during the last fifty years , progress in imaging 
 systems using penetrating radiation for biomedical purposes has brought about 
 new topics in mathematics and fueled intense research activities with far 
 reaching results .
transmission computer assisted tomography ( cat - scanning ) 
 is based on an integral transform in two dimensions discovered in the sixties 
 by cormack [ 2 , 3 , 4 ] , who did not realize that j. 
 radon had already introduced and studied it in his seminal paper   in 1917 .
subsequently , in an 
 effort to reconstruct directly in three dimensions an object without having to 
 assemble its two - dimensional sections , one is led to consider the so - called 
 x - ray transform or cone beam transform .
this transform is an off - spring of the 
 radon transform and maps a function in  to its 
 ( straight ) line integrals in . one way to reconstruct an object is to convert the 
 line data into planar data in  using the 
 grangeat technique
. then application of the inversion formula of the 
 three - dimensional radon transform   yields the answer
. a further generalization of the 
 radon transform in emission imaging , called the attenuated x - ray transform , 
 accounts for the radiation loss in the traversed medium .
the problem of its 
 inversion has been a mathematical challenge for decades and has been solved 
 only in 2001 , 
 thanks to complex analysis methods applied to the stationary photon transport 
 problem .
the 
 success of this branch of mathematics , coined by gelfand as integral geometry , 
 goes far beyond the imaging science scope as it has brought significant 
 progress in group representation theory , partial differential equations , 
 boundary value problems , and so forth .
the standard radon transform and its variants are 
 related to a process of data collection in an interaction free propagation of 
 radiation through matter , possibly affected by attenuation .
however , a sizable part of 
 traveling radiation does suffer an interaction with matter through compton 
 scattering . 

generally as the data quality is lowered , compton scattering effects have been 
 treated as noise and must be eliminated .
so far , in most image processing 
 methods , the aim is to deal away with it , for example , by filtering or by 
 geometric rejection using an antiscatter grid of miniature lead septa . in 
 conventional projection imaging ,
most of the radiation have been scattered in 
 the patient body , the scatter - to - primary ratio can be as high as 10 . but throwing out all the 
 scattered radiation may not be a smart move since this also means a loss of 
 sensitivity and certainly a loss of valuable information .
this is why we have recently advocated the use of 
 scattered radiation to improve quality in image processing   as well as to construct new 
 imaging principles .
this proposal generalizes the concept of a compton 
 camera , proposed by several workers some thirty years ago [ 1315 ] , in which the idea of 
 electronic collimation is implemented .
when radiation is detected at a lower 
 energy than the originally emitted energy , there must be at least one compton 
 scattering occurring along its propagation path .
but the vast majority of 
 scattered radiation is due to mainly single scattering events . to an emerging ray detected at a definite energy
there corresponds an ensemble of incoming rays distributed 
 on a circular cone of definite opening angle .
the measured data consisting of 
 collected emerging radiation at some detection point is viewed as the integral 
 contribution of a source function on specific circular cones ; it may be also 
 called a conical projection of the source .
the name of conical radon transform 
 is attributed to such integral transform and is added to the list of already 
 known radon transforms on geometric manifolds in  : paraboloids , spheres , special surfaces , second - order surfaces , and so forth .
other 
 proposals for use of scattered radiation in x - ray imaging which do not rely on 
 conical projections have been made independently by .
thus , the main topic addressed here is the 
 mathematical framework defining the working principles of a new imaging 
 technique . to keep the discussion transparent ,
we have adopted an ideal working 
 assumption whereby attenuation is not taken into account .
a well - known 
 difficulty is that the nonuniform attenuation along the radiation path leads to 
 tremendous mathematical complication . for a given projection , attenuation 

it should be pointed 
 out that , in the case of the attenuated x - ray transform in emission imaging , a 
 comprehensive analytic solution has been attained only recently .
in conventional emission imaging , the data collected 
 for object reconstruction is formed by linear projections ( cone beam 
 projections ) of unscattered radiation filtered by an energy acquisition window 
 which has already discarded from 70% to 80% of the incident 
 photon flux . moreover , among the retained some have suffered scattering in the 
 collimator and hence must be discarded .
it has been reported that collimator 
 scatter increases as the energy of the photopeak of interest increases from a 
 low of 1.9% for tc-99 m 
 ( 141 kev ) to a high of 29.4% for i-131 ( 364 
 kev ) with the usual high - energy collimator .
therefore , correction for photons that penetrate through , 
 or scatter in , collimator septa is hardly important at all for tc-99 m tracers 
 . in the proposed imaging technique ,
data for 
 reconstruction are provided by the so - called conical projections , which gather 
 radiation from point sources lying on large surfaces inside the emitting 
 object .
although contamination of the scatter component by multiple scatter 
 events may be as high as 30% of the total 
 scatter according to monte carlo investigations [ 16 , 24 ] , we believe that the signal 
 of single scattered radiation is largely sufficient to make the new imaging 
 principles work , in particular when advanced semiconductor - based detectors with 
 better energy and spatial resolution and sensitivity will become available
. 
 these interesting issues will be explored in future work . in this paper , we present a unified treatment of 
 conical radon transforms relevant in emission imaging by scattered radiation . 
 in fact , we will be concerned with two classes of conical radon transforms 
 originating from image formation by compton scattered radiation on a gamma 
 camera with and without collimator .
the first class of conical radon transform 
 uses circular cone sheets with fixed axis direction ( 1-cones ) whereas the second class deals with circular 
 cone sheets with axis swinging around a point ( 2-cones ) .
note that if this point goes to infinity in 
 a given spatial direction , then the 2-cone goes over 
 the 1-cone .
in each 
 case , we will start by showing how the image formation process leads to an 
 integral transform and how this transform is related to the conical radon 
 transforms .
each conical transform will be introduced and its relevant 
 properties for imaging purposes , in particular their invertibility , discussed . 

let f be a real nonnegative integrable , smooth , with compact support function on . the same function in cylindrical ( spherical ) coordinates is noted f(f ) .
the definitions of various transforms of f(x , y , z ) are as follows : f^i : conical i - radon transform of f , i = 1 , 2 ; fj : j - dimensional 
 fourier transform of f , j = 1 , 2 , 3 .
special functions   are as follows : y ( x ) : heaviside unit step function ; jl(kr ) : spherical bessel function of order l and variable ( kr ) ; pl(cos  ) : legendre polynomial of order l and of variable cos ; yl , m(k ) : spherical harmonic ( l , m ) with argument k , solid angle in the direction of the unit vector k.
in this article , we consider the emission imaging 
 problem , that is , the problem of reconstructing in  a gamma - ray 
 radiating object from its compton scattered radiation data .
this object is 
 described by its activity volume density function f. detection of scattered radiation is performed by a 
 gamma camera in two instances : with or without collimator .
the recorded data 
 consists of the coordinates of the detection site , the surface flux density of 
 photons at this site ( pixel ) , and the value of their energy ( list mode ) . 
 between the radiating object and the detector stands a scattering medium : it 
 may be a volume or a layer as illustrated by   figure 1 . note that the object 
 itself may also be a scattering medium , and for photon energies above 25 kev , 
 over 50% of the 
 interactions in biological tissues are scatterings .
higher - order scattering 
 events ( of much lower probability of occurrence ) will be the object in future 
 studies .
as compton scattering plays a key role , we will recall 
 some of its properties . the compton effect discovered in 1923  
had served to confirm the 
 particle ( photon ) nature of radiation , as proposed by a einstein .
thus , energetic 
 radiation under the form of x- or gamma - rays behave like particles and scatter 
 with electrically charged particles in matter . in biomedical domains , x- or 
 gamma - photons scatter electrons in the biological media they traverse .
this 
 scattering process has cylindrical symmetry around the direction of the 
 incoming photon and the energy of the outgoing photon is given by the compton 
 formula ( 1)e = e011+(1cos),where  is the 
 scattering angle as measured from the incident photon direction , e0 , the photon initial energy ,  = e0/mc , and mc the rest energy of the electron .
equation ( 1 ) shows that single - scattered photons have a continuous energy spectrum in the range e0/(1 + 2 )  e  e0 .
thus an emerging photon with an energy e in a direction 
 of unit vector n may originate 
 from an incoming photon of energy e0 emitted from a 
 site on a sheet of a circular cone of axis direction n and an opening 
 angle . at a scattering site m , the number of particles dnsc scattered in a 
 solid angle dsc along a 
 direction making an angle  with the 
 incident direction follows from the definition of the differential scattering 
 cross - section ( see   figure 2)(2)dcd,when the following quantities 
 are given : in , the incident photon flux density , ne(m)dm , the number of scatterers ( electrons ) around the 
 scattering site m with volume dm .
we 
 have(3)d2nsc=inne(m)dm(dcd)dsc.the compton differential cross - section has been computed in 1929 by klein and nishina .
it appears as the product of the area of a disk of radius re , the classical radius of the electron , and a probability factor p( ) , that is , 
 ( 4)(dcd)=(re2)p( ) , 
 where re = 2.82  10 m and 
 ( 5)p()=121[1+(1cos  )]2(1+cos2+(1cos2  )1+(1cos   ) ) . 

thus , we see that , for a given incident energy , the angular distribution of scattered photons is no longer 
 isotropic .
hence , the final form of this number of scattered 
 photons in the direction given by the angle  is 
 ( 6)d2nsc=inne(m)dmre2p()dsc , this number is basic to the 
 image formation by scattered radiation . in standard computer assisted tomography ( cat ) ,
the 
 data is gathered under the form of line projections or integrals of a function 
 ( attenuation or activity ) along straight lines . here following the scattering 
 mechanism
, the data would appear as conical projections or integrals of a 
 function f on a circular 
 cone sheets .
the integration is carried out with the lebesgue measure of the 
 cone in a chosen coordinate system .
the result is a function of the coordinates of the cone vertex , the parameters of the unit vector of the cone axis , the opening angle of the cone , that is ,  the scattering angle .  
figure 3 displays the representations of line 
 projection and conical projection . in the text we will have two types of 
 conical projections corresponding to the two cases of image formation mentioned 
 earlier .
the question is now how to use the conical projections to reconstruct the 
 source function f. we will treat the two cases separately in the coming 
 sections .
in this section , we consider the possibility of imaging a three - dimensional object by collecting data on its scattered 
 radiation on a gamma camera equipped with a collimator and show how the 1-conical radon transform arises .
figure 1 shows the experimental arrangement with the location 
 of the radiating object , the scattering medium , and the collimated gamma camera . by scattered 
 radiation to concentrate on the scattered imaging principle ,
we 
 make some simplifying assumptions [ 12 , 30 ] : absence of attenuation for the propagating radiation , constant density of the electrons ne in the scattering medium , isotropic emission from original radioisotopes in object . to compute the photon flux density at a detection site 
 ( pixel ) d on the collimated camera we start from ( 6 ) .
radiation is emitted at point source s , will propagate to scattering site m and reach 
 detection site d. the incoming photon flux density in on scattering 
 site m is now computed 
 from the emission data from point source .
let f(s)ds be the number 
 of gamma photons emitted per unit time by a volume ds in the object around 
 site s. the emission being isotropic , the number of photons 
 emitted in the direction sm in a solid 
 angle ds is 
 ( 7)f(s)ds4ds .
therefore , the incoming photon flux density at scattering site m is 
 ( 8)f(s)ds41sm2=in , 
 where sm=|sm| .
next , the number of scatterers around site m in a volume dm is ne dm .
the net number of photons emerging from the 
 scattering is 
 ( 9)f(s)ds41sm2ne  dmre2p()dm , 
 this means that the detected photon flux density at site m is 
 ( 10)f(s)ds41sm2ne  dmre2p()1md2 .
now , all the contributing point sources s , for given scattering center m , lie on a circular cone sheet of axis identified with md and opening angle  , thus we must integrate with the measure (cone)ds first .
next , we must take into account all the scattering sites in the scattering medium 
 situated on the line parallel to the collimator axis at site m. hence , we must perform a second integration with the 
 measure (line)dm .
to sum up the detected photon flux density at d is 
 ( 11)g(d,)=(cone)f(s)ds41sm2ne  dmre2p( )  (line)1md2 . in the cylindrical coordinate system of figure 4 ,
the 
 integration measure on the cone is r sin  dr d and the measure along the line is simply dzm , ( 11 ) becomes 
 ( 12)g(d,)=re2p()ne4r  sin  drdf(xd+r  sin    cos  ,yd  + r  sin    sin  ,zm+r  cos  )dzmzm2 . 

remark 1 in practice as f and ne are volume 
 densities , to keep the physical dimensions right , we should think of the cone 
 sheet as having a small thickness e and the line on 
 which m moves as having 
 a tiny section s. these are constants and do not affect the 
 mathematical reasoning , they will be dropped for the sake of expression 
 simplicity , but should always be kept in mind . 
  in practice as f and ne
are volume 
 densities , to keep the physical dimensions right , we should think of the cone 
 sheet as having a small thickness e and the line on 
 which m moves as having 
 a tiny section s. these are constants and do not affect the 
 mathematical reasoning , they will be dropped for the sake of expression 
 simplicity , but should always be kept in mind .
clearly this imaging equation is a compounded integral 
 transform . assuming that integration interchange is valid ,
if we define the 
 first integral transform as 
 ( 13)h(xd+r  sin    cos  ,yd+r  sin    sin  ,r  cos  )=dzmzm2f(xd+r  sin    cos  ,yd  + r  sin    sin  ,zm+r  cos   ) , 
 we see that the imaging equation ( 12 ) is just 
 a conical projection of the function h on the 1-cones with 
 vertex on the plane xoy .
let us define an interaction factor which also 
 includes e and s : 
 ( 14)k()=re2p()ne4es , then ( 12 ) 
 reads 
 ( 15)g(d,)k()=r  sin  drdh(xd+r  sin    cos  ,yd+r  sin  sin,r  cos   ) , 
 or in terms of our notations ( see section 2 ) 
 ( 16)g(d,)k()=h^1(xd , yd, ) .
the 1-conical radon 
 transform has been discussed in , where some of its properties have been studied . 

we 
 will not go through the method of decomposition of functions in circular 
 components but will show that there exists a variant of the central slice theorem   which provides the grounds to invert the transform as it is done in the standard 
 radon transform .
first , we observe that by definition in the chosen 
 cylindrical coordinate system the 1-conical radon 
 transform of f , the cones having vertex on the xoy plane , 
 is 
 ( 17)f^1(xd , yd,)=r  sin    dr  df(xd+r  sin    cos  ,yd+r  sin    sin  ,r  cos   ) .
this can be rewritten under the form of a fredholm 
 integral equation of the first kind with a delta function kernel concentrated 
 on the sheet of a circular cone 
 ( 18)f^1(xd , yd, )  = dx  dy  dzk1(xd , yd,|x , y , z)f(x , y , z ) , 
 with 
 ( 19)k1(xd , yd,|x , y , z )  = (cos  (xxd)2+(yyd)2z  sin   ) .
we use now the fourier representation of the delta function 
 ( 20)(cos  (xxd)2+(yyd)2z  sin   )  = d  exp  2i(cos  (xxd)2+(yyd)2z  sin  ),and the two - dimensional fourier transform of f(x , y , z , 
 ( 21)f(x , y , z)=dpdqe2i(px+qy)f2(p , q , z ) .
we can perform the integration over z , which restores a three - dimensional fourier transform f3(p , q , r ) of f and yields a new form of 
 f^ 
 ( xs , ys ,  ) , 
 ( 22)f^(xs , ys,)=dx  dyd  exp  2i(cos  (xxs)2+(yys)2)dp  dqe2i(px+qy)f3(p , q,  sin   ) .
let f^2 
 ( p , q ,  ) be fourier 
 component with respect to the coordinates xs and ys of f^ 
 ( xs , ys ,  )
, then 
 ( 23)f^2(p , q,)=df3(p , q,  sin  )dx  dy  exp2i[  cos  (xxs)2+(yys)2+(p(xxs)+q(yys ) ) ] .
the last integral of ( 23 ) can be computed 
 using polar coordinates in ( x , y ) and ( p , q ) spaces , that 
 is , ( x  xs ) =  cos  ( y  ys ) =  sin  , dx dy =  d d , and 
 p = k cos  , p = k sin  , k dk d. therefore , 
 ( 24)dx  dy  exp  2i[  cos(xxs)2+(yys)2+(p(xxs)+q(yys))]=  d  de2i[  cos+k  cos( ) ] .
integration on  yields a bessel 
 function of order zero in the last integral of ( 24 ) : 
 ( 25)  d  de2i[  cos  +k  cos  (  ) ]  = 0  de2i  cos  2j0(2k ) .
thus , in two - dimensional fourier space , the 1-conical radon 
 transform appears as 
 ( 26)f^2(p , q,)=d0  d2j0(2p2+q2)f3(p ,
q,  sin  )e2i  cos  . now , we perform the integration on  which brings 
 back the three - dimensional fourier transform of f,(27)df3(p , q,  sin  )e2i  cos  =1sin  f2(p , q,  cot   ) . this step reduces the 1-conical radon 
 transform to a hankel of order zero:(28)sinf^2(p , q,)=0d2j0(2p2+q2)f2(p , q, 
, we 
 should switch to an appropriate variable , that is ,  =  cot . but care must be exercised as far as the range of  is concerned . 

we will distinguish two cases . 
 0   <  < /2 , then  > 0 as well as tan  > 0 and ( 28 ) can be rewritten 
 as 

 ( 29)sin    cot  2f^2(p , q,<2 )  = 0d2j0(2  tan  p2+q2)f2(p , q,)./2 <  <  , then  < 0 as well as tan  < 0 and ( 28 ) can be rewritten 
 as(30)sin    cot2  f^2(p , q,>2 )  = 0d2j0(2  tan  p2+q2)f2(p , q,).application of the hankel identity 
 ( 31)1k(kk)=0rdr2jl(2kr)2jl(2kr ) 
 leads to the inversion of the hankel transforms and yields the three - dimensional fourier components of the 
 object f. as the dual variable to  is p2+q2 , with  = tan  , we get 
 ( 32)f2(p , q,)=(p2+q2)0d2j0(2  tan  p2+q2 )  1||1+2(y(2)f^2(p , q,<2 )  + y(2)f^2(p , q,>2) ) . 
  0   < 
< /2 , then  > 0 as well as tan  > 0 and ( 28 ) can be rewritten 
 as 

 ( 29)sin    cot  2f^2(p , q,<2 )  = 0d2j0(2  tan  p2+q2)f2(p , q, ) .
/2 <  <  , then  < 0 as well as tan  < 0 and ( 28 ) can be rewritten 
 as ( 30)sin    cot2  f^2(p , q,>2 )  = 0d2j0(2 
application of the hankel identity 
 ( 31)1k(kk)=0rdr2jl(2kr)2jl(2kr ) 
 leads to the inversion of the hankel transforms and yields the three - dimensional fourier components of the 
 object f. as the dual variable to  is p2+q2 , with  = tan  , we get 
 ( 32)f2(p , q,)=(p2+q2)0d2j0(2  tan 
hence , f can be 
 recovered by inverse three - dimensional fourier transform . 
 

remark 2 in order to reconstruct the object by inverting the 
 compound integral transform given by ( 12 ) , we need to invert 
 ( 13 ) .
this is 
 quite easy since it can be viewed as a convolution in the variable zd between the two 
 functions 1/zd and f (  ,  , zd + r cos  ) . as this operation is not important to the topic of 
 this paper ,
we refer the reader to [ 12 , ] . 

  in order to reconstruct the object by inverting the 
 compound integral transform given by ( 12 ) , we need to invert 
 ( 13 ) .
this is 
 quite easy since it can be viewed as a convolution in the variable zd between the two 
 functions 1/zd and f (  ,  , zd + r cos  ) .
as this operation is not important to the topic of 
 this paper , we refer the reader to [ 12 , ] . 

that compton effect which has been proposed as 
 mechanism for imaging is known ever since the fifties . however , there are many 
 ways to do the experimental setups .
most proposals are systems with collimated 
 point source and point - like detector , see , for example , .
in fact the compton effect 
 is used to probe the electron density of matter and applied often to nondestructive 
 material control .
here we are interested in data collected by a gamma camera . 
 in the previous section
we have examined the case of a gamma camera with a lead 
 collimator which has the disadvantage of rejecting many of the scattered 
 photons .
so to improve drastically detection sensitivity , the idea of a compton 
 camera has been proposed as early as 1974 by many workers [ 1315 , 32 ] .
the concept of a compton camera is analogous to the 
 scheme of section 4 except that the scattering medium is now a thin scattering 
 layer parallel to the face of a gamma camera without collimator .
the data 
 consists of 2-conical 
 projections , the cone sheet axis converging to the detection site d ; see figure 5 .
note that when d   in a given 
 direction we recover the 1 cones .
following the image formation process in a compton 
 camera , we will show how a new conical radon transform , the 2-radon 
 transform , comes up and sketch a proof of its invertibility under specific 
 conditions .
the true conical radon transform of a compton camera is not yet an 
 analytical inversion formula .
its primary rays hit the scattering layer and will be absorbed by the 
 planar camera ( see figure 5 ) .
if only photons of energy e below the 
 energy e0 of the primary photons are recorded , then each detection site collects all possible conical 
 projections coming from all directions in half space , delimited by the photon 
 absorbing detector .
this is the principle of electronic collimation which has 
 been designed to improve sensitivity of gamma cameras . to reconstruct an object described by a source 
 function f(x , y , z ) ,
we need a set of data consisting of conical 
 projections depending also on three variables .
ideally one could select one 
 detection site d , and consider all the projections along circular 
 cones of opening angle  and axis 
 swinging around d but with cone 
 vertex constrained to be on a plane . with these conditions
a conical projection 
 will depend on three parameters : the scattering angle  and the two 
 coordinates of the cone vertex on the scattering plane .
thus , we obtain a 
 mapping of f onto a function 
 of three variables .
the inverse mapping , when explicitly worked out , would 
 yield a correct imaging procedure by a compton camera .
following the assumptions of section 4.1 , the photon 
 flux density at detection site d is evaluated in 
 the same manner as for the case with collimator : 
 ( 33)g(d,)=e(cone)f(s)ds41sm2  nedmre2p()s(line)1md2 . now in the chosen coordinate system ( see figure 5 ) , it 
 has the expression(34)g(d,)=k()(cone)f(s)ds1sm2  dxmdym1l2+xm2+ym2 , l being the 
 distance od , the explicit integration on the cone sheet (cone)ds will be given 
 later in section 5.3 since it is related to the 2-conical radon 
 transform we will be examining . up to
cree and bones   were the first to consider 
 conical projections on a compton camera which still has a collimator .
later on 
 basko    as well as parra   have designed inversion 
 techniques based on properties of spherical harmonics as they consider conical 
 projections as made up of cone beam projections , but have not touch really upon 
 the problem of converting cone beam data into conical projection data , a 
 problem similar to the one solved by grangeat   for planar projections in . there are numerous approximate reconstruction 
 methods , most of them using some back projection techniques ( or numerical 
 algorithms ) and search for point sources as intersections of cone sheets 
 reconstructed from coincidence measurements on the compton camera .
lastly let 
 us also cite some other original approaches based on statistical physics 
  as well as 
 algebraic methods [ 3739 ] .
in this situation , it is of interest to find an analytic inversion of the special 2-conical radon transform responsible for the imaging process in compton camera . before tacking 
 this problem
, we discuss here a more general problem in which the vertex of the 
 scattering cone is not constraint to lie on a plane ( or a surface ) as in the 
 compton camera case .
this would give us some freedom to find an inversion 
 formula for another class of 2-conical radon 
 transform .
thus , the 
 mathematics of the compton camera imaging process have led to a new class of 
 conical radon transform .
the 2-cone is 
 generated by the rotation of a straight line making an angle  around an axis 
 direction n and meeting 
 this axis at vertex m ; see figure 6 .
let s be a running 
 point of the cone sheet and denote this point by ds=s = rk .
let the angle  be defined by 
 ( k  n ) = cos . thus , by considering trigonometric relations in the 
 triangle dms , one can write down the relation 
 ( 35)r = psinsin()=psinsin(cos1(kn ) ) , which may be regarded as the 
 cone equation in a meridian section within polar coordinates ( r ,  ) and polar axis n. at  = /2 we recover the equation of a plane as a degenerate cone .
when evaluating a 2-conical 
 projection , we integrate a nonnegative function f on one sheet of 
 the cone .
this is equivalent to saying that , giving a  mass  density , we 
 compute the
 mass  of a piece of cone surface limited by the intersection 
 curve of the support of f with the cone . 

the 
 area element of the cone is the product of the arc element of the line by the 
 element of circle in a plane perpendicular to the cone axis .
the arc element is 
 ( 36)ds=(dr)2+(rd)2=p  sin    sin  ( )d  sin  (  ) .
now the element of circle is r sin  d. hence , the cone area element is 
 ( 37)da = r  sin 
dpsin  sin  ( )dsin()=  sin  d(p  sin    sin  ( ))2dsin( ) . in the same coordinate system , a 2-conical 
 projection of f is expressed as 
 the integral of f on the cone 
 sheet with the integration measure da .
f has then the 
 expression f 
 ( r , k ) , with k = ( , ) , r = p sin / sin (    ) and angular ranges 0 <  <  and 0 <  < 2. thus , 
 ( 38)f^2(pn,)=f(psin  sin(),k)sin  d(p  sin  sin())2dsin( ) , 
 or alternatively 
 ( 39)f^2(pn,)=(f(r,k)r2)r = p(sin  /sin  ( ))sinddsin().one may use the integration over 
 a one - dimensional delta function to express the substitution of r by the cone 
 equation ( 35 ) : 
 ( 40)dr(rpsin  sin( ) ) . 
 since 
 ( 41)(rpsin  sin  ( ))1sin()=(p  sin  r  sin  (  ) ) , 
 and r dr sin  d d = r dr 
 dk = ds , we can replace the original integration 
 range:(42){r ,  0<< ,  0<<2 } , 
 by 
 ( 43){r+ ,  =[0<<<<+],s1=[0<<2 ] } , 
 which fits in the chosen spherical coordinate system with n as polar axis .
the -conical 
 projection of f has now the 
 fredholm form of the first kind with a delta function kernel . in intrinsic 
 vector notations
, it reads 
 ( 44)f^2(pn,)=dsf(s)k2(pn,|s),where 
 ( 45)k2(pn,|s)=(p  sin  r  sin  ( cos1(kn))).(compare with the previous case , 
 see ( 18 ) , 
 ( 19 ) ) .
finally , at  = /2 we recover 
 precisely the planar radon projection in :(46)f^2(pn,2)=dsf(s)(pr(kn ) ) .
again as in section 4 , we may use the fourier 
 decomposition of the delta function 
 ( 47)(p  sin  r  sin())=d  exp2i(p  sin  r  sin  ( )),together with the decomposition 
 of f into spherical 
 components(48)f(x , y , z)=f(r,k)=l , mflm(r)ylm(k),where , in the chosen spherical 
 coordinate system , k denotes the 
 direction parameters of the unit vector k hence , inserting ( 47 ) and ( 48 ) in ( 44 ) we obtain(49)f(pn,)^=l , mde2i(psin )  0r2drflm(r)s1dkylm(k)e2irsin( ) . recall cos 
= cos ( /2   +  ) , we introduce the decomposition of a plane wave in 
 space in spherical components ( see , page 471 ) : 
 ( 50)e2ircos(/2+)=nin(2n+1)jn(2r)pn(cos(2+ ) ) , where jn ( x ) is the 
 spherical bessel function of order n. 
  let u be a fixed unit vector , defined by u , such that cos ( /2   +  ) = ( k  u ) .
then the legendre polynomial pn ( cos ( /2   +  ) ) can be further 
 expanded in terms of spherical harmonics : ( 51)pncos(cos(2+))=42n+1m=nm=nynm(k)ynm(u ) .
putting relation ( 51 ) in ( 50 ) and integrating on d k , since the range of  is  , we can write(52)s1dkylm(k)ynm(k)=mmklnm( ) , 
 with |m| < inf(l , n ) .
right - hand side of ( 53 ) expresses f(pn,)^ as a spherical 
 component expansion of the -conical radon 
 transform in terms of the variable p sin u , instead of pn . so if the -conical data 
 can be rewritten under the form of a function of   p sin n and of  , with a spherical component decomposition , that 
 is,(54)f(pn,)^=g(p  sin  u,)^=lmglm(p  sin  ^,)ylm(u),then we can extract the new data 
 spherical component as 
 ( 55)glm(psin^,)=de2i(psin)4il0r2dr  ljl(2r)klnm()flm(r ) .
now by fourier inverting with respect to q = p sin  the two sides 
 of ( 55 ) , 
 ( since sin  > 0   for 0 <
recall 
 that for spherical bessel functions ( see , e.g. , ) this identity is of the 
 form(57)1k2(kk)=02d4jl(2k)4jl(2k).this new identity allows to 
 extract the spherical component flm(r ) of f and thereby 
 achieve inversion ( details will be presented elsewhere ) .
equation ( 54 ) relates implicitly 
 p and  and may suggest a new type of gamma camera realization , which should be investigated . finally 
 for 
= /2 , ( 44 ) shows that this 2-conical radon 
 transform is just the radon transform in . the whole procedure goes through with drastic 
 simplifications since the natural choice for u is n , thus f(pn,)^=g(pu,)^.
the radon transform has enjoyed tremendous popularity 
 in imaging science as it has been extended , generalized in pure mathematics ( see , 
 e.g. , [ 19 , 41 , 42 ] ) as well as exploited in 
 many fields of applications [ 17 , 43 , 44 ] . in this paper
we have presented two further 
 generalizations of the radon transform , namely , two classes of conical radon 
 transforms which originate from imaging processes using compton scattered 
 radiation .
the first class , called 1-conical radon 
 transform , is related to an imaging principle with a collimated gamma camera 
 whereas the second class , called 2-conical radon 
 transform , contains a special subclass which models the compton camera imaging 
 process .
we have also shown that inversion of 2-conical radon 
 transform can be achieved under a special condition which is not yet , for the 
 moment , implemented in gamma - ray emission imaging science . exploiting scattered 
 radiation to reinforce sensitivity as well as enlarging field of view and 
 cutting down operating time may lead to employing large scattering medium but 
 without collimator for gamma cameras . the mathematics behind this perspective 

will be based on a yet little known radon transform : the torus radon transform , 
 which may bring even more exciting mathematical topics to be explored in the 
 years to come .
 the mathematical principles of tomographic imaging using detected ( unscattered ) x- or gamma - rays are based on the two - dimensional radon transform and many of its variants . in this paper , we show that two new generalizations , called conical radon transforms , are related to three - dimensional imaging processes based on detected compton scattered radiation . the first class of conical radon transform has been introduced recently to support imaging principles of collimated detector systems . the second class is new and is closely related to the compton camera imaging principles and invertible under special conditions . as they are poised to play a major role in future designs of biomedical imaging systems , we present an account of their most important properties which may be relevant for active researchers in the field . 
1. INTRODUCTION
2. NOTATIONS
3. WORK SETTING
4. THE
5. THE 
6. CONCLUSIONS AND PERSPECTIVES