Anoiseproperty analysis of single-photon emission computed tomography data

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1 INSTITUTE OF PHYSICSPUBLISHING Inverse Problems 20 (2004) INVERSE PROBLEMS PII: S (04) Anoiseproperty analysis of single-photon emission computed tomography data J-PGuillement and R G Novikov CNRS, UMR 6629, Département de Mathématiques, Université denantes, BP 92208, F-44322, Nantes Cedex 03, France guillement@math.univ-nantes.fr and novikov@math.univ-nantes.fr Received 8 July 2003, in final form 13 October 2003 Published 28 November 2003 Online at stacks.iop.org/ip/20/175 (DOI: / /20/1/011) Abstract We give formulae describing the simplest statistical properties of single-photon emission computed tomography (SPECT) data modelled as the attenuated ray transform with Poisson noise. To obtain some of these formulae we obtain and use an inequality relating the even and odd parts of the attenuated ray transform without noise. Precise equations relating the even and odd parts of this transform are also discussed. Using these results we proposenew possibilities for improving the stability of SPECT imaging based on the explicit inversion formula for the attenuated ray transformation with respect to the Poisson noise in the emission data. Numerical examples illustrating some of the theoretical conclusions of the present work are given. 1. Introduction We consider the two-dimensional (2D)attenuated ray transformation P a defined by the formula P a f (γ ) = exp( Da(sθ + tθ,θ)) f (sθ + tθ)dt, Da(x,θ)= R γ = (s,θ) R S 1, θ = ( θ 2,θ 1 ) for θ = (θ 1,θ 2 ) S 1, (1.1a) + 0 a(x + tθ)dt, (x,θ) R 2 S 1, (1.1b) where a and f are real-valued, sufficiently regular functions on R 2 with sufficient decay at infinity, a is a parameter (the attenuation coefficient), Da is the divergent beam transform of a, f is a test function. In this definition, we interpret R S 1 as the set of all oriented straight lines in R 2.Ifγ = (s,θ) R S 1,thenγ ={x R 2 x = sθ + tθ,t R} (modulo orientation) and θ gives the orientation of γ. The transformation P a is abasic transformation of the single-photon emission computed tomography (SPECT); see, for example [NaW] /04/ $ IOP Publishing Ltd Printed in the UK 175

2 176 J-P Guillement and R G Novikov In SPECT (as soon as the problem is restricted to a fixed two-dimensional plane identified with R 2 ), f 0isthe density of emitters of photons (the radionuclide distribution), a 0is the linear photon attenuation coefficient of the medium, and (in some approximation) CP a f is theexpected emission data (the expected sinogram), where C is a positive constant depending on detection parameters. More precisely, considering the emission data in 2D SPECT, we assume that a(x) 0, f (x) 0 for x R 2, a(x) 0, f (x) 0 for x R (1.2) and consider in R S 1 adiscrete subset Ɣ of the form Ɣ ={γ ij = (s i,θ(ϕ j )) s i = R + (i 1) s,ϕ j = ( j 1) ϕ, s = 2R/(n s 1), ϕ = 2π/n ϕ, i = 1,...,n s, j = 1,...,n ϕ }, (1.3) where θ(ϕ) = (cos ϕ,sin ϕ), R is the radius of the image support of (1.2), n s, n ϕ are sufficiently large natural numbers, and n ϕ is even. We say that Ɣ is a detector set. In 2D SPECT, in some approximation, the emission data for the detector set Ɣ are given by a function p on Ɣ, where p(γ ) is a realization of a Poisson variate p(γ ) with the mean Mp(γ ) = CP a f (γ ) for any γ Ɣ (1.4) and all p(γ ), γ Ɣ, are independent. In addition, we assume that C = C 1 t,wheret is the detection time per projection and C 1 is independent of t. Wesaythat the aforementioned p is the attenuated ray transform CP a f on Ɣ with the Poisson noise. We consider the following inversion formula for the transformation P a : Cf = N a g, (1.5) where g = CP a f, N a q(x) = 1 ( K (x,θ)θ 2 dθ + ) K (x,θ)θ 1 dθ, (1.6a) 4π x 1 S 1 x 2 S 1 K (x,θ)= exp[ Da(x, θ)] q θ (xθ ) (1.6b) q θ (s) = exp( θ (s)) cos( θ (s))h (exp( θ ) cos( θ )q θ )(s) +exp( θ (s)) sin( θ (s))h (exp( θ ) sin( θ )q θ )(s), (1.6c) θ (s) = 1 2 Pa(s,θ), θ(s) = H θ (s), q θ (s) = q(s,θ), (1.6d) where q is a test function, P = P 0 is the classical two-dimensional ray transformation (i.e. P 0 is defined by (1.1a)with a 0), H is the Hilbert transformation defined by the formula Hu(s) = 1 π pv u(t) dt, (1.7) R s t where u is a test function, x = (x 1, x 2 ) R 2, θ = (θ 1,θ 2 ) S 1, θ = ( θ 2,θ 1 ), s R,dθ is the standard element of arc length on S 1. In a slightly different form (using complex notation) the formula (1.5) was obtained in [No1]. The formula (1.5) was successfully implemented numerically in [Ku2] via a direct generalization of the (classical) filtered back-projection (FBP) algorithm (see [Na1] for the description of the classical FBP algorithm). In a similar way, the formula (1.5) was also implemented numerically in [Na2]. However, the aforementioned generalized FBP algorithm turned out to be less stable, in general, than its classical analogue. The first application of

3 A noise property analysis of SPECT data 177 the formula (1.5) to real SPECT imaging was given in [GJKNT] using the generalized FBP algorithm of [Ku2] and some techniques of [Ku2] and [GJKNT] for improving the stability of this algorithm with respect to the Poisson noise in the emission data g. In the present work we continue the studies of [Ku2, No2, GJKNT] on improving the stability of SPECT imaging based on the formula (1.5) with respect to the Poisson noise in the emission data g. To this end we develop a noise property analysis of the SPECT data modelled as p of (1.4). More precisely, the studies of the present work can be summarized as follows. We give formulae describing simplest statistical properties of the SPECT data modelled as p of (1.4) (see sections 2 and 5). To obtain some of these formulae we get and use aninequality relating the even and odd parts of the transform P a f without noise (see sections 2, 3 and 5). Using results of [No1, No2] on the range characterization of P a,wealso discuss precise equations relating the even and odd parts of P a f (see section 4). Using these formulae, inequality, and equations, we proposenew possibilitiesfor improving the stability of the reconstruction of Cf from p via (1.5) with respect to the Poisson noise in p (see sections 2 and 6). Finally, we give numerical examples illustrating some of the theoretical conclusions of the present work (see section 7). A detailed presentation of the main results of the present work is given insection2. As regards other results given in the literature for the transformation P a in the framework of SPECT imaging, see [SV, Na1, MIMIKIH, Ku1, HL, Br1, KLM, MP, ABK, Br2, No1, Ku2, NaW, No2, GN, PSKZM, SP, GJKNT] and references therein. As regards other results given in the literature on the noise property analysis of images subject to Poisson statistics, see, for example, [BCB] and references therein. 2. Main results Let Mξ, Dξ = M(ξ Mξ) 2,andVξ = (Dξ) 1/2 / Mξ denote the mean, dispersion, and variation coefficient (respectively) of a variate ξ. (Weassume that V ξ = + if Mξ = 0.) Let ( q L α (Ɣ) = s ϕ ) 1/α q(γ ) α, (2.1) γ Ɣ where q is a test function on Ɣ defined by (1.3), α N. Let ζ = p Mp L 2 (Ɣ), (2.2) Mp L2 (Ɣ) where p is the Poisson field of (1.4). (We assume that Mp 0.) We say that ζ is the noise level for p. Let us start with the following formulae (obtained in section 5 of the present work): M p L1 (Ɣ) = g L1 (Ɣ), D p L 1 (Ɣ) = s ϕ g L 1 (Ɣ), (2.3a) (2.3b) M p 2 L 2 (Ɣ) = g 2 L 2 (Ɣ) + g L 1 (Ɣ), D p 2 L 2 (Ɣ) = s ϕ(4 g 3 L 3 (Ɣ) +6 g 2 L 2 (Ɣ) + g L 1 (Ɣ)), (2.4a) (2.4b) Mζ 2 = g L 1 (Ɣ) g 2, (2.5a) L 2 (Ɣ)

4 178 J-P Guillement and R G Novikov Dζ 2 = s ϕ(2 g 2 L 2 (Ɣ) + g L 1 (Ɣ)) g 4, L 2 (Ɣ) (2.5b) V ζ 2 = ( s ϕ(2 g 2 L 2 (Ɣ) + g L 1 (Ɣ))) 1/2, g L1 (Ɣ) (2.5c) where g = Mp = CP a f.(weassume that g 0onƔ.) In a typical situation arising in SPECT, g has such propertiesthat V p α L α (Ɣ) 1, α {1, 2}, (2.6a) V ζ 2 1, (2.6b) g α L α (Ɣ) < g α+1 L (Ɣ), α {1, 2}. (2.7) In particular, for the numerical example considered in section 7, we have that V p L 1 (Ɣ) = , V p 2 L 2 (Ɣ) = , V ζ 2 = , and g L1 (Ɣ), g 2 L 2 (Ɣ), g 3 are in the ratio L 3 (Ɣ) of approximately 1:11:148. The assumption (2.6b)and Chebyshev s inequality written in the form (V ξ)2 Prob{ ξ Mξ >ε Mξ } ε 2 (2.8) imply that ζ (Mζ 2 ) 1/2 with a probability close to 1. (2.9) In addition, the assumptions (2.6a) forα {1, 2}, (2.7) for α = 1, and Chebyshev s inequality (2.8) imply that ( ) (Mζ 2 ) 1/2 p 1/2 L1 (Ɣ) p 2 L 2 (Ɣ) p L 1 (Ɣ) with a probability close to 1. (2.10) In particular, for the numerical example considered in section 7, we have that (Mζ 2 ) 1/2 = 0.3, ζ = 0.298, and the right-hand side of (2.10) is equal to We use (2.9), (2.5a) for modelling SPECT data with a given noise level. We use (2.10), (2.9) for determining the noise level ζ of given SPECT data. In the SPECT imaging considered in the present work and based on the formula (1.5), we use the noise level ζ of the emission data for determining filter parameters for these data (see equations (2.24), (2.27)). To obtain further formulae describing statistical properties of SPECT data modelled as p of (1.4), we obtain and use the following basic inequality for the transform P a f without noise. Suppose that a γ L 1 (γ, [0, + [), f γ L 1 (γ, [0, + [) (2.11) for some γ R S 1.Then Pa f (γ ) ρ a (γ )P a + f (γ ), (2.12) where P a ± f (γ ) = P a f (γ ) ± P a f ( γ), (2.13) γ = ( s, θ) for γ = (s,θ), (2.14) 1 exp( Pa(γ )) ρ a (γ ) = 1+exp( Pa(γ )), (2.15) where P = P 0 is the classical two-dimensional ray transformation. This inequality is proved in section 3.

5 A noise property analysis of SPECT data 179 If a γ 0, then the inequality (2.12) implies that Pa f (γ ) = 0. Previously, the fact that a γ 0 Pa f (γ ) = 0wasobserved and used, in particular, in [PSKZM, SP]. As a rule, if (2.11) holds and a γ 0, f γ 0, then Pa f (γ ) is considerably smaller than ρ a (γ )P a + f (γ ) and, roughly speaking (especially if a γ is not strong), one can say that Pa f (γ ) P a + f (γ ). (2.16) If P a + f (γ ) and P a f (γ ) are known, then (2.12) gives an estimate for ρ a (γ ), and, as a corollary of (2.15), gives an estimate for Pa(γ ). Therefore, (2.12) can be used also in the framework of the identification problem consisting in (approximately) finding a from P a f. The identification problem is considered, in particular, in [Na1, Br1, Br2, NaW, GN]. In section 4, in addition to the inequality (2.12), we discuss exact equations relating Pa f and P a + f on R S1.Theseequations follow immediately from results of [No1, No2] on the range characterization for the transformation P a. In section 5, using the inequality (2.12) we obtain, in particular, the following statistical formulae. Suppose that (2.11) holds for some γ Ɣ (defined by (1.3)). Then V p + (γ ) ρ a (γ )V p (γ ), (2.17) ( ) 1 ρa (γ ) 1/2 ( ) V p(γ ) V p + 1+ρa (γ ) 1/2 (γ ) V p(γ ), (2.18) 2 2 where p ± (γ ) = p(γ ) ± p( γ). (2.19) (We assume that ρ a (γ )V p (γ ) = + if V p (γ ) = + and ρ a (γ ) = 0.) In a completely similar way to (2.12), (2.16), if (2.11) holds and a γ 0, f γ 0, then, as a rule, V p + (γ ) is considerably smaller than ρ a (γ )V p (γ ) and V p + (γ ) is considerably closer to (V p(γ ))/ 2than follows from (2.18) and, roughly speaking (especially if a γ is not strong), one can say that V p + (γ ) V p (γ ), (2.20a) V p + (γ ) (V p(γ ))/ 2. (2.20b) The inequalities (2.17), (2.20a) showthat the even part p + of the emission data p of (1.4) is considerably less sensitive to the Poisson noise in p than the odd part p of p,where p ± (γ ) = p(γ ) ± p( γ), γ Ɣ. (2.21) In addition, the formulae (2.18), (2.20b)showthat p + is about 2 times less sensitive to the Poisson noise in p than the data p themselves. In section 5, we also consider ζ ± = p± Mp ± L2 (Ɣ). (2.22) Mp ± L 2 (Ɣ) We say that ζ ± is the noise level for p ±. We give for p +, ζ + the formulae (5.11) (5.13), (5.26), (5.27) similar to (2.3) (2.5), (2.9), (2.10). We give for p, ζ the formulae (5.14), (5.15), (5.28). In addition, we give for (M(ζ ± ) 2 ) 1/2, (Mζ 2 ) 1/2 the formulae (5.29), (5.30), (5.33) similar to (2.17), (2.18), (2.20). To continue the presentation of the main results of the present work, we now introduce some additional notation. Consider X ={x ij x ij = ( R + (i 1) s, R + ( j 1) s), s = 2R/(n s 1), i = 1,...,n s, j = 1,...,n s }, (2.23) where n s and R are the same as in (1.3).

6 180 J-P Guillement and R G Novikov Given q on Ɣ, weassume that N a q is defined on X and denotes a numerical realization of (1.6) (as in [Ku2, Na2] without any regularization). Consider a low-frequency filter W defined by Wq(s,θ(ϕ)) = 1 Ŵ (k, l) ˆq(k, l)e isk e iϕl dk, s R, ϕ [0, 2π], 2π R l Z ˆq(k, l) = 1 2π (2.24) q(s,θ(ϕ))e isk e iϕl dϕ ds, k R, l Z, 2π R 0 where q is a test function on R S 1, θ(ϕ) = (cos ϕ,sin ϕ), Ŵ(k, l) approximates to 1 for small (k, l) and Ŵ (k, l) approximates to 0 for large (k, l), where is an appropriate norm. (It is not assumed that W is an ideal low-frequency filter.) Given q on Ɣ, weassume that Wq is defined on Ɣ and denotes a numerical realization of (2.24) (by means of 2D discrete Fourier transformation). Given q on Ɣ, we assume that W ω q is defined as Wq,where Ŵ (k, l) = Ŵ ω (k, l), (2.25) ( ( ) ( )) πk πl 2 Ŵ ω (k, l) = sinc sinc for k ωk nyq, l ωl nyq, ωk nyq ωl nyq (2.26) Ŵ ω (k, l) = 0 for k ωk nyq or l ωl nyq, where sinc(z) = z 1 sin(z), ω>0isafilter parameter, k nyq = π/ s and l nyq = π/ ϕ are thenyquist frequencies of the discretization in s and ϕ (of (2.24)), respectively. Let ζ(q 2, q 1 ) = q 2 q 1 L2 (Ɣ), (2.27) q 1 L 2 (Ɣ) where q 1, q 2 are test functions on Ɣ. Using the aforementioned results of sections 3, 4 and, especially, 5, in section 6 we consider some new possibilities for improving the stability (to the Poisson noise in p) of SPECT reconstruction based on the formula (1.5). The first of these possibilities consists in approximately reconstructing Cf from p as Cf p,ω = N a W ω p, (2.28) where the filter parameter ω is determined from the equation p W ω p L2 (Ɣ) = ζ(p, Mp), (2.29) W ω p L 2 (Ɣ) where ζ(p, Mp) is defined by meansof(2.27) and is the noise level of p;inaddition, ζ(p, Mp) is approximately determined by the formula ( ) p 1/2 L ζ(p, Mp) 1 (Ɣ) p 2 L 2 (Ɣ) p. (2.30) L 1 Note that an approximate reconstruction of Cf from p in the form (2.28) was, actually, already used in [GJKNT]. However, the new point of the reconstruction based on (2.28) (2.30) consists in an optimal choice of the filter parameter ω on the basis of (2.29), (2.30). Equation (2.29) is motivated by the assumption that one may expect that W ω p Mp for optimal ω and by the fact that (2.29) is equivalent to the equation ζ(p, W ω p) = ζ(p, Mp) (due to the definition of ζ(p, W ω p) by means of (2.27)). The formula (2.30) is based on (2.9), (2.10). An algorithm for determination of the filter parameter ω on the basis of (2.29), (2.30) from the

7 A noise property analysis of SPECT data 181 emission data p is given in section 6. This algorithm is based on solving equation (6.9) by the bisection method and the formulae (6.8), (6.10) (6.12). The second of the possibilities proposed in section 6 consists in approximately reconstructing Cf from p as Cf p,ω +,ω = 1 2 N a(w ω + p + + W ω p ), (2.31) where p ± are defined by (2.21) and the filter parameters ω +, ω are determined from the equations p + W ω + p + L2 (Ɣ) W ω + p + L 2 (Ɣ) = ζ(p +, Mp + ), (2.32a) ((ζ(p, Mp)) 2 (ζ(p +, Mp + )) 2 ) W ω + p + 2 L 2 (Ɣ) = p W ω p 2 L 2 (Ɣ) (ζ(p, Mp))2 W ω p 2 L 2 (Ɣ), (2.32b) where ζ(p, Mp), ζ(p +, Mp + ) are defined by means of (2.27) and are the noise levels of p, p + (respectively). In addition, ζ(p, Mp) is approximately determined by (2.30) and ζ(p +, Mp + ) is approximately determined by the formula ( ζ(p +, Mp + p + ) 1/2 L1 (Ɣ) ) p + 2. (2.33) L 2 (Ɣ) p+ L 1 (Ɣ) In addition, as a corollary of (2.9), (5.26), (5.33b), ζ(p +, Mp + ) ζ(p, Mp)/ 2. (2.34) Given p,wedetermine ω + and then ω on the basis of (2.32) (2.34), (2.30) in a completely similar way to the aforementioned determination of ω on the basis of (2.29), (2.30). In section 6, the reconstruction based on (2.32) (2.34), (2.30) is discussed in detail. In section 6 we give also a partial explanation of an experimental numerical result of [GJKNT]. Finally, in section 7 we give numerical examples illustrating some of theoretical conclusions of the present work. In particular, section 7 includes examples of (simulated) SPECT imaging based on (2.28) (2.30) and (2.30) (2.34). 3. Basic inequality for P a f Proposition 3.1. Let the conditions (2.11) be valid for some γ = (s,θ) R S 1.Thenthe inequality (2.12) holds. Proof of proposition 3.1. The definitions (1.1), (2.13) and the conditions (2.11) imply that Pa f (γ ) = (exp( Da(sθ + tθ,θ)) exp( Da(sθ + tθ, θ))) f (sθ + tθ)dt, R (3.1) P a + f (γ ) = (exp( Da(sθ + tθ,θ)) +exp( Da(sθ + tθ, θ))) f (sθ + tθ)dt, R Pa f (γ ) ρ a (γ )P a + f (γ ), (3.2a) where ρ a (γ ) def exp( Da(sθ + tθ,θ)) exp( Da(sθ + tθ, θ)) = max t R exp( Da(sθ + tθ,θ)) +exp( Da(sθ + tθ, θ)). (3.2b) It remains to prove that (2.15) holds. Using (3.2b), the formula exp( Da(sθ + tθ,θ))exp( Da(sθ + tθ, θ)) = exp( Pa(s,θ)), (3.3)

8 182 J-P Guillement and R G Novikov and (2.11) for a,weobtain that ρ a (γ ) = u u 1 e p max e p u 1 u + u 1 e, p where p = Pa(γ ), u = exp( Da(sθ + tθ,θ)). (3.4) One can see that d u u 1 e p 4ue p = > 0, (3.5) du u + u 1 e p (u 2 +e p ) 2 where e p u 1, p 0. Using (3.4), (3.5) we obtain (2.15). Proposition 3.1 is proved. Remark 3.1. At fixed γ R S 1,the inequality (2.12) implies the inequalities exp(pa(γ ))P a f (γ ) P a f ( γ), exp(pa(γ ))P a f ( γ) P a f (γ ). (3.6) 4. Exact relations between P + a f and P a f Suppose that a, f L,1+ε (R 2, R) for some ε>0, (4.1) where L,σ (R 2, C) ={u L (R 2, C) u 0,σ < + }, u 0,σ = ess sup(1+ x ) σ u(x), σ 0. x R 2 Then, as shown in [No1], the following identity holds: (4.2) exp[ Da(x, θ)] g θ (xθ ) dθ = 0, S 1 x R 2, (4.3a) g θ = H a,θ g θ, (4.3b) g θ = P a,θ f, (4.3c) where Da is defined by (1.1b)andH a,θ is the operatorsuchthat H a,θ u(s) = exp( θ (s)) cos( θ (s))h (exp( θ ) cos( θ )u)(s) +exp( θ (s)) sin( θ (s))h (exp( θ ) sin( θ )u)(s), (4.4) where u is a test function, H is defined by (1.7), θ, θ are defined by (1.6d), θ S 1, s R, dθ is the standard element of arc length on S 1.Asisshown in [No2], in addition to (4.4), the following formula also holds: Ha,θ 1 u(s) = 1 2 (H (exp( 2 θ)u)(s) +exp( 2 θ (s))hu(s)), (4.5) where u is a test function, θ S 1, s R. Consider g θ ± = g θ(s) ± g θ ( s), (4.6a) g ± θ = g θ(s) ± g θ ( s), (4.6b) where g θ, g θ are defined by (4.3c), (4.3b) (respectively), θ S 1, s R.

9 A noise property analysis of SPECT data 183 Proposition 4.1. Let the conditions (4.1) be valid. Then the following relations hold: (exp[ Da(x, θ)]+exp[ Da(x,θ)]) g θ + (xθ ) dθ S 1 + (exp[ Da(x, θ)] exp[ Da(x,θ)]) g θ (xθ ) dθ = 0, S 1 x R 2, (4.7a) g θ ± = H a,θ gθ, (4.7b) g ± θ = H 1 a,θ g θ, (4.7c) where we usethenotation of (4.3) (4.6). Proof of proposition 4.1. Therelation (4.7a)follows from (4.3a), (4.6b). The relations (4.7b) follow from (4.3b), (4.6) and the formula JH a, θ q θ = H a,θ Jq θ, (4.8) where q θ (s) = q(s,θ), q is a test function, J is the operator defined by Ju(s) = u( s), (4.9) where u is a test function, θ S 1, s R. Inturn, to obtain (4.8), we use (4.4) and the formulae JHu= HJu, (4.10a) JP θ a = P θ a, (4.10b) where u is a test function. To obtain (4.7c)weproceed from (4.5), (4.6). Further, the proof of (4.7c)issimilar to the proof of (4.7b). Proposition 4.1 is proved. The formula (4.7a)givesan exact equation relating g + and g.therefore, as a corollary of (4.7b), (4.7c), (4.3c)and (2.13), (4.6), proposition 4.1 gives exact equations relating Pa f and P a + f on R S1. 5. Some statistical properties of the attenuated ray transform with the Poisson noise Let Mξ, Dξ = M(ξ Mξ) 2,andV ξ = (Dξ) 1/2 / Mξ denote the mean, dispersion, and variation coefficient (respectively) of a real variate ξ. (Weassume that V ξ = + if Mξ = 0.) Let covar(ξ 1,ξ 2 ) = M(ξ 1 Mξ 1 )(ξ 2 Mξ 2 ) denote the covariance of real variates ξ 1 and ξ 2. Consider p(γ ), γ Ɣ, definedasin(1.4). Consider ζ defined by (2.2). Proposition 5.1. Let (2.11) hold for all γ Ɣ. Thenthe formulae (2.3) (2.5) hold. Proof of proposition 5.1. The formulae (2.3) follow from the properties of p of (1.4), the definition (2.1), and the formulae n n M ξ j = Mξ j, M(cξ) = cmξ, (5.1a) D j=1 n ξ j = j=1 j=1 n Dξ j, D(cξ) = c 2 Dξ, (5.1b) j=1 where ξ 1,...,ξ n and ξ are real variates, c is a real constant, and in (5.1b) itisassumed that ξ 1,...,ξ n are independent.

10 184 J-P Guillement and R G Novikov To obtain (2.4), (2.5) we also use that Mξ 2 = µ 2 + µ, Mξ 3 = µ 3 +3µ 2 + µ, Mξ 4 = µ 4 +6µ 3 +7µ 2 (5.2) + µ, where ξ is a Poisson variate with Mξ = µ. The formulae (2.4) follow from the properties of p of (1.4), the definition (2.1), the formulae (5.1), and the following formulae obtained using (5.2): Mξ 2 = µ 2 + µ, Dξ 2 = 4µ 3 +6µ 2 + µ, (5.3) where ξ is a Poisson variate with Mξ = µ. The formulae (2.5) follow from the properties of p of (1.4), the definition (2.1), the formulae (5.1) and the following formulae obtained using (5.2): M(ξ Mξ) 2 = µ, D(ξ Mξ) 2 = 2µ 2 + µ, (5.4) where ξ is a Poisson variate with Mξ = µ. Proposition 5.1 is proved. Consider p + (γ ), γ Ɣ, defined by (2.19). Suppose that (2.11) holds for (some) γ Ɣ. Then Mp + (γ ) = CP a + f (γ ), Dp+ (γ ) = CP a + f (γ ), (5.5a) Mp (γ ) = CPa f (γ ), Dp (γ ) = Dp + (γ ) = CP a + f (γ ), (5.5b) covar(p + (γ ), p (γ )) = CPa f (γ ), (5.5c) where P a ± f (γ ) are defined by (2.13), C is the constant of (1.4). In addition, p + (γ ) is a Poisson variate (with Mp + (γ ) = CP a + f (γ )) and (under the condition that (2.11) holds for all γ Ɣ) all p + (γ ), γ Ɣ +, are independent, (5.6) where Ɣ + is obtained from Ɣ by identifying γ with γ, γ Ɣ. The formulae (5.5a), (5.5b)follow from the definitions (2.19), (2.13), the properties Mp(γ ) = CP a f (γ ), Dp(γ ) = CP a f (γ ) (5.7) (following from (1.4) and the fact that Mξ = Dξ if ξ is a Poisson variate), and the formulae (5.1). The formula (5.5c)isobtained as follows: covar(p + (γ ), p (γ )) = M(p + (γ )p (γ )) Mp + (γ )Mp (γ ) = M(p 2 (γ ) p 2 ( γ)) ((CP a f (γ )) 2 (CP a f ( γ)) 2 ) = Dp(γ ) Dp( γ)= CPa f (γ ). The properties (5.6) are obtained using the properties of p(γ ) of (1.4) and using, in particular, that the sum of independent Poisson variates is again a Poisson variate. Proposition 5.2. Let the conditions (2.11) be valid for some γ Ɣ. Then the inequalities (2.17), (2.18) hold. Proof of proposition 5.2. The inequality (2.17) follows from (5.5a), (5.5b) and the inequality (2.12). The inequalities (2.18) follow from (5.5a), (5.7), the formula P a f (γ ) = (P a + f (γ ) + P a f (γ ))/2, (5.8) and the inequality (2.12). Proposition 5.2 is proved.

11 A noise property analysis of SPECT data 185 Note that V p(γ ), V p ± (γ ) show the sensitivity of (respectively) p(γ ), p ± (γ ) (where p(γ ), p ± (γ ) are defined by (1.4), (2.21)) to the Poisson noise in p(γ ) at fixed γ Ɣ. Indeed, consider ζ γ = p(γ ) Mp(γ ) Mp(γ ), ζ γ ± = p± (γ ) Mp ± (γ ), γ Ɣ. (5.9) Mp ± (γ ) Then: V p(γ ) = (Mζγ 2 )1/2, V p ± (γ ) = (M(ζ γ ± )2 ) 1/2, γ Ɣ. (5.10) In addition to ζ defined by (2.2) and ζ γ, ζ γ ± defined by (5.9), we also consider ζ ± defined by (2.22). (If some denominator in (2.2), (5.9), (2.22) is zero, then we assume that the related ratio is +.) Proposition 5.3. Let (2.11) hold for all γ Ɣ. Then M p + L 1 (Ɣ) = g + L 1 (Ɣ), D p + L1 (Ɣ) = 2 s ϕ g + L1 (Ɣ), M p + 2 L 2 (Ɣ) = g+ 2 L 2 (Ɣ) + g+ L 1 (Ɣ), D p + 2 L 2 (Ɣ) = 2 s ϕ(4 g+ 3 L 3 (Ɣ) +6 g+ 2 L 2 (Ɣ) + g+ L 1 (Ɣ)), (5.11a) (5.11b) (5.12a) (5.12b) M(ζ + ) 2 = g+ L1 (Ɣ) g + 2, (5.13a) L 2 (Ɣ) D(ζ + ) 2 = 2 s ϕ(2 g+ 2 L 2 (Ɣ) + g+ L1 (Ɣ)) g + 4, (5.13b) L 2 (Ɣ) M p 2 L 2 (Ɣ) = g 2 L 2 (Ɣ) + g+ L1 (Ɣ), (5.14) M(ζ ) 2 = g+ L 1 (Ɣ) g 2, (5.15) L 2 (Ɣ) where g ± = Mp ± = CP a ± f. Proof of proposition 5.3. Consider Ɣ 1/2 ={γ ij i = 1,...,n s, j = 1,...,n ϕ /2}, (5.16) where weusethenotation of (1.3). Note that if γ Ɣ 1/2 then γ Ɣ\Ɣ 1/2. (5.17) Note also that Ɣ 1/2 can be identified with Ɣ + of (5.6). The definition (2.22), the property (5.17), and the symmetries p ± (γ ) =±p ± ( γ), g ± (γ ) =±g ± ( γ), γ Ɣ, (5.18) imply that p ± α L α (Ɣ) = 2 p± α L α (Ɣ 1/2 ), g± α L α (Ɣ) = 2 g± α L α (Ɣ 1/2 ), (5.19a) ζ ± = p± Mp ± L2 (Ɣ 1/2 ), (5.19b) Mp ± L 2 (Ɣ 1/2 ) where q Lα (Ɣ 1/2 ) is defined by (2.1) with Ɣ 1/2 in place of Ɣ. Further, using (5.6) andthe expression (5.19b) forζ +,weprove (2.3) (2.5) with p +, Ɣ 1/2, ζ + in place of p, Ɣ, ζ,

12 186 J-P Guillement and R G Novikov respectively. This proof repeats the proof of proposition 5.1. The formulae (5.11) (5.13) follow from (2.3) (2.5) with p +, Ɣ 1/2, ζ + in place of p, Ɣ, ζ and the formulae (5.19a), (5.1). The formula (5.14) follows from the definition (2.1), the formula M(p (γ )) 2 = (g (γ )) 2 + g + (γ ), γ Ɣ, (5.20) and the formula (5.1a). In turn, (5.20) follows from (2.19), the properties of p of (1.4), the expression of (5.2) for Mξ 2,whereξ is a Poisson variate, the formula M(ξ 1 ξ 2 ) = Mξ 1 Mξ 2, (5.21) where ξ 1, ξ 2 are independent variates, and the formula (5.1a). The formula (5.15) follows from the definitions (2.22), (2.1), the formula M(p (γ ) Mp (γ )) 2 = g + (γ ), γ Ɣ, (5.22) and the formula (5.1a). In turn, (5.22) coincides with the formula for Dp (γ ) of (5.5b). Proposition 5.3 is proved. As a corollary of the assumptions (2.6), (2.7), we also have that V p + α L α (Ɣ) 1, α {1, 2}, (5.23a) V (ζ + ) 2 1, (5.23b) g + L1 (Ɣ) < g + 2 L 2 (Ɣ). (5.24) The properties (5.23), (5.24) follow from (2.6), (2.7), (2.3) (2.5), (5.11) (5.13) and the formula 2 g α L α (Ɣ) g+ α L α (Ɣ) 2α g α L α (Ɣ), α {1, 2, 3} (5.25) (where g = Mp = CP a f, g + = Mp + = CP a + f ). In turn, to obtain (5.25), we use (2.13), the non-negativityof g, and the Minkowski inequality. The property (5.23b) andchebyshev s inequality (2.8) imply that ζ + (M(ζ + ) 2 ) 1/2 with a probability close to 1. (5.26) In addition, the properties (5.23a), (5.24) and Chebyshev s inequality (2.8) imply that ( (M(ζ + ) 2 ) 1/2 p + ) 1/2 L1 (Ɣ) p + 2 L 2 (Ɣ) p+ L1 (Ɣ) with a probability close to 1. (5.27) Note that the formulae (5.26), (5.27) are similar to the formulae (2.9), (2.10). In view of (5.15), (5.14), (5.11), in some approximation one can also use the formula ( ζ p + ) 1/2 L (Ɣ) p 2. L 2 (Ɣ) p+ L 1 (Ɣ) (5.28) In particular, for the numerical example considered in section 7, we have that ζ = 1.29, whereas the right-hand side of (5.28) is equal to 1.3. However, note that (5.28) is obtained from (5.15), (5.14), (5.11) without an analysis of V p 2 L 2 (Ɣ) and V (ζ ) 2 and, therefore, (5.28) is less reliable than (2.9), (2.10), (5.26), (5.27). Proposition 5.4. Let the conditions (2.11) hold for all γ Ɣ. Then (M(ζ + ) 2 ) 1/2 ρa max (M(ζ ) 2 ) 1/2, (5.29) ( 1+(ρ 2 1/2 (Mζ 2 ) 1/2 (M(ζ + ) 2 ) 1/2 max a ) 2 ) 1/2 (Mζ 2 ) 1/2, (5.30) 2 ρa max = max ρ a(γ ) (5.31) γ Ɣ (and we assume that ρa max (M(ζ ) 2 ) 1/2 = + if (M(ζ ) 2 ) 1/2 = + and ρa max = 0).

13 A noise property analysis of SPECT data 187 Proof of proposition 5.4. The inequality (5.29) follows from (5.13a), (5.15), and the inequality (2.12). The inequalities (5.30) follow from (2.5a), (5.13a), (5.15),the formulae P a + f L 1 (Ɣ) = 2 P a f L1 (Ɣ), P a + f 2 L 2 (Ɣ) + P a f 2 L 2 (Ɣ) = 4 P a f 2 L 2 (Ɣ), (5.32) and the inequality (2.12). Proposition 5.4 is proved. In a completely similar way to (2.12), (2.16), if the conditions (2.11) hold for all γ Ɣ and a Ɣ 0, f Ɣ 0(whereƔ is interpreted as the unification in R 2 of all lines γ Ɣ), then, as a rule, (M(ζ + ) 2 ) 1/2 is considerably closer to 2 1/2 (Mζ 2 ) 1/2 than follows from (5.30) and is considerably smaller than ρa max (M(ζ ) 2 ) 1/2,and, roughly speaking (especially if a Ɣ is not strong), one can say that (M(ζ + ) 2 ) 1/2 (M(ζ ) 2 ) 1/2, (5.33a) (M(ζ + ) 2 ) 1/2 2 1/2 (Mζ 2 ) 1/2. (5.33b) The formulae (5.29), (5.30), (5.33) for (M(ζ ± ) 2 ) 1/2, (Mζ 2 ) 1/2 are similar to the formulae (2.17), (2.18), (2.20) for V p ±, V p. Theformulae (5.29), (5.30), (5.33), (2.17), (2.18), (2.20) show that the even part p + of the emission data p of (1.4) is considerably less sensitive to the Poisson noise in p than the odd part p of p and that p + is about 2 times less sensitive to the Poisson noise in p than the data p themselves. 6. Applications to SPECT imaging based on the formula (1.5) In this section we propose, in particular, some new possibilities for improving the stability (to the Poisson noise in p of (1.4)) of SPECT imaging based on (1.5). The first of these possibilities consists in the following. Possibility 6.1. To improve the stability (to the Poisson noise in p of (1.4)) of SPECT imaging based on (1.5), one can approximately reconstruct Cf as Cf p,ω of (2.28), where the filter parameterω is determined from equation (2.29), where the noise level ζ(p, Mp) is determined from p by (2.30). The possibility 6.1 was commented on already in section 2. Adding to these comments, we now describe an algorithm for determination of ω on the basis of (2.29), (2.30) from the emission data p. Westartwith some additional notation and formulae. Consider p (defined on Ɣ by (1.4)) as a function on Ɣ ={( j 1, j 2 ) Z 2 0 j 1 n s 1, 0 j 2 n ϕ 1}. (6.1) We suppose that p j1, j 2 = p(γ j1 +1, j 2 +1), (6.2) where ( j 1, j 2 ) Ɣ, γ j1 +1, j 2 +1 Ɣ, whereγ ij is defined in (1.3). We identify Ɣ with Ɣ. Let us suppose that n ϕ and n s of (1.3), (6.1) are even. Let F denote the 2D discrete Fourier transformation defined by (Fq) k1,k 2 = 1 ns n ϕ ( ( j1 k 1 q j1, j 2 exp 2πi ( j 1, j 2 ) Ɣ n s + j 2k 2 n ϕ )), (k 1, k 2 ) ˆƔ, (6.3) where { ˆƔ = (k 1, k 2 ) Z 2 n s 2 k 1 n s 2 1, n ϕ 2 k 2 n } ϕ 2 1, (6.4) q is a test function on Ɣ identified with Ɣ.

14 188 J-P Guillement and R G Novikov Let the low-frequency filter W ω of (2.28) be realized by the formula W ω q = F 1 Ŵ ω Fq, (6.5) where (Ŵ ω Fq) k1,k 2 = Ŵ ω (k 1, k 2 )(Fq) k1,k 2, (6.6) ( ( ) ( )) 2 2πk1 2πk2 Ŵ ω (k 1, k 2 ) = sinc sinc for k 1 ω n s ωn s ωn ϕ 2, k 2 ω n ϕ 2, Ŵ ω (k 1, k 2 ) = 0 for k 1 >ω n s or k 2 >ω n (6.7) ϕ 2 2, q is a test function on Ɣ identified with Ɣ, (k 1, k 2 ) ˆƔ. Let ζ ω = p W ( ) ω p L 2 (Ɣ) p 1/2 L, ζ appr = 1 (Ɣ) W ω p L2 (Ɣ) p 2 L 2 (Ɣ) p. (6.8) L 1 (Ɣ) Let us write the equation (2.29) and the formula (2.30) for determining ω from p as ζ ω = ζ appr. (6.9) The following formulae hold: ( ) 1/2 ( ) 1/2 ζ ω = Ŵ ω (k 1, k 2 ) ˆp k1,k 2 2 (1 Ŵ ω (k 1, k 2 )) ˆp k1,k 2 2, (6.10) (k 1,k 2 ) ˆƔ ζ ω 0 asω +, ( ζ ω M = ( ˆp 0,0 ) 1 (k 1,k 2 ) ˆƔ\{(0,0)} (k 1,k 2 ) ˆƔ ˆp k1,k 2 2 ) 1/2 as ω 0, (6.11) ζ ω1 ζ ω2 for 0 ω 2 ω 1, (6.12) where ˆp = Fp. The formula (6.10) follows from the definition of ζ ω (of (6.8)), the formulae (6.5), (6.6), and the isometry property of F. The formulae (6.11), (6.12) follow from (6.10) and the following properties of Ŵ ω : Ŵ ω (0, 0) = 1, (6.13a) Ŵ ω (k 1, k 2 ) 1 asω +, (6.13b) Ŵ ω (k 1, k 2 ) 0, (k 1, k 2 ) (0, 0), as ω 0, (6.13c) Ŵ ω1 (k 1, k 2 ) Ŵ ω2 (k 1, k 2 ) for 0 ω 2 ω 1, (6.14) where (k 1, k 2 ) ˆƔ. Given p, our algorithm for finding ω from (6.9) consists in the following. (1) We compute ˆp = Fp on ˆƔ. Ifζ appr M,whereM is defined in (6.11), then p cannot be considered as correct data. If ζ appr < M, thenwecontinue as follows. (2) Using the formulae (6.10) (6.12), we find ω from (6.9) by the bisection method. For our case this method consists in the following. We set ω 0 = 0, ω 1 =, ω 2 = 1, where is considerably greater than 1, say = 4. If ζ ζ appr,thenthis means that the data p may contain only a very weak noise component and we determine ω as ω =. If ζ <ζ appr,thenusing ω 0,ω 1,...,ω j we compute ω j+1 = 1 ( ) max ω 2 i + min ω i. (6.15) i j,ζ ωi >ζ appr i j,ζ ωi <ζ appr

15 A noise property analysis of SPECT data 189 We stop the computations (6.15) as soon as we obtain ω J such that ζ ωj ζ appr or ω J ω J 1 0 (for example, ω J ω J 1 (n s n ϕ ) 1/2 < 1). (6.16) Then we determine ω as ω = ω J. Let us estimate the complexity of this algorithm. The computation of ˆp (by means of a FFT algorithm) requires O(n s n ϕ log(n s n ϕ )) operations. The computation of ζ ω j for fixed j (by means of (6.10)) requires O(n s n ϕ ) operations. The formula (6.15) and the stopping condition ω j ω J 1 (n s n ϕ ) 1/2 < 1implythat J < 3+log 2 ( (n s n ϕ ) 1/2 ). Asacorollary, under the assumption that c 1 n s n ϕ c 2 n s for some fixed positive c 1 and c 2, (6.17) our algorithm for solving (6.9) requires O(n 2 s log n s) operations. This complexity is negligible in comparison with O(n 3 s ) operations required for the algorithm (of [Ku2, Na2]) computing N a q on X from q on Ɣ under the assumption (6.17). To motivate the second of the possibilities proposed in this section, consider the decomposition Cf = Cf + + Cf, Cf ± = 1 2 N ag ±, (6.18) where g ± (γ ) = g(γ ) ± g( γ), g = P a f, γ R S 1.Dueto(2.17), (2.20a), (5.29), (5.33a), the odd part p of the emission data p of (1.4) is considerably more sensitive to the Poisson noise in p than the even part p + of p. Therefore, Cf reconstructed as N a p is considerably more sensitive to the Poisson noise in p than Cf + reconstructed as N a p +.Therefore, to improve the stability (to the Poisson noise in p) ofspect imaging based on (1.5) we consider Cf approximately reconstructed from p as Cf p,ω+,ω of (2.31), where ω <ω +. Note that W ω filters more strongly when ω decreases. Note also that if ω = ω + = ω,thenthe reconstruction ansatz (2.31) is reduced to (2.28). Possibility 6.2. To improve the stability (to the Poisson noise in p of (1.4)) of SPECT imaging based on (1.5), one can approximately reconstruct Cf as Cf p,ω+,ω of (2.31), where the filter parameters ω ± are determined from the equation (2.32), where ζ(p, Mp) is determined from p by (2.30), ζ(p +, Mp + ) is determined from p by (2.33) or (2.34). Equation (2.32a)forω + is motivated in a similar way to equation (2.29) of possibility 6.1. Equation (2.32b)forω follows from: (1) the equation p + + p (W ω + p + + W ω p ) L 2 (Ɣ) W ω + p + + W ω p L2 (Ɣ) (2) the property that = ζ(p, Mp), (6.19) (p +, p ) L2 (Ɣ) = 0, (W ω + p +, W ω p ) L2 (Ɣ) = 0, (6.20) where (q 1, q 2 ) L 2 (Ɣ) = s ϕ γ Ɣ q 1 (γ )q 2 (γ ), (6.21) where q 1, q 2 are real-valued test functions on Ɣ, (3) the Pythagorean theorem in L 2 (Ɣ), and (4) the relation (2.32a)between p + W ω + p + L 2 (Ɣ) and W ω + p + L 2 (Ɣ).

16 190 J-P Guillement and R G Novikov In turn, equation (6.19) is motivated by the assumption that one may expect (W ω + p + + W ω p )/2 Mp for the optimal ω + and ω and by the fact that (6.19) is equivalent to the equation ζ(p,(w ω + p + + W ω p )/2) = ζ(p, Mp) (in view of the definitions (2.21), (2.27)). The property (6.20) follows from the symmetries p + (γ ) = p + ( γ), p (γ ) = p ( γ), γ Ɣ, (6.22) Ŵ ω (k, l) = Ŵ ω ( k, l), k R, l Z, and the definition of W ω.theformulae (2.30), (2.33) are based on (2.9), (2.10), (5.26), (5.27). Given p, wedetermine first ω + and then ω on the basis of (2.32), (2.33), (2.30) in a completely similar way to the (described above in this section) determination of ω on the basis of (2.29), (2.30). Note that, in addition to (2.32b), the following equation: where p W ω p L2 (Ɣ) W ω p L2 (Ɣ) = ζ(p, Mp ), (6.23) ( ζ(p, Mp p + ) 1/2 L1 (Ɣ) ) p 2, (6.24) L 2 (Ɣ) p+ L1 (Ɣ) can be also used for the determination of ω.equation (6.23) is motivated in a similar way to equations (2.29), (2.32a). The formula (6.24) is based on (5.28). However, as was already mentioned in section 5, the formula (5.28) is less reliable than (2.9), (2.10), (5.26), (5.27). Therefore, in the present work (in section 7) we determine the parameter ω from equation (2.32b). Note that the relation ω <ω + is not imposed explicitly in possibility 6.2. Actually (see section 7), this relation arises automatically from equations (2.32b)andthePoisson noise model (1.4). The possibility of using the reconstruction ansatz (2.31) with ω <ω + in order to improve the stability (to the Poisson noise in p of (1.4)) of SPECT imaging based on (1.5) was not observed in the preceding works. Nevertheless, the following decomposition was successfully used in [GJKNT]: Cf = ε 1 + ε 2, (6.25a) ε j = N a, j g, j = 1, 2, (6.25b) where g = CP a f, N a, j q(x) = 1 ( K j (x,θ)θ 2 dθ + ) K j (x,θ)θ 1 dθ, 4π x 1 S 1 x 2 S 1 j = 1, 2, (6.26a) K 1 (x,θ)= h θ (xθ ), (6.26b) K 2 (x,θ)= (exp(da(x,θ) 1 2 Pa(xθ,θ)) 1)h θ (xθ ), (6.26c) h θ (s) = exp( 1 2 Pa(s,θ)) q θ (s), (6.26d) where q is a test function on R S 1, q θ is defined by (1.6c), x R 2, θ S 1, s R. Given q on Ɣ, weassume that N a, j q, j = 1, 2, are defined on X and denote numerical realizations of (6.26). In [GJKNT], it was found experimentally that ε 1 computed as N a,1 p is considerably less sensitive to the Poisson noise in the emission data p than ε 2 computed as N a,2 p.theoretically, using results of the present work, these noise properties of N a,1 p and N a,2 p can be partially

17 A noise property analysis of SPECT data 191 explained as follows. The formulae (6.26a), (6.26b), (6.26d), (4.7b) with q in place of g,(4.8), (4.10b)implythat N a,1 q = 1 2 N a,1q +, (6.27) where q ± (γ ) = q(γ ) ± q( γ), γ R S 1, q is a test function. The formula (6.27) implies that N a,1 p is, actually, independent of p. This and the strong noise sensitivity of p (see sections 2 and 5) give a partial theoretical explanation of the aforementioned noise properties of N a,1 p and N a,2 p. In [GJKNT], to improve the stability (to the Poisson noise in p of (1.4)) of SPECT imaging based on (1.5), the function Cf was reconstructed (in the first approximation) as Cf appr = N a,1 W 1 p + N a,2 W 2 p, (6.28) where W 2 is astronger filter that W 1 and it also assumed that Ŵ j (k, l) = Ŵ j ( k, l), k R, l Z, j = 1, 2, (6.29) in the definition (2.24). Note that, due to (6.27), (6.29), N a,1 W 1 p = 1 2 N a,1w 1 p + in (6.28) (and also N a,1 p = 1 2 N a,1 p + ). In addition, the similarity of (2.31) and (6.28) consists in the fact that in both cases p + is filtered more weakly in some sense that p. Finally, in addition to possibilities 6.1 and 6.2, we can also propose the following new possibilities. Possibility 6.3a. To improve the stability of the SPECT reconstruction based on (1.5) with respect to the Poisson noise in the emission data g, one can use the inequality (2.12) for correcting the (less stable) odd part g of the emission data g in terms of the (more stable) even part g + of g. Possibility 6.3b. To improve the stability of the SPECT reconstruction based on (1.5) with respect to the Poisson noise in the emission data g, one can use (4.7a) asanequation for finding the (less stable) even part g + of the filtered emission data g from the (more stable) odd part g of g. In connection with possibility 6.3b, we recall that the g ± are related to g by (4.7b), (4.7c) and that g is used for back-projecting in (1.5). Examples of (simulated) SPECT imaging using possibilities 6.1 and 6.2 are given in section 7. Restrictions in time prevent us from trying possibilities 6.3a and 6.3b in the present work. 7. Numerical examples We consider a version of the elliptical chest phantom (used for numerical simulations of cardiac SPECT imaging; see [HL, Br2]). The major axis of the ellipse representing the body is 30 cm. Figure 1(a) shows the attenuation map; the attenuation coefficient a is 0.04 cm 1 in the lung regions (modelled as two interior ellipses), 0.15 cm 1 elsewhere within the body ellipse, and zero outside the body. Figure 1(b) shows the emitter activity f ; f is in the ratio 8:0:1:0 in myocardium (represented as a ring), lungs, elsewhere within the body, and outside the body. The functions a and f (and all reconstructionsof f )areconsidered on X defined by (2.23), where n s = 129 and the radius of the image support R = 17 cm. Figures 2(a) (c) show the transform P a f and its even and odd parts P a ± f.

18 192 J-P Guillement and R G Novikov (a) (b) Figure 1. Model distributions: (a) attenuation map a;(b) emitter activity f. (a) (b) (c) Figure 2. The attenuated ray transform P a f (a) and its even and odd parts P + a f (b) and P a f (c). (a) (b) (c) Figure 3. Noisy emission data p (a) and their even and odd parts p + (b) and p (c). Figures 3(a) (c) show the emission data p (modelled according to (1.4)) and their even and odd parts p ±.Inaddition, the constant C in (1.4) was specified by the equation (Mζ 2 ) 1/2 = κ (7.1) where κ = 0.3 andmζ 2 is given by theright-hand side of (2.5a). The functions P a f, P a ± f, p, p ± are considered on Ɣ defined by (1.3), where n ϕ = 128, n s = 129, R = 17 cm (n s and R are the same in (1.3) and (2.23)). Notice that all two-dimensional images of the present work are drawn using a linear greyscale, in such a way that the dark grey colour represents zero (or negative values, if any) and white corresponds to the maximum value of the imaged function.

19 A noise property analysis of SPECT data 193 (a) (b) Figure 4. The activity Cf reconstructed from the emission data g = CP a f without noise: (a) Cf g = N a g;(b)theprofile of Cf g for j = 64. We calculate that R 1 g L 1 (Ɣ) = 30.63, R 1/2 g L 2 (Ɣ) = 18.45, R 1/2 g + L 2 (Ɣ) = 36.36, R 1/2 g L 2 (Ɣ) = 6.00, p(γ ) = , R 1 p L1 (Ɣ) = 30.63, R 1/2 p L2 (Ɣ) = 19.25, γ Ɣ R 1/2 p + L 2 (Ɣ) = 37.18, R 1/2 p L 2 (Ɣ) = 9.82, (7.2) (7.3) ζ(p, g) = 0.298, ζ(p +, g + ) = 0.21, ζ(p, g ) = 1.29, (7.4) where g = CP a f, g ± = CP a ± f, (7.5) (where C is the constant of (1.4)), the norms L2 (Ɣ), L1 (Ɣ) are defined by (2.1) (where s, ϕ are defined in (1.3)), ζ(q 2, q 1 ) is defined by (2.27). Note that g + L 1 (Ɣ) = 2 g L 1 (Ɣ), p + L1 (Ɣ) = 2 p L1 (Ɣ). The value of ζ(p, g) of (7.4) shows that our emission data p contain 29.8% of the Poisson noise. That is, we consider strongly noisy emission data. The numbers of (7.2) for the L 2 -norms of g +, g confirm (2.16). The numbers of (7.3), (7.4) confirm (2.30), (2.33), (2.34), (5.28) and (5.33). Given q on Ɣ, weassume that N a q is defined on X and denote the numerical realization of (1.6) as in [Ku2, Na2] without any regularization. As coordinates on X we take the indices i, j of (2.23). Figures 4(a), (b), 5(a), (b) and 6(a), (b) show Cf g = N a g, Cf g + = (1/2)N a g + and Cfg = (1/2)N a g with their profiles for j = 64, where g, g ± are defined on Ɣ by (7.5). Note that the maximum values of the profiles shown in figures 4(b), 5(b) and 6(b) are equal to 188.3, and 7.8, respectively. We have that λ(cf g +, Cf g) = 0.39, (7.6) where λ(u 2, u 1 ) = u 2 u 1 L2 (X)/ u 1 L2 (X), (7.7) ( ) 1/2 u L2 (X) = s u(x) 2, (7.8) x X where u, u 1, u 2 are test functions on X.

20 194 J-P Guillement and R G Novikov (a) (b) Figure 5. The component Cf + reconstructed from the emission data g = CP a f without noise: (a) Cf g + = (1/2)N ag + ;(b)theprofile of Cf g + for j = 64. (a) (b) Figure 6. The component Cf reconstructed from the emission data g = CP a f without noise: (a) Cfg = (1/2)N ag ;(b)theprofile of Cfg for j = 64. Figures 7(a), (b), 8(a), (b) and 9(a), (b) show Cf p = N a p, Cf p + = (1/2)N a p + and Cfp = (1/2)N a p with their profiles for j = 64. Note that the maximum values of the profiles shown in figures 7(b), 8(b) and 9(b) are equal to 199.5, and 61.3, respectively. We have that λ(cf p, Cf g ) = 1.57, λ(cf p +, Cf+ g ) = 0.95, λ(cf+ p, Cf g) = (7.9) The relative errors of (7.9) show that Cf p + is noticeably less sensitive to the Poisson noise in p than Cf p.thisnumerical result is a corollary of the properties (2.20b), (5.33b). Moreover, (7.9) shows that Cf p + is closer to Cf g in the sense of the L 2 -norm than Cf p. The reconstruction of Cf g as Cf p + can be considered as Cf p,ω +,ω of (2.31), where ω+ = +, ω = 0. Figures 10(a) (c) with 11(a) (c) show Cf p,ω = N a W ω p for ω = 3/4, 5/8, 1/2with their profiles for j = 64, where W ω is the low-frequency filter defined in section 2. We have that λ(cf p,ω, Cf g ) = 0.38, 0.37, 0.39 for ω = 3/4, 5/8, 1/2, respectively. (7.10) The filter parameter ω = 5/8was found from equation (2.29),where ζ(p, Mp) was determined from (2.30) as 0.30 (the precise value of ζ(p, Mp) is 0.298). Figures 12(a), (b) show Cf p,ω +,ω = (1/2)N a(w ω + p + + W ω p ) for ω + = 11/16, ω = 13/32 with its profile for j = 64. We have that λ(cf p,ω +,ω, Cf g) = 0.33, ω + = 11/16, ω = 13/32. (7.11)

21 A noise property analysis of SPECT data 195 (a) (b) Figure 7. The activity Cf reconstructed from the noisy emission data p with no filtration: (a) Cf p = N a p;(b)theprofile of Cf p for j = 64. (a) (b) Figure 8. The component Cf + reconstructed from the noisy emission data p with no filtration: (a) Cf p + = (1/2)N a p + ;(b)theprofile of Cf p + for j = 64. (a) (b) Figure 9. The component Cf reconstructed from the noisy emission data p with no filtration: (a) Cfp = (1/2)N a p ;(b)theprofileofcfp for j = 64. The filter parameters ω + = 11/16, ω = 13/32 were found from equations (2.32), where ζ(p, Mp) was determined, as above, from (2.30) and ζ(p +, Mp + ) was determined by (2.34). Note that for our example of the functions a, g, andp, therelative error λ(cf p,ω, Cf g ) has its minimumexactly when ω = 5/8 andthatthe relative error λ(cf p,ω+,ω, Cf g), atleast for ω + = 11/16, has its minimum exactly when ω = 13/32.

22 196 J-P Guillement and R G Novikov (a) (b) (c) Figure 10. The activity Cf reconstructed as Cf p,ω = N a W ω p from the noisy emission data p: (a) ω = 3/4; (b) ω = 5/8; (c) ω = 1/2. (a) (b) (c) Figure 11. Profiles of Cf p,ω for j = 64: (a) ω = 3/4; (b) ω = 5/8; (c) ω = 1/2. (a) (b) Figure 12. The activity Cf reconstructed as Cf p,ω +,ω = (1/2)N a(w ω + p + + W ω p ) from the noisy emission data p with ω + = 11/16, ω = 13/32 (a) and its profile for j = 64 (b). Finally, one can see from (7.10)and (7.11),that the algorithm based on (2.30) (2.34)and filtering p as (1/2)(W ω + p + + W ω p ) (i.e. using a two-component filtration) gives a better reconstruction result (inthe sense of the L 2 -norm relative error with respect to the noiseless reconstruction) than the algorithm based on (2.28) (2.30) and filtering p as W ω p (i.e. using a one-component filtration). Therefore, in a subsequent work, in the framework of SPECT imaging based on (1.5) (and PET imaging based on the classical Radon inversion formula), proceeding from results of the present work, we plan to develop a multi-component filtration of the emission data.

23 A noise property analysis of SPECT data Conclusions We obtained formulae describing the simplest statistical properties of SPECT data modelled as the attenuated ray transform with the Poisson noise or, more precisely, as p of (1.4). In particular, we obtained the formulae (2.30), (2.33), (6.24) and (2.17), (2.18), (2.20), (2.34). The formulae (2.30), (2.33), (6.24) determine the noise levels of the emission data p and their even and odd parts p ± (and are based on the formulae (2.9), (2.10), (5.26) (5.28)). The formulae (2.17), (2.20a) showthat the even part p + of the emission data p is considerably less sensitive to the Poisson noise in p than the odd part p of p. In addition, the formulae (2.18), (2.20b), (2.34) show that p + is about 2 times less sensitive to the Poisson noise in p than the data p themselves. To obtain the formulae (2.17), (2.18), (2.20), (2.34) we got and used the inequality (2.12) relating the even and odd parts of the attenuated ray transform P a f without noise. In addition to the inequality (2.12), we also presented the exact equations (4.7) relating the even and odd parts of P a f.theequations (4.7) are obtained by reformulating results of [No1, No2] on the range characterization for the transformation P a. Further, using the formulae (2.30), (2.33), (6.24), (2.20), (2.34), the inequality (2.12) and equations (4.7) we proposed new possibilities for improving the stability of SPECT imaging based on the formula (1.5) (obtained in [No1]) with respect to the Poisson noise in the emission data. The first of these possibilities (possibility 6.1) consists in approximately reconstructing Cf from p as Cf p,ω of (2.28), where the filter parameter ω is determined on the basis of (2.29), (2.30). The second of these possibilities (possibility 6.2) consists in approximately reconstructing Cf from p as Cf p,ω+,ω of (2.31), where the filter parameters ω + and ω are determined on the basis of (2.32) (2.34), (2.30). The third of these possibilities (possibility 6.3a, 6.3b) consists in correcting the (less stable) odd part of the emission data in terms of the (more stable) even part of these data by means of the inequality (2.12) and equations (4.7). Finally, we illustrated some of theoretical conclusions of the present work with numerical examples. In particular, we presented examples of (simulated) SPECT imaging based on (2.28) (2.30) and on (2.30) (2.32), (2.34). Acknowledgment We thank C Comtat, F Jauberteau and R Trebossen for useful discussions. We also thank the referees for their useful recommendations. References [ABK] Arbuzov E V, Bukhgeim A L and Kazantsev S G 1998 Two-dimensional tomography problems and the theory of A-analytic functions Sib. Adv. Math. 8 (4) 1 20 [BCB] Beis J S, Celler A and Barney J S 1995 An automatic method to determine cutoff frequency based on image power spectrum IEEE Trans. Nucl. Sci [Br1] Bronnikov A V 1995 Approximate reconstruction of attenuation map in SPECT imaging IEEE Trans. Nucl. Sci [Br2] Bronnikov A V 2000 Reconstruction of attenuation map using discrete consistency conditions IEEE Trans. Med. Imaging [GN] Gourion D and Noll D 2002 The inverse problem of emission tomography Inverse Problems [GJKNT] Guillement J-P, Jauberteau F, Kunyansky L, Novikov R and Trebossen R 2002 On single-photon emission computed tomography imaging based on an exact formula for the nonuniform attenuation correction Inverse Problems 18 L11 9

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