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Field of th Q _Ivention
This invention is concerned with gamma camera imaging and, more
particularly, with methods and systems for obtaining images having
reduced artifacls du~ -to multiple photopeak and unwanted events.
An event is herein defined as a pho-ton strikins. the gamma camera
detector and causing a scintillation that is ac~luired as data for
use in constructing an image. This is an improvement to the
invention entitled "Compton-Free Gamma Camera lmages" filed in
Israel on June 11, ~990, and which received ~erial No. 094691.
Back~round of_tle Invention
In passin~ thro-lgh the human body, gamma pholons have a certain
probability o~ scattering due to the Compl-~n effect. Such
scattering changes the direc-tion and energy o~ e photons. When a
photon that has been scattered is detect:ed by Ihe gamma camera,
false position informa-tion is derived from the ~.ca-ttered photons.
Thus, the scattered pho-tons cause events tha-t a-e unwarl-ted for use
in cons-tructing the ima~e. ~lther unwantecl ~vents exist. For
example, the radiation emitted from -the patient -ften excites lead
(K~ X-rays from the collimator and other lead narts. These X-rays
also impinge on the detector and may be regi!-lered as events.
These X-ray photons constitute an additionill source of image
blurrin~.
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The problem Or X-I ay induce(i ~v~nts arises esr~ ially for radio
isotopes emitting photons in the energy rang~ of ~8-1~0 KEV. In
-this range, tlle lead X-ray e~cita-t;orl probabilil-~ is high and the
spectrum of these photons coincides with a relevant part of -the
isotopes spec-trum, by partially o~erlapping the pho-topeak. Thus,
the unwanted part of the spectruln in each pixel has two terms: one
made up of -the Conlp-ton scatterecl phol:ons, and t~ other made up of
the lead X-ray photons.
In principle the events caused by unwan-ted photons should be
discarded. Howeve.r, it is not easy to arrive at ,~.riteria that are
e.fficient ancl efEective for discarding such events. For example,
an energy level criterion i5 not efEective because although the
photon loses par-t of its ellergy in the scatlering process, the
energy resolulion of the -typical .gamllla camera jr such that there
is a large amount o~ overlap betueen the energ~ oE unscatterecl and
scattered photons.
The invention of the previously mentioned ~-atent application
provided methods and means ~o~ qualitativel~ and quantitatively
improving the recorded images by signiEica~ ly reducing the
contribution of Compton scattered photons to the final image to
thereby provicling a practically Comp-ton-free .image within seconds
after acquisiticn. The invention accomplishes ~he task oE reducing
the number of` events caused by Compton scaltered photons by
locally determining the energy spectrum and fi-tling -the determined
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energy spectrum wi-th a "trial" function comr~3sed of a photopeak
component of known energy shape but unknown magnitude and a
Compton scatter component having a theore~ically derived energy
shape and an unknown magnitude for each pixel o[ -the image.
The true physical charac-teris-tics of the Comp-ton process are used
in the previously mentioned Patent Applicatiol)-to derive Compton
multi-sca-tter functions which are subsequently llsed -to cons-truct
the Compton sca-tter component energy spectra. Ihus, the previous
Patent Application uses the following inputs to determine the
unknowns; (i.e., the magnjtude of -the photopeak component and the
magni-tude of -the Gompton multi-scat-ter componenl:s):
1. -the measured energy spectrum E)er piY~I. This includes
counts due -to sca-tterecl and unscattere-:l pho-tons, and
the measured system energy spread function for -the
isoto~3e centerline which provide the photopeak energy
shape.
The shape of -the ~ompton Gomponent of tl~ trial function is
analytically derived in the prior application hy conver-ting the
Nishina-Klein Eguation that describes -the physical relativistic
scattering of photons wi-th electrons int-o a probability
distribution for a photon to scat-ter from a given energy -to a
lower energy in a single in-teraction with an electron. Repeated
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convolutions arc llsed to obtai~l Ihe probabil.it:y distribution for
the higher order scatter terms.
By locally fit-ting -the -trial function to t.he measured energy
spectrum of acquired data the values of the multi-scattered
Compton co-ef~i.cients and the ~hotopeak magnilllde were obtalned.
This enables the removal of Compton contalllination from the
acquired data.
The prior invention however assumed a single photopeak. In
certain isotopes -there is more than one photopeak. If a single
peak is assumed when more -than one pealc a~ ually exists the
removal of scat-tered events ftom -the image will t!e incomplete.
Accordingly the invention of this Applica-tion is an improvement
over the invention of -the pri.or mentioned ~pl~lications in tha-t
among other -thin~s it takes in-to account radi(~ isotopes having
more than one peak and also takes in-to account all unwanted events
due to Compton sca~tered photons and photons de~rived from such
phenomena as X-rays caused t~y gamma radiatioll interacting with
lead components.
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Brief De~ t.ion_of the Inverltioll
The present invention represents an improvement over -the inventi.on
oE -the Israel Paten-t Application, Serial No. 09~1691. The present
invention reduces events caused by unwanted pho-t:ons including, but
not limited to, Compton scat-ter photons and also -takes in-to
account multiple photopeaks, such as are )I>tained when using
certain radio isotopes. Thus, -the image provided by utilization of
the present invention improves over -the image o~ the invention of
the prior mentioned Patent Application.
In accordance with the present invention, thele is provided a
method of reducing the contribution of unwanted photons to an
image produced by a gamma ray imaging system, said method
including the s-teps of:
detecting photons impinging on a ga.mma ray detector as event
counts,
measuring the energy of said ;mpingin~ phot-~rls and an X, Y
location for each photon according to the location of the
impingement of the photons on the detector,
grouping each detected photon according to the measured energy and
the X, Y location,
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accumulating counts of said ~hotons at ea~ X, Y location
according to the determined energy level of the photons,
constructing a rneasured energy spectrum a-t each X, Y location
using the accumulated counts oE -the determined erlergy levels, said
measured energy spectrum including counts oE wanted and unwanted
photons,
calculating the energy distributions of unwanted photons,
determining the energy spread ~unction of the f~amma ray imaging
system bein~ used,
.
o~taining a sys-tem dependent energy distribution of the unwanted
photons per location by using the energy distribution of the
unwanted photons and the energy spread function of the system,
constructin~ a trial function compri~ing -the system dependent
energy spread function mult;p];e~l by an unkno~1n coefficient of
wanted photons plLIs unknown ,-oeflicients ol: unwanted photons
convolved with the system's energy spread function.
solving Eor the unknown coeff.icient of the wanted photons by
locally fitting the measured energy distrihlltion to the trial
energy distribution of photons, and
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using the cOIJnt of -the wanted pi~otons -to produce an image
prac-tically free of unwanted photons.
According to a feature of the inverltion, the unwanted photons
include Gompton scattered photons origina-ting from single or
mul-tiple radio iso-tope photopeaks.
According -to another feature of the invention, the unwanted
photons further include photons such as those due to lead X-rays.
Brief ~escript on of the ~ a in~s
The a~ove men-tioned obiec-ts and features of the present invention
along with addi-tional obiec-ts and ~eatu.res will be best understood
when considered in -the light of the following description made in
con~unction wi~.h t}-~e accompanying clrawings; wherein:
Fig. 1 is a block diagram showing of a gamma radiation imaging
system for providing improved images by elimir)ating blurs caused
in the pas-t by multiple pho-topeak isotopes and Ihe inclusion of
unwanted even-ts genera-ted by Comp-ton scattered photons and other
unwanted photons, and
Fig. 2 represents details of the preparatiorls, compu-tations and
operations used in the system shown in Fig. 1.
.
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General Descri~tion
Fig. 1 at 11 generally shows in block diagram form the inventive
gamma camer~a system for producing improve-l images. Fig.
comprises a measured energy spectrum s-tage 12, a -trial function
preparation stage 14 and a curve fi-tting or compu-tation stage 15
which provides an unwanted pho-ton-Eree image (~ I) 16.
The measured energy spectrum stage 12 comprises a gamma radiation
detector 17. The gamma radiation de-tector 17 provides electrical
signals responsive to events; i.e., photons imp;nging on the Eac.e
thereof, such as indicated a-t 1~. When an event occurs, electrical
signals are provided on conductors 19, 21 and 22. These conductors
19, 21 and 22 are directed immediately -t:o a coordinate computer 23
which determines X and Y loca-tion of tlle impingelllent of the pho-ton
18 onto the detector 17.
Conductors 22 and ~4 carry an electrical rel!resentation of the
energy oE the photon. The electrlcal represerlt~tion of the energy
is provided to an energy (Z) correction Cil~CUit ~5. An energy
processing circuit 26 divides -the range of energy de-tected into a
number of energy windows prede-termined by the s~stem operator.
When the energy is within certain limits, the energy correction
circuit sends an enable signal over conductor ~ which enables the
coordinate computer to determine the X and Y coordinate location~
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of the event. This informa-tion is direc-ted -to an image corrector
and digitizer circuit 31 which correc-ts and digi-tizes the X Y
coordinates of the event. The information on tlle number o~ events
is placed into a plurality of matrices ~2 clependent on the
photon s energy. Each of the matrices is a memory that retains the
counts of even-ts per X Y location for a partic~llar energy window
such as for example a window tha-t extends frolll 22 KEV to 25 KEV
for window No. 1 and ~5 KEV to ~8 KEV for window No. 2 etc. The
windows are shown as W1 W.2 W3 e~tending -to Wn where n is the
predetermined number of energy windows.
The matrices are thus divided into X Y locations that correspond
to the co-ordinate loca-tion of the event on the cle-tector. The X Y
locations also correspond to pixels in the final image. An imaging
preprocessor 33 receives the clata pixel-by-pixe.l from each of -the
winclows and computes a measured or an acquired energy spectru~ N~
per pixel as shown in block 34. This acquiled energy spectrum
includes both the counts due to unwanted photons and wanted
pho-tons. The unwanted photons include Compton scat-ter photons and
other or additional unwanted photons. No-te -tha-t the energy
spectrum may include more than one energy peal~ as shown in block
34.
The trial function s-tage 14 of Fig. 1 prepares a -theoretical or a
trial energy distribu-tion n(X Y ~) including wanted and unwanted
events herein:
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~(XlY~)=Np(X~Y)~ m (x~y)Jd~F~ )~m(~ ~(X ~Y) R(~) (1)
m=1
here:
~ = E/m~C2 the photon ener~y in units of electron rest energy,
m~C~,
u)
P(~)=,EWlP(~ ); it is -the sys-tem energy spread Eunction at
~ 2...; (P(~) can also be measured in an envil~onment free of all
unwanted photons).
(k) is a superscript denoting the number of discrete energy lines
in the source,
m is a subscript indicating -the numbe:r of the C.ompton scat-ter
order, and
M is a script indicating the chQsen number of Compton scatter
orders included in the computa-tion.
~m(~ ) iS -the energy distri~ution of events caused by photons
scattered m times from original energies ~ ith known relative
intensities W~ to intermediate energy ~' (i.e., the shape of the
energy probability distribution of pho-tons scattered m times),
~m(~ Wl~ ) with ~[~, for m~ being calculated
recursively,
' :
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W1 are the known relative intensities of ~ Wi=1.
N~tX,Y) is the spa-tial distribu-tion (counts/pixel) of events
caused by unscattered photons.
Qn.rX,Y) is the spatial dis-tribution (counts/pixel) of events
caused by photons scattered m times,
is the original energies of the pho-tons emitted from a
radioactive source,
is -the measured energy of the photon,
~' is an intermediate energy of a photoll,
R(~) is the measured energy spectrum of aclditional unwanted
photons such as by way of example photons from lead X-rays. tNote
R(~) can also be calculated using published tables and convolvi.ng
with -the system spread function).
Ko(X~Y) is the spatial distribution tcounts/pixel) of the events
caused by the addi-tonal unwanted radiation.
~n important purpose of the invention is -to determine the spatial
distribution of the wanted events N~ (X, Y).
-- ~ ~ 7 ~ 3 1
Y~i95
To determi.ne -tlle coun-t of events per pixel, block 15 fits the
measured value6 that i8 the measurecl energy spectrum per pixel and
the system energy spread function with unknowns; i.e., the
magnitude of the photopeaks and the shape and magnitude of the
unwan-ted photon spectrum -to the values of the trial distribution
n(X,Y,~). 'rlle fit provides the wanl:ed spa-tial dis-tribution N~ (X,
Y). With the knowledgè of the spatial distribution of the wanted
photon, the scattered and other unwan-ted photon-free image is
pr~cluced as indicated at 1~.
Details of the computations that oscur a-t -the trial function
preparation sectiorl 14 of Fig. 1 are inclicated ln Fig. ~. More
particularly, as shown in l~ig. 2, values based on the system
energy spread Eunction shown in block ~,1 of Fig. 1 and Fig. 2, are
entered into bloclcs 36 Figs. 1 and 2. In addition, values based on
acldlti.onal unwanted (photons) racliation such as, for example, lead
X~rays are determined (either by measurement or by compu-tation) as
6hown in block 40 oE Figs. 1 and 2.
The energy spread function of the system i6 a.~.sumed to be known.
It is measured once and ls kep-t in -the memory of -the sy~tem. The
measurement is easily accomplished by providin~ sources of gamma
radia-tion oE known energy and detecting the racliation with -the
equipmen-t 11 oE Fig. 1, for example. The de-tecl:ion is made without
any Compton scatter media or X-ray providing lead between the
energy 50urce and the detector. This provides an energy spread
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function for a monoenergy source or a multi-energy force due to
the detector energy resolution without unwanted photons as shown
in block 41. The preparation block 36 comE~u-tes ~m i.e., the
energy distribution of the unwanted photons including Compton
photons, for example, and further including Compton photons for
eac.h scattering order. This is done by using the Nishina-Klein
equation to derive -the differen-t orders of scat-tered unpolarised
photons; i.e.:
-~-rl (c~ t ~
; elsewhere
~ ) is the weighted com~ination of the fil~st order Comp-ton
energy distribution for each oE tt-e k photopeaks, or
~t~ W~ a)
i=l
The higher orders of sca-tters are derived recursively by repeated
convolution usin~ the equa-tion:
,,~ 6~ (3 ~
~ ~ ; elsewhere
Where ~ is the maximum of all ~ ... k).
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Note that -the equations are solved recursivel~ in that each hi~her
order equation requires knowledge oE the lower prior orders.
The energy distribution of Compton scatter photons provide a curve
independent of the system for each order of the scatter. However,
-this system independent curve is acted upon by the system energy
spread function to provide -the system clependent Compton
multi-scattered energy distributions denoted by ~m(~). The
shapes of the C~ ) dis-tributions are obtained by convolving
~m with the system ~nergy spread function P(~ ); i.e.:
(k) ~ (k)
C~ (~)=) d~'~m (~')P(~ ) (4)
This set of equations provides the shape of the Compton energy
distributions for each order of scatter af-ter bein~ operated on by
the system energy spread function.
Fig. 2 indicates the computations resulting in the ~m values using
the Nishina-Klein equa-tion in blocks 42, 4~ and 44 for ~. and
~)
consequently ~2 . . . m.
The shapes of ~kl ~2 and ~ in blocks 42, 43 and 44 are shown as
being convolvecl with the sys-tem energy spread furlction of block 41
in blocks 46, 47 and 48 respectively, thereby providing the shapes
C1~ C2, etc . The computations to determine ~ 2, e-tc., are
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indicated as being recursive by the arrows going from ~ to ~2,
etc.
Hereafter the superscript (k) denoting the number of discrete
energy lines in the source is omit-ted from the Cs.
A method for drastically reducing the number of computations is
useful in this system. The reduction in the number of computations
is accomplished by orthonormalization of the se-t Cm(~). The
orthonormiaization is provided by constructing an orthonormal
function (vector) set ~ using the Graham-Schmidt procedure:
~, = Cl/ J~C12>
(~2 = ( C2 ~ C2 > ~ < C2~2 > - < ~ l ~ C2 > 2
~) M ¦ ~ M ~
= (C - E ~C >~D/I<c - ~<~- C ~2 (5)
M-~1 M+l ~=1 M+1 ~ Y M+1 ~ =1 M-~1
Where for convenlenc,e C~ ) is defined as being identical to
R(~).
Where sums (integrals) over energy are defined by:
E F(E) - ~F~
E
P~.95
Note that -the array set C~} obeys:
(~,i=j
< ~ = ~ ~ J
o,i=i .
The or-thonormalization is accomplished in computer 49 and the
results; i.e., ~ 2.. .~m-~l are shown in blocks 51, 52 53, for
example.
The Compton sum (EQ(4)) can be rewritten using the ~k'S:
QmCn~ = ~ am~ < Cm~31c ~.~)1': ( )
E t ~ < Gm ~ ~)k > Qm ) ~ ~)k ( 6 )
=~ qle ~ ~k
where:
.
q~ < Cm ~)k > am
and: m = 1,2...M+1
k = 1,2...M+1.
*[with an orthonormal base ~"~} any vector v can be represented
as a superposition of an array oE ~m'
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2~7~8~
v = ~<v~m>~m.] (7)
The trial distribution now reads:
n(X,Y;E) = Np(X,Y) P(Eo~ C(X,Y;Eo,E) ~8)
where:
C(X,Y;Eo,E) = Lq~(X,Y)~k(EO,E) (9)
Hereafter the known energy spread function, P is normalized such
that <P> = 1.
In a preferred implementation, a least squares fit is used. More
par-ticularly, with the trial function n(X,Y;~) of equation (1) and
the multi--window acquisition resul-ts N~(X,Y) from block 34, a
solution is sought for -the number of counts caused by unscattered
photons N~(X,Y) that will minimize the sum of the squares of
differences for each pixel ~(X,Y):
~ (X,Y) = ~n(X,Y;E) - N~(X,Y]2> (10)
More particularly, in the block 15 the followin~ "fit" operation
is performed, i.e.,
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- = o, and (11)
N~
~
= O, where k = 1,2,....................... (12)
It can be shown that the solution of these equations is:
N~(X,Y) = <N~(X,Y) J(E)> (13)
qk(X,Y~ = ~N~(X,Y) G~(E)> (14)
where:
P(Eo,E) - <P-Q.c>.Qk(E~,E)
J(E) - - (15)
~ p2> -- ~ p~ Q~e >2
k
Gk(E) = ~k(E~,E) - <P-Ok> ~J~f ~ (16)
Note that since J(E) and Gk(E) are data independent, they can be
apriori derived as indicated in Fig. 2 by block 54 which shows
computing J(E) using ~ ,02 ....Q~c and P. The "per-pixel"
operations entail only the evaluation of the scalar product
Np = <N~ ~J(E) ~ . (17)
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The fit, therefore, determines the per pixel Compton scatter free
count N~.
To compensate ior low statistics per pixel per energy window, the
relative constancy of scatter distributions over large spatial
domains is put to advanta~e by use of a "quasi-local" solution.
More particularly, an expanded or "large" pixel is preferably
used. Thus, if the top pixel is (XOIYO) the spatial window W is
defined as:
(XO-W)~X~(XO~W) ; (YO-W)~Y~(YO+W) (18)
of area s = (2Wtl)~
When the Compton component of the entire window s, C~ is computed,
the "per-pixel" Comp-ton ac-tivi-ty, C~ can be approximated by its
average:
C ~ - C~/s . ( 1 9 )
The measured activity in the spatial window 5 ( symmetric around
the coordinates (X,Y)) is deno-ted as N~(X,Y), and the (X,Y)-pixel
activity is denoted as N~(X,Y). A single parameter fit is done to
find the local pho-topeak count. It can be shown that this i5 Oiven
by:
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S s
Np>(X,Y) = <N~X,Y) A,~ + Nr~(X,Y) .A,E> (20)
where:
A~ = P(E)/<P2s (21)
s
Al = [J(E) - Al ]/s (22)
Solving for N~X,Y) gives the count/pixel of the Compton free
image.
An alternative fitting method is the Maximum Likelihood Me-thod.
Given the measured ac-tivities {Nl } the ioin-t Poi5son probability
with respect to the parameters of the -trial function n(E) are
maximized; i.e., find n(E) such that
= ~ Çn(E) -n(E~
~= maximum (23)
Nl! J
or, since ~ is positive:
ln ~ = <N~ ln n(E) - n~E) - ln N~!> - maximum (24)
t can be shown that for the maximum likelihood solution:
<n(E)> - ~N~>
which enables eliminating N~ from the n(E) function.
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(llereaf-ter (X,Y) are implicit, e.g., n(X,Y;E) = n(E). Thus from
equations (8) and (9) it follows tha-t:
n~E) = < N~>P ~qk(~C - ~k>~P) (25)
Calculating the derivative~ of ln ~ wi-th respec-t to q,~ and settlng
the resulting equation -to o as required b~ -the maximum condition,
the following equations are obtained:
N~
< (~ <~.c~P)> = ~ ; k = 1,2,.. t26)
n(E)
This is a set of non-linear coupled equations and canno-t be solved
in closed form. IJsing the mul-ti-gradient method, an iterative
solution Eor the q,cs can be obtained. Denoting oq,c as the
difEerence between q.c before and q',~ after the i-teration
oq~ = q k - q.~ . ( 27
The coupled set of equations is linearized and soluble
~Mi~oq. = Ul (28)
N~
M1J = < (~)1 <01>P) (~ -<~ ~P) > (29)
nZ(E)
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N~
U = < ( ~ - < ~ P ) > ~ 3Q )
n(E)
Af-ter proper convergence of the solution for the array q,c has been
attained, the Compton free activity Nv can be obtained from:
N~ - <Ni> - ~qk<~l~> (31)
1~
Yet another alternative fit is the partial Maximum Likelihood
Solution. Suppose that the least square solution provides the
approximate functional structure of the Compton component.
However, it is desired to in-troduce the Poisson statistics by
changing the ratio of the photopeak to Comp-ton events Eraction in
order to optimize the ioint distribution. The trial func-tion,
n(E), then is
n( E) = <NOE ? t f~.P + (1-f~)C] (32)
where:
N~ .
f~ = is the photopeak fraction (33)
~N~>
and C is the least square Compton solution, normalized to t;
i.e.:
C = C /<C? (34)
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Now, the Maximum Likelihood ~qua-tion is maximized with respect -to
a single parameter, the photopeak fraction, f~. Once f~ is
calcula-ted, the scatter-free event distribution N~ can be found
using the equation:
N~ = fv,~N~> (3~)
The optimization equation resulting from the differentiation with
respect to fF, reads:
N~tp-~)
~_ ) = O (~)
~fF~(P-)
It is soluble by an iterative New-ton-Raphson method:
<N~.~>
f F~ = fF~ (37)
<N~ .~2
where:
P-C
_ (38)
C+f~(P-C)
Yet another related me-thod of obtaining the value of NF~ involves
the semi-local Maximum Likelihood Fi-t Solution. As for the
quasi-local solution, this is implemented here as follows: First a
solution is ob-tained for the square called S surrounding the pixel
XO~Yo~ i.e.:
- .
- ~5 -
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5=(XO-W)~X~(XO~w), (Y~-W)~Yc(YO~w) = (2w+l)2 t39)
Once the Compton free component of the entire window has been
o~tained, el-ther by the full or by the partial Maximum Likelihood
method; i.e., N~(X,Y) is known, the slow Compton spatial variance
is used by assumlng:
Cl = C~/s (40)
-to obtain the (X,Y)-pixel Compton free activity-
N,~ N
N ' = -- ~ <N ' - `~
p s E s (41 )
If it i5 desired to eliminate the events caused by additional
unwanted photons then block 40 is used to include such additional
unwanted photons ori~inated events in the "trial" equation.
In operation, the inventive system locally analyzes the energy
spectrum which may comprise mu].tiple energies and fits it with a
trial function comprising a combination of the unscattered
pho-topeak function and a function, representing the Compton
scattered spectrum, and a function representing other unwan-ted
photons. The function representin~ the Compton scattered spectrum
is derived usinO the Nishina-Klein formula. The Compton sca-tter
spectrum shape, therefore, inherently reflects the true
-
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rela-tivistic distributions of the Compton sca-tter, unlike the
previously used arbitrary polynomlals. The function representing
other unwanted photons can either be measured or computed. If
measured, the system spread function is automatically included in
-the result. If computed, the result must be convolved with the
system spread function.
The Nishina-Klein formula is recursively used to generate the
multi-scat-tered Compton distribution ~c. Then each ~(~)is convolved
with the system energy spread function to obtain C~m~ the system
dependent Compton scatter distributions. The convolved functions
are averaged or integrated for discrete windows to obtain discrete
arrays required for the calculations. The set of discrete
~u~
functions Ci is then preferably orthonormalized to reduce the
number of computations necessary and to assure that the inventive
system can provide practically Compton free ima~es within seconds
after acquisition. The coefficients of the orthonormalized
functions are parameters that provide the scattered counts per
pixel. The parameters are determined by fitting between the final
trial function comprised of the photopeak component or components,
the scatter component and on other unwanted photon components to
the measured ener~y distribution which includes both the
scattered, unscattered photons and other unwanted photons. Local
(and quasi-local) fitting can be used to expedite obtaining the
coefficients of the fit functions.
- ~7 -
P4~5 ~7~8~9
A unique approach of the invention is tha-t the parame-ters -to be
determined are coefficients of the physical Compton scatter
functions. The said func-tions have the correct high energy
threshold behavior ensuring correct fit at every point.
The inventive method also preferably improves the statistics for
the calculations for the Compton fit by a me-thod that takes
advantage of the smoothness of the Compton distribution
(quasi-local method). That is, the data in a preferred embodiment
is summed over (2n ~ 1) square pixels for the fit where n is an
integer. The values are then a-ttributed only to the central pixel
of the square. Similar calculations are done for each pixel.
Preferably a leas-t square fit is used to solve the unknown
c.oefficients, i.e., the amount per pixel of the unscattered events
(and if desired for the amount per pixel of the scattered events).
However, several variations employing the maximum likelihood fit
are also described and are within the scope of this invention.
While the invention has been described with regard to specific
embodiments, it should be understood that the description is made
by way of example only and not as a limi-tation of the scope of the
invention which is defined by the accompanying claims.
