fed onto the delay line at the corresponding connection of that anode wire with the delay line 56. The pulse of current travels in opposite directions along the delay line 56 to the ends thereof and thence via the amplifiers 58 into the discriminators 57.

The discriminators 57 produce corresponding output pulses corresponding to the leading edge of the respective current pulses. The time difference between successive pulses is proportional to or otherwise representative of the position of the ionizing event as detected by the closest anode wire 53. The pulses are thence fed into the time-to-amplitude converter 59 for producing an output potential corresponding to the position of the ionizing event. This potential is thence converted to digital data in the analog-to-digital converter 61 and fed to the computer 62. The computer stores the ionizing event in a respective channel corresponding to the position of the ionizing event. Subsequent ionizing events detected during the measurement of one shadowgraph for each angle of .theta. are stored in their respective channels. Thus, the computer has stored in its memory, after one rotation of the source through 255.degree., 255 sets of shadowgraph data. The computer will then utilize these sets of shadowgraph data for reconstructing a 3-D tomograph of the section of the body 11 under examination, as more fully described below.

One of the problems with the delay line type of position sensitive detector 21, as shown in FIGS. 7 and 8, is that it is limited to a counting rate of approximately 10.sup.5 counts of ionization events per second. It is desired to employ a detector which is capable of counting at a rate of 10.sup.8 events per second or higher. For example, in order to obtain a 3-D tomograph having a 1/2 percent precision in the density, approximately 10.sup.9 counts per second are required. It is desirable that the 3-D tomograph data be acquired during one breath-holding period, i.e., a time of approximately 15 seconds or less. This then leads to a desired counting rate of at least 10.sup.8 per second.

The counting rate can be increased to at least 10.sup.8 per second by deleting the delay line 56 and connecting each of the individual anode wires 53 to a respective amplifier 65 and counter 66, as shown in FIG. 9. The outputs of the counters 66 are fed to an input of a multiplexer 67 such that upon the completion of a shadowgraph for each angular position of .theta., the data in the counters 66 is read out via the multiplexer 67 into the computer 62 via a computer interface 68. The computer 62 may comprise, for example, a PDP 11/45 minicomputer provided with a random access memory 69 and a disc memory 71. In addition, the minicomputer 62 provides a keyboard terminal 72, and a color display terminal 73, wherein contours of a given density in the 3-D tomograph are displayed with different given colors, such that density differentiation is enhanced to the human eye. In addition, the minicomputer includes a line printer 74 for printing out a 3-D density tomograph in terms of numbers corresponding directly to density.

Referring now to FIGS. 10-19 and to the flow charts of FIG. 22 the computer method for reconstructing the 3-D tomographs from the sets of angularly displaced shadowgraph data will be explained in greater detail. The sets of shadowgraph data, as detected by the position sensitive detector 21, are generated by absorption of penetration radiation by the body under analysis as taken along an array of divergent paths or rays. The preferred 3-D tomograph reconstruction method requires that the shadowgraph data correspond to absorption of penetrating radiation along an array of parallel paths or rays.

It has been found that the detected divergent path shadowgraph data can be reordered into sets of shadowgraph data representative of that obtained by arrays of parallel rays. This reordering process for reordering the divergent ray shadowgraph data into parallel ray shadowgraph data is illustrated in FIG. 10. In position .theta..sub.i the source 13 projects a fan-shaped beam over a continuous distribution of divergent paths contained within the angle .psi. subtended by the fan-shaped beam. If we consider the central path or ray 75 which is identified r.sub.i,0 it is seen that this ray passes through the axis of revolution 33 to the detector. Other rays denoted by r.sub.i,.sub.-37, r.sub.i,.sub.-36 . . . r.sub.i,.sub.+37 are spaced at 1.degree. intervals within the fan. When the source 13 of the fan-shaped beam is rotated in the positive .theta. direction by one degree about the axis of rotation 33 and from the initial position of .theta..sub.i to .theta..sub.i+ 1, it will be seen that there is a new central ray identified as r.sub.i.sub.+1,0 which is displaced from r.sub.i,0 by 1.degree. and which passes through the axis of rotation 33. In addition, there is a ray r.sub.1,1 which is parallel to r.sub.i,0. Likewise, when the source 13 is rotated 2.degree. to .theta.i+2 there is a ray r.sub.i.sub.+2,2 parallel to both r.sub.i,0 and r.sub.i.sub.+1,1. Following this rationale it will be seen that there are sets of parallel rays according to the series indicated in FIG. 10 where r is a ray or path and has subscripts .theta. and .psi.where .theta. is the angular position of the source 13 and .psi. is the angular displacement of the ray from the center ray of the fan-shaped beam.

Rays may be labelled as r.sub.i,j where i is an index defining the position of the source (.theta..sub.i = i.DELTA..theta.) and j is an integer denoting the position of each ray within a particular fan. The central ray of the fan passes through the center of rotation and is denoted by j=0. Adjacent rays are numbered consecutively. Referring to FIG. 10 it can be seen that it is possible to obtain at least two arrays of parallel rays as shown. If we denote a series of parallel rays inclined at .theta..sub.i, i=0 . . . 180/.DELTA. .theta., by r'.sub.ij then the reordering process can be generalized by the following transformation where .psi. fan is an odd multiple of .DELTA..theta.:

where j.sub.max =(.psi. fan.sup.-.sup.1)/2.DELTA..theta.. For the particular case .DELTA..theta.=1.degree. and .psi.fan=75.degree., then 180 sets of parallel rays are formed with i=0, . . . , 179.

Although in the above discussion for convenience the position of the source 13 was depicted as located at particular points, and the rays represented by lines, it should be understood that the source 13 and detectors 21, etc., rotate at constant angular velocity and data is accumulated during intervals during which the source moves continuously from one position to the next, so that .theta. represents a mean source position during a particular interval in time. Likewise, the detector 21 is sensitive to the continuous distribution of transmitted radiation so that rays really represent the average transmission in a region of narrow width bounded by neighboring rays.

For relatively high resolution it is desired to obtain 180 sets of parallel rays at one degree .theta. intervals. It can be shown that if 180 sets of such parallel rays are to be obtained the source 13 must be rotated through a total angle .theta. of 180.degree. plus the fan angle .psi.fan. In the case of a fan angle .psi.fan of 75.degree., the total angular displacement .theta. is 255.degree.. Thus, the 255 sets of divergent path or ray shadowgraph data are reordered by the computer 62 into 180 sets of parallel ray shadowgraph data. The reordering may be accomplished by the computer 62 after the divergent ray shadowgraphic data is stored in the respective channels of the memory or the data, as it is obtained at the detector 21 and multiplexed into the memory of the computer 62, is addressed in accordance with the desired reordering address method so that the data, as initially stored, is stored in sets of parallel shadowgraphic data.

In order to optimize the spatial resolution that can be achieved with a given finite number of measurements it is found that the fan rays must be more closely spaced than the rotation step angle .DELTA..theta.. If this spacing is chosen to be a fractional value of .DELTA..theta., say .DELTA..theta./n where n=2,3,4 then the above reordering process can still be used providing a slant approximation is introduced. A preferred value of n is 2 yielding .DELTA..psi.=1/2.degree. if .DELTA..theta.=1.degree.. FIG. 11 illustrates how sets of parallel rays are obtained from fan rays in this case. The slant approximation is introduced as follows. Ray r.sub.i.sub.+1,1 is selected to be a member of the series of rays parallel to r.sub.i,0 with spacing midway between r.sub.i,0 and r.sub.i.sub.+1,2. It has been found that this approximation introduces a negligible loss of spatial resolution in the reconstruction. Two sets of parallel rays are indicated in FIG. 11 at .theta..sub.i and .theta..sub.i.sub.+1. Thus for n=2 and using the slant approximations the reordering transformation becomes:

and ##EQU1## Here j* refers to an even integer which may be either j or j+1.

Also it can be shown that the sets of reordered parallel paths or rays are not of equal lateral spacing. The spacing decreases with distance away from the central ray. This is depicted in FIG. 12, where the x abscissa scale represents the spacings of the reordered sets of parallel rays. The preferred 3-D method of reconstruction employs data based upon uniform lateral sapcing between parallel rays of the set. Therefore, it is desired to modify the sets of reordered parallel ray shadowgraphic data into such data having equal lateral spacing between all parallel rays of the set.

Referring now to FIGS. 12 and 13 there is shown the method for transforming the parallel ray shadowgraphic data into such data having equal lateral spacing. A set of parallel rays of unequal spacing is shown at 70 in FIG. 12. In this example n=1, i.e., the slant approximation is not used, and .DELTA..theta.=1.degree.. The x designated abscicca scale shows the unequal lateral spacing where x.sub.o =0, x.sub.1 = Rsin 1.degree., x.sub.2 =Rsin 2.degree. . . . x.sub.j =Rsin j.DELTA..theta. where R is the radius of the circle of revolution of the source 13 relative to the body 11 and j is the number of the ray from the centray ray. The abscissa scale for equal lateral spacing of the rays 70 is that indicated by y, where y.sub.1 =a, y.sub.2 =2a, y.sub.3 =3a . . . y.sub.n =na where ##EQU2## In the case of .psi.=75.degree. then ##EQU3##

The detected radiation intensity I.sub.1, I.sub.2. . . I.sub.n is based upon parallel rays of unequal lateral spacing, i.e., they have the x abscissa scale as shown in FIG. 13. Here the radiation intensity is assumed to be uniform within bins bounded by the midpoints between rays as shown, and the area of bins represent the measured radiation intensities I.sub.i. The x scale intensities I.sub.1, I.sub.2, I.sub.3 . . . I.sub.n may be transformed by a rebinning process to derive parallel ray shadowgraphic intensities I'.sub.1, I'.sub.2 . . . I'.sub.n of equal lateral spacing, as follows.

The new intensities are determined by the amount of area in old bins overlapped by new bins. For example, ##EQU4## and, ##EQU5##

This process may be generalized by the equation: ##EQU6## where f.sub.ij are the coefficients in the above equations (Eq. 1-3) and represent the fractional overlap of new bins with old bins as determined by simple geometry as illustrated above. For speed and convenience these may be calculated in advance and stored in a disk file which may be used by the reconstruction program. Although usually the above series of equations contain only two terms, occasionally three terms will be present corresponding to the case in which a new bin as illustrated in FIG. 13 (y axis) is overlapped by three old bins. In the case of use of a slant approximation with n=2,3,4, rebinning may proceed as above except that the values of the coordinates are given by x.sub.j =Rsin j.DELTA..theta./n and ##EQU7##

The coordinates of the boundaries of the old bins illustrated in FIG. 13 may be calculated by an alternative method used by the computer program referred to in the flow chart of FIG. 22. The x and y axis are taken as a line passing through the center of rotation 33 in FIG. 6, and perpendicular to a particular series of parallel rays. The new bins are equally spaced as before. The old (x axis) are defined by the intersection of the bounds of the actual fan rays with this line. For n=2, these bounds are typically given by lines at +1/4.degree. with respect to a perpendicular line from each particular central source position 13 included in a parallel series. Typically the bins defined in this way are not immediately adjacent as in FIG. 13 but are separated by gaps. The bins representing slant rays are introduced by using rays bounded by lines at -3/4.degree. and -1/4.degree. with respect to a perpendicular line from each source position, and will be seen to fit into the spaces between the perpendicular rays, although insignificantly small gaps will remain. Bin boundaries defined in this way closely conform to those of the previous method, with small differences which improve the accuracy of the slant approximation. The f.sub.ij coefficients are now calculated by determining the amount of overlap of old bins with new bins as before.

The preferred computerized method for reconstructing the 3-D tomographs from the angularly displaced reordered sets of parallel ray shadowgraphs is a method disclosed in an article titled "Three Dimensional Reconstruction from Radiographs and Electron Micrographs: Application of Convolutions Instead of Fourier Transforms" appearing in the Proceedings of the National Academy of Sciences, U.S.A., Vol. 68, No. 9, pages 2236-2240 of September 1971. Briefly this method consists of transforming the parallel ray shadowgram data into shadowgram data corresponding to the natural logarithm 1n of the intensity of the unmodified detected radiation as a function of y, namely I' (y), normalized to the beam intensity I.sub.0 (y). I.sub.0 (y) is measured before the shadowgrams are made, by detecting the unabsorbed beam on each of the detector wires 53, and applying the reordering and rebinning transformations, and that information is stored in the computer for use in these calculations.

FIG. 14 shows a typical shadowgraph for the function 1n I'(y)/I.sub.O (y) which may be referred to as g(na,.theta.). The linear shadowgraphs for different angles .theta. are each scanned at intervals a and these data are then convoluted with a function q(na) to obtain g'(na;.theta.) using the following algorithm: ##EQU8## where ##EQU9##

The function q(na) is shown in FIG. 15 and the product of the function of FIG. 15 with the point 78 of the shadowgraph function of FIG. 14 is shown in FIG. 16. As can be seen from the shape of the function of FIG. 15, this function has zero value for even numbered intervals and drops off relatively quickly with interval number n so that the convolution of the function of FIG. 15 with that of FIG. 14 need only be evaluated at a reasonably small number n of intervals a away from the point on function 14 being evaluated. The individual products of the function of FIG. 15 with g(na,.theta.) for each value of y or na of FIG. 14 are summed to derive the function g'(na;.theta.) of FIG. 17 this process is known mathematically as the convolution of g(na,.theta.) with q(na) as expressed in Eq. (5). In other words, the result of Eq. (5) is shown in FIG. 17 for a given value of .theta.. Thus, there is generated by the algorithm of Eq. (5) 180 sets of the function of FIG. 17, one for each angularly spaced set of parallel ray shadowgraph data. These 180 shadowgraphs are then back projected for calculating the resultant 3-D reconstructed tomograph employing the following algorithm: ##EQU10## where t and N are integers and r and .phi. are the polar coordinates of the individual reconstruction matrix elements. The interval for .theta. is .DELTA..theta. = (180/N).degree., where N is the number of shadowgraphs recorded at regular intervals over the range of -.pi./2 to +.pi./2, typically 180. In Eq. (7), the value of r cos (.phi.-t.DELTA..theta.) will not in general be a multiple of a; therefore a linear interpolation between the calculated values of g'(na;.theta.) is made so that the resolution of the final 3-D reconstructed data obtained for f(r,.phi.) will depend upon the fineness of the interval a at which the shadowgraph data are available and the consequent accuracy of the interpolation.

This method of back projection is schematically indicated in FIG. 18. More particularly, the slice of the body 11 to be examined and for which a 3-D tomograph is to be reconstructed is considered to comprise a two dimensional matrix of elements 80 of equal size. In a typical example, the dimensions of the matrix elements are chosen equal to the spacing between adjacent anode wires 54, i.e., 2.5 mm. The reconstruction algorithm Eq. (7) consists of projecting the individual values of g'(na,.theta.) back across the matrix along lines perpendicular to the y axis of the particular shadowgram.

This back projection process is conveniently accomplished as follows: The coordinates r, and .phi. for the center of a particular matrix element is calculated. The value of y corresponding to the point on the axis of a particular projection intersected by a perpendicular line from the point r,.phi. is calculated. This is given by r cos (.phi.-t.DELTA..theta.). The value of g'(na,.theta.) at that value of y is calculated by means of linear interpolation between the two values of g'(na,.theta.) for which na is nearest y. Hence ##EQU11## where k is the nearest integer lss than y/a. This process is repeated N times for each value of .theta. and the sum of each projected value of g(y,.theta.) yields the value of f(r,.phi.) at that grid point. The value of f(r,.phi.) at other grid points is computed sequentially in a similar manner.

Referring now to FIG. 19 there is schematically indicated the problem of position uncertainty encountered when detecting a fan-shaped beam with a rectilinear array of position sensitive detecting elements 89. More particularly, as shown in FIG. 19 it is assumed that the detector 21 has some depth d in the direction of the incoming rays 88. These rays are divergent and in addition intercept the rectilinear array at an acute angle. Assuming that the ray is divided into a multiplicity of detecting bins 89 it is seen that near the outer ends of the detecting array 21 a given ray may pass through more than one bin 89. Therefore, some uncertainty is introduced relative to the position of the detected ray.

In addition, the spacing s taken along the length of the detector 21 between rays of equal angular spacing .psi. increases toward the outer ends of the detector 21. Thus, each bin 89 near the ends tends to detect less radiation than bins near the center ray r.sub.O,O of the fan-shaped beam. Therefore, it is desirable to provide an improved position sensitive detector which will eliminate or substantially reduce the positional uncertainty factor and unequal spacing between rays as intercepted by a rectilinear detecting array of equal spacing between detector bin 89.

Referring now to FIGS. 20 and 21, there is shown an improved position sensitive detector 91 to replace the position sensitive detector 21 in the embodiment of FIGS. 1 and 6. In detector 91, the detector includes a gas-tight housing 92 formed by an arcuate channel structure including a pair of parallel arcuate sidewalls 93 and 94, as of stainless steel, closed on the bottom by a relatively narrow arcuate end wall 95. The open end of the channel structure is closed by means of a high strength thin metallic foil 96, as of nickel or stainless steel, which is brazed at 97 along one marginal side edge to an inside shoulder of sidewall 93 and to an arcuate rib portion 98 of side wall 94.

Opposite ends of the channel structure 92 are closed via end walls 103 and 104. A removable side wall portion 90 is secured via cap screws 102 to the bottom and end closing walls 95, 103 and 194 and to the arcuate rib 98 which bridges across between the end walls 103 and 104. An indium wire seal 101 extends around the periphery of the removable cover plate portion 90 for sealing same in a gas tight manner.

The conductive housing 92 forms the cathode electrode of the detector 91 and the anode electrode comprises an array of radially directed anode wires 53 centrally disposed of the chamber 92. Each anode wire 53 is supported between a pair of glass insulating terminals 105 and 106. Insulators 105 are supported from the removable cover plate 90 and the insulators 106 are feedthrough insulators for feeding the anode potential to the individual anode posts 107 through the envelope for connection to the respective amplifiers 65.

The chamber 92 is filled with an ionizable gaseous medium such an xenon to a pressure above atmospheric pressure, such as 5 atmospheres. The individual anode wires 53, as of stainless steel, have a diameter of 12.5 microns and a length of, for example, 10 centimeters. The anode wires 53 are spaced apart at 1/2.degree. intervals of .psi. with a total of 151 wires 53. The projected center of the radial array of wires 53 is the source 13 so that the individual anode wires are arrayed parallel to the rays of penetrating radiation to be detected. This substantially reduces the uncertainty and unequal spacing problems as previously alluded to with regard to rectilinear detector arrays.

For a position sensitive detector 91 based upon the concept of capturing between 50 and 100% of the penetrating radiation of up to 100keV incident thereon, the product of the gas fill pressure, in atmospheres, times the length of the individual anode wires 53 should equal 50 atmosphere-centimeters. This means that the detector wires may be 1 centimeter long if the pressure fill is at 50 atmospheres. Alternatively, the gas fill may be 5 atmospheres if the length of the individual anode wires is 10 centimeters. Thus, the detector 91, as contrasted with the linear detector 21, provides increased spatial resolution and improved high efficiency operation for X-ray or .gamma.-ray energies of 100keV and higher. Improving the spatial resolution, simplified the reconstruction of the X-ray shadowgraphic data into a 3-D tomograph.

The flow diagram for the computer program for carrying out the 3-D reconstruction according to the process described above with regard to FIGS. 10-18 is shown in FIG. 22 and actual computer reconstruction programs in Fortran language, are as follows: ##SPC1##

The advantage of the fan beam penetrating ray 3-D tomograph apparatus of the present invention, as contrasted with prior systems utilizing both angular rotation and transverse rectilinear translation, as exemplified by the aforecited U.S. Pat. No. 3,778,614, is that the lateral translation is eliminted and the resultant apparatus is substantially less complex. As a result, the time required to obtain the amount of shadowgraphic data required for high resolution, i.e., 1% accuracy 3-D reconstruction, is reduced to times less than a breath-holding period so that portions of the body subject to movement with breathing and the like can be obtained without blurring due to body movement. For example, the present invention permits 3-D tomographs to be obtained of the lungs without blurring due to movement.