CA1139458A - Tomographic scanner - Google Patents

Abstract

ABSTRACT OF THE DISCLOSURE
A method and apparatus for computerized axial tomography (CAT) image reconstruction applicable to X-ray scanning of the human body. Successive series of calculations determine the values of a characteristic in defined segmented areas of an examination plane. A reduction in the time and equipment require-ments for reconstruction calculations may thereby be effected. Images representa-tive of the difference between the value of the characteristic at a reconstruction point and the average value of the characteristic in adjacent regions (.DELTA.µ) may be calculated and displayed from measurements taken within a localized contiguous region of the examination plane. Radiation dose to patients and computation time in X-ray computerized axial tomography scanning systems is thus reduced.
Differential displays of the type described may be adjusted to image boundaries, in which case they do not suffer from gray scale resolution problems which are typical of prior art displays.

Description

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This application is a division o-f our Canadian patent application Serial No. 316.117, filed November 10, 1978.
The general field of this invention is tomography, that field that relates to obtaining an image of internal body parts in a plane through the body.
Specifically, the field of this invention, called transverse axial tomography, relates to a method and apparatus for calculating the value of physical character-istic at a point within a body from the values of a plurality of line integrals derived from an incoherent propagation process through the body. The invention is particularly suited for applying a plurality of X-ray or gamma ray beams through a plane of a body, measuring the attenuation of each beam as it passes through the body, and using the measurement information obtained to construct individual attenuation coefficients for each element of a defined element matrix in the body plane.
A prior art method and apparatus for transverse axial tomography is described in United States Patent 3,778,614 of Hounsfield issued December 1, 1973.
That patent describes a technique to reconstruct a cross-sectional view of a body from a series of transmission measurements obtained by translating a radiation source and detector across the body section and repeating this translation motion at a nwnber of angular orientation ;n the plane of the section.
The obj ective of these measure1nents is to ob-tain, after computer analysis of thousands of pieces of raw information about beam attenuation through the body plane, the attenuation coefficient associated with each element of a matrix defined in the body plane. The method is useful for internal description of any body, but is primarily useful for identification of internal human body abnormalities.
The attemlation coefficients are different for normal body tissue, tumors, fat, etc. and consequently provide identifying information about soft tissues in a human body. ~specially useful for identification of brain disease and abnormal-ities, tomography by computer reconstruction eliminates obvious disadvantages of patient discomfort and morbidity normally associated with brain investigations using pneumography, angiography and radioactive isotope scanning.
In one prior art method, the scan signals are processed to yield visual information and local values of the beam attenuation coefficients over the body section. Detector scan signals are applied to an analog/digital converter to convert the analog scan signals which are proportional to each beam attenuation to digital form and subsequently are recorded in a storage unit. Computer analysis of the entire matrix of scan signals, typically about 28,000 points, yields attenuation coefficients associated with a element matrix defined for the body.
These attenuation coefficients are related to the local physical properties in the body plane. After they are computed, the attenuation coefficients are recorded in a storage unit, and subsequently converted to analog signals by means of a digital/analog converter. These signals drive a viewing unit, typically a CRT, with the information content to pictorially display the attenuation coefficient for each matrix element. A permanent record of the display is achieved by means of a camera.
A scanning motion consisting of translation followed by separate rota-tion is usually clumsy and subject to mechanical vibration and wear. Because of the mechanical problems involved, it is often difficult to speed the sequence of translation and rotation movement to reduce scanning time. Purther problems are related to the complexity of prior art computer programs necessary for reconstruc-tion and the sophistication of the programs that are required.
A thin cross-section or plane through a body is examined by passing X-or gamma ray beams through a body plane. The body plane is depicted for examina-tion purposes as a two-dimentional matrix of elements defined by a plurality of concentric circles which create concentric rings. The outermost ring is denoted as the R ring, the next inner ring to the outermost ring described as the R-l :, :
- ~ ' ring~ and so on. Elements in the rings are created by di-viding each of the rings into N elements. In this manner, the notation NR represents any number equal to two or greater equally angularly spaced elements of the outermost R ring, the R-l ring being divided into NR 1 elements~ and so on.
The method of determining individual attenuation or coefficients for each element in the defined e]ement matrix begins by rotating X- or gamma ray beams around the outside of the body, where a bearn is provided for each concentric ring and is so directed in the plane under investigation as to be continuously tangent to its associated ring.
From each beam emerging from the body, at Nr d;screte angular intervals during the beams' rotation, a d:iscrete output signal is recorded representing the sum of the attenuation of the eleme7lts in each respective concentric ring inter-sected by the respective beam.
For the outmost R ring, the NR discrete output signals from the beam tangent to the R ring are used in deriving signals proportional to the individual absorption or transmission elements associated with each of the NR elements in the R ring.
In response to the NR ldiscrete output signals from the beam tangent to the R-l ring and the signa]s proportional to the individllaL atte7lucltion co-

2~ efficients from the beam tangent to the R-l ring and the signals proportional to the individual attenuation coefficients associated with the elements in the R
ring through which the beam tangent to the R-l r;ng passes at each of the NR 1 discrete angular intervals, signals are derived proportional to the individual attemlatio7l coefficients associated with each o-F the NR 1 elements in the R-l ring.
rhis method is repeated for each succeeding ring in turn for ring R-2 toward the center of the concentric circles. For each concentric ring, signals 5~

proportional to the individual attenuation coefficients associated with each of the Nr elements in the ring are derived in response to the Nr discrete output signals from the beam tangent to that ring and the previously derived signals proportional to the individual absorption or transmission coefficients associated with the elements in all other rings through which the beam passes at each of the Nr discrete angular intervals.
According to the invention the value of the attenuation coefficient at the center of rotation may thus be derived from equations which, in form, resemble the know general convolution-backprojection algorithm, but which include filter (weighting) functions which are derived from the concentric ring model. The computed radial origin of the ring model may~ effectively, be shifted to permit calculation of the attenuation coefficient at any point in the plane.
~ ilter (weighting) functions utilized in the above-described computa-tional method have positive values for measurements passing through the origin of the computational system and negative values for measurements in the adjoining ring. A numerical reconstruction of an image necessarily involves interpolation between these positive and negative values, which interpolation may introduce significant image artifacts. The introduction of an additional weighting func-tion, preferably a Gaussian function, can substantially reduce interpolation errors and errors associated with the propagation o-f statistical noise through the image without materially affecting image resolution.
'I'he reconstructed value of the attenuation coefficient calculated at any image point is substantially influenced by the actual value of the attenua-tion coefficient at all other points in the plane. The effect of the attenuation coefficient at a remote point upon the calculated value of an attenuation co-efficient at a reconstruction point is substantially a function of the reciprocal of the distance between the points. Generally, clinical requirements dictate ~ ~ ~3~ 5~

that all points in a body plane be utilized for the reconstruction of attenuation coefficients, at any other points in the plane.
The methods of computed tomography generally yield images having a far wider range of values than it is possible to display on a conventional image out-put device. A radiologist examining computed tomography images must, therefore, often shift the range and threshold of a gray-scale display to allow imaging of significant clinical features. A detailed discussion of the nature of the gray-scale resolution proble~ and a prior art device for overcoming that problem is described in United States Patent 4,030,119 Or Ellis~ issued June 4, 1977.
A novel apparatus is disclosed for performing the method. A rotating frame is provided supportecl w-ith respect to a fixed frame by means of a bearing and is rotated therein. A source of X- or gamma rays is mounted on a first arm attached to the rotating frame. The source generates one or more beams in a plane perpendicular to the axis of rotation. The beams are intercepted by a system of detectors mounted on a second arm attached to the rotary frame. The beams are defined by collimators associated with both the ray sources and the detection system and directed so as to be effectively tangent to concentric rings defined about the axis o-f rotation of the rotating ~Erame in a plane of a body placed in or neclr the .IX;S C)l` rotation between tile sourcc and cletector system.
In an embodiment of the detector system a reference crystal detector and a plurality of measurement crystal detectors are provided in groups, which may hc moved in position on a track so as to intercept different beams passing through the body on different rotations of the rotating frame. Photomultiplier tubes are provided, one for each measurement crystal detector, to generate electrical signals proportional to the corresponsding beam intensity. Means are provided to magnetically store the heam attenuation signals in digital form. A

stored program digital computer is provided for deriving signa]s proportional to attenuation or transmission coefficients for the defined element matrix in the body plane. These signals are stored, and are then useful to provide a representation of the absorption characteristics of the body plane.
Thus, in accordance with one broad aspect of the invention, there is provided a method of examination, by penetrating radiation of a planar body section, comprising the steps of rotating beams of penetrating radiation through a series of concentric rings defining a planar body section, generat-ing attenuation values for each of said rings over a plurality of tangential positions for each successive concentric ring until the entire body plane is scanned, assigning weighted values to each of said attenuation values so as to compensate each beam attenuation value for attenuations at other than one of said tangential positions, reconstructing all of said weighted values over each concentric ring so as to reconstruct said body plane in terms of the individual attenuation values at each of said tangential positions, and displaying said reconstructed values for imaging said planar body section.
In accordance with another broad aspect of the invention there is provided an apparatus for examination, by penetrating radiation of a planar body section, comprising means for rotating beams oF penetrnting racliation through a series of concentrk: r:ings defining a planar body section, means for generating attenuation values for each of said rings over a plurality of tangential positions for each successive concentric ring until the entire body plane is scanned, means for assigning weighted values to each of said attenuation values so as to compensate each beam attenuation value for attenuations at other than one of said tangential positions, means for recon-structing all of said weighted values over each concentric ring so as to re-construct said body plane in terms of the individual attenuation values at each of said tangential positions, and means for displaying said reconstructed a~

values :Eor imaging said planar body section.
This invention, as well as its objects and features, wi:Ll be better understood by references to the following detailed descriptions of the pre-ferred embodiments of this invention taken in conjunction wi.th the accompany-ing drawings in which:
Figure 1 is an X- or gamma ray source/detector orientation, con-structed in accordance with this invention, for rotation of a beam pattern about a body in which an element matrix is defined by concentric circles and equally spaced radii;
Figure 2 shows in more detail the defined element matrix, construct-ed in accordance with this invention, for measurement of absorption coeffici-ents in a body plane;
Figure 3 is a perspective of physical apparatus, constructed in accordance with thi.s invention, for rotating a beam pattern through a plane of a body and the measurement of beam attenuations after the beams pass through it;
Figure 4 is an X-ray tube beam spread, constructed in accordance with this invention, as it rotates about the body ~mder invest:Lgation;
E'igure 5 is a schematic cl:Lagraln o.E beam generat:~ou and detection in accordance with this invelltion;
Figure 6 is a schematic diagram of an alternative embodiment of beam generation and detection in accordance with this invention;
Figure 7 is a schematic diagram of measurement data collection, recording and processing in accordance with this invention;

-6a-, ~ ~
r~

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Figure 8 schematically describes a general scanning procedure;
Figure 9 shows a change o-E coordinates between scanning and reconstruc-tion frames of references;
Figure 10 is a plotting of the values of Fj;
Figure 11 is a plotting of the values of Gj;
Figure 12 defines a scan transition region;
Figure 13 is a plotting of the value of ~~<~> across a plane interface between uniform media;
Figure 14 is a partial scanning procedure;
Figure 15 illustrates the geometry of artifacts generated in a partial scanning approach;
Figure 16 is a plotting of values of u and ~-<~> within a partial scan ; region;
Figure 17 illustrates the generation of interpolation artifacts duri.ng backprojection;
Figure 18 shows the effect of a Gaussian weighting function on the spatial resolution of a reconstruction of a uniform cylinder;
Figure 19 is a reconst-ruction of the vallles of ll :in a reg:ion oE interest which was calculated ~from data obt.l:i.ned :Erom a fn:ll scan o:f a body plane;
Figures 20-23 are reconstructions of ~values i.n a region of interest which were calculated from data scanned in a limited part of the body plane.
I.IS'I` OF PRINCIPAL SYM~OLS
In the followi.ng discussion the principal symbols are summarized and defined as follows:
f weighted projection value for ~ reconstruction defined by Equation (2.22).

Fj weighting coefficients to construct f; defined by Equation (2.23).
Fj weighting coefficients to construct f; defined by Equation (3.2~.
g weighted projection value for (~-<~>) reconstruction ; defined by Equa-tion (2.28~.
Gj weighting coefficients to reconstruct g; defined by Equation (2.29~.
Gj weighting coefficients to reconstruct g; defined in Equation (3.6).
Hj weighting coefficients for non-uniform radial sampling intervals; de-fined by Equation (6.21).
I X-ray beam intensity Q index of radius of averaging circle for <~>.
N number of angular sampling increments r radial coordinate of polar (r, ~ system in the image plane.
rl radial increrrlent x coordinate of Cartesian (x,y) system.
y coordinate of Cartesian (x~y) system.
1 geometrical factor relating interception of beam in rings.
X-ray beam attenuation.
~> intensity-averaged measured value of rj weighting coefficients; defined by Equation (6.11).
, ~ incremental difference operators.
angular coordinate of polar (r, ~ system in the image plane.
~1 angular increment.
~i j parametric coefficients; defined by Equation (2.6~.

5~

generalized averaging dimension; defined by Equation ~6.13).
local value of linear attenuation coeffici.ents.
<~> average value of ~ over a circle of radius ' about each reconstruction point.
distance along a ray path from source to detector p radial coordinate of polar (p,~) system in the scanning plane.
1~ a,ngular coordinate of polar (p,~) system in the scanni.ng plane.

~1 angular increment.
angular sector scanned in an incomplete scanning.
weighting function; defined by F.quation (6.1).
Concentric Ring Scanning Figure 1 shows a slcetch of a body plane 111 to be exami.ned by trans-verse axial tomography according to this invention~ The body 111 is assumed to be placed between a source 300 of X- or gamma rays arl(l a cletoctor 301, which may be a sci.ntillator and a photomulti.l)lier an-l whicll llroferab:ly also includes a collimator. For illustrative purposes, detector 30:1 is asswned to be movable on a track 302 such that beams may be detected which pass at various angles from the source through body 111. Multiple detectors, each with an associated collimator can oE course be provided as detectors 301, 301', 301'', etc., or multiple de-tectors may be movable on track 302. The X-ray source 300 and detectors 301, are attached to a rotating ring 303 which is rotatable about an axis 0 perpendi-cular to the body plane 111. Body 111 is shown in Figure 1 coexistent with axis lt~

0, but it may be placed anywhere within the beam range of source 300 and detec-tor 301.
As shown in Figure 1 a series of concentric circles is defined about axis of rotation 0. As ring 303 rotates about the axis of rotation 0, the X-ray beam or beams is continuously directed (as shown at one orientation angle of rotation) perpendicular to subsequent radii from axis 0 at point P at all times as ring 303 rota-tes about axis 0. As a result, a beam such as 310 is at all -times tangent to the outer ring about center 0 as the source-detector system rotates.
Figure 2 shows in more detail the concentric system defined about axis of rotation 0. Beam 310 is shown at a particular orientation during its rotation about body 111 and is perpendicular to a particular radius vector r at point P.
By appropriate collimation, the beam width w can be made to approximate the concentric ring width rl. The example depicted in Figure 2 shows beam 310 pass-ing tl~rough the outermost concentric ring (i). Perpendicular to radius vector r, beam 310 is depictcd as passing through elements labelled t = ni 1~ ni~ 1, 2 and 3. These elements are among those elements in the i ring, totalling ni elements.
Ln order to descril)c tllc intcric):r c)f llody LIl according to the matrix Or elements throllghou1: the concent:r-ic ring-radius vector system shown in Figure 2, each smal] element is assigned an unknown value of attenuation coefficient.
For example, thc attenuation coefficient for element t=l in the i th ring is designated ~i l; -for elemen-t t=2, lli 2; for the t th element, ~i t. The measur-ed beam attenuation for beam 310 shown will be given by the sum of the average value of the linear attenuation constants ~ for each element through which the beam passes.

During rotation about axis 0, the beam attenuation between source 300 and detector 301 is obtained at ni different positions, only one of which is shown in Figure 2. Beam attenuation for each measurement, designated ~i z is simply the sum of the linear attenuation constants for each element through which the beam passes, multiplied by an individual geometrical factor determined by the interception of the beams with each cell. The rotation-measurements steps of the beam 310~ as source 300 and detector 301 rotate about 0, are identified by an index z. This index z runs from 1 in steps of l until z = ni, equal to the number of elements in ring i. Thus, the measurement of the beam attenuation at each position of the first intercepting ring leads to the equations, hi (~i, t-z)(~ t) ~i,z (1.1) t=l where z= 1, 2, -- ni Ihe term ~i t z represents the geometrical factor determined by the interception of the beam 310 with each element t as it rotates in z steps about ring i.
S:ince t is taken eqllal to z, tllat is, tlle nulllber of elcmerlts in ring i is t, and the numher of mcasurements around ring i is equal to z, equation (1.1) represents a system of equations z=ni in number, having t=ni unknown parameters ~i t. I`he solutioll of the system of equations (1.1) yields the values of ~ associated with each element on the i = 1 ring.
In the next scanning ring the i-l ring, the measurement of the beam attenuation leads to the new system of equations, p~
hi-1 hi ~ l, t-z)(~i-l,t) ~i-l,z ~ ~l,t-z)(~i,t) (1.2) -t=l for, z = 1, 2, . . ni_l wherein ~'1 t is the geometrical factor determined by the interception of the beam (e.g. beam 311, Figure 1) in the new ring, i-l, with the elements of the outer ring i.
The values ~i t have been determined by the solution of Equations (1.1);
the solution of the system of Equations (1.2) provides the -values of ~i 1 t in the ring i-l. The measurement in each scanni:ng ring with decreasing radii provides a system of equations similar to (1.2) with terms on the right hand side contain-ing known values of ~ in the elements pertaining to the outer rings. It is apparent that the number of elements of each outer ring which contributes to the attenuation along an inner ring decreases rapidly as the scanning radius approaches zero i.e. as the scann:ing beam approaches the center of rotati.on.
Thus~ the local properties are fully determined upon completion of each scanning ring without havi.ng to wa-it -for thc total scann:in~ ol. thc body section.
The number of equations in each set, similar to equation (1.2), is relatively small and can be arranged to decrease as the interior rings with small-er radii are measured. Assuming for example a scanning radius of the outer ring of the order of 150 mm and an element width on the order of 3 mm, each independent equation set for the outer rings consists of only several hundred equations. The solution for the unknown ~'s for each ring sequenti.ally from the outside ring toward the inside rings, requires fa:r less computational time than prior art X-ray tomographic systems. As the inner rings are measured, it is possible to 9~58 decrease the number of measurements taken around the ring ~i.e. define ni to be less for the inner rings than for the outer rings, thereby keeping the ; element size approximately constant) with the result that the equation set size is reduced. Computational time is correspondingly reduced for solution of inner ring ~'s.
Concentric Ring Scanner Illustrated in Figure 3 is a perspective drawing of a concentric ring scanning apparatus. A fixed frame 600 supports a rotating frame 601 which is free to revolve about an axis of rotation 602. A motor drive 614 is provided in fixed frame 600 to propel rotating frame 601. Attached to rotating frame 601 are two arms 603, 604 spaced approximately 180 degrees rom one another. Arm 603 supports an X-ray tube 605 and an associated X-ray tube collimator control 606. Arm 604 carries a detector assembly 607 and associated detector collimators.
A couch 608 is provided to allow a human body 111 to be positioned between X-ray tube 605/-X-ray tube collimator control 606 and detector assembly 607. Couch 608 is supported by couch support 609. A couch control system 610 is provided which translates the couch 608 parallel to the axis of rotation 602, thereby positioning body 111 to a point where beams from X-ray tube 605 may intersect a desired plane through the body 111. In addition, the couch control system 610 translates the couch 608 in any direction in a plane perpendicular to the axis of rotation, thereby positioning the axis of rotation close to the desired area of the body 111.
Since the X-ray tube 606 is rotatable about center line 602 means are provided to cool it and provide it with high voltage electrical power while it is rotating. These means, shown in modular form, are a cooling water ?~

rotating assembly 61] and a high voltage slip ring assembly 612. Means must also be provided to send command and control signals to X-ray tube 605 and its associated collimator assembly and collimators associated with detectors 607 while they are rotating. Command and control slip ring assembly 614 is provided for that purpose. Likewise data transmission slip ring assembly 613 is provided to provide a means for transmission of data signals from detectors 607 which they are rotating.
Figure 4 shows a preferred orientation of X-ray tube 656 and its associated collimator control 606 with respect to detector and detector collimator apparatus 604.
As indicated in Figure 3, X-ray tube 656 and detector assembly 604 are rigidly connected to each other by arms 603, and 604 on rotating frame 601. Rotation of the frame 601 about center line 602 ~point 0 to Figure 4) causes the X-ray beam 700 to sweep out a fan-shaped pattern, which substantially covers any body placed within an aperture 701. In a preferred embodiment, the fan shaped beam subtends an approximately 30 degree arc as the X-ray tube-detector assemblies are rotated at speeds of up to one complete rotation per second for approximately 10 revolutions. The aperture opening 701 is approximately 26 inches in diameter. 'l'he arms 603, 604 attaching the X-ray housing 605 and detector system 607 are approximately thirty inches long.
The rotating frame 601 is supported with respect to fixed frame 600 by a single, thirty-:Eive inch diameter, precision ball bearing.
Figure 5 illustrates the multiple beam scanning aspects of this invention. The X-ray tube 605 emits a continuous fan-shaped array of X-rays, but this continuous array must be divided into beams in order for the methods described previously in this specification to be applicable. Collimators 606 and 800 are provided to create a plurality of beams passing through a cross section of a body 111 placed within aperture 701. For illustrative purposes three detector system pairs consisting of crystal scintillators and photo-multipliers (811, 820; 812J 821; 813, 822) are shown in position 1. A refer-ence scintillator 810 and its associated photomultiplier 823 are stationary.
The detector pairs remain in position 1 for the first rotation of rotating frame 601 (Figure 3). At the start of the second rotation, the detector system pairs are shifted along track 302 to position 2 for detection of beams at that position. The detectors are shifted to position 3 at the start of the third revolution, and so on. This shifting of detectors at the end of one rotation and the beginning of another rotation assures that the entire body placed within aperture 701 may be scanned.
In practice the translatory mo-tion of the X-ray source and/or detectors need not be accomplished in stepwise fashion after each rotation.
~atherl the translatory and rotational motions may be accomplished simultaneous-ly so that the point of tangency of each X-ray beam moves in a smooth spiral.
The data thus obtained may then be interpolated, for example by a linear interpolation in a digital computcr, to calc~ -tc ccluivalcnt datcl at points in the concentric ring coordin~tc system.
A preferred embodiment of the scanning system of Figure 5 consists of an arrangement capable of scanning a test object contained within a 20 inch diameter circle about axis of rotation 0. T}lirteen detector units are provided one of wh;ch is the reference pair 810, 823, the other twelve of which are movable to ten positions along detector track 302. Each detector system is used -to scan a 2. 5 degree sec-tor of the total scanning area, ten revolutions of the X-ray tube/detector system 60~ being used to scan the entire body.

Detector 810 and photomultiplier 823 are used to generate a reference beam attenuation signal for all the other detectors to account for any variations with time in beam strength eminating from X-ray tube 605. As shown in Figure 5 a particular beam 855 is collimated by tube collimator 606 and passes through an attenuator 850 located outside the location of the body being examined. The absorption characteristics of attenuator 850 are preferably selectecl to be similar to that of the body being examined. Tissue equivalent plastic is an example of an attenuator material suitable for this purpose.
Detector pair 810, 823 generates a signal, the intensity of which is proportion-al to the strength of the X-ray beam after absorption by attenuator 850.
Each detector pai:r for the beams passing through the body under investigation generates a signal proportional to a particular beam's intensity after it passes through the body. The crystal scintillators produce a high-frequency signal (visible light spectrum) proportional to the number of photos in the X- or gamma ray beams impinging on them. The photomultiplier tubes associated with each crystal scintillator, react to the light energy from their respective scintillators to generate an electrical signal proportion-al to beam strength impinging on the scint;llators. For example> an electrical signal proportional to the beam strength of bealll 856 is generatecl at the output of photomultiplier tube 820. Similarly, crystal scintillator/photo-multiplier pairs generate output signals proportional to the stength of other 'beams at position 1, position 2, etc. for the entire beam pattern after successive rotations of system 604.
In a preferred embodiment of this invention, the X-rays generated by X-ray tube 605 are collimated by means of a 15 cm long collimator 606 at the X-ray tube source, and a 20 cm long collimator 800 at the detector system 604~

This collimation at the X-ray source and detector defines radiation beams having a rectangular profile of l mm by 5 mm width as measured by scanning a lead edge at the mid-point of the beam path.
The range of values for which the photomultiplier must respond can be reduced by covering the body being examined with a material, the absorption of which is known, so that beam intensities received by the detectors are kept as constant as possible as they pass through the body.
Figure 6 shows an alternate embodiment of detector orientation.
Detectors 910 and 911 are located on track 9013 and detectors 920 and 921 are ].ocated. on track 902. As shown, detectors 910 and 911 measure beam attenuation through circ-llar r:ings, defined about rotation axi.s 0, different from those measured by detectors 920 and 921. Multiple positions on each track can be established and the detectors shifted in position with each rotation until a defined ring matrix i.s entirely scanned and detected. Collimators 606 are provided at the X-ray source and collimators 930 at the detectors a.re also provided.
An X-ray tube appropriate for the particular embodiment discussed above is a modi.fied vers;.on of a Philips 160 kV l3eryl.1:ium W:indow Tube, Model MCN 160.
Appropriate detectors include scintillation detectors such as NaI, CaF2, BG0 and proportional counters such as high pressure xenon detectors and solid statc detectors.
Figure 7 indicates how the beam attenuation data measured by the detector systems, including the photomultipliers lO00l 10002, ... 10003, are processed during the rotational scanning of a body. An information signal is generated in each photomultiplier at each defined increment for each rotation of the X-ray source/de-tector system. These signals are individually amplified by amplifiers 101~1~ 10102~ ... 10103, are each taken up in turn by serializer 1020, converted to digital form by analog to digital converter 1030, and stored in a data storage medium 1040 such as magnetic tape, disk, or drum or solid state memory. lhis data collection process continues for each detector position~ for each defined increment step~ Eor the complete rotation. During or after the data collection process, a computer 1050 under direction of a stored reconstruction code program, processes the collected data according to the methods discussed elsewhere in this specification. The output of the computer 1050 is a sequence of digital signals proportional to the attenuation coefficients of each element in the defined circular ring matrix. These signals are stored in a data storage unit 106 which may be identical to unit 1040 or similar to it. The output digital signals can then be printed and/or converted to analog form and used to drive a display on a cathode-ray tube, thereby pictorially indicating the attenuation coefficients -Eor the defined matrix in the cross section of the body being investigated.
GENERAL RECONSTRUCTION METHOD
The concentric ring reconstruction may be general-i~ed for image reconstruction at any point. In the plialle x, y oF Iigure 8, the cross-section of a body is confined within the boundary S. Line ~ -represents the axis of an X-ray beam, which ideally is assumed to be of negligible cross section.
The total attenuation of the beam passing through the body is given by I

= ln -I (2.1) ~ ~ ~3C~

where Io and I are the beam intensity at the entrance and exi.t of the body section respectively. The -total attenuation ~ can be written in terms of the local value of the attenuation coefficient ~ as the line integral P = J ~Id~ (2.2) Assume now that at each point P of the body section the values of P are available for any ~ passing through P in any angular direction. From these values of ~3 it is possible to compute the value of ~ at each point of the body section. 0 is an arbitrary point where ~ is to be computed. Define a family of circles with center 0 and radii rj = jrl ~j = 0, l, 2, ~) (2.3) where rl is an arbitrary dimension which is small compared to the body section dimensions. Thus, the body section is divi.ded into a large number of circular sectors.
At a point Pj over the circle of radius rj let Pj be the measured attenuation along a line tangent to the circle as indicated in Figure 8.
Define ~h as the average value of 1l in the region between the circle of radius hrl and the circle of radius (h+l)rl. Thus, ~0 is the average value of ~ within the first ci,rcle of radius rl. 13y avcrag,.inK ovcr 2Ir the values of p,0 measured wi.th the ~-ray beams passing through r one has 27r 0 130d~ = 4~rl 0 ~h (2.4) Similarly the average over 2Ir of the values of ~j is related to the average values of ~h (for h ~ j by the equation O o ~ 4~r1 h j t3j~h~ h (2.5 w}lere t3j k = ~ [J(j+k)~ (j+k~1)~ (2.6) From Equation (2.4), with the aid of Equation (2.5), one obtains ~o 4 ~ ~ (~O ~ ~ J~ ~j) d~ (2.7) where K = 1 -- (1 - t)1 j Kl - t)2 j-l K2 ~ -- tj-1,2 j-l with 1 t~l 1 J~ (2.9 The asymptotic value of coefficients Kj is Kj = - l. (j > > 1) (2.lO) The numerical values of Kj up to j=100 are presented in 'I'able :[. The imnledi.l.te question which arises in cxamin:;ng l:quation (2.7) :i.s how m.Llly terms oE the sum have to be included in the comput,Ltion of the value of ~O in orde~ to perform the reconstruction within a gi-ven error. In other words how far from r has the body secti.on to be scanned over the sequence of circular orbits to re-construct the distribution of ~ in a limited region around r ? The answer to this question largely depends upon the dynamic range of values of ~j. Assume, for example, that the di.stribut:ion of values of ~j is not Ear from uniform (asit would be approximately in the case of a water filled compensation bag en-closing the body under scrut:iny). Tlle contribution of the terms outside of the ~ ~.Q $~

range j = j would be Kj ~ 2 j J j ~r ~j . (2.1l) As a consequence of this slow rate of decrease wi~h J it is apparent that, with the values of rl in the millimeter range, both scanning and computation must include almost the entire body section if the error in the reconstruction has to be maintained within a small l;mit, for example 1%.
PARTIAL SCANNING
The result expressed by Equation (2.11) is equivalent to saying that the contribution of the scanning of an area of the body section located at a distance r from 0 affects the computation of ~ as r 1. Thus, if one computes the differences of the values of ~ at two points close to each other, the scan-ning of a surrounding area affects the difference of ~ as _1 1 (2.12) where rl, r2 are the distance of the area from the two points. Thus, for large values of rl, r2 the scanning of an area of the body section affects the dif-ferences of values of ~ essentially as rl 2 . By using a differential-like method in the image reconstruction, ;t is poss;ble to conf;ne t)oth scamling <and computat;on to a limited area of thc body sectiorl. As one examl)le of a mathe-mat;cal analysis of this method, consider first the average value of ~ within a circle of radius Qr. The average < ~ > is given by Q-~
Q2 h 0 [(h+l) - h21 ~h (2.13) By virtue of Equation (2.5), Equation (2.13) transforms to r [ ~ ~ ] d~ (2.1~) 5~

where -K = 1 [ 2~ Kl ~ ~)29j-l 2 j-1,2 j 1 ]

K _ 2 _ 2 ~ 1 1 ~ ~ (2 . 15) ,j ~3j 1 j Q,j-9~+1 O,Q+l,j-QKO9Q-l-----(~j_l 2Ko j 1 ~

+~1 j.KI + ~)2 j-1-K2 ~ -- + (~Q-l,j-Q+2 KQ-l ~2.15) The coefficients Kj in Equation (2 . 7) and Ko j in Equation ~2 .14) satisfy ~heasymptotic condition ~loo(jKo j~ = Q2 l;lm (j Kj) (2.16) Thus, from Equations (2 . 7) and (2 .14) one obtains 21~ 1 Q-l ~
~ ~o ~ 4~rl 1 [( 1-92)~o -~ e j~j ] d~ (2.17) where ' j ] j Q2 Kj ] ~e j = [ Kj - 1 K ] (2.18) Asymptotically the coefficicnt ~l~ j decreascs as j ~ allcl this rap;d rate of decay is the basis for a localizcd scLmning and imagc reconstruction.
An important property of both equations (2 . 7) and (2 .17) is the uniform averaging process of the attenuation measurements over each circle of the image reconstruction sequence, as a result of the integration over 2~.
Thus, the effect of the statistical fluctuations of the individual measurements of ~ is minimized lmiformly over the entire reconstruction area.
Both Equatioils (2 . 7) and (2 .17) providc the solution of the recon-struction problem. The reconstruction point r is an arbitrary point in the x,y plane and Equations (2.7) and (2.17) assume that the values of the atte-nua-tion data ~ have becn measured over the family of circles concentric with r.
Assume that the attenuakion measllrements have been conducted ;n the polar systenm of coordinates p,~ of Figure 9 in such a way that the values of ~ are kno~n over the fami.ly of circles concentric with the origin O. From Equation (2.7), the value of 7,1 at a point O of polar coordinates r,~ is given by 2~ ~ K.
~r,0) = 4 - ~ {~[rcos~7~ ] - ~ [rcos(7~-~)+jrl,7~]}dl~ (2.19) Figure 9 shows the locus of the points of tangence of the circles of center 0 with the X-ray beams which are located at a distance rj = jrl from the recon-structi.on point r~ From Equation (2,17) the value of 7~ - < 7~ > at 0 is 1 271 1 a7 7,1-~> = 4~rrr r ~ 2)~[rCos~ 7~] ~ ~ [rC0s~7~ jrl~l~]} d ~ 2.20) where ~i,j, j < Q - 1 ~2,21) ~j ~e,j' j ~

Both equations ~2.19) and (2.20) can be written in the samc form o:E the re-construction solution obtai.ncd w-ith a convoluti Oll apl)r().lCII. (ons:i.cler the :I'llllC-tion ~ 0 Fj {~[(h+j) rl7~] -~ ~[(h-j) r~ ]} (2.22) where the coef:Eicients 7.7j are 1 j = 0 K. (2,23) ~ O

Coefficients Fj are the wei.ghting f7m cti.ons and they satisfy the condition ?'~

oo ~ Fj = O (2.2 In Equation (2.22) one has ~[(h-j) rl,~]=~[~ -j) r~ ] ~2.25) when h - j < O (2.26) Equation (2.19) transforms to ~(r~3 = 4~r r f[¦rcos(~ ] d~ (2.27) The coefficients Fj are plotted in Figure 10 and the numerical values of F
are given in Table II for j ~ 100.
In a similar manner Equation (2.20) can be written again with the definition of the function OD
g(hrl~O Gj{~[(~l~j) rl,~] + ~[(h-j) rl,l~]} (2.28) where ~ 1 ~ Q12 j = O
Gj = ~ (2.29) ~ O
and the following condition is satisfied ~ ~ = o ~2.30) j=O
A plotting of Gj is shown in Figure 11 and the values of Gj are given in Table III for j < 100 in the particular case of Q = 11. One observes the inversion of sign of the terms Gj for i ? Q.
In a way similar to Equation (2.11) one can compute in Equation (2.28) the order of magnitude of the contribution of the terms of the sum outside of j = j with the assumption of a quasi-uniform distribution of values of ~j.
One has ~ Gj{~h-j)ri,~] + ~[(h-j)ri,~]}~ ~> 3~ .3 (2.31) Equation (2.17) can be wr tten again in the form ~ )r ~ ~ 4~r ~ g~¦rcos( ~ ] d~ (2.32) By virtue of Equations (2.1~), Equation (2~32) reduces to Equation (2.27) in the limit Q -~ ~ . Thus Equation (2.32) may be considered a more general solution of the reconstruction problem.
Equation (2.32) defines the approach of the localized scanning. Due to the rapidly diminishing value of the contribution of the terms of the second sum in Equation (2.17), a progressively larger error in the measurement of ~
can be tolerated for increasing values of j (i.e. for increasing distance from the region of reconstruction of the ~ image). Beyond a given distance the measurement of ~ becomes unnecessary and assumed ~ values can be substituted for the actual measured data without introducing a signiEicant error ;n the calcula-tion of a~.
In the limit of radius of Q approaching unity~ the recor.structed value ~ acquires the property of the local average of the second derivative of ~l.
This is illustrated in Figure 3 which shows the value of ~ as a function of the distance from a plane interface M between two uniform media. The value of A~
is zero at the interface and at a large distance from the interface. Finite values of ~ are confined to the region ~ or-Qrl from the interface.
Thus the relationship between a ~ image and a ~ image is controlled by the relation of the parameter Q.

5~3 If one has to extract the local value of ~ from the ~ image an in-dependent knowledge or measurement of the value of ~ within each radius Qrl is required. This average may be known beforehand or it may be obtained from a total scanning of the body section. However, the latter need only have a low spatial resolution if Q is large compared to unity and, as a consequence, the total scanning has less stringent requirements on the stability of the attenua-tion measurements as compared to a high spatial resolution scanning.
Two categories of clinica] intentions can be identified where, in principal, the ~l~ image is of diagnostic value per se.
If the images are used to diagnose localized density perturbations in essentially uniform areas (the liver is an example) and the value of Q is such that the pertubations fit well within the averaging circle, the information con-tained in the ~ image provides the full diagnosis of the anomaly or departure from usual tissue properties. A scanning localized to the area of interest provides the sct of ~ measurements required for the a~ image. The precision of the reconstruction is placed in the difference between local and normal values rather than the absolute values of ~.
The second category corresponds to images of the interface of body organs and boundaries o tissue cmomo1ies. 'I'hesc imagcs corrcs[~ond to small values of Q, for example Q = 2, where the boundaries are identified by a dis-tribution of positive and negative values of ~ as shown in Figure 13 (as long as the radius of curvature of the boundary is larger than several pixal sizes).
EFFECTS TilE' SCANNING P ~
To compute either function g(hrl,~) or f~hrl,~, it is necessary to extract the values of ~ from the measurements of the X-ray beam intensity ac-cording to Equation (2.1). In an actual scanning and reconstruction procedure the values f To and Ie in Equation (2.1) are measured at the entrance and exit of the body section respectively. ~t a first glance it would then appear that the reconstruction calculation is dependent upon the values of intensity Io out-side of the body. However, it is easy to write solutions of ~ and ~-<~> in a form which shows that a knowledge of lo is not required. To do so, write f(hrl,~) in the form + ~
f(hrl,~ F ~[(h+j)r1,~] (3.1) j=_~ ]
where ~,~ ~
F j = F+j = F~ O) ~ (3.2) F = 2F
_ o o By virtue of 2.24, Fj satisfies the condition + co ~
~ Fj = 0 (3.3) Consequently in Equation (3.1), the value of f does not change if the values of are changed into a new distribution ~' such that ~' [(h+j)rl,~ ] = ~ [(h+j)rl,l~] + ~O(~) (3-4) where ~ is an arbi-trary constant independent o~ j. In a Si.lll~il a-r way g(hr~ ) can be writte]l as -1' co ~
g(hr~ Gj ~[(h+j)rl,~] ~3~5) J=_~
whe-re f ~
G j = Glj = Glj~ ) (3.6) Go = 2(1- Q-2~ ) ~

Gj also satisfies the condition + co ~
~ G = 0 (3.7) 5,~

Thus g~hrl, ~) is also indepenclellt of an additive arbitrary constant ~ in the values of the attenuation measurements.
ARTIFACTS GENERATED BY PARTIAL. SCANNING
Tn the case of a partial scanning, the attenuation data are collected only within a circle of radius r , resulting in an error in the reconstruction of either ~ or ~ >. This error is essentially due to the superposition of artifacts generated by the incomplete scanning of points located outside of the circle of radius rS.
Assume a ~-like object located at a point P at a clistance r > r as shown in Figure 14 and reconstruct the value of either ~ or ~-<1.l> within the circle of radius rS with no other object in the scanning plane.
If thc distance of each scanning ray from the center O never exceed rs, the attenuation data are collected between the angles ~ O formed by the two lines which are perpendicular to the tangents al, a2 Erom P to the circle of radius rS as shown in Figure 14. Thus point P is scanned only within the fraction ~O/rr of the total scanning cycle, regardless of the scanning pro-cedure. Accordingly the presence of the ~-object outside of the scanning circle generates an artifact distribution which is essentia.Lly oriented along the two lines al and a2. In the l:i.m;.t o-f rp -~ rs, ;.e., -For ~ ap[~rolcll:ing thc point Ps on the scanning ci.rcle, one has ~ r and the two lines al, a2 coverage in a single line at Ps tangent to the circl.e.
As a consequence the partial scanning i.ntroduces an error in the image reconstruction within the circle of radius rS, which depends primarily upon the distance of the reconstruction po:int from the lines al, a2 rather than the dis-tance from the location P of the object left outside of the scanning circle.
llence the maximum error is found close to the points o-f tangence of al, a2 with the circle.

~ ~ r,~r~

The reconstructed value of 1I within a region of the scanning circle close to either al, or a2, is clescribed by an approximate solution written in the form:

~ (r ~ l Fj (4.1) 47rr -i2 ~l-j2 ( 1 )2 where r is the distance of the reconstruction point from P and il, i2 are given by il ~ r- ¦ sin ~ ¦ ; i2~ r ¦ sin ( n - ~0 ) I (4.2) being the angle between al and thc line POr as shown in Figure 15. The value of ~-<~> at the same point is obtained simply by substituting ~-<~> to ~ and Gj to Fj in E~uation (4~1), i.e.

_ <~>~ ~_ ~ Gj (4-3) 47rr -i2 ~_j2 ( ~ 2--The maximum values of both ~ and ~-<~> are found within a distance frcm al, a2 of the order of rl, and the order of magnitude of the maxima are ~max - ; ~ ) Ill'lX _ ~ I Q2) (4. ) 27rr 2rrr Thus the two maxima are of the same order of magnitude regardless of the value of the parameter Q, and decrease slowly with the distance of P
from the points of tangence. As indicated by the shaded regions of Figure 15, only a small frac~ion of the circle area, close to the periphery of the scanning circle is affected by the maximum amplitude of the artifacts given by Equations (4-4) As the distance from either al or a2 increases, -the reconstructed value of 1I decreases rather rapidly.
For values of n close to O and ~O and in the intervals rl rl n'' ~ 0 - n>> -- ~4 . 5) the value of ~ is ~- _ (4.6) 2~2 r2~l~2 where = r~ ; ~2 = I~O - n (4- 7) ~- <1l> decreases also and attains a negative value at values of n whose order of magnitude is given by Qrl n ~ ~0 - n~ - (4.8) and for values of n n ~ (4.9) r r The magnitude of 1l- <~> decreascs very rapidly with either ~1 or ~2 according to the equation ~rl3 (Q _ l)2 _ <~>~ _ ~ (4.10)

3~r2 r4~1 2 i.e., the magnitude of ~- <~> decreases inversely to the third power of the distance from either al a2. As a consequence, within a distance Qrl from the periphery of the circle the ~ ~ image reconstruction is virtually un-affected by the presence of the ~-object outside of the scanning circle. A

plotting of both ~ within the angular interval 1~ is shown in Figure 160 RECONSTRUCTION COMPUTATION
___ . _ The image reconstruction will, in general, be implemented numerically in a general purpose digital computer which may, additionally, include dedicated array processing hardware. The specific solutions are, of course, highly de-pendent on such factors as the required computational speed, accuracy, and capital investment. The following discussions are,therefore, intended to en-able those skilled in the computer programming art to effectively implement numeri.cal solutions wi.thout undue experimentation.
The general solution to the reconstruction problem is stated by Equation (2.32), since :in the :Limit Q -~ > -~ O and Gj -~ Fj which are the weighting functions for reconstruction of ~. Numerical implementation of the reconstruction algorithm can be discussed in terms of Equations (2.28), (2.29) and (2.32), recognizing that Equations (2.22), (2.23) and (2.27) are recovered as a special case. Therefore, the differences between a partial scanning and reconstruction and a total scanning and reconstruction reside in the magnitude of the requirements for information storage, reconstruction speed, and inter-pretation of the solui:ion, hut not ;.n tlle form of thc equcl.t:i.olls or thc :logic of the instructions to :implelllent their solut:;.on. Accordi.ngly, a single re-construction code may be deve:lo~ped in which specifica.tion of the value ~ is the only parameter which clistinguishes a "partial reconstruction", producing the solution for ~ >, from a "total reconstruction", producing the solution for .
The reconstruction code may be structured in four basic modules. The first module calculates and stores the weighting functions (Equations 2.23 and 2.29).

Fj if Q > j max = ~ (j O 1 ~ j ) (5.1) ~Gj it Q < ~ max and an auxiliary function;

~ k (j = 1, 2~ 3~ jmax) (5.2) Selection of the value of j must be made on the basis of accuracy and com-putational speed; these considerations will be elaborated in the following discussion.
l'he second module of the reconstruction code calculates the function f or g (depending of the value of Q) in accord with Equations (2.22) or (2.28):

( f(hrl'~) j h jS+h [(h+j)rl,~l] + ~ [lh jlrl,~ ]
~g(hrl ,~) o,l ( ) ~(js h 1) + ~o,2 (~ ~(js+h+l) (h j l-j 2-j , ...-1, O, 1, ...jo~2, jO ~jo where the scanning is carried ou~ over a maximum radius rs = jSrl and the reconstruction is to be carried out within a eircle of radius rO = jOr. The values of ~(r,l~) may be obtained from direct measurement as provided in prior art scanning instrumentatioll. The functions ~o 1 (~) and ~o 1(~) represent either (a) the background attenuation measured outside the body in the case of a total scanning i.e., ~o 1 = ~0 2 = ~o (~ or (b) suitable approximations to the at-tenuation in the compensation region (see Figurc 12) in the case of a partial scanning, e.g., ~o 1 = ~(rS,Ij) and ~0 ~ = ~(rS,~ ). The transformation stated by Equations (2.25) and (2.26) should be recalled in connect~on with Equation (5.3).

113~4S8 ;

If the values of ~ are reasonably uniform, the value of jmax re-quired to achieve a specified degree of accuracy can be estimated from Equation (2.30) or ~2.24). The error incurred by truncating the summation after imaX

terms is simply ~max j. Thus, ~ > ~ however, as will be demonstrated below, ~ =0 an "exact" solution, in the sense that ~ [(h+j) rl,~] = O, can be achieved with a finite value of j ax~ dictated only by j and Q.
It can be seen from Equation (5.3) that jmax = js + jO is required to complete the indicated summation in this equation. Consider first the case of a total scanning of the body section; jO = js is usual in this case.
As pointed out above, an arbitrary constant ~ (~), may be added to or substract-ed from the measured attenuation values without affecting the solution for f(r,~) (assuming jmax > ~) Therefore, the values of ~lr,~) in Equation (5.3) may be normali7ed to ~(r,~ O(~), yielding;

s js+h ~=O j [(( j)r~ o(~)] + ~ j3[(¦h-j¦r + ~ O(~) (5-4) ~ in which case the auxiliary function rj is not needed, and jmax = 2js is ; required to complete an "exact" solution. However, for typical values of js neededto carry out a total scan of a body section with a high degree of precision, the computational requirement to include 2js terms in the calculation of f(frl,~) at each of 2js ~ 1 radial positions, for each angle ~, may still pre-sent an unacceptable limitation on reconstruction speed. On the other hand, use ; f jmax ~ 2js introduces an error of order 2js j into the solution, which must be assessed in terms of the required ~max+l precision of the reconstruction.

.

In the case of a partial scanning, the reconstruction region should be confined to a circle ot radius rO = rr ~ Qr1 = ~j5 ~ Q~rl. Thus jmax = 2js -Q is required to carry out the indicated sl~mations in Equation (5.3). If the values of p(r,l~) are normalized with respect to a linear function of r in this case, ~n(r'~) ~0,1 + (~0,1 ~ ~0,2) (r-rS)/2I-S (-r2 < r ~ rS) (5.5) Equation (6.5) becomes:

jS-h j5+h g(hr ,~ [(~~~n)(h j) 1'~ ] j_0 (lh-jlrl~ll)+~) (5.6) which only requires jmax = 2js-Q terms for an "exact"solution and, again, deletes the auxili.ary function yj.
~ ince the maximum scanning radius rS = j r for a partial scanning is presumably much smaller than that required for a total scanning, the required value of jmax for an "exact" solution is correspondingly reduced. Therefore, calculation time for the function g, which is roughly proportional to j , should not be a limiting factor in achievement of acceptable reconstruction speed in the case of a partial reconstruct;.on.
It should be pointed out that si.llcc the auxi.l:iary :fullction yj may be computed, and storcd, for arbi.trar;ly large values of j x in the first module of the reconstruction code, at very modest computational expense, Equation (5.3) may be preferred over (5.4) or (5.6), since normalization of the data i.s not required. Equations (5.4) and (5.6) demonstrate that the minimum number of weighting terms required for exact total and partial reconstruction are 2jS
and 2j - Q respec-tively, if the attenuation values are normalized to ~0,1 ~0,2 The third module of the reconstruction code is the "backprojection"
or reconstruction step, per se~ as given by Equation (2.32) or ~2.27):

~(r, I = 4~r r ~ rcos(~ )d~ (5-~ >) (r~ ~) ) 1 ~g~

wherein the transformation given by Equations (2.25) and (2.26) has been utilized. The integration may be carried out by the si.mple trapezoidal rule procedure using equally spaced angular intervals, typically the same as the angular increment in the scanning data. The requi.red values of the integrand are linearly interpolated from ~he calculated values.
The fourth, and fi.nal module of the reconstruction code is the image display. The reconstructed distribution of ~ or ~ ~ over the scanning plane may be displayed by either (a) assigning a grey scale to the range of values of ; ~ or (~ - <~>) and a pixel size to each coordinate point (x,y) = (rcos~, rsin~) to produce a photographic type image, or (b) searching the distribution for the contour lines ~ = constant or (~ - <~>) = constant which can be plotted as continuous functions of (x,y) or (r,~). Details of both techniques are well known in the art; however, i.t should be pointed out that gencrat;.oll of a greyscale i.mage is relatively f.a.st and quali.tatively informative, whcreas the con-tour :line technique provides more quantitative detail, both in terms of spatialand density resolution, but at a much greater computational expense. Obviously, plots of ~ or ~ ~ as a f-mcti.on of position along selected lines within the scanning plane may also be obtained in place of, or in addition to, the image display.
_ORRECTION OF ARTIFACTS DUE TO FINITE _AMPLING
The reconstruction algorithm is based on the assumption that ~ is a continuous function of position which changes slowly over the e].emental step 3 ~

rl. Because of the interpolation procedures, artifacts are generated in the presence of discontinuities, such as interfaces between regions with largely different values of ~. Errors and artifacts are also generated by the finite radial and angular sampling intervals of any scan procedure.
In Figure 17 r is a reconstruction point. During the summation or backprojection of data that numerically replaces the integration in Equation 2.27 some interplation must be made between proximal values of f, such as f[jrl,~] and f[~j*l)rl,l~]. Tnterpolation artifacts will be generated if f suffers a large change in this interval. Specifically the sign inversion of Fj and Gj from a positive value at j=O to negative values at j=_l is responsible for the large interpolation errors in the presence of discontinuities of the distribution of ~. The reconstruction algorithm may be optimized by modifying the weighting functions in such a way as to minimize the interpolation error and the resulting reconstruction noise. A modification of the algorithm leading to a reduction of the computational reconstruction noise will also have a bene-ficial effect on image artifacts generated by statistical noise in the data acquisition system.
Obviously, any change ;n the shape of the weight;ng funct;ons ;ntended to minimize the reconstruction noise must be allalyzed in terms of its effect on the reconstructcd values of both ll and ~ > and ;n particular on the spatial resolution of the reconstructed images. To discuss this problem it ;s conveni-ent to modify the reconstruction approach discussed above by substituting the calculation of ~ at each point Or~r,~) with the calculation of a weighted average ~ of the attenuation coefficient as defined by the equation 2~ ~
~(r,~ d~ s,~) ~(r) r dr (6.1) wllere r, ~ are the polar coordinates relative to -the reconst~lction point r s = Ir2 + r2 + 2r rcoS(~ ~)] 1/2 (6.2) s;n~ - ) sin(~

and ~(r) is a continuous function of the distance r from r' which satisfies thecondition 2~ r ~ ~r) r dr = 1 (6.3) Assume a family of circles concentric with r and radii rh = hr ~6.4) Equation ~8.1) can be written in the form ~ h ~h ~6.5) where ~h is the average value of ~ between the circles of center r and radii hrl and (h+l)rl, and (h-~l)r Mh = 2~ r ~ (r~ r dr (6.6) In the manner describcd above, onc obtains tl~c avcrago V.llllCS ~h 1 2~ ~ K
h 4~rl 0 l~,h~h[rCoS(l~-o)~ [rcos(~ }d~ (6.7) where the coefficients Kj 1 a,re related to the parameters ~j k defined in E',quation (2.6) by Kh h =

,l h,j-h-~l h,h ' ~~j 1 2Kj 1 h] (6.8) d ~

Thus the value of ~ gi.ven by Equati.on (6.5) can be wr;.tten as ~ ~ ~7Trl r f[rlCS~ ]d~ (6-9) where +co f(hr~ rj ~[(h-~j)rl,~] ~6.10) : and r = 2M

,l (6.11) 2,1 1,2 1 rj = r_j j~ [Mj - Mo - (j-l) ~j_l,2rj_l-----~j,jrl The relationship between the reconstructed value of ~ and the actual value of the attenuation coefficient ~ depends upon the selection of functi.on ~(r) which determines the parameters Mj in the coefficients Ij. Assume for instance that it is a Gaussian function - 1 (r )2 ~(r) = - e o (6.12) whe:re the dimension rO is related to rl by rO = ~rl (6.13) ~ being an arbitrary positive number. With the particular function ~ gi.ven by Equation (6.12), in a first approximation, the value of ~ maintains the signifi-cance of an average value of the attenuation coefficient within a circle of radius ~rl and the coefficients M. become - 3~ _ +l~2 ~1j = e ~2 -e ~2 (6.1~

The resulting values of rj are shown in Table IV for several values of ~.
In the limit of ~ small compared to uni.ty, lim M = 1 ; lirn Mj~o= O
~ ~' (6.15) and rj reduces to the value of Fj, as is apparent from Table IV for ~ = .25.
Conversely in the limit of ~ large compared to unity, rj is positive for j<~, consistent with the behaviour of the coefficients Kj, Ko j w}l;ch determine the average value of ~ within the circle of radius ~rl, in Equation (2.14). Of particular :interest is the smooth transition from positive -to negative values of rj for ~>>l, with a mi.nimum value of rj found at j larger than ~ as shown in Table IV. Thus a value of ~ of the order, or smaller than unity, leads to a value of ~ close to the local value of ~ as given by Equation (2.27), without altering the spatial resolution of the reconstructed image in any significant manner. An example of the effect of changîng the parameter ~ Equation (6.14) is provided by the reconstruction of the ~ image of a uniform cylinder as shown in Figure 18, (based on a computational simulation of attenuation data in a cylinder coaxial w:ith the axis o:F scanning). Tllc cylincle:r racl-i.us :is equal to 10rl; F;.gure 18 shows the va]ues oF ll versus the rad-ial d:istance f:rom the axis for values of ~ equal to .25, 1, 1.5, 2, 4. The reconstruction of ~ across the boundary of the cylinder, at r = 10rl, shows the increasing loss of spatial resolution above ~ = 1. Outside of the cylinder the reconstruction error fluctuates about zero and Table V shows the effect of ~ on the values of ~
within the radial interval 70<r/rl<99. The effect of ~ is particu]arly pro-nounced on the large error of ll of the order of 10 2 which is found in the proximity of r/rl = 96 for ~ = .25. Table V shows that the error at r/rl - 96 decreases rapidly with increasing values of A, and in particular the error is approximately halved for A = 1. I-le~ce a substantial improvement of the re-construction interpolation error is achieved without a significant loss of spatial resolution. Equation (6.12) is only an example of a continuous function ~, which yields an optimum form of the reconstruction algorithm as a trade-off between a value of ~ sufficiently close to ~l and a minimum amplitude of the computational noise.
In this formulation of the reconstruction algorithm, the d;fference between local value ~ and average value <~>, can be readily computed from Equation (6.9) as the diEference between two values of ~ for ~~1 and A-Q>l, i.e.

~ > ~ 4~r- r If~-l A-QI (6.16) Asymptotically for j>>Q, one has rj(A~l)~rj(~~Q)~ 2 _l_ (6.17) and the difference between the two values of rj for A~l, A-Q decreases as j Thus solution (6.16) maintains the same propert;es of solut;on 2.23 of thc problem of loca]ized scannillg. In thc numorical applic.ltions o-f E(luation (6.1$), the reconstruction interpolation errors are generated primarily by the first term of the integrand (A ~ 1). Consequently, the a-rtifacts generated in the numerical reconstruction procedure of the ~l - <~> image have essentially the same amplitude of the artifacts generated in the reconstruction of the local value of the attenuation coeEficient.
Equation (6.14) represents the solution o-E the direct problem of computing the values rj of the weighting function from a specified function of

- 4~ -to be reconstructed in the image plane. The i.nverse problem can be stated:
i.f a particular shape of the we;ghting function is specified, one may compute the :Eunction o in Equation (6.11) and determine the relationshi.p between the reconstructed value ~ and the actual attemlation coefficient.
From Equation (6.11) one obtains:

M = z r ... .... (6.18) Mj = 2 rO + ~1 j rl + 2~)2,j lr2 + - + ~ lrj Thus from Equation (6.6) the average value ~h of ~ between the circles of radi.i hrl and (h+l.)rl is - h _ (6.19) h (2h~ r 2 10and finally~ by means of Equation (6.19), Equation (6.6) provides the value of at each reconstruction point O .
EXAMPLES OF A a ~ IMAGE RECONSTRUCTION
Figures 19 - 23 illustrate the transiti.on from a ~ to a a ~ image obtained with different values of the paramcter Q. F;gllre 19 :is .a reconstrllc-tion of ~ val~les from dilt.a obta:i.ncd -in a l'l~ s trallslilti.onill scanncr (Tomoscan R 200 manufactured by Phi.lips Medical Systems, :rnc. Shelton, Connecticut). Figure 19 is a partial reconstruction withi.n a c:ircular region which includes the li.ver, obtained with a full set of scanning data both inside and outside of the circ:le. The radius o:E the circle, normali~ed to rl , is 20equal to l27.
Figures 20 - 23 are a set of a ~ images for values Q = 60, 40, 20 and 5 obtained by ignoring scann;.ng data outside of the circle of Figure 19.

- 41 _ The gray scale of each image consists of sixteen equally spaced gray levels;
the middle level corresponds to ~ ~ = 0. Thus the images of Figures 20 - 23 present the full range of values of ~ ~ with t,he negative values corresponding to the darker half of the gray scale and the positive values corresponding to the lighter half. The images of Figures 14 - 23 are obtained by assuming that outside of the circle the value of ~ and each radial line is constant and equal to the value measured on the circl.e for that particular line. The difference between the Q = 60 (Figure 20) and Q = 40 ~Figure 2].) images is minor and both are close to the conventional image of values shown in Figure 17. The overall range of ~ ~ values of the reconstructed image decreases with decreasing value.s of Q and the decrease in Q results i.n a sharper transi.tion across the interfaces as shown in the Q = 20 (Figure 22) image. This becomes even more apparent in the Q = 5 (Figure 23) image which reduces to the outline of the body organs with values of ~ - <~> small everywhere else, (almost within the noise level);
within the bone and in the soft tissue area as well.
It is worthwhile pointi,ng out that the lack of actual scanning data outsi.de of the reconstruction circle has a negligible effect on the reconstruc-tion of the i.mages of Figures 20 and 21 and the :i.mage di,stort:ion is confined to a very sma]l annular close to thc boundclry of thc ci.rc:lc cven for -the klrger values of Q. This is t~le main reason why the ~ ~ algorithnl allows a partial scanning of the area of interest.
The value of Q represents an additional parameter in the displ.ay of a reconstructed image wh:ich can be used, ~for example~ to enhance the geometry of interfaces in the area under scrutiny. ~in th;.s connection it is worthwhile pointing out that for small values of Q Equation (2.32) acquires the essential property of a local average of the second derivative of ~. The value of - <~ is zero at the interface between two uniform media. Thus in a general <~x~

situation of image reconstruction across sharp boundaries between media of dif-ferent physical properties ~like soft tissue-air interface of soft tissue-bone interface) the boundary would be described by one of the family of equations.

11 - <11> --provided that the radius of the averaging circle is smaller than the local radius of curvature of the interface in the scanning plane. Hence ~ ~1 images provide a very convenient tool to outline either bone or soft tissue interfaces without the need of computing the local values of the attenuation coeffic;ent.

Claims (6)

THE EMBODIMENTS OF THE INVENTION IN WHICH AN EXCLUSIVE
PROPERTY OR PRIVILEGE IS CLAIMED ARE DEFINED AS FOLLOWS:

1. A method of examination, by penetrating radiation of a planar body section, comprising the steps of rotating beams of penetrating radiation through a series of concentric rings defining a planar body section, generat-ing attenuation values for each of said rings over a plurality of tangential positions for each successive concentric ring until the entire body plane is scanned, assigning weighted values to each of said attenuation values so as to compensate each beam attenuation value for attenuations at other than one of said tangential positions, reconstructing all of said weighted values over each concentric ring so as to reconstruct said body plane in terms of the individual attenuation values at each of said tangential positions, and displaying said reconstructed values for imaging said planer body section.

2. An apparatus for examination, by penetrating radiation of a planar body section, comprising means for rotating beams of penetrating radia-tion through a series of concentric rings defining a planar body section, means for generating attenuation values for each of said rings over a plurality of tangential positions for each successive concentric ring until the entire body plane is scanned, means for assigning weighted values to each of said attenuation values so as to compensate each beam attenuation value for attenuations at other than one of said tangential positions, means for recon-structing all of said weighted values over each concentric ring so as to re-construct said body plane in terms of the individual attenuation values at each of said tangential positions, and means for displaying said reconstructed values for imaging said planar body section.

3. The apparatus of claim 2, wherein said means for assigning weighted values and for reconstructing function to calculate the attenuation values at said tangential points in accordance with the formula .

4. The apparatus of claim 3, wherein said assigning and reconstructing means further function to apply a weighting function in said calculations for reducing interpolation errors in said calculation.

5. The apparatus of claim 4, wherein said weighting function is a Gaussian function.

6. The apparatus of claim 1, wherein said assigning and reconstructing means include a general purpose digital computer including a stored program for affecting said calculation.