Stochastic Aspect of the Tomographic Reconstruction Problems in a Transport Model

1 Abstract . The stochastic differential and integral equations describing the system of particles weakly interacting among themselves which a re absorbed and scattered by particles of a medium are considered. The time-dependent transport equation with scattering is studied taking into account s tochastic nature of parameters in nuclear i maging. Using dynamic att enuated Radon transform the solution of transport equat ion may b e d erived taking into account of the scattering as perturbation. We ana lyze the influence of the random variables upo n the ima ge reconstruction both general ly a nd in more deta ils for the case of point source. It is sh own by t he example of the method of the filtered back projection (FBP) that unaccount ed small f luctuations of attenuation coefficient can c ause ess ential distortions o f image texture and degradation of the resolutio n at image reconstruction in single-photon emission computerized tomography (SPECT) and less in X-ray computerized tomography (CT). The mechanism o f these distortions is analyzed. The way fo r their elimination is shown for po int sources. We demonstra te th at fo r the practical purposes it is enough to define averaged at tenuation coefficient in investigated area when its difference fro m true attenuation coefficient has ce rtain s tochastic prop erties. It is shown that for pos itron emission tomography (PET) stochastic components in parameters of a transport model without the scattering can be take n into account of the corrections of projections. Keywords : Image Reconstruction, Nuclear Imaging, PET, SPECT, Transport Equation, X-ray CT, Radon Transform. I. I NTRODUCTI ON NE of major proble ms of image recon stru ction from the proje ction d ata in a to mography is the problem of noise eliminatio n. T he physical sources of a n oise can condit ionall y be sectio ned into two basic types. Moreo ver there are the nois e which is bounded wit h the uncertainty of detection methods, internal noise of electronics et c. This cl ass of t he noise sources is “the noise of de vices”. Th e s econd kind of a noise is generated ph ysical processes having stochastic natu re su ch as processes of radi ation, absor ption, scattering and detection in abstract sense. I. G. Kharin is with the Nati onal Science Center " Kharkov Institute o f Physics and Technolo gy", Ak ade mi cheskaia 1, Kharkov, 61108 U kraine, (e-m ail: igor@isc.k harkov.com, http://isqom .org/usr/igor) In practi ce th e most widesprea d reconstructio n m ethods are based on analytical methods of th e solution of p roblems in CT. Anal ytic methods are comp utationall y i nexpensive but they can b e n o e ffective in t he p resence of a no ise [ 1]-[3]. I n systems of rec onstructive t omograph y t he nois e in eac h recons tructed element d epends on t he noise of each pr ojection passin g throu gh the gi ven infor mation element and comput ational noise which bou nds with al gorithm of the recons truction. Thus with usage of int egral transfor mations the change o f th e noise fr om the el ement t o elem ent influences upon the r econstruction which depen ds on con volution core [4] because o f this c entral areas are more corrupted b y t he nois e t han p eripherals. Statisti cal t echniques of t he recons truction have some attractive features for accounting a nois e [ 3], [ 5], [6 ] t hough s ometimes t hey req uir e prohibitivel y (for practical applications) lo ng compu tation tim es. Besides in som e cases t he probl em becomes more complicated when it is necess ary i n p rinciple to t ake into account depend ence of param eters on ti me in dy nami c CT [5 ], on en ergy as for exampl e in polyenergetic X-r ay CT [ 6]. It is obvio us that any recons truction methods are appli ed in c ontext of ce rtain ph y sic omathematical model of pr ocesses in CT. The conse cutive ac count of diffe rent sto chastic aspects within t he framew ork of corresponding m odel is necess ary f or the creation of effecti ve algorithms. If scatter is in cluded the transpo rt model is mo st appropriate. For X-Ray CT the the oretical results with account of scatter ing [7] and n umerical m ethods for special choices of scatter ing term [8] were o btained for the models based on transpo rt formulation. In most practical application s of emissi on compu ted tomo graphy (ECT) in co mpariso n with X- ra y CT t here is additional co mplexit y as sociated with the unkno wn attenuation coefficien t. On t he b asis o f t ransport equat ion, it is possible to f ind the distribution of radiatin g sour ces i f on e formulates the h y poth esis about atten uation coeffici ent µ : µ is a constant in known ar ea [9], µ is a functio n which is in accord with defi ned conditi ons a s in [10] and [ 11]. But us eful i n practice t he definiti on way of µ from th e emission data has b een not found . The con tribution of disp ersed p articles at the d etection of proje ction data can be re ache d with the using collimation on energy and angl e [12]. At the sam e time we can do n othing simil ar with other the noise of second t yp e a s some uncert ainty of att enuation coef ficient is . How in p ractice it i s S tochastic Aspect of the T omographic Reconstruction Proble m s in a T ran sport Model I gor Kharin O 2 met besides t hat we onl y a ssume t he defined form of µ dependence on coordinat e, energy and t ime else resear ching objects are ra ndom inhomogeneous me diums as a rule. Moreover the quantum nat ure of processe s i s t he r eason becaus e of which we can not sa y fr om what t he fluctuation of projection data i s depended on at t he given concrete measurem ent: eit her the den sity fl uctuati on o f radiation sourc es or numbe r of emitted photons or attenuatio n coefficient or s cattering p arameters etc. Appar en tl y , i t will be useful to re sear ch when ever poss ible a ll aspects influ ence upon image r econstru ction of th e fl uctuation of parameter s for the development of practical ly usef ul methods of the solution so the composi te problem s (both p hysical and mathematical sense). In t he g iven paper the rec onstruction problems bo unded with account o f the s econd t ype of a noise i n SPECT, P ET a nd X-ray CT are stu died within the frame work of transport model. H ere we investigate sto chastic asp ects of t ransport model on b ase of the fu ndamental kinetic equ ation f or distributi on f unction of particles when we can n eglect t heir intera ction with each oth er. Then tr ansport equation and correspondin g integra l transformation of th e distribution of emitted part icles ar e s tudied for photons in case of t he contribution of fluct uations o f p arameters. We an alyze th e influence of the ra ndom components upon image reconstructio n i n CT both generall y and in more details for t he case of a point sou rce when att enuation coef ficient is ass umed to have a p rior p robabilit y distribution, called a Gaussian Markov rando m field. II. S TOCHASTI C A SPECT OF A T RANSPORT M ODEL Let's con sider the e quation boun ding distrib ution function of particles ( ) , I t p x with impulse p , di stribution function ( ) , ; F t x p of particles wit h the i mpulse p r adiated from sourc e at t he point x in area Ω and time t . Inside Ω absorption cross-s ection ( ) , ; t E σ x a nd s catterin g kernel ( , ; , ') W t x p p are not equal to zero. In this cas e it is w ell known fro m statistical phy sic s the general kinetic equati on is written in the followin g form ( ) ( ) ( ) ( ) 3 , ( , ; ) ( , ; ) , ( , ; , ') , ( , ; ', ) , ', V I t t F t t I t W t I t W t I t d p σ   ∂ ∂ ∂ ∂ + +   ∂ ∂ ∂ ∂   = − +   −   ∫ p p p' p v x x x p x p x p x x p p x x p p x (1) where V is interactio n of particles each other and external field, v =d x /dt . Further we will stu dy the s ystems consisting of medium and particles wh en the ener gy in teraction V can be neglected, as it i s for exam ple i n the case of phot ons o r neutrino. K ernel ( , ; , ') W t x p p and abs orption cross -section ( ) , ; t σ x p can be param eterized depending on th i component of the medium ( , ; , ') ( , ) ( , ') ( , ) ( , '), ( , ; ) ( , ) ( ), i i i i i i W t N t w N t t N t β σ σ → → = = = ∑ ∑ x p p x p p x p p x p x p (2 ) where and , ( , ') i i i N σ β p p i s the den sity, cro ss s ection of t he absor ption and scattering functi on for th i component of the mediu m i n Ω ac cordingl y. In troducing at tenuation coefficient 3 ( , ; ) ( , ; ) ( , ; ', ) ' t t W t d p µ σ = + ∫ x p x p x p p and taking into account the above mention ed we can rewrit e (1) in form of the transpo rt equation ( ) ( ) 3 ' ( , ; ) , ( , ; ) ( , ; , ') , ' . t I t t F t W t I t d p µ ∂   + ∇ +   ∂   = + ∫ p p v x p x x p x p p x (3) Let us study th e case wh en last t erm in (3) is a s mall value. And let us suggest t hat we know how to solve (3) wit hout the scatter ing term . In th is cas e we will denot e th is solutio n as 0 ] [ I p . It is known from the functional analysis we can write th e solution (3) conc erning f unctional ] [ I w → p with acc urac y up to the first order o n small value w in th e following f orm: ( ) ( ) [ ] 3 ( , ' ) 0 , [ ] ' ( , ') , 0 ( , ') i i i i I t w exp d p I t ε δ β δε → = =         ∑ ∫ p p p p x p p x p p (4) with th e equations for functional derivati ves 3 3 2 ' ' " 0 0 0 1 ) ' ( , ' ) ( , ' ) ' ( , ' ) [ 0 ] , 2 ) ( , ' ) ( , ' ' ) , ( , " ) ( , ' ) . . . i i i i j i j j i i I d p t N d p I I t I I N N ε ε ε δ µ β δ ε β δ µ δ ε δ ε δ δ δ ε δ ε = = = ∂   + ∇ +   ∂   = ∂   + ∇ +   ∂     = +       ∫ ∫ p p p p p v p p p p p p v p p p p p p p p       (5) which i s got fro m (3). The for m of ( 3) without the scattering term an d ever y eq uation fro m (5) ar e identi cal. Therefore when we kn ow ho w to solve (3) witho ut th e scatteri ng ter m s o we can solve the ch ain of the equatio ns from (5). When we take i nto ac count dependence o n t ime only for F (as for e xample in [5]), kinetic ener gy of particl e depends onl y on absolute valu e of impulse that is su fficient th e general case so 3 ( ) ( ) 3 ' 1 ( ; ) , v ( ) v ( ) ( , ; ) ( , ; , ' ) , ' , v ( ) v ( ) v ( ) | | , / | | . I t t F t W t I t d p E µ   ∂ + ∇ + =   ∂   + ∂ = = = ∂ ∫ p p x p ω x p p x p x p p x p p p v p p ω p (6) At using boundary conditions ( ) , 0 I t = ω x a t ∈ ∂Ω x a nd 0 < ν ω ( ν is the ext erior n ormal to surface Ω ∂ ), neglecting sc attering the integr ating (6) gi ves: ( ) ( ) v v ( ; ) , [0] , ' ; ( ), ' , v( ) x ω p x ω x y p x y p y x ω y ω p s ds I t F t t d l t e µ − ∞ − = − − = ∫ ∫ , (7 ) where v v( ) A A = p , ] ] , − ∞ x ω x ω m eans th e ray at po int x a nd direct − ω , the integration on , d s d l is y ielded along the direction ω . With account o f th e fi rst order over scatteri ng term we h ave ( ) ( ) [ ( ) 3 v ' ' v v ' ' v ' ' v " ( ; ) ( '; ' ) ' [ ] ( ') ' , ' ; ' ' , ' " ; ' ' ' " ' v ( ) v ( ' ) , ( ', ; , ') '' ' ' ( ' ' ) , ' , '' , s d s s d s w dl t t t d p t t I t F W t t d l t t F e e µ µ − ∞ − ∞ − − ≈ − + − − − −   −    = = ∫ ∫ ∫ ∫ ∫ x ω p x ω x ω x ω x x x x p p x x x ω x ω x ω x ω p p x p x p p x p x   (8) here the integration on , ds dl and ', ' ds dl is yielded alo ng the direction ω and ' ω accordingl y. In general case for t he real mediums which we are goin g to investigate in this pap er the funct ions i N is stochastic value s becaus e of ran domness is unco ntrolled inter actions of consid ering system as well as i nformation losses about micros copic moving at measurement s processe s and the initi al station of the s ystem. How i t is for st andard situation when we do not take into account th e time depen dence in detection process. Thus we can consi der (6) as a stoch astic equation. At least the two com ponent mediu ms ( , ; ) t µ x p and ( , ; , ') W t x p p are s toch astic fun ctions dependin g on i N . Next we wil l b e more int erested in t he ca se when at an y param eters the d ensity of radiation sources i s either l ess then th e d ensity other abso rbing and dissipatin g medium component s, or radiation so urces are outside m edium bou nds. It means t hat we may negl ect b y con tributio n from th ese sour ces to scattering and absorp tion, t his means ind ependence on i N of r andom comp onents of F . Takin g to account t hese views aft er the a veraging (8) o ver th e stochasti c functions i N ( ) ( ) [ ( ) 3 v v v ' ' ' ' v v ' ' ' ' ( ; ) ( ' ; ') ' ' , ; [ ] ' , ' ; ( ' , ' ; , ' ) " ' " ' ' , , s d s s d s d p w d l t t W t t t t dl I t F F t e e µ µ − ∞ − ∞ − − ≈ − + − − −      ∫ ∫ ∫ ∫ ∫ x ω p x ω x ω x ω x x x x p p x x p p x p x p   (9) we can s ee th at essential ch anging h as been taken pl ace i n this appro ach onl y for term includi ng t he a ttenuation coefficient becaus e in general case an average o f exponen t fun ction of a stoch astic value is n ot exp onent function of th e average. Herein after we will study wh at this w ill give for the recons truction of F in practice. Let us conti nue our investigati on of transport model for photons with th e energies (b elow 1 MeV) of int erest in nuclear imaging. Usu ally in su ch meas urements the sources o f isotropi c radiation a re used and we wish to es timate theirs intensit y in general to gether with isot ropic attenu ation coeffici ent whereas for t ransmission stat ement we r econstruct onl y isotropi c att enuation coefficient. On the base of general transpo rt equation (3) the d ynamic transport equ ation can be written in the follo wing form: ( ) ( ) , ' , ( , ; ) , ; ( , ; ) ( , ) ( , , ' ) , ; ' ( , , ' ) ' , ω ω ω x x x x ω ω x ω ω i i j j i j t E I t E F t E t N t E I t E E d µ β ω ∂   + ∇ + = +   ∂   ∑ ∫ (10) where i t means the pro pagation velocit y of p hotons is eq ual to 1. He re ( , ; ) F t E x is the unknown distribution of photons radi ated from the poi nt x and at ti me t , ( , ; ) t E µ x is the isotropi c attenuation coefficient which decomposes fo r th i component of the m edium as , ( , ; ) ( , ) ( ) ( , ) ( , ' , ) ' i i i i i j j t E N t E N t E d µ σ β ω = + ∑ ∫ x x x ω ω in general case with th e unknown densit y ( , ) i N t x of th i component and known terms , ( ) ( , ', ) ' i i j j E E d σ β ω + ∑ ∫ ω ω representi ng all ph y si cal 4 processes which end flights of photons at energy E with i nitial energy ' j E E ≥ . For th e energy relevant to nuclear imagin g these comp rise photo eff ect, Rayleigh scatte ring and Compton scattering [1 3]. We may model scattering kernel for th i component and th j scatt ering mechanis m as , , ( , ; , , ') ( , ) ( , , ') i j i i j W t E N t E β = x ω ω x ω ω . For exampl e Compton scattering is d escribed th e known Klein-Nis hina scatterin g cross s ection with ' ( , , ' ) / 1 ' E E E E m   = −     ω ω ωω , where m is the electron mass, . Notice that the energy of photons used in n uclear i magine is often monoch romatic but only Ra yleigh scat tering do es not cause energ y los ses o f photons. Assu ming th at our detector has enough perf ect e nergy r es olution and we are capab le o f recognize photons w hich have scattered. Rayleigh s cattering can b e ne glected with r espect t o other s cattering mechanism [13]. For m onochromatic sources ( ) ( ) ' , ; , ; ' j I t E I t E ω ω x x ≫ for ' j E E ≠ . Ther efore the scatterin g term we can cons ider as perturbati on wit h re spect t o th e ter m wi th t he attenuation coefficient in (1 0) si nce ( ) ( ) , , ' , ; ( , ' , ) ' ( , , ') , ; ' ( , , ' ) ' i j i j j I t E E d E I t E E d β β ∫ ∫ ω ω x ω ω ω ω ω x ω ω ω ≫ and (4) ar e applied. The soluti on of the simplified dynamic transport equ ation ( ) ( , ; ) , ; ( , ; ) t E I t E F t E t µ ∂   + ∇ + =   ∂   ω ω x x x (11) will be ini tial approach f or solution of ( 10). I mportance of t he account o f th e scatterin g term is gr eater or lesser dependin g on monochro maticity degree of sou rces. The det ector resolution is finite that implies data is collected over energy window 0 E E ± ∆ . Therefor e we observe the valu e ( ) ( ) 0 0 , , ; . E E E E I t I t E dE + ∆ − ∆ = ∫ ω ω x x In (10) we can take into account this integrat ing over the energy for wi ndow 0 E E ± ∆ accounti ng ( ) ( ) 0 0 ( , ) , ( , ; *) , ; , E E E E t I t t E I t E dE µ µ + ∆ − ∆ = ∫ ω ω x x x x where [ ] 0 0 * , E E E E E ∈ − ∆ + ∆ . Let's no te that ( , ) t µ x is stochasti c function and th e more of the ind ependent random factors define the attenuatio n coefficient the more the probabilit y distribution of stochastic compo nent of function ( , ) t µ x is nearer to Gaussian distributi on. Finally we can write th e following stochastic d ynamic transpo rt equation ( ) ( , ) , ( , ) , t I t F t t µ ∂   + ∇ + =   ∂   ω ω x x x (12) where ( , ) t µ x and ( , ) F t x are indepe ndent r and om functions . Wh en we i nter est only ave rage va lues of the distrib ution fu nction fr om ti me 0 t to t (i .e. ( ) ( ) ( ) 0 0 , , , I t I t I t − ω ω ω x x x ≫ ) aft er int egratin g (12) over t we o btai n t ranspo rt equ ation wid espre ad usin g in n uclear im agin g [ ] ( ) 0 0 0 ( ) ( ) , 1 1 ( ) ( , ) , ( ) ( , ) , ( ) ( , * ) , * [ , ] . t t t t I F I I t d t F F t d t t t t t t t µ µ µ ∇ + = = = ∆ ∆ ≡ ∈ ∫ ∫ ω ω ω ω x x x x x x x x x (13 ) The omitt ed t im e d epend ence adds an uncer taint y to t he atte nuati on coe fficient at suc h meas urement s. III. SPEC T, P ET AND X- RAY CT Let u s con tinue our i nvestigat ion s ome stochast ic aspe cts o f tra nsport m odel for problems of ECT and X–ra y CT. In fact th e p roblem of ECT is t he d efiniti on of t he avera ge d ensi ty ρ of radia tion s ource s (w hich does not coin cide with F gener all y spe akin g) du ring the proje ctions measu rement o f distrib ution fu nction of radi ated pa rticles. T he ran dom compo nent in F is det ermined by flu ctuation s of the num ber of emit ted p articles and f luctu ations o f the d ensit y o f radi ation so urces whi ch hap pen for tim e le ss t han t ime of measure ment of t he given pr ojection . F or e mission to mograph y it is cle ar t o suppos e that ρ i s eq ual to the a veraged F wit hin propor tionali ty f act or. The pr oblem of X-ray C T is the definition of average value of ran dom var iabl e µ . Let's s tud y the p roble ms of ECT and X-ra y CT wi th account of erro rs in definiti on µ , st atisti cal hete rogen eity of a me diu m and fluctuation s of number of ph otons which are emitted duri ng the proje ction me asurem ent. In the stati c case when at time of proj ection measur ement th e avera ge va lues of ran dom fu nctions a re chan ging sm all we can u se (13 ). This equ ation is def ined in area n R ⊆ Ω f or an y 1 n S − ∈ ω ( see F ig. 1) . If we con sid er a case o f abs ence of radi ation so urces ou tside of Ω : Ω ω x Fig.1. Sche me of measurem ents and some denotes f or CT. ω is direction in which em itted at x phot ons are detected. 5 ( ) 0 I = ω x , ∈ ∂Ω x , 0 < ν ω , ( ) ( ) , I g = ω x ω x , ∈ ∂Ω x , 0 ≥ ν ω , where ν is th e exterior normal t o s urface Ω ∂ . Without losing general ity we can consider the transmissio n case as distributi on sources which locate at points o f boundar y . When th e function µ ( x ) i s known th e sol ution of (13 ) lo oks like the follo wing in ne glecting scattering: ( ) ( ) exp ( ) ( ) I s ds F d l µ − ∞   = −       ∫ ∫ x ω x ω x ω y x y y (14) where the integrati on on dl ds , is yielded along the directi on ω . Thus in case o f the stat ement of EC T pro blem the random function ( ) , g ω x d escribing num ber of phot ons, whi ch can be detect ed i n a point x a nd wit h velocity i n th e direction ω , expresses th rough random fu nctions F and µ ( ) ( ) , exp . g ds F dl µ −∞   = −       ∫ ∫ x ω x x ω y ω x y ( 15) When t he s ource emits p articles mutuall y in o pposite directions gamma-quant a is detected on coincidence (PET). Then expon ent function c an be taken out from int egral sign ( ) ( ) , exp g ds F d l µ −∞ − ∞   = −     ∫ ∫ x ω x ω x ω x ω ω x y but for X-ra y CT ( ) 0 , exp . g I ds µ −∞   = −     ∫ x ω x ω ω x (16) Furt her for using the proj ection da ta obt ained i n practice we have to average the formul as for projections o ver fluctuations F ( x ) ( ) ( ~ ) ( ) ( x F x F x F + = ) bou nded with qu antum natu re of the radiation process and den sity fluct uations of radiation sourc es during measurement, and over rando m field ( ) µ x ɶ ( ) ( ) ( ), ( ~ ) ( ) ( x x x x x µ µ µ µ µ = + = ). Takin g into account that the average o ver fluct uations of the attenu ation coeff icient does not affect F (how we could see a bove) we shall obt ain , . I I f f F µ ∇ + = = ω ω ω ( 17) Taking i nto acco unt the averaging o ver all r andom parameter s we obtain fr om (14) ( ) ( ) ( ) exp I s ds f dl µ − ∞   = −       ∫ ∫ x ω x ω x ω y x y (18) for ECT an d ( ) ( ) 0 e x p I I s d s µ − ∞   = −     ∫ x ω ω x ω x (19) for X -ray CT. So i t is obtai ned that practicall y we can recons truct aver age densit y o f rad iation sources ( ( ) ( ) c o n s t f ρ ≈ x x ) in CT after certain st atistical processing o f d etection result s which are realizatio n of random f unction ( ) , g ω x . And we have t o t ake into account the distorti on of Radon t ransformatio n be cause of th e averaging over fluctuations ( ) µ x ɶ of t he facto r e x p [ ] d s µ − ∞ − ∫ x ω x ω . In t he case of PET th e factor before f ( x ) is taken out of integral. But for SPECT and X-ray CT we have som ething new - in gener al case (18) and (19) ar e not r educed to Ra don t ransform. F urther we s hall st udy t hose i ntegral transform ations which ta ke in to account rando m components of at tenuation coeffici ent in SP ECT, X-ra y CT and consequ ence of this for nuclear im aging. For full s tatistic description of random f unction µ it is enough to know its ch aracteristic function al: [ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 0 0 e xp , , ' . ' k k k i k x x d x x i k x x x k x k x µ δ µ δ δ µ µ δ δ +∞ −∞ = =   Φ =     Φ = Φ = − ∫ (20) Hence w e obtain: ( ) ( ) { } ( ; , ) exp [ ] , ; , ( ) ( ) ( ) , ' , k i s s ds k s s s s θ µ µ θ δ θ θ ⊥ ⊥ =   − = Φ       = − − − − = ∫ x y x ω y ω x ω y ω x y x ω y ω y ω ɶ (21 ) here and furth er ⊥ a denote s the p rojection of vector a onto pl ane which is perpen dicular to ω . So, it is possible to r ewrite (18) in th e following form 6 ( ) ( ) ( ) ( ) ( ) ( ) ; , [ ; , ] , [ ] ( ) k i s I i s f d l k I f d l i k µ µ θ θ δ µ δ − ∞ = − ∞ = Φ Φ = ∫ ∫ ω ω x ω x ω x ω x ω y ω x ω x x ω y ω y x x y ɶ ɶ (22) Herein after under I ω and ( ) , g ω x we und erstand the values averaged o ver fluctuations F ɶ . It is necessary t o note, that when we can construct c haracteristi c functional for the rando m functi on µ ( x ), (2 2) i s convenient for p ractical definition f ( x ) using obs ervable ( ) , g µ ω x ɶ . At th e same time (17 ) can be written by the followin g form: I I I f µ µ µ µ µ ∇ + + = ω ω ω ω ɶ ɶ ɶ ɶ (23 ) With acco unt of the well known propert y [ ] [ ] ( ) ; , ; ( ) l n ( ) ( ) k k k l k k l k k i k l q s q s i q l δ δ δ δ       Ψ + = Λ Ψ +         Λ = Φ ɶ ɶ ɶ ɶ ɶ (24) we obtain t he following rel ation ( ) ( ) ; ( ) ( ') I I i µ µ δ µ δ µ   = Λ     ω ω x x x x x ɶ ɶ ɶ , ( 25) which allow s to exclude ra ndom function µ ~ from (23) , ( ) ( ) ( ) ; ( ) ( ) ( ' ) I I I f i µ µ µ δ µ δ µ   ∇ + + Λ =     ω ω ω ω x x x x x x x ɶ ɶ ɶ .(26 ) In particular for Gaussian stochastic functio n µ ~ we get ( ) ( ) ( ) ( ) ( ) ( ' ) ' ( ) , ( ' ) n I I I d x f δ µ µ µ δ µ + ∞ − ∞ ∇ + + = ∫ ω ω ω x ω x x x x x x x ɶ ɶ (27) here and f urther we omit index µ ~ in sign of th e averaging. Thus after the averagin g of (13) we have got integro - differential eq uation t hat m eans presence co ordinate derivati ves of ord er n>1 in the equation fo r I ω and consequentl y t he dissipation. Besides there is the renormali zation of a ttenuation co efficient a nd the particles velocit y , th at correspond s t o z ero and first derivati on accordingl y . IV. G AUSSIAN N OISE Let us consider t hat attenuati on coefficient µ is invol ved with flu ctuations w hich are cal led Gaussian noise. The distrib utions with G aussian charact eristic function al m eet in man y si tuations. We s hall use t he most common form of Gaus sian characteristi c functional [14 ] for ( ) x µ ~ : ( ) ( ) ( ) 1 [ ] ex p ' , ' ' 2 k k x k x A x x dx dx   Φ = −     ∫∫ , (28 ) where ( ) ( ) ( ) ' ~ ~ ' , x x x x A µ µ = is cor relation function. I n concret e cases th e form of the f unction A can be d etermined on th e base of the physical situation . So, from (22) we have ( ) ( ) ( ) 1 e x p , ' ' ( ) ( ) , 2 I A s s d s d s s d s f d l µ − ∞   = −       ∫ ∫ ∫ ∫ x ω x x ω ω x ω y y x y x y y (29) where ( ) ( ) 1 1 , ' ' , '' ( ' ) ( ' ' ) ' '' , ' '' 0, ' , ' ' ' . n n A s s A d d s s δ δ ⊥ ⊥ ⊥ ⊥ − ⊥ − ⊥ ⊥ ⊥ ⊥ = − − = = = = − = − ∫ ω z z x z x z z z x ω z ω z ω z ω y ω z ω y ω Further using (26) and (29) we obtain ( ) ( ) ( ) ( ) ( ) ( ') ( ' ) ( ' ) ( ') ' . n I I f I d x µ µ µ θ δ +∞ ⊥ ⊥ −∞ ∇ + = + − − ∫ ω ω ω ω x x x x x x x ω x ω x x x ɶ ɶ (30) It is wel l visible that t he pres ence o f Gaussian n oise component causes the appearance of ( after the a veraging in right side of (13) ) the ter m which has th e same f unctional form as scattering ter m from (3). Fro m (29) it is po ssible to concl ude that we can r estore f ( x ) even at l ocally gr eat magnitu de of Gaussian nois e i f the c orrelation function ( ) , ' A s s ω is known. Let's i llustrate the influence of ra ndom components on the recons truction on example of the simpl e asymptoti c case called Gaussian Marko v process ( "col ored" noise) ( ) ( ) ( ) ( ) 2 2 2 0 , ; ' ' e x p [ | ' | ] , , li m e xp [ | ' | ] 2 ( '), A h h α µ µ µ σ α σ α σ α δ → ∞ = = = − − = − − = − x x x x x x x x x x x ɶ ɶ ɶ (31) where σ i s dis persion of ) ( ~ x µ , 0 ≤ h < ∞ , 1 α is radius correlat ions. At α→∞ , w e o btai n from (30 ) t he equatio n with th e terms of t he ord er whic h is n o less than 2 1 α , 7 2 2 ( ) ( ) ( 1 ) ( ) ( ( ) ) ( ) ( ) . h h I I h I f µ α α − ∇ + − ∇ + − = ω ω ω ω x ω x x x x (32 ) Thus th e d issipati on ( first term (32)) ha s been appeared. We can se e that the renormalizati on o f the attenu ation coefficient to h µ − and the velocity t o 1 / h α − ta kes place in (32). If α >1/ ∆ , wh ere ∆ is necess ary reso lution (i.e. radius of correlation t end to zero and in the limi t t he "c olored" nois e becomes " white" when radius o f correlation is much less o f characteristi c dista nces in the s ystem) and / 1 h α ≪ we o btain the situation t hat been reduc ed t o attenuated Radon transform ation wit h renormali zed attenuation coefficient * h µ µ µ → = − . At other values of α and h we have the cas e which is not reduced to Radon t ransformation with acc ount of the attenuation. Further we will investigate thi s situation i n details. From (29) we obtain ( ) ( ) ( ) ( ) ( ) ( ) ( , ) ( , ) * 1 , e x p [ , ( 1 ) ] ( ) e x p ( 1 ) ] ( ) , s d s s d s g h h f d l f d l e e e e α τ µ α τ µ τ α α ⊥ − ∞ − ∞ ⊥ − − ⊥ − − − ∫ − ∫   = − + −       × = −     × ∫ ∫ x y x y x ω x ω x ω x ω ω x y ω ω x y ω ω x ω x y ω y y (33) here ( ) , τ ⊥ ⊥ + x ω x ω is cross point of th e ray , whi ch is going out fr om ⊥ x i n th e direction ω , with the boundar y of the area Ω at co ndition 0 ≥ ν ω . Let's remark t hat ( ) , 0 τ ⊥ − ≥ ω x y ω thus factor ( ) ( , ) ex p ( 1 ) h e α τ α ⊥ − −   −     ω x y ω ( the cause of t he difference of ( 33) from Radon transform with the attenuation) from the core of the integral tran sformation is l ess or equal to 1. Here it is necessary to note that having resp ect t o physical sense the inequality ( ) * 0 s ds µ ≥ ∫ x y has t o be true. In case of X-ray CT ( ) ( ) ( ) * 0 , , ex p ( 1 )] s ds L h g I e e µ α α −∞ − ⊥ −   ∫ = −     x ω x ω ω x ω x (34) where ( ) , L ⊥ ω x is t he s ize of Ω at ⊥ x in the di rection ω . Using ( 34) we have t o correct the projections then different methods for t he reconstruction can be applied as well as it is in the general case for PET. How w e can see from (19) an d (29) in case of Gaussi an noise for X-ray CT the inverse Radon t ransform with the correction of projections is t he solution of (19) with r espect to renormalized average attenuation coefficient * µ . Wh ereas for SPECT on this way we have obt ained t he el ementary conditions f or t he noise parameters li miting t he application of Radon transformation with the att enuation without the account of stochastic phenomena: the co rrelation radius has to be less than ne cessary resol ution, the square of the product of noise dispe rsion by th e co rrelation rad ius has to be much l ess t han one and i n turn t he i nequality ( ) * 0 s ds µ ≥ ∫ x y has t o be true for self-consis tency of the approach. V. P OINT S OURCE The influen ce of errors upon reconstruction at the definition of the proj ections is i nvestigated quite [3 ], [ 9]. Let's consider the im age reconstruction at the presence of "colored" noise i n attenu ation coefficient on example of t he point sou rce with intensit y 0 ( ( ) ( ) ) I d I δ = − y y y . We shall estimate th e errors bounded w ith unaccounted nois e compone nt in t he case of invers e exponential Radon t ransformation used widely i n practice, whic h in our approach co rrespond to ren ormalized averaged coeff icient * µ . Let 's stop o n the factor ( ) ( , ) e xp ( 1 ) h e α τ α ⊥ − −   −     ω x y ω fr om the core of i ntegral transform ation (33) which i s t he r eason of d ifference (33) from ex ponential Radon t ransformation. In this case for 2 R Ω ⊂ we have for averaged projection data from (33) 0 * ( , - ) 0 0 ( , ) 0 0 ( , ) ( ) ( ) , ( ) exp ( 1 ) , g u I u G h G e e µ τ α τ δ α     ⊥         ⊥ − ⊥ − − = −     = −       ω y ω y ω ω y ω y ω ω ω y ω ω (35) here ( ) , u u τ ⊥ + ω ω ω is cross point with the boundary o f th e x ( , ) u τ ω ω Ω ⊥ ω θ Fig. 2 Coordinat e system and param eters use in cal culations. 8 area Ω of t he r ay wh ich is going out from u ⊥ ω in th e direction ω (see Fig.2). In this case th e invers e transformati on for exponential Radon transf ormation looks like as the foll owing [9] ( ) ( ) 1 1 * 1 * ^ ^ -1 * , 0 * , 0 * ( ){ ( , ) } ( ) , ( ){ ( , ) } ( , ) , 1 1 ( , ) ( , ) , ( 2 ) . 2 2 S i i u R g u d R g u I N d g d g g u du N e e e e e µ µ µ η µ τ µ τ µ η η η η η η π π − − ≤ < + ∞ ∞ −∞ ⊥ ⊥ ⊥ − − = = = = ∫ ∫ ∫ ω y ω ω y ω x ω x ω x ω x x ω ω ω ω ω ɶ (36) It i s well known th at th e d ifferent filters of low freq uency ( ) η Φ [9] are u sed for th e algorith ms o f FBP. T aking into account it w e obtain for p oint source 1 0 0 * ( ) ( ) ( ) ( ) ( ) , lim ( ) ( ), b S b b i d I N d G d d d e e µ η µ η η η ≤ < ∞ →∞ ⊥ − − − = × Φ = ∫ ∫ x y ω x y ω x ω ω x x ɶ ɶ ɶ (37 ) where 1 0 ( ) | | 2 η η π ≤ Φ ≤ and ( ) η Φ =0 for | | b η > . Since 0 ( ) 1 G < ≤ ω this integral sat isfies the followin g inequalit y 1 0 * ( ) ( ) 0 0 | ( ) | | ( ) | | ( ) | , l i m ( ) ( ), b S b b b i d I N d d I R R e e µ η µ η η η δ δ δ ≤ < ∞ → ∞ ⊥ − − − ≤ Φ ≤ − = ∫ ∫ x y ω x y ω x ω x y ɶ (38 ) here R bR bJ R b / ) ( ) 2 ( ) ( 1 1 − = π δ , 0 | | R − = x y . To calculate ( ) b d x ɶ at 1 ( ) | | 2 η η π Φ = and ( ) η Φ =0 f or | | b η > le t’s consid er (37) i n coordinate system (s ee.Fig.2) where 0 0 0 ( ) co s( ) , ( ) si n ( ) ( c o s , s i n ) , ( c o s , si n ). R R R R θ ϕ θ ϕ ϕ ϕ θ θ ⊥ ⊥ − = = − − = = − = − = x y ω x ω x y ω x ω ω x y First we shall inte grate over η usin g identity ( ) s i n ( ) c o s ( ) ( ) , n n i n n n R i R i r J R e e θ ϕ µ θ ϕ η θ ϕ η − − − + − = ∑ ɶ (39) where 1 / 2 2 2 , r η µ η η µ η µ   + = = +   −   ɶ . Furth er, i n (37) the integral s appear th e following t y pes: ( ) ( ) ( , ; ) ( ) b n n n n n K R b J R d µ µ η µ η µ η η η η + + − = ∫ ɶ ɶ , (4 0) it can see m surprising whi ch are explicit ly i ntegrable with Bessel functions. At b >>1 we obta in (1) 2 1 ( 2 ) 2 ( , ; ) ( , ) ( 1 / ) , ( , ; ) ( 1 ) 4 ( ) ( , ) ( 1 / ), m m m b m m K R b K R O b K R b R K R O b µ µ µ π δ µ − = + = − + + (41) where m =1,2… and 1 (1) 3 (1) 2 ( 1) , 1 1 ( 2 ) 2 ( 2 ) 2 , 2 0 ( , ) 2( 2 1 ) ( ) , 1 ( , ) 4 ( ) , m k m m k k m k m m k k K R m R c R R K R m c R R µ µ µ µ µ µ µ − − = − = = − + = + ∑ ∑ 1 2 m- 1 (1) , 1 2 m ( 2 ) , 1 2 !( 2 1 ) ! , ( 1 ) ! ( 1 ) !( 2 ) !( 2 1 2 ) ! 2 !( 2 ) ! . ( ) ! ( 1 ) !( 2 ) !( 2 2 ) ! k m k j m k k m k j m k j m c m k j k m j m j j m c m k j k m j m j − = − − − = − − − = + − + + − − − = + + + − − ∑ ∑ With account o f (41) rec onstructin g function of point s ource at 0 y is 0 2 1 1 ( 1 ) 2 1 1 ( 2 ) 2 1 2 0 2 ( 2 1 ) 2 ( ) [ 2 R e ( ) ] ( ) ( 1 ) ( , ) I m ( ) ( 1 ) ( , ) R e ( ) (1 / ) , 1 ( ) . 2 b b m m m m m m m m m m n i m i m i m i n d I G G R K R G K R G O b G G d e e e e π θ θ θ ϕ δ µ µ ϕ π = + − = = − − = + + − + − + = ∑ ∑ ∑ ∫ x ω ɶ (42) As ( , ; ) 0 lim n n K R b µ →∞ = at l east for 0 < Rb >1 1 ( ) [ ( ) ( ) ] ( ) ( 1 / ) 2 b b d I G G R O b θ θ π δ = + + + x ɶ . (43 ) So, we can see t he decreasing and d ependin g on angl e θ of radiation int ensity of the point source i n comparison with th e initial whi ch is e qual to I . I n additi on the cas es are po ssible when at d efinite angles t he intensit y o f reconst ructed so urce is strongly r educed t hat evidentl y causes fluctuations of brightness on reconstru cting image. It is well known [9 ] i f the nois e is not t aken into account at finite b th e re constructed point s ource has width of order 2 / b π ( b >>1) this is determin ed b y the first term in (43). But with account of the noise th e ad ditional widening of reconstruct ed functi on app ears b ecause o f seco nd term in (43), which at / R b π ≈ is equal to 2 ( ) I M b θ ( M ( θ )<1) on an ord er of ma gnitude. So the contri bution t o d ~ of the first t erm (in decreasin g orde r of Fourier co efficients) i s e qual t o 2 2 2 0. 1 2 R e ( ) i I b e G θ − (s ee (41 ) and (4 2)). Th at i s co mpa rabl e value with the maxim u m of the first term of (43) at R =0, whic h is equal to 2 1 [ ( ) ( ) ] /4 2 I G G b θ θ π π + + . Let 's pay att ention at the fact that be cause of the noise at usin g o f th e met hod of i nvers e pr ojecti on w e c an r eliably resto re m inim um d etails b y si ze D ( )> 2 / b θ π d epen ding o n the angle. T here are the best o f t he resoluti on for centr al ar ea and degrad ation to edge o f i nvesti gated a rea fo r this ki nd of a nois e. In addition the maximal res olution D 2 / b π ≈ is possi ble onl y w hen t he inves tigat ed area is circle with radi ating sou rce in cent re. Whil e at realizatio n of a Nyqu ist conditi on (at the specif ied va lue of dis cretiz ation step / h b π ≤ ) i n prin ciple it is po ssibl e to reco nstr uct det ails of size 2 / b π [9] i ndep endentl y of th e pl ace of radi ating s ource . It is clear that at t he p resence of the s et of point sourc es t he interf erenc e arises beca use of th e dep enden ce on angl e causi ng t he st ructur e di stortion o f ima ge de tails a nd t he resol ution degr adatio n. In som e ca ses the ap pearan ce o f arti facts is p ossibl e. VI. C ONCLUSION We inve stigat ed the transp ort eq uation wh en its solution ca n be expr ess ed b y e xpan si on into a fun ctiona l Ta ylor ser ies i n terms o f the sca tterin g fun ctions. The f unction al d erivati ves are dete rmine d from th e chain of coupl ed equatio ns which have the similar type. Us ing dynamic att enu ated R adon trans form t he e very fun cti onal de rivative is d etermin ed from th e inho mogen eous term which is deri ved fr om t he p revious equ ation . I n detail we con sidered first-or der corre ction on the sc attering t o the standar d Radon tran sforms and th e changi ng of the form atten uate d Rad on t rans form after t h e avera ging o n th e rand om val ues. It has bee n sh own t hat ne glectin g the sc attering we ca n cons ider attenu ation coef fici ent and the dist ribution of emitted particles as t h e ind epend ent st ochas tic fu nctions . For t he case p hotons w ith the en ergies (belo w 1 M eV) o f int erest in nucl ear im agin g the co nditio ns ha s be en anal yzed whe n we can n eglect scatt ering t erm in t he stocha stic i nte gral equ ation , tha t is sat isfied in most condi tions for stat ements of th e CT probl ems i n conse quence o f t he mo noch rome sourc e of phot ons. It i s i nteresti ng t hat averagi ng o f these equ ations over time sm oothe s out th e fluctu ation s of th e flo w d ensit y o f det ected p articles and the de nsit y of r adiati on so urces , but add s u ncer taint y i n a tt enuation coeffici ent. So here is s hown, th at at th e define condi tion s using the reco nstru ction al gorithms it is import ant to t ake i nto acco unt prima rily sto chasti c n ature of at tenu ation coef ficient to obtai n n ecess ary qual ity of ima ge. Th e eq uation (9 ) pro vides a good persp ective fo r the im age reco nstruc tion wit h acc ount of scatteri ng. In addi tion it is necessar y to invest igate ( ) , I t p x with acco unt of th e c ontrib utions of next or ders of function al derivat ives wit h res pect t o the s catterin g fun ction fr om (3) an d t o describ e the con ditions of st aging of prob lems of im age re constr uction , whe n it is impo rtant in appli cations . In this wa y s tud ying nu clear ima ging we h ave o btai ned th at in SPECT, PET o r X-ra y CT on e deals wit h some effective atte nuati on coe ffici ent * µ wh ich l ooks l ike h µ µ ∗ = − in t he el ementa ry cas e of “ whit e” n ois e. Beside s additi onall y fo r “col or” noise th ere a re e ffecti ve ve locit y ( 1 / h α − ) o f p hotons and d issipati on in (32). W e have shown f or E CT that it is pr acticall y useful to di vide atte nuation coeffi cient on weakl y chan ging fu nction in comp arison with a consta nt in a kn own regi on ( or in gen eral ca se the funct ion having be for ehand guess ed f or m) a nd th e sm all add end consi dered as the noise whi ch par ameters c an be e xperi ment ally o r theo retic ally det ermined . P erhaps it is quite enou gh t o estimat e bot h dis persio n and correl ation r adius for pra ctical applicati ons. Wit hin t he fram ewo rk of t he simpl est tr ansp ort m odel (s ee (1 3)) f or P ET t he ac count of st ochastic asp ects can onl y r esu lt in th e c orrectio n of pr ojecti ons a s well as for X-ra y CT in the cas e of th e presenc e of Gaussi an noise in attenuati on coef ficie nt. W he reas for SPECT wit h pres ence of Gaussian noi se in attenuati on coeffici ent the corre ctnes s of t he inverse Rado n tr ansform is d efin ed t he f ollowi ng condit ions: t he cor relatio n radi us is l ess than ne cessar y resolut ion an d the squ are of t he prod uct of noise disp ersion b y corr elati on radius is much l ess than one. In ot her ca ses the devi ation o f th e actual mec hanis m of t he generation of emission data from Rado n tra nsform ation with the atte nuatio n caus es the sen sible di fferenc e of the rec onstruct ed fro m real densi ty o f radi ation sou rces in SP ECT. If t he approxim ation of Gaus sian n ois e is not true th ese diffe renc es can be ess ential for X -ray CT t oo. Besi des th e u naccount ed nois e co mpon ent in attenuati on 10 coefficient ca uses t he distortion of im age t exture and degrades the resolution depending on the boundar y form of investigatio n area an d may be t he so urce of artifacts in some cases. For th e furth er soluti on of CT pr oblems, i t i s perspective to stud y methods of corrective act ions on the b asis of modificati on of the integral transformation for SPECT (15) and for X-ray CT (16) to i mproving th e reconstru ction methods b eing based on inverse integral tr ansformation . In other side we can make correcti ons evidently knowing the expressions for the distortion o f reconstr ucted fun ction for point sou rce such a s (43) when t he approximation of ran dom component s of attenuation coe fficient by Gaussi an Markov rando m field is correct. By the sam e way o f th e study ing of the modification of ot her methods of im age reconstruction is interesting wit h account o f above mention ed in this paper . It i s interesti ng that ther e i s exp licit expressi on for t he integrals in (40 ). I do not find an y consi deration of this integrals f amily elsewhere. A CKNOWLEDGMENT The autho r would like to thank V.V. 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