Explicit Circular Harmonic Inversions of Exponential Radon Transform
Jiangsheng You, Geyang Du, Gengsheng L Zeng, Zhengrong Liang
EExplicit circular harmonic inversion
Cubic Imaging LLC – Preprint 2012
Explicit Circular Harmonic Inversions of Exponential Radon Transform Jiangsheng You , Geyang Du , Gengsheng L Zeng and Zhengrong Liang Cubic Imaging LLC , 18 Windemere Dr., Andover, MA 01810 Department of Mathematics , Peking University, Beijing 100871 Department of Radiology , University of Utah, Salt Lake City, UT 84108 Department of Radiology , State University of New York, Stony Brook, NY 11794 [email protected] [email protected] [email protected] [email protected] Abstract
Using Plemelj formula we obtain three circular harmonic inversion formulas of the exponential Radon transform with complex coefficients. We also derive two different range conditions and prove that Novikov’s range condition does imply the traditional range condition for real coefficients.
I. Introduction and Preliminary
We denote by R the two-dimensional plane with the point representations of ),( yxr in the Cartesian coordinates and ),( rr in the polar coordinates, respectively. We frequently use these notations: )sin,(cosθ , )cos,sin(θ , )1,1( I , }1||:{ rrD , }1||:{ rrS , I and D for the closure of the corresponding open sets, and C for the complex plane. For C , the exponential Radon transform (ERT) of ),( yxf is defined by dtetsfsp t )θθ(),,( . (1.0) The evenness condition of the ERT is expressed as ),,(),,( spsp . (1.1) Without specifically mentioning throughout this paper, we use assumption ( A1 ) stated as (A1) ),( yxf is a smooth complex function with support in D , is a complex constant and its ERT ),,( sp is a smooth function with support in SI . The ERT has many practical applications as discussed in [1-4] and references therein. One of the most important applications is the single photon emission computed tomography (SPECT) [1, 2], in which ),( yxf is the density distribution of the radioactivity tracer injected in the human The paper was previously submitted to
Inverse Problem in 2012 with requesting a revision. Due to schedule conflicts without submitting a revision, the paper was withdrawn. Lately the authors revisited the subject and surveyed recent development in the literature. It is very exciting that there are some new works published by several researchers. Also it is worth noting that the argument of using Plemelj formula, formula (2.1) are new. (2.24), (2.25) and (3.13) are the extensions of exterior formulas by Cormack and Puro to complex coefficients. xplicit circular harmonic inversion
Cubic Imaging LLC – Preprint 2012 body tissue, represents the linear attenuation of human body tissue to Gamma rays and ),,( sp is obtained through modifying the counts of emitted photons reaching to the detector. The goal of SPECT is to estimate ),( yxf from measurement ),,( sp . All the parameters in SPECT take non-negative real values. However, in some other applications such as nuclear magnetic resonance (NMR) imaging, ultrasonic tomography and Doppler tomography described in [3, 4], may take pure imaginary values. Nonetheless, the main task in these applications is to find ),( yxf from measurement ),,( sp provided that is known. Mathematically, this task is equivalent to inverting (1.0), which is often called image reconstruction in tomographic imaging. The ERT can be regarded as a linear operator from functions of D to functions of SI . The range condition is to describe a set of relations under which a function ),,( sp is the ERT of certain ),( yxf . The study of the explicit inversion and range conditions of the ERT has been a research topic for many years from both mathematical interests and practical needs. A substantial review on the known results and open issues can be found in [5, 6] and many review papers from journals such as
Inverse problems , IEEE Transactions on Medical Imaging , and
Physics in Medicine and Biology . The fundamental work on the explicit inversion formulas and range conditions for the ERT are from [7-10]. Recent works [11-14] provide a new way to study this topic, for example solving transport equation is the key technique in Novikov's work. Inspired by Cormack's historical works [15, 16], in this paper, we continue to explore the circular harmonic approach to study the explicit inversion and range conditions of the ERT. The circular harmonics concern the Fourier series expansion of a function with respect to the angular variable. In the expression of the Fourier series expansion, ),( rf and ),,( sp can be written as inn erfrf )(),( , (1.2) inn espsp ),(),,( . (1.3) Throughout this paper ),( sp n always stands for the circular harmonics defined in (1.3). The goal of the circular harmonic approach is to establish explicit relations between )( rf n and ),( sp n . The circular harmonic approach has been investigated in [7-9, 15-20] and many other papers. The relation derived in [7, 8] is through the Fourier transform of ),( sp n with respect to s . An explicit formula was obtained in [9] for the case of )(),( rfyxf . The algorithm in [17] first used the relation in [7, 8] to obtain the circular harmonics of the Fourier transform of ),( yxf and then found )( rf n through inverse Fourier transform. The numerical method in [18] is the direct application of the Fourier series expansion to the convolution kernel without using an explicit formula. In [19], Puro derived two types of closed-form inversion formulas for pure imaginary coefficients. The main idea of [19] is to use complex analysis to calculate the singular integrals. Later on Puro and Garin extended work [19] to the case of complex axially attenuation coefficients in [20]. More work on this subject can be found in [21-24]. In this paper, for the complex , we use Plemelj formula to calculate the singular integrals to derive the inversion formulas. The first type of inversion is the explicit and stable circular harmonic formulas without Chebyshev polynomials by using two equivalent inversion formulas from [4]. The second type of inversion includes two formulas derived by the range condition and orthogonal relations of the exponential Chebyshev functions. We also analyze the relations among several range conditions. For R , we show that Novikov’s range condition yields the traditional range condition in [8, 10]. xplicit circular harmonic inversion Cubic Imaging LLC – Preprint 2012
In Cormack's works [15, 16], the Chebyshev polynomials play a very important role in establishing the relations between )( rf n and )0,( sp n . For n and Rs , let )( sT n and )( sU n stand for the Chebyshev polynomials of the first and second kind, respectively. The detailed orthogonal relations and numerical properties of the Chebyshev polynomials can be found in [25]. To study the ERT, we introduce the exponential Chebyshev functions previously suggested in [19]. For , Rrs , r , integer n and C , the exponential Chebyshev functions ),,(~ rsT n and ),,(~ rsU n are defined as ])()([21),,(~ nrsinrsin r rsser rssersT (1.4) ])()([2),,(~ nrsinrsin r rsser rssers rrsU (1.5) Notice that )0,,(~ rsT n and )0,,(~ rsU n become the conventional Chebyshev polynomials )( rsT n and )( rsU n , respectively. For ),,(~ rsT n and ),,(~ rsU n , we collect several simple relations ),,(~)1(),,(~ rsTrsT nnn , ),,(~),,(~ rsTrsT nn , (1.6) ),,(~)1(),,(~ rsUrsU nnn , ),,(~),,(~ rsUrsU nn , (1.7) .)(),,(~),,(~ nrsinn r rssers rrsTrs rrsU (1.8) Throughout this paper, for integer n we will use the following definition
11 01)sgn( nnn
With aforementioned notations, we state several known facts from [5, 17, 19] as Proposition 1 with proofs. The moment condition is extended to an exterior expression.
Proposition 1 . Under assumption ( A1 ) , )( rf n and ),( sp n satisfy || 22 )(),,(~2),( s nnn rdrrfsr rsTsp , (1.9) ),()1(),( spsp nnn , (1.10) || )sgn( rs nsnim dsspes , r , ||0 nm , (1.11) )sgn( dsspes nsnim , ||0 nm . (1.12) Proof . Both (1.9) and (1.10) have been derived in the early literature, for completeness we provide the proof in this paper. For r , (1.11) was previously derived in [5, 19]. Here we prove that the range condition (1.11) holds for r . The derivation of (1.12) is straightforward and is necessary to derive the inversion formulas in next section. Calculating the Fourier series of (1.0), we obtain (1.9) as follows. xplicit circular harmonic inversion Cubic Imaging LLC – Preprint 2012 ])θθ([])θθ([ ])θθ([),,( dtdetsfedtdetsfe dedtetsfdesp intint intin (1.13) Let str , srt , )/arccos( rs and ),()θθ( rfsrsf , then || 2222|| 220 2220 )(0 2220 220 20 )()( ]),([ ])θθ([])θθ([
22 22 22 22 s nnsrs ninsr ininsr insrint drsr rrfr srise drsr rrfee drsr rderfee srddesrsfedtdetsfe (1.14) || 2222|| 22|| 2220 )(0 2220 220 20 )()( ]),([ ][])θθ([])θθ([
22 22 22 22 s nnsrs ninsrs ininsr insrint drsr rrfr srise drsr rrfee drsr rderfee srddesrsfedtdetsfe (1.15) Combine identities from (1.13) to (1.15) we have || 22|| 222222|| 22222220 )(),,(~2 )( )(),,(),( s nns nnrsinrsis nnsrnsr inn drsr rrfrsT drsr rrfr rsser rsse drsr rrfr sriser srise despsp (1.16) Relation (1.10) is straightforward from (1.6) and (1.9). Let ts , )/acos( s , it follows that cos s and sin t . We rewrite the left hand side of (1.11) as xplicit circular harmonic inversion Cubic Imaging LLC – Preprint 2012 r mnenimiiinmr iniminnr nimininr nimin rs snitminrs nsnim dfdeeee ddeef ddeedef ddefde dsdtestsfde dsspes ni
20 1)sgn(20 ]sincos)[sgn(20 ]cos)sgn([sin20 )(20 cos)sgn(sin20 || )sgn(20|| )sgn( )(])([21 )cos()( )cos(]),([ ])cos)(,([ ])θθ([ ),( )sgn( (1.17) Let i ez , we have the series expansion ))sgn((!1)()( )sgn( k knkmnenimiiin znikzzzeeee ni . (1.18) For ||0 nm , (1.18) only contains either negative or positive powers of z , thus
20 )sgn( )sgn( deeee ni enimiiin . (1.19) Combining preceding relations from (1.17) to (1.19), we obtain (1.11). For ),( sp n , ||0 nm , using the integration by parts, we have )sgn(1)sgn( dsspesnimsdsspes nsnimmnsnim . (1.20) This completes the proof. II. Explicit Inversion Formulas of (1.0)
We will use the complex analysis method to evaluate several singular integrals in deriving explicit inversion formulas of (1.0). For the general knowledge on the complex analysis, we refer to [26]. We also use the Plemelj formula to evaluate the Hilbert transform on the unit circle, we refer to [27] for more details on the Plemelj formula. The main result in this section is the closed-form formulas to express )( rf n as an integral operator of ),( sp n . We also obtain an exterior inversion formula and a new range condition. Theorem 1 . Under assumption ( A1 ) , let ),( sp n be the derivative of ),( sp n with respect to s, then )( rf n can be obtained by the following formula rs nnnnn dssprsTrs rsrdssprsUnrrf || 221 ),(),,(~)sgn(21),(),,(~)sgn(21)( (2.1) Proof . Equation (2.1) closely mimics Cormack’s inversion formula for the Radon transform. We cite two types of inversion formulas from Theorem 2 and Corollary of [5] as follows
20 θ)(θ2 |]),,([41),( ddsspslelerf rlslir . (2.2)
20 θ)(θ2 |]),,([41),( ddsspslelerf rlslir . (2.3) xplicit circular harmonic inversion
Cubic Imaging LLC – Preprint 2012
Rewrite (2.2) to .),(]cos[41 ]),()cos([41 ]),,()cos([41),(
20 )cos(sin2 20 ))cos(()sin(2 20 ))cos(()sin(2 dsspdesree ddsespsree ddsspsreerf ninsririn innsrir srir (2.4) Then (2.4) becomes dsspdeseer eerf ninii resiin i ),(]2)(21[1),(
20 )( . (2.5) Similarly, we rewrite (2.3) as dsspdeseer eerf ninii resiin i ),(]2)(21[1),(
20 )( . (2.6) Define two functions ,1)/(2 )](exp[21 2)(21),,(
220 )( S ninii resin dzzzrsz rzsiri deseer esrK i (2.7) .1)/(2 )](exp[21 2)(21 2)(21),,(
220 )(20 )( S ninii resi inii resin dzzzrsz rzsiri deseer e deseer esrK ii (2.8) With such notations, equations (2.5) and (2.6) become .),(),,(1 ),(),,(1)( dsspsrK dsspsrKrf nn nnn (2.9) For r and Rs we define two functions r rssrsa ),( , (2.10) r rsssrsb )sgn(),( . (2.11) Notice that ),(),( rsarsa nn and ),(),( rsbrsb nn (2.12) )],()][,([ )],()][,([1)/(2 rsbzrsbz rsazrsazzrsz (2.13) To derive closed-form formulas for (2.9) we need evaluate ),,( srK n and ),,( srK n . For this purpose, we rewrite ),,( srK n to two different expressions as xplicit circular harmonic inversion Cubic Imaging LLC – Preprint 2012 ),(1),( )](exp[21 )),(1),(1(2 )](exp[21),,( S nS nn dzzrsbzrsbz rzsiri dzzrsazrsazrs rrzsirisrK (2.14) If rs || , the integrand of (2.14) has only one pole ),( rsb inside the unit disk. If rs || , the integral of (2.14) contains two Hilbert transforms on the unit circle with singular points of ),( rsa and ),( rsa . It is straightforward to verify the following identities ]exp[)],((exp[ rsirsrasi , (2.15) ]exp[)],((exp[ rsirsrasi , (2.16) ])sgn(exp[)],((exp[ rssirsrbsi (2.17) Applying the residual theorem for rs || and the Plemelj formulas (A-6) for rs || , we obtain an explicit expression for ),,( srK n when n .|| ||),,(~)sgn(),,(~ ),,(~21 ||),(])sgn(exp[)sgn( ||)],(),([221),,( rs rsrsTrs rsrsU rsUr rsrsbrssirs rs rsrsaersaers rrsrK nnn n nrsinrsin (2.18) We mention that for n , ),,( srK n has an extra pole at z , (A-6) is no longer applicable to evaluate ),,( srK n . Our method to obtain )( rf n for n is to use ),,( srK n . We rewrite ),,( srK n to ),(1),( )](exp[21 )),(1),(1(2 )](exp[21),,( S nS nn dzzrsbzrsbz rzsiri dzzrsazrsazrs rrzsirisrK (2.19) For n , applying the residue theorem and the Plemelj formulas (A-6) to (2.19), using (2.12), (2.16) and (2.17), we obtain an explicit expression rsrsTrs rsrsU rsrsUr rsrsbrssirs rs rsrsaersaers rr rsrsbrssirs rs rsrsaersaers rrsrK nnn n nrsinrsi n nrsinrsin ||),,(~)sgn(),,(~ ||),,(~21 ||),(])sgn(exp[)sgn( ||)],(),([221 ||),(])sgn(exp[)sgn( ||)],(),([221),,( (2.20) Combining (2.18) for n and (2.20) for n , we have a unified expression as follows xplicit circular harmonic inversion Cubic Imaging LLC – Preprint 2012 .|| ||),,(~)sgn(),,(~)sgn( ),,(~)sgn(21),,(ˆ rs rsrsTrs rsrsUn rsUnrsrK nnnn (2.21) Using (2.9), (2.1) can be symbolically written as .),(),,(ˆ1)( dsspsrKrf nnn (2.22) This completes the proof of (2.1). Remark 1 . As shown in the derivation of (2.18) and (2.20), functions ),,( srK n and ),,( srK n are the Hilbert transforms on the unit circle and can be evaluated by Plemelj formula (A-6). Notice that ),,( srK n and ),,( srK n are the same as ),( srI and ),( srI defined in [19], respectively. In [19], ),( srI and ),( srI are evaluated by the generator of Chebyshev polynomials )(1)/(2 1 n nn zrsUzrsz . (2.23) Series expansion (2.23) converges if z and rs || . For Sz , (2.7) and (2.8) are understood as principal integrals and it does not seem to be obvious that (2.23) can be used to calculate the Hilbert transform on the unit circle for singular points Sr sris . In this paper formula (4.6) in [19] will be proved by using range condition (1.12) and (A-5) instead of (2.23) . Theorem 2 . Under assumption ( A1 ) , let ),( sp n be the derivative of ),( sp n with respect to s, then )( rf n can be reconstructed by the following formulas ,),(),,(~)sgn(21 ),()0,,(~! ))(sgn(21)( || 22 1|| 0 1||)sgn( rs nnrr nnk knksnin dssprsTrs rsr dssprsUk rinerrf (2.24) .),(),,(~)sgn(21 ),()0,,(~! ))(sgn(21)( || 22|| 1|| 0 1||)sgn( rs nnrs nnk knksnin dssprsTrs rsr dssprsUk rinerrf (2.25) Moreover ),( sp n satisfies the following range condition .),()0,,(~! ))sgn((),(),,(~ rr nnk knksninn dssprsUk rniedssprsU (2.26) Proof . Formulas (2.24) was previously derived in [19] with help of (2.23). In this paper we use (A-5) to derive (2.24). From (2.9) we have dsspsrKsrKrf nnnn ),()],,(),,([21)( (2.27) The main task is to obtain an explicit expression of (2.27) through evaluating ),,( srK n for n and ),,( srK n for n . First we consider ),,( srK n for n . In case of rs || , we rewrite (2.7) to xplicit circular harmonic inversion Cubic Imaging LLC – Preprint 2012 ]! )(! )([1)/(2 )exp(21 ])(!1)(!1[1)/(2 )exp(21),,( ||1|| 02 ||1|| 02 S nk nkknk nkkS nnk knk kn dzzk rizk rizrsz siri dzzrzikrzikzrsz sirisrK (2.28) Notice that ||1|| 0 ! )(! )( nk nkknk nkk zk rizk ri is the series expansion of )exp( ri , by (A-5) we obtain .)]0,,(~! )(2),,(~[21 )0,,(~! )(4)],(),([4 )exp( )0,,(~! )(4)],(),([4 )exp( )],(),([! )(2)],(),([! )(4 )exp( )],(),([! )()],(),([! )(4 )exp( ]! )(! )()[),(1),(1(4 )exp( ),,( nk knksin nk knknrsinrsisi nk nkknrsrainrsrai nk nknkkk nknkk nk nknkknk nknkkS nk nkknk nkkn rsUk riersUr rsUk rir rsrsaersaeersr rsi rsUk rir rsrsaersaersr rsi rsarsak rirsarsak rirsr rsi rsarsak rirsarsak rirsr rsi dzzk rizk rirsazrsazrsri rsisrK (2.29) In case of rs || , 0 and ),( rsb are the poles inside the unit circle. If n , by the residue theorem we have ),(])sgn(exp[2 )sgn( ),(),(),( ))],((exp[1 ),(1),( )](exp[21),,( rsbrssirs s rsbrsbrsb rsrbsir dzzrsbzrsbz rzsirisrK nnS nn (2.30) If n , we have an extra term )exp( si because of the pole at 0 )exp(),(])sgn(exp[2 )sgn(),,( sirsbrssirs ssrK n . (2.31) By range condition (1.12) with m , )exp( si has no contribution to (2.9). In summary ),,( srK n of (2.9) for n can be expressed as rs rsrsTrs rsrsU rsUk rinersUrsrK nn nk knksninn || ||),,(~)sgn(),,(~ )0,,(~! ))(sgn(2),,(~21),,(
221 1|| 0 1||)sgn(1 (2.32) Next we consider ),,( srK n for n . If rs || we rewrite (2.8) to xplicit circular harmonic inversion Cubic Imaging LLC – Preprint 2012
10 of 19 ]!)(!)([1)/(2 )exp(21 ])(!1)(!1[1)/(2 )exp(21),,(
102 102
S nk nkknk nkkS nnk knk kn dzzkrizkrizrsz siri dzzrzikrzikzrsz sirisrK (2.33) Notice that ]!)(!)([ nk nkknk nkk zkrizkri is in the format of series expansion, by (A-5) we obtain ].)0,,(~!)(2),,(~[21 )0,,(~!)(4)],(),([4 )exp( )0,,(~!)(4)],(),([4 )exp( )],(),([!)()],(),([!)(4 )exp( ]!)(!)()[),(1),(1(4 )exp( ),,( nk knksin nk knknrsinrsisi nk nkknrsrainrsrai nk nknkknk nknkkS nk nkknk nkkn rsUkriersUr rsUkrir rsrsaersaeersr rsi rsUkrir rsrsaersaersr rsi rsarsakrirsarsakrirsr rsi dzzkrizkrirsazrsazrsri rsisrK (2.34) If rs || , 0 and ),( rsb are two poles inside the unit circle. If n , by the residue theorem we have ),(])sgn(exp[2 )sgn( ),(),(),( ))],((exp[1 1),(1),( )](exp[21 1)/(2 )](exp[21),,( rsbrssirs s dzrsbrsbrsb rsrbsir dzzrsbzrsbz rzsiri dzzzrsz rzsirisrK nS nS nS nn (2.35) For the case of n , we have an extra term )exp( si because of the pole at 0 )exp(),(])sgn(exp[2 )sgn(),,( sirsbrssirs ssrK n . (2.36) By range condition (1.12) with m , )exp( si has no contribution to (2.9). In summary ),,( srK n of (2.9) for n can be defined as rsrsTrs rsrsU rsrsUkriersUrsrK nn nk knksninn ||),,(~)sgn(),,(~ ||)0,,(~!)(2),,(~21),,(
221 1|| 0 1||)sgn(1 (2.37) Combining (2.18), (2.20), (2.32) and (2.37), for all integers n , we obtain rsrsTrs rs rsrsUk rinersrKsrK nnk knksninn ||),,(~)sgn( ||)0,,(! ))(sgn(1),,(),,(
22 1|| 0 1||)sgn( (2.38) xplicit circular harmonic inversion Cubic Imaging LLC – Preprint 2012
11 of 19 rsrsU rsrsUk rinersUrsrKsrK n nk knksninnn ||),,(~ ||)0,,(! ))(sgn(),,(~1),,(),,( (2.39) The combination of (2.27) and (2.38) leads to (2.24). Using range condition (1.12), (2.24) can be rewritten as (2.25). Identity (2.26) is derived by combining (2.9) and (2.39). This completes the proof. Remark 2 . Recently (2.24) was extended to radius-dependent attenuation in [20] but some extra terms are needed to reflect the differentials of backprojection weights. We have not been able to derive simpler formulas like (2.24) by using Plemelj formula.
III. Exterior Inversion Formula of (1.9)
For pure imaginary , in [19] Puro obtained the following exterior formula r nnn dssgsrs rrsTcrf ),(),/(1)( . (3.1) The derivation of (3.1) is through (5.2) of [19]. Following the same idea used in [19], we prove that (3.1) holds for complex constant . Lemma 1 (Orthogonality of exponential Chebyshev functions).
For ts and C , we have ts nn rdrsr rsTrt rtT (3.2) Proof . It is easy to verify the following equation .),,(~),,(~4
22 22)(22 22)( 22 22)(22 22)( nrsrtinrsrti nrsrtinrsrti nn rss rtterss rtte rss rtterss rtte rsTrtT (3.3) In ],[\ ttC , we define a complex function ].[),,,(
22 22)(22 22)( 2222 nzsztinzsztin zss zttezss ztte ztzs ztsz (3.4) In [19] taking the real part of the complex function was used. The definition of (3.4) allows us to avoid taking the real part of ),,,( tsz n in calculating the path integrals. We chose the branch cuts of zt and zs such that ),,,( tsz n is analytical in ]),[],([\ tstsC and iz zsz zt zz limlim , (3.5) )(lim rtirt , trs || , (3.6) )sign()(lim rsrirs , trs || . (3.7) xplicit circular harmonic inversion Cubic Imaging LLC – Preprint 2012
12 of 19
The cuts are described in Fig. 1: Fig. 1 Illustration of cuts for zt and zs . Taking the path integral on the large unit circle along the clockwise orientation, we obtain || ztsz nz , idztsz dz nd || . (3.8) It follows that .4 )],,,(),,,([)],,,(),,,([lim i drtsirtsirdrtsirtsir ts nnst nn (3.9) For ),( tsr , we have ].[ ][ )],,,(),,,([lim
22 22)(22 22)(2222 22 22)(22 22)(222200 nrsrtinrsrti nrsrtinrsrti nn rss rtterss rttersrt r rss rtterss rttersrt r tsirtsir (3.10) For ),( str , we have ].[ ][ )],,,(),,,([lim
22 22)(22 22)(2222 22 22)(22 22)(222200 nrsrtinrsrti nrsrtinrsrti nn rss rtterss rttersrt r rss rtterss rttersrt r tsirtsir (3.11) Combining equations from (3.9) to (3.11), we obtain
00 2222 2222 ts nnst nnts nnts nn drtsirtsirdrtsirtsiri rdrrs rsTrt rtTi rdrsr rsTrt rtT (3.12) -t t -s s 0 rs rs rs rs rt rt rs rt rs rt rt rt xplicit circular harmonic inversion Cubic Imaging LLC – Preprint 2012
13 of 19
This completes the proof. In the case of R , from [28-31], ),( rf can be reconstructed from ),,( sp in ],0[ R , which is often called the half-scan in SPECT. To our knowledge, no result has been obtained to reconstruct ),( rf from ),,( sp in ]2,0[ R or ]2,0[ R . For s , ),( sp n can be obtained from ),,( sp in ]2,0[ R , and ),( sp n can be calculated from ),,( sp in ]2,0[ R by (1.10). Here we prove that (3.1) holds for complex . Theorem 3 . If ),,( sp is a smooth function in SI . Define )( rf n as r nnn dttprt rtTrf ),(),,(~1)( , (3.13) Then )( rf n satisfy (1.9) in ]2,0[ R . Proof . The proof of (3.13) uses the same arguments in [5, 19] as follows ).,( ),(]),,(~),,(~[2 ]),(),,(~[),,(~2 )(),,(~2 sp dttprdrsr rsTrt rtT rdrdttprt rtTsr rsT rdrrfsr rsT n s nts nns r nnns nn (3.14) Thus the ERT of inn erfrf )(),( is ),,( sp on ]2,0[ R . This completes the proof. Puro and Garin extended some inversion formulas of [19] for the imaginary attenuation to the radius-dependent complex attenuation in [20]. Hereafter )( r denotes the radius-dependent complex coefficients. To be consistent with the notations used in this paper, we reformulate several definitions of [20] as follows
10 2222220 22 ))1(()(),,( drsrsidttsrsa sr , (3.15)
10 2||0 2 dssdsssb s , (3.16) ])()([21),,(~ nrsanrsan r rsser rssersT , (3.17) ])()([2),,(~ nrsanrsan r rsser rssers rrsU , (3.18) dtetsfsp tssat ),,()sgn( )θθ(),,( , (3.19) inn espsp ),(),,( , (3.20) || 22 )(),,(~2),( s nnn rdrrfsr rsTsp . (3.21) xplicit circular harmonic inversion Cubic Imaging LLC – Preprint 2012
14 of 19
Here we point out that both (3.15) and (3.16) are changed to path integrals so that ),,( rsa and ),( sb can be easily extended to complex functions in C if )( r is extended as a complex function. Next we present our theorem 4 with sketchy proof. Theorem 4 . Assume that )( r is a complex smooth function on I and can be extended to an analytical function )( z in ],[\ ttC and continuous on C with the following growth condition zsazta when || z . (3.22) Then for ts , we have ts nn rdrsr rsTrt rtT , (3.23) and )( rf n satisfies (3.21) in ]2,0[ R , here )( rf n is defined as r nnn dttprt rtTrf ),(),,(~1)( . (3.24) Proof . The arguments mainly follow the proof of Lemma 1 and Theorem 3. We only provide the key steps. From the assumption, )( r can be understood as the restriction of a continuous function )( z on I , thus )( z is continuous in the region that contains ],[ tt . In ],[\ ttC , we modify the complex function of (3.4) as ].[),,,(
22 22)],,(),,([22 22),,(),,( 2222 nzsaztanzsaztan zss zttezss ztte ztzs ztsz (3.25) We choose the same cuts as for ),,,( tsz n satisfying conditions from (3.5) to (3.7). The growth condition (3.22) and continuity of )( z on C yield (3.8) and (3.12) for ),,,( tsz n . By using the same arguments in the proof of Lemma 1 and Theorem 3, we obtain (3.23) and (3.24). This completes the proof. According to the calculations in [20], Novikov’s inversion formula can be rewritten as
20 θ),,(2 |]),,()),(),(cos([41),( ddsspsl sblbelrf rlrla . (3.26) Compared with (2.2, 2.3), (3.26) requires evaluating the differentials of ),,( rla and ),( sb . If )( r is not a constant, (2.2) and (2.3) don’t hold due to the differentials of the backprojection weights ),,( rla e and convolutional kernel )),(),(cos( sblb , then formula (3.10) of [20] includes extra terms. In [20], the generating series (2.23) was used to calculate the singular integrals in the derivation. After some calculations, we have not been able to obtain simplified inversion formulas like (2.1) and (2.24) for the radius-dependent complex coefficients )( r . IV. Range Conditions
A review on the range conditions for the attenuated Radon transform can be found in [6] and the references therein. The classic works on the range condition for the Radon transform are from [32, 33]. The range condition includes the evenness condition (1.10) and the moment condition (1.11). We point out that (1.11) is new for r . The sufficiency of the combination of (1.10) and (1.11) xplicit circular harmonic inversion Cubic Imaging LLC – Preprint 2012
15 of 19 has been proved for the case of . However, if , it has not been proved that the evenness condition (1.10) and moment condition (1.11) are sufficient to uniquely reconstruct ),( rf from ),,( sp . Finch mentioned that it is desirable to have a simple proof to show that Novikov’s range condition can yield the existing range conditions for the (exponential) Radon transform. In this paper, for R , we will prove that Novikov’s range condition does imply the existing range conditions. Recall that Novikov’s range condition is expressed in an integral identity by
20 θ ddsspsr srie r . (4.1) A direct proof of (4.1) can be found in [4] and the sufficiency of (4.1) was proved in [12] in the context of attenuated Radon transform. Subtracting (2.3) from (2.2), we obtain
20 θ ddsspsr srie r . (4.2) From the proof of Theorem 2, (2.26) is the circular harmonic expression of (4.2). So far, we have three range conditions of (1.11), (4.1) and (4.2) derived through different methods. It may be interesting to investigate the relations among these range conditions. In the circular harmonic expression, by (2.38), Novikov’s condition (4.1) becomes .),()0,,(! ))sgn((),(),,(~)sgn( rr nnk knksnirs nn dssprsUk rniedssprs rrsTs (4.3) Under condition ( A1 ), if r , the left hand side of (4.3) becomes zero, then we have
11 1|| 0 1||)sgn( dssprsUk rnie nnk knksni . (4.4) We rewrite (4.4) in the series of q r , ||11|| nqn , as follows qn nq q rc , r . (4.5) It follows that q c . Comparing the coefficients in (4.4), if , we obtain
11 )sgn( dsspes nsnim , ||0 nm . (4.6) This implies that under condition ( A1 ), (4.1) leads to the moment condition (1.11) for r . Let ),,(~ p be the Fourier transform of ),,( sp with respect to s dsespp is ),,(),,(~ . (4.7) Expanding ),,(~ p in its circular harmonics, we have inn epp ),(~),,(~ . (4.8) From [7, 8], we have the following evenness identity ),(~)(),(~)( nnnn pp , |||| . (4.9) The sufficiency of (4.9) was proved in [10] for R . Next we prove that for R , (4.1) does yield the evenness condition (4.9). Theorem 5
Assume that R , ),,( sp meets (4.1) and ),,( sp is continuously differentiable with support in SI , then ),(~ n p satisfies (4.9). xplicit circular harmonic inversion Cubic Imaging LLC – Preprint 2012
16 of 19
Proof . In the distribution sense, the Fourier transforms of si e and s are )(2 and i , respectively. The convolution of )( and )sgn( is )sgn( . Using the polar coordinate expressions of r , )cos(θ rr and )sin(θ rr , we obtain |||| )cos( )cos( ),,(~)sgn( ),,(~))sgn()(sgn(2 ),,(θ ))θ(cosh( depi depi dsspsr sri ri ri (4.10) Substitute (4.10) in (4.1) and perform the Fourier series expansion with respect to , we have
20 |||| )cos()sin(20 ddepede ririn . (4.11) Equation (4.11) is equivalent to |||| 20 )cos(sin dpde nrnir . (4.12) We cite one relation from the Appendix in [8] as follows )()(21 abJab baide nnnbnia , |||| ab . (4.13) With (4.13), (4.12) becomes || 222222 drJpp nnnnn , (4.14) where )( n J is the n th-order Bessel function. For more details of the Bessel functions and Hankel transform, we refer to [34]. Changing variable , (4.14) becomes .)(1),(~)( )(1),(~)( drJp drJp nnn nnn (4.15) Applying the n th-order Hankel transform to (4.15) with respect to variable r , we have ),(~)(),(~)( nnnn pp . (4.16) Let for |||| , (4.16) becomes ),(~)(),(~)( nnnn pp . (4.17) This completes the proof. V. Conclusion and Discussion
With the help of Plemelj formula, we have derived several explicit inversion formulas (2.1), (2.24) and (2.25) based on the inversion formulas of [4] for the complex constant . Formula (2.1) can be regarded as the circular harmonic expression of two FBP formulas in [4]. Formula (2.25) is in the expression of exterior formula and may be used to reconstruct the function if only exterior data xplicit circular harmonic inversion Cubic Imaging LLC – Preprint 2012
17 of 19 is available. The exterior formula (3.13) was first proved by Puro for pure imaginary and in this paper it is proved to hold for complex constant and the radius-dependent complex attenuation )( r . The formula (3.13) R did not appear in the literature from our knowledge. There are four different range conditions. In Section IV, we analyzed the relations among multiple range conditions and then used the circular harmonic expansion to show that Novikov’s range condition (4.1) yields the traditional range condition (4.9) if R . It is desired to further investigate the relations among these range conditions. Appendix.
Properties of the Plemelj formulas
Assume that )( is a Hölder continuous function on S . We define )( z as the Cauchy integral )(21)( S dziz . (A-1) It is easy to see that )( z is analytical in \ SC . Denote by )( z and )( z the restrictions of )( z inside and outside of S , respectively. On S , (A-1) is understood as the principal integral. We expand )( in the Fourier series nn a )( , S . (A-2) Three properties of the Plemelj formulas are summarized below: )( n nn zaz , z . (A-3) )( n nn zaz , z . (A-4) )(21)(21 n nnn nnS aadi , S . (A-5) For the details on the derivation of (A-3), (A-4) and (A-5) and other properties about the Plemelj formulas, we refer to [27]. Let )( zf be analytical in a region that contains D , for Sz , an immediate result from (A-5) is )(21)(21 zfdzfi S . (A-6) xplicit circular harmonic inversion Cubic Imaging LLC – Preprint 2012
18 of 19
References [1]
Anger HO, “Scintillation camera,”
Rev. Sci. Instrument , vol. 29, pp. 27-33, 1958. [2]
Muehllehner G and Wetzel RA, “Section imaging by computer calculation,”
J. Nucl. Med. , vol. 12, pp. 76-84, 1971. [3]
Natterer F, “Inversion of the Fourier transform without low frequencies,” Preprint, 2007. [4]
You J, “The attenuated Radon transform with complex coefficients,”
Inverse Problems, vol. 23, pp. 1963-1971, 2007. [5]
Natterer F,
The Mathematics of Computerized Tomography , Wiley-Teubner, New York, 1986. [6]
Finch D, “The attenuated X-Ray transform: Recent developments,”
Inside Out: Inverse Problems and Applications , Uhlmann, G, Ed., Cambridge University Press, 2003. [7]
Bellini S, Piacenti M, Caffario C, and Rocca F, “Compensation of tissue absorption in emission tomography,”
IEEE Trans. on Acoust., Speech, Signal Processing , vol. 21, pp. 213-218, 1979. [8]
Tretiak O and Metz CE, “The exponential Radon transform,”
SIAM J Appl Math , vol. 39, pp. 341-354, 1980. [9]
Clough A and Barrett HH, “Attenuated Radon and Abel transforms,”
J. Opt. Soc. Amer. , vol. 73, pp. 1590-1595, 1983. [10]
Kuchment PA and L’vin SY, “Paley-Wiener theorem for the exponential Radon transform,”
Acta Appl Math vol. 18, pp. 251-260, 1990. [11]
Novikov RG, “An inversion formula for the attenuated X-ray transformation”, (
Preprint in May, 2000 ) Ark. Math. , Vol. 40, pp. 145-167, 2002. [12]
Novikov RG, “On the range characterization for the two-dimensional attenuated X-ray transformation,”
Inverse Problems , 18 (2002), 677-700. [13]
F. Natterer, “Inversion of the attenuated Radon transform,”
Inverse Problems , vol. 17, pp. 113-119, 2001. [14]
J. Boman and J. O. Strömberg, “Novikov’s inversion formula for the attenuated Radon transform – A new approach,”
Journal of Geometric Analysis , vol. 14, pp. 185-198, 2004. [15]
Cormack AM, “Representation of a function by its line integrals, with some radiological applications,”
J. Applied Physics , vol. 34, pp. 2722-2727, 1963. [16]
Cormack AM, “Representation of a function by its line integrals, with some radiological applications, II,”
J. Applied Physics , vol. 35, pp. 2908-2913, 1964. [17]
Hawkins WG, Leichner PK and Yang NC, “The circular harmonic transform for SPECT reconstruction and boundary conditions on the Fourier transform of the sonogram,”
IEEE Trans. Med. Imaging , vol. 7 pp. 135–148, 1988. [18]
You J, Liang Z, and Zeng GL, “A unified reconstruction framework for both parallel-beam and variable focal-length fan-beam collimators by a Cormack-type inversion of exponential Radon transform,”
IEEE Trans. Med. Imaging , vol. 18, pp. 59–65, 1999. [19]
Puro A, “Cormack-type inversion of exponential Radon transform,” Inverse Problems , vol. 17, pp. Puro A and Garin A, “
Cormack-type inversion of attenuated Radon transform, ” Inverse Problems , vol. 29, 065004, 2013. [21] Wen J. and Liang Z, "An inversion formula for the exponential Radon transform in spatial domain with variable focal-length fan-beam collimation geometry," Med Phys. Vol. 33, 792–798, 2006. xplicit circular harmonic inversion
Cubic Imaging LLC – Preprint 2012
19 of 19 [22]
Yarman CE and Yazıcı B, "A new exact inversion method for exponential Radon transform using the harmonic analysis of the Euclidean motion group," Inverse Problems and Imaging, vol. 1, No. 3, 457–479, 2007. [23]
Rigaud G , “
On the inversion of the Radon transform on a generalized Cormack-type class of curves, ” Inverse Problems , vol. 29, , 2013. [24] Andersson F, Carlsson M and Nikitin VV , “
Fast Laplace transforms for the exponential Radon transform, ” J. Fourier Anal Appl. Vol. 24, pp. 431–450, 2018 . [25]
Mason JC and Handscomb DC,
Chebyshev Polynomials , CRC Press, 2003. [26]
Conway JB,
Functions of One Complex Variable, (2nd ed.). New York: Springer-Verlag, 1995. [27]
Muskhelishvili NI,
Singular Integral Equations: boundary problems of function theory and their applications to mathematical physics , Wolters-Noordhoff, The Netherlands, 1953. [28]
Noo F and Wagner JM, “Image reconstruction in 2D SPECT with 180 o acquisition,” Inverse Problems , vol. 17, pp. 1357-1371, 2001. [29]
H. Rullgård, “An explicit inversion formula for the exponential Radon transform using data from 180 degrees,” (
Preprint in Sep, 2002 ) Ark. Math. , vol. 42, pp. 353-362, 2004. [30]
Noo F, Defrise M, Pack JD and Clackdoyle R, “Image reconstruction from truncated data in SPECT with uniform attenuation,”
Inverse Problems , vol. 23, pp. 645-668, 2007. [31]
Huang Q, You J, Zeng GL, and Gullberg GT, “Reconstruction from uniformly attenuated SPECT projection data using the DBH method,”
IEEE Trans. Medical Imaging , vol. 28, pp. 17-29, 2009. [32]
Helgason S, “The Radon transform on Euclidean spaces, compact two-point homogeneous spaces and Grassmann manifolds,”
Acta Math . vol. 113, pp. 153-180, 1965. [33]
Solmon DC, “Asymptotic formulas for the dual Radon transform and applications,”
Math. Z . vol. 195, pp. 321-343, 1987. [34]
Piessens R, The Hankel Transforms, in