Fast Laplace Transforms for the Exponential Radon Transform

Transcription

1 J Fourier Anal Appl (018) 4: Fast Laplace Transforms for the Exponential Raon Transform Frerik Anersson 1 Marcus Carlsson 1 Viktor V. ikitin 1 Receive: May 016 / Revise: 18 December 016 / Publishe online: 10 March 017 The Author(s) 017. This article is publishe with open access at Springerlink.com Abstract The Fourier slice theorem for the stanar Raon transform generalizes to a Laplace counterpart when consiering the exponential Raon transform. We show how to use this fact in combination with algorithms for the unequally space fast Laplace transform to construct fast an accurate methos for computing both the forwar exponential Raon transform an the corresponing back-projection operator. Keywors Exponential Raon transform Fast Laplace transform Incomplete ata Mathematics Subject Classification 65R10 65R3 1 Introuction One of the most popular ways of computing the stanar two-imensional Raon transform for parallel beam geometry relies on Fourier transforms via the Fourier slice theorem. A stanar iscretization with equally space samples in the spatial variables yiels a relation between the sample spatial ata an its Fourier transform on a polar lattice. This can be rapily evaluate by using a combination of stanar Communicate by Eric To Quinto. B Frerik Anersson fa@maths.lth.se Marcus Carlsson mc@maths.lth.se Viktor V. ikitin nikitin@maths.lth.se 1 Centre for Mathematical Sciences, Lun University, Box 118, 100 Lun, Sween

2 43 J Fourier Anal Appl (018) 4: FFT an algorithms for unequally space FFT. In this way the time complexity is essentially reuce by one orer, from O( 3 ) to O( log ). The main purpose of this paper is to show that a similar reuction in computational cost can be obtaine for the exponential Raon transform, by relying on algorithms for unequally space fast Laplace transforms instea of the counterpart for Fourier transforms. The same hols for the ajoint operator, sometimes calle the exponential back-projection operator, which in particular enables us to implement a fast algorithm for the inverse exponential Raon transform, base on a formula by Tretiak an Metz [1]. The nee for fast transforms is particularly important when employing iterative schemes for the solution of (ill-pose) inverse problems, arising e.g. from incomplete angular coverage in the measurements, or for employing noise reuction techniques. Upon iscretization, the Raon transform becomes an operator on finite imensional space, having a matrix representation. The same goes for the inverse an ajoint Raon transform, an these iscretizations are associate with approximation errors which epen on the choice of specific parameters etc. On top of that, there are errors that arise when replacing the iscrete operator with its fast counterpart, which is the topic of this paper. We show that the errors that come from the latter part can be controlle to arbitrary precision level ɛ at a computational cost that is in practice proportional to log(ɛ). In this way, reconstruction issues concerning for instance noise sensitivity can be isolate to the (stanar) sampling setup, with only minor influence coming from the propose fast computational algorithms. The inverse problem of reconstructing ata moele by the exponential Raon transform is ill-pose, meaning that aitional regularization techniques often nee to be applie in orer to obtain suitable reconstructions of ata. There are several ifferent regularization approaches in the literature, an the particular type of regularization will typically epen on the structure of the sample measure on. We therefor restrict our attention to how to construct fast algorithms for computing the exponential Raon transform an its ajoint, an only briefly mention some regularization techniques that coul be useful. In particular, we inclue a escription on how to inclue a iscretize version of the filter operator present in the filtere back-projection type of reconstruction formula for the inversion of Raon ata, as it is the most commonly use reconstruction metho use for stanar Raon ata. Let s R an θ S 1, where S 1 enotes the unit circle. The stanar twoimensional Raon transform is efine as R f (s,θ)= f (x) x = f (sθ + tθ ) t, x θ=s an the parameters s an θ are use to parameterize the set of lines in R.Theattenuate Raon transform in R escribes the mapping from functions f an μ to line integrals of the form R at f (s,θ)= f (sθ + tθ )e t μ(sθ+τθ ) τ t. Integrals of this type appear for instance in meicine, where f escribes the intensity istribution of isotopes or raiopharmaceuticals insie a tissue, an where the function μ(x) 0 escribes the attenuation of the tissue. For the case where μ is constant within

3 J Fourier Anal Appl (018) 4: a known convex boy, then the ata obtaine from the attenuate Raon transform can be moele by the simpler exponential Raon transform [0], R μ f (s,θ)= f (sθ + tθ )e μt t. (1) The ajoint operator associate with the exponential Raon transform (1) can be written as R μ g(x) = S 1 e μx θ g(x θ,θ)θ, () cf. [13,1]. It is a weighte integral of g over lines passing through the point x, an with μ = 0 this operator is typically referre to as the back-projection operator. In [13] a generalization of the Fourier slice theorem is erive for the exponential Raon transform along with an integral equation for the reconstruction problem of obtaining f given R μ f. A similar integral equation is also erive in [5]. There also exist integral equations for the attenuate Raon transform, where the function μ has certain constraints. An explicit inversion formula is given by ovikov [15], an a generalization is given in [7] for a somewhat larger class of functions. In [14] a result closely relate to ovikov s formula is erive, using a ifferent approach. Yet another approach for inversion for arbitrary realistic attenuation was presente by Arbuzov et. al [4]. In [1] properties of the exponential Raon transform are erive, along with an inversion formula of filtere back-projection type. In Sect. we recast this formula in a particularly simple format, see Theorem.1. Since then, the exponential Raon transform an the associate inversion problem have been iscusse by several authors. For instance, in [1] range conitions of operator R μ are shown using Paley-Wiener type theorems. Cormack-type inversion formulas are base on the circular harmonic expansion of the transform an solving integral equations of special type (generalize Cormack equations) an they are iscusse in [16] with generalization for the attenuate Raon transform in [17]. An explicit integral formula for 180-egree ata reconstruction is obtaine in [18] along with numerical tests for the approximation of the integrals. Another approach to reconstruct ata from 180-egree measurements is propose in [11], the approach is analytical except for the pre-calculate inverse kernel, where the least-squares metho is utilize. In [] the exponential Raon transform is extene to various spaces of istributions. In [,3] it was shown that the exponential Raon transform inversion can be expresse as a convolution on Eucliean motion group (the rigi motion on R ). Fourier-base approaches are propose in [10,1]. Here, the Fourier series expansion in the angular variable allow to increase the stability of the inverse transform. In this paper we aress a metho for the exponential Raon transform, i.e., for the case where the attenuation parameter μ is constant. The Fourier Laplace Slice Theorem an Inversion in the Continuous Case The Fourier slice theorem which relates the one-imensional Fourier transforms of Raon transforme ata with the two-imensional Fourier transform of ata is fairly

4 434 J Fourier Anal Appl (018) 4: straightforwar to generalize to the exponential Raon transform. To see this, we introuce the transform ˆR μ as the one-imensional Fourier transform of the exponential Raon transform with respect to the first variable s of R μ f, i.e. ˆR μ f (σ, θ) = R μ f (s,θ)e isσ s= f (sθ + tθ )e isσ +μt s t = = R f (sθ + tθ )e i(sθ+tθ ) (θσ )+μ(sθ+tθ ) θ s t f (x)e ix (θσ )+μx θ x, where the ot enotes scalar prouct in R. Since f by assumption has support in the boune omain, its Fourier transform extens to an analytic function in C, an hence we can express the outcome of the above calculation as ˆR μ f (σ, θ) = f ( σθ + iμ ) θ. (3) which is a generalization of the Fourier slice theorem, sometimes referre to as the Fourier Laplace slice theorem. For more etails, see [13,1]. ote that ˆR μ maps compactly supporte smooth functions to rapily ecaying smooth functions in L (Z) where Z = (R \{0}) S 1 is a polar-type coorinate system, except for the fact that we allow negative raii. We now wish to express this as a transform acting into a space of functions on Cartesian-type coorinates. We therefore let R enote the two copies of the puncture plane on top of each other. Concretely, we set R = (R \{0}) {1, 1} an efine a bijection ι : Z R via ι(σ, θ) = { (σ θ, 1) if σ>0, (σ θ, 1) if σ<0. We will use the notation ξ = (ξ, α) for the inepenent variable in R, where ξ is the Cartesian part of ξ an α simply the sign of σ in the ientity ξ = ι(σ, θ). Fora function g on Z or R we will employ the stanar convention of suppressing ι in the notation, writing g(σ, θ) when the former (polar type) coorinate system is use, an g( ξ) when the latter (Cartesian type) coorinate system is use. Given any non-zero x = (x 1, x ) R we introuce notation x = ( x,x 1 ) x an moreover we efine ξ = αξ, ξ R. ote that with this notation we always have ξ = θ whenever ξ = ι(σ, θ). Finally, we let H be the measure on R that coincies with the Lebesgue measure on each of the two planes. For μ 0set E(x, ξ) = e μx ξ +ix ξ

5 J Fourier Anal Appl (018) 4: an E (x, ξ) = e μx ξ +ix ξ. Then ˆR μ ( f )( ξ) = f (x)e μx ξ ix ξ x = f, E(, ξ). (4) Using this notation, we shall show that the inversion formula by Tretiak an Metz takes a simple form given in the following theorem: Theorem.1 Let f be a compactly supporte smooth function. Then for μ 0, we have f (x) = 1 f, E(, ξ) E (x, ξ)h( ξ) (5) or, expresse ifferently, f (x) = ξ > μ ξ > μ f (y) cosh(μ(x y) ξ )e i(x y) ξ yξ. Proof The inversion formula by Tretiak an Metz (see Corollary 1 of [1]) is written as an integral formula involving a family of kernels epening on two parameters σ, ɛ, an a limit as these approach zero. The reason for the parameters σ, ɛ is that the corresponing integrals for σ = ɛ = 0 may be ivergent. For compactly supporte smooth functions it is easy to see that this is not the case an the formula takes the following simpler form f = R μ WR μ f, (6) where W is a convolution (in s) with the inverse Fourier transform of the function w(σ) = σ/ 1 { σ > μ }, an 1 { σ > μ } enotes the characteristic function of { σ > μ }, (the normalization of the Fourier transform is ifferent in [1], so the kernel looks slightly ifferent there). This leas to f = R μ F M w FR μ f = ˆR μ M w ˆR μ f (7) where M w is multiplication by w. ote that, for a smooth rapily ecaying function g on Z we have ˆR μ f, g = f (x)e ix θσ μx θ xg(σ, θ)σ θ R = f (x) e ix θσ μx θ g(σ, θ)σ θ x R so ˆR μ (g)(x) = e ix θσ μx θ g(σ, θ)σ θ,

6 436 J Fourier Anal Appl (018) 4: which gives ˆR μ (M wg)(x) = S 1 = 1 = 1 σ > μ ξ > μ ξ > μ σ g(σ, θ)e μx θ +iσ x θ σ θ g( ξ)e μx ξ +ix ξ H( ξ) g( ξ)e ( ξ)h( ξ). (8) The formula (5) now follows by (4), (7) an the above computation. The subsequent formula is an immeiate consequence since H equals Lebesgue measure on each copy of the puncture plane, an ξ = (ξ, α) has opposite irections epening on whether α equals 1 or 1, so the evaluation of (5) reuces to f (y) eμ(y x) ξ i(y x) ξ yξ ξ > μ + = ξ > μ ξ > μ f (y) e μ(y x) ξ i(y x) ξ yξ f (y) cosh(μ(y x) ξ )e i(y x) ξ yξ. By the formulas (4) an (5) it is clear that both R μ an the inversion formula can be evaluate by using Laplace transforms. To use the continuous formulas erive in this section, we will nee to make use of iscretization schemes, an this will be iscusse in the next section. To illustrate how the fast Laplace transform work, let us consier a one-imensional iscrete counterpart of the Laplace integrals above, an specifically the rapi evaluation of sums of the form J f j e ζ j x j=1 where ζ j = (a j iξ j ).Letϕ be a Gaussian with a fixe with, an efine ϕ a through the Fourier relation ϕ a (x) = e ax ϕ(x). It then hols for a single exponential that e iξ j x = 1 ϕ a ϕ a j (x) j (ξ ξ j )e ixξ ξ, which can be expresse as e (a j iξ j )x = 1 ϕ a ϕ(x) j (ξ ξ j )e ixξ x.

7 J Fourier Anal Appl (018) 4: By applying this relationship to the iscrete Laplace sum above, we obtain that J f j e (a j iξ j )x = 1 J f j ϕ a ϕ(x) j (ξ ξ j ) e ixξ ξ. j=1 Due to the presence of the (moulate) Gaussians, the integran above will be smooth an it can be approximate (to arbitrary precision) with the trapezoial rule. This implies that it can be evaluate rapily at equally space points in x by using FFT. To make this work in practice, the Gaussian ϕ must be comparatively wie in orer to avoi loss of accuracy (it cannot be too close to zero). This means that the ϕ (an ϕ a j ) will have comparatively short support. An aitional practical requirement is to use some amount of oversampling compare to the set of equally space point (x) at which the iscrete Laplace transform is evaluate. We will make use of an oversampling factor ν to escribe this level of oversampling. A typical value is ν =. Moreover, the inner sum in the expression above contains iscrete convolution between the ata values f j an the moulate Gaussians ϕ a j. As these Gaussians ecay rapily, only a few of terms will contribute (within given precision), an we may therefor truncate the Gaussians. We will use the parameter M to enote where we can truncate the Gaussian. Specifically, this means that the iscrete convolution above is performe over at M + 1 terms, an the contribution from the other terms can be iscare. The with parameter M (an the choice of the Guassian ϕ) is relate to the esire level of accuracy as well as the oversampling parameter ν. For more etails, see Appenix. j=1 3 Discretization We will now consier iscretize versions of the exponential Raon tranform an its ajoint. The iscretization will be fairly stanar, using an equally space sampling in the spatial coorinates as well as in the (s,θ)-coorinates. We assume that f has compact support. Without loss of generality we assume that it has support on the isc {x : x < 1/}, an we represent it by sampling on the Cartesian lattice [ X =,..., 1] [,..., 1] for even, an let III X enote the operator that samples functions on X. Wealso iscretize R μ f (s,θ)at equally space points in θ that cover the whole interval [0, ). Specifically, let us assume that we have samples at a gri with number of points in the s-irection an θ points in the θ-irection an efine s m = m, m =,... 1, θ l = l, θ l = 0,... θ 1.

8 438 J Fourier Anal Appl (018) 4: S = { (s m,θ l ) : m =,... } 1, l = 0,... θ 1. Via the iscrete Fourier transform, the noes s m are associate with frequency noes σ k = k, k =,... 1 as well as a corresponing polar gri, = { (σ k,θ l ) : k =,... } 1, l = 0,... θ 1. We let III S an III enote the sampling operators on the two sets. As us customary in this fiel, we ientify θ l with the point (cos(θ l ), sin(θ l )) on S 1, wherever convenient. 3.1 Fast Computation of the Discretize R µ We now compute a iscretize version of the exponential Raon transform (1)intwo steps using two-imensional Laplace transforms [3] an FFT. A iscrete counterpart for computing III f (σ θ + iμ θ ) using III X f is given by the operator L : R X C, efine by L (III X f )(σ k,θ l ) = 1 f n 1,n = ( n1 ),n e ( μθ n l iθ l σ k ) ( 1, n 1 ). (9) If μ = 0 then sums of the above type can be approximately evaluate (with precision ɛ)ino( log + θ log(1/ɛ)) by using unequally space fast Fourier transforms (USFFT): [6,8,9]. To account for the case where μ = 0, we will instea use unequally space fast Laplace transforms [3] (USFLT), at the same computational cost an the same precision epenence as for the USFFT. We will enote the corresponing operator L USFLT. See Appenix for more information on this topic in two imensions. Once f (σ k θ l + iμ θ l ) is (approximately) compute, a iscrete version of R μ ( f )(s m,θ l ) can be obtaine by a one-imensional iscrete inverse Fourier transform R μ f (s m,θ l ) 1 1 k= f (σ k θ l + iμ θ l )e iσ ks m 1 1 L (III X f )(σ k,θ l )e iσ ks m. k=

9 J Fourier Anal Appl (018) 4: We enote the one-imensional iscrete inverse Fourier transform acting on the first variable by F : C C S, i.e. F g(s m,θ l ) = 1 k= g(σ k,θ l )e iσ ks m. Summing up, we can approximately compute III S R μ f by F LUSFLT III X f. An analysis of the errors involve in this approximation is foun in Appenix. 3. Error Analysis The total error of our iscretization/approximation is III S R μ f F LUSFLT III X f (10) which by the triangle inequality an the fact that DFT is isometric can be estimate by the sum of the error cause in the iscretization of the one-imensional Fourier integrals, III S R μ f F III f (σ θ + μθ ) ; (11) the iscretization error for the Laplace transform III f (σ θ + μθ ) L (III X f ) ; (1) an the error ue to the fast evaluation of the iscrete Laplace transform (L L USFLT )(III X f )). (13) The first two errors are ue to the iscretization, an the thir one is ue to the propose fast computational algorithm. As mention above, the error in (13) will be boune by the choice of precision parameter ɛ, see Appenix. We will now show that the errors in (11) an (1) bear similarities to stanar sampling results about essentially banlimite functions. To this en we introuce B μ, ( f ) = sup max( ω 1, ω )<, max( σ 1, σ ) μ n 1,n Z, n 1,n =0 f ( ω 1 + σ 1 i + n 1,ω + σ ) i + n.

10 440 J Fourier Anal Appl (018) 4: as a measurement of the frequency content of f outsie [ /, /]. ote that for banlimite functions whose frequency support is containe in the central square with sie length it hols that B 0, ( f ) = 0. Proposition 3.1 Assume that f has support in [ 1/, 1/]. The errors (11) an (1) are then both boune by a constant times B μ, ( f ). Proof By (3) it follows that ( ) R μ f (s m,θ l ) F III f (σ θ + μθ ) (s m,θ l ) ( ) ( ) = Fσ s f (σ θ + μθ ) (s m,θ l ) F III f (σ θ + μθ ) (s m,θ l ). (14) Set g(σ ) = f (σ θ +μθ ) an note that its frequency support lies within ( 1/, 1/). The Poisson summation formula yiels F σ s g(s) = k= g (k) e iks where the summation from /to/ 1correspons to F. It is now easy to see that the ifference (14) is boune by B μ, ( f ), yieling an l estimate of (11). The corresponing l estimate follows by the equivalence of finite imensional norms. The estimate for (1) is similar. By the two-imensional Poisson summation formula (applie to f (x)e μθ l x )wehave 1 = n 1 = n = n 1 = n = f f ( n1, n ) e ( μθ n l iθ l σ k ) ( 1, n ) 1 ( θ l σ k + μθ ) l + (n 1, n ). i By the assumption on the support of f, the first ouble sum can be replace by / 1 / 1 n 1 = / n = /, so the first line equals L (III X f )(σ k,θ l ).Itfollowsthat f (σ k θ l + μθl ) L (III X f )(σ k,θ l ) ( f n 1 =0 n =0 θ l σ k + μθ l i )) + (n 1, n, which yiels a pointwise estimate of the ifference in (1). The esire conclusion again follows by the equivalence of finite imensional norms.

11 J Fourier Anal Appl (018) 4: Fast Computation of the Discretize Ajoint The back-projection operator R μ in () can be iscretize in a similar way. ote that R μ g(x)= g(s,θ)e isσ x (iσθ+μθ s e ) ) σ θ,=( ˆR S 1 μ F s σ g (x) (15) which follows from () an the fact that g(x θ,θ) = F σ x θ F s σ (g(s, θ)). In the iscretize setting, the inner integral (i.e. F s σ ) correspons to F whereas the two outer integrals (i.e. ˆR μ ) correspon to (LUSFLT ). We remark that the latter o not correspon to a two-imensional Fourier Laplace integral (since the factor σ is missing), but it can still be evaluate fast, i.e., in O( log ) time. In the terminology of the Appenix, the USFLT operation changes from a Z C operation (for L USFLT )toac Z operation (for (L USFLT ) ), esigne to rapily evaluate sums of the type F k,l e (μθ l n iσ k θ l ) ( 1, n ), (16) k,l for all values ( n 1, n ) in X, where Fk,l can be arbitrary numbers. For the evaluation of III S R μ g via (L USFLT ) F III S g, we obviously have F = F III S g, but we shall see in the next section that other choices also will be relevant. An error analysis of this approximation can be one in a similar way as in Sect. 3., but we omit this. 3.4 Inversion from Complete Measurements To invert measurements of the exponential Raon transform on the polar gri S, we use the formula (7); f = R μ F M w FR μ f = ˆR μ M w ˆR μ f (17) A iscrete approximation g of ˆR μ f on is easily compute by applying F to the available ata III S R μ ( f ), an the operation ˆR μ correspons to (LUSFLT ). Since M w is multiplication by the weight w, the inversion coul in principle be one via f (L USFLT ) M w g, but the iscontinuities of w will introuce large errors for low-frequency components of f. To compensate for this, we will evaluate ˆR f on a polar gri with a more ense raial sampling near the singularities σ =± μ. The operator (LUSFLT ) computes sums of the type (16) over unequally space gris, an in particular there is no nee to have equally space sampling in σ. However, if we increase the sampling near the singularity, the corresponing sum (16) will not be a goo approximation of the

12 44 J Fourier Anal Appl (018) 4: corresponing integral in ˆR μ, an we therefore nee to compensate for this by moifying the weight w, using composite rules for quaratures. We briefly outline this in more etail below. In practice, the bottleneck of the reconstruction algorithm lies in evaluating (L USFLT ), an hence it is esirable to keep the total number of gri points low. We therefore propose to use a ense sampling in σ near the singularities ± μ.we remark that values of μ typically are less than, an that the original gri is sample with raial sampling istance of unit length. An increase sampling near μ can thus be achieve by increasing the amount of gri points in near the origin. Recall that f is assume to have support on the unit circle, which implies that R μ ( f )(s,θ) is supporte on [ 1/, 1/] in the s-variable. By zero-paing of R μ ( f )(s,θ)in the s-irection an application of FFT, an oversample representation of g(σ, θ) can be obtaine (on an equally space gri in σ, but not all values will be use). To erive appropriate weights, it is sufficient to consier accurate quaratures for σ > μ σ g(σ, θ)e μx θ +iσ x θ σ = σ > μ σ h(σ ), where we fix θ an set h(σ ) = 1 g(σ, θ)e μx θ +iσ x θ. Concerning the region aroun the singularity σ = μ, we construct a composite quarature by consiering b a σ h(σ )σ, (18) for several small intervals [a, b], containing, say, M < 10 gri points. On each such interval there are coefficients c ={c m } m=0 M 1 such that h(σ ) m c m σ m (19) is an Mth-egree polynomial approximation of h on the interval. These coefficients can be foun by solving h(σ k ) = c m σk m (for k ranging in the relevant range for [a, b]), or in matrix form h = Vc, where matrix V = V (m, k) is the Vanermone matrix built up by σk m. Upon inverting V, which is inepenent of h, we o not nee to compute the coefficients explicitly. Inserting this approximation in (18) gives b a σ h(σ )σ b a = k V 1 (m, k)h(σ k )σ m+1 σ m k ( b V 1 m+ ) (m, k)h(σ k ) m + am+ = m + m k w k h(σ k ). In this way, we obtain quarature weights w k for the equally space approximation of (18). The weights can be combine to form composite quarature rules that take

13 J Fourier Anal Appl (018) 4: Fig. 1 Weights for the approximation of the integral σ μ σ h(σ )σ, scaling by layers. Left panel weights after trapezoial rule correction near singularities σ = μ. Right panel total weights with unersampling for high frequencies. The black an re ots use values at every fourth sample (corresponing to the first two terms in the approximation (0)), while the blue ots use every secon sample (corresponing to the thir term in (0)) (Color figure online) the singularity at σ = μ into account. The same proceure can be performe when ealing with the singularity at σ = μ. The left panels of Fig. 1 emonstrate the effect of the correction with the weights w k near both the singularities. In orer for the approximation step in (19) to be accurate, h has to be sufficiently oversample. However, this is only neee in a vicinity of the singularity, an a regular sampling is sufficient away from the singularity. A irect usage of a trapezoial rule woul, however, use a sampling that is etermine by the neee sampling rate aroun the singularity. We therefore split the integral as b μ σ h(σ )σ = c μ σ h(σ )v(σ )σ + b σ h(σ )(1 v(σ))σ, where < c, v is a smooth function satisfying 0 v(σ) 1, v(σ) = 1ifσ<, v(σ) = 0ifσ > c. ow, the erive quarature rule with a ense sampling can be use for the left integral in the right-han sie above, while a reuce sampling ensity can be use on the right integral. This proceure can also be use several times to graually ecrease the sampling rate. In the numerical simulations presente in Sect. 4, we use an oversampling factor of four to account for the singularity at an make sampling reuction in two steps, i.e., by a quarature rule of the form σ h(σ )σ w 4k h(σ 4k ) + w 4k+ h(σ 4k+ ) + w k+1 h(σ k+1 ), σ μ k <K 1 k <K k <K 3 (0) where K 3 K << K 1. The setup is illustrate in the right panels of Fig. 1. μ,

14 444 J Fourier Anal Appl (018) 4: umerical Simulations In this section reconstruction results an computational spee tests are presente. C++ routines were written for the fast exponential Raon transform an the fast Laplace transform along with MEX interfaces to MATLAB. For accelerations we use the tools from Intel Composer XE 016, incluing the Intel Math Kernel Library (MKL), Intel Performance Primitives (IPP), an OpenMP API. For the simulation we use a stanar esktop with an Intel i7-380k processor an eight OpenMP threas, an the computations were performe in single precision. To begin with, accuracy tests were performe on the Shepp Logan [19] phantom, where we use the moifie version available in MATLAB. The phantom consists of linear combinations of characteristic functions of ellipses, an its support is inscribe in a circle, see the left panel of Fig.. The re line is epicte with corresponence to a specific angle θ an polar sample s. Values through this line multiplie by factor e μt are plotte in the left panel of Fig.. The upper part of Fig. 3 show the exponential Raon transform for ifferent values of the exponential parameter μ. In orer to make accuracy comparisons, we o not carry out comparisons with the original function, but use instea a slightly smoothe out version of the phantom. This is to account for the fact that the characteristic function of the ellipses oes not ecay rapily in the frequency omain, an hence high-frequency errors woul be ominant in the error plots. A reconstruction metho using only a finite frequency range shoul have as aim to recover that frequency range accurately, an it is therefore natural to measure the accuracy in this way. The minor smoothing effect can be note in the plots of Fig.. The lower part of Fig. 3 illustrates reconstruction errors for ifferent values of μ chosen to be the same as for the corresponing projection from the upper part of the same figure. To show how the choice of μ affects the reconstruction we compute reconstructions using the fast Laplace base filtere back-projection. In the filtere back-projection metho the high-frequency components are booste, an boosting of high frequent noise is also cause by the exponential factor in the back-projection Fig. Weighting with factor e μt for a line through the phantom image. Exponential weight y(t) = e μt is plotte by the ashe line, ratios between maximal an minimal weights equal to 1, 10, 100, 1000, respectively

15 J Fourier Anal Appl (018) 4: Fig. 3 Exponential Raon ata of the Shepp Logan phantom for ifferent values of the exponential parameter μ. Corresponing reconstruction with the FBP metho Fig. 4 Influence of the Gaussian noise (SR = 0) on the inversion with the FBP metho for ifferent attenuation parameter μ operator. In particular for high attenuation parameters, this reconstruction technique becomes sensitive to noise. This is illustrate in Fig. 4, which shows filtere-backprojection reconstruction results for varying values of the attenuation parameter μ. The Raon ata for these ifferent attenuation values are contaminate with white Gaussian noise with a signal-to-noise ratio (SR) of 0 B. Let us now turn our attention to the computational spee of the reconstruction metho. Tables 1 an show the total times of the simulations an the times per operation (to verify the time complexity) of applying the exponential Raon transform an backprojection, respectively, for ifferent image sizes. The time complexity of USFLT is O(ν log + (M + 1)(M + 1) ) (see Appenix), an that is the main computational part of the exponential Raon transform. Typically M is about 8 10 for single precision in the case where μ represents variations of up to a factor of ln(1000), therefore in practice the factor (M + 1)(M + 1) will

16 446 J Fourier Anal Appl (018) 4: Table 1 Computational times for the exponential Raon transform for images of size ( ) with θ = π Time Ratio t USFLT t Direct t USFLT /( log ) t USFLT / t Direct / e-0 1.4e-01.3e e e e-0 7.4e e e e e e e e e e e e e e e e+0 7.0e e e e+00 3.e e-08 5.e e e+01.7e e-08 5.e e-08 Table Computational times for backprojection of the exponential Raon transform for images of size ( ) with θ = π Time Ratio t USFLT t Direct t USFLT /( log ) t USFLT / t Direct / e-0 1.e e-07.6e e e-0 7.6e-01.5e e e e e e e e e-01 5.e e e e e e+0 9.1e e e e e e e e e e e e e-08 ominate the ν log factor. This effect can be seen in the ratio columns of Tables 1 an. We also verify that the USFLT-base metho outperforms the irect metho as expecte. 5 Conclusions We have shown how to construct fast algorithms for the computation of the exponential Raon transform an the associate (ajoint) back-projection operator by using algorithms for fast Laplace transforms. In aition, we have inclue estimates for the iscretization errors that arise, an shown how to separate them into parts where one bears similarities to stanar sampling arguments of (almost) banlimite functions, an where the other part is ue to the approximation errors arising in the fast Laplace transform. The latter part can be controlle at arbitrary numerical precision. Acknowlegements This work was supporte by the Crafoor Founation ( ) an the Sweish Research Council ( , ). Open Access This article is istribute uner the terms of the Creative Commons Attribution 4.0 International License ( which permits unrestricte use, istribution,

17 J Fourier Anal Appl (018) 4: an reprouction in any meium, provie you give appropriate creit to the original author(s) an the source, provie a link to the Creative Commons license, an inicate if changes were mae. Computation of L USFLT For the sake of completeness, we present algorithms for fast evaluation of twoimensional iscrete Laplace transforms. These algorithms are analogous to the one-imensional ones presente in [3], but in this appenix they are aapte for computing the exponential Raon transforms. Let n 1, n an ξ j be as in Sect. 3. We escribe how to fast evaluate the operations f = L C Z (F) : f n 1,n = j F j e (μ ξ 1 j iξ 1 j ) n 1 +(μ ξ j iξ j ) n, (1) F = L Z C (f) : F j = n 1,n f n1,n e (μ ξ 1 j iξ 1 j ) n 1 +(μ ξ j iξ j ) n. () where f n1,n an F j represent any complex numbers. ote that with f n1,n = f ( n 1, n ) ( ) an F j = f ξ j + iμ ξ j, the operation () correspons to the evaluation of (9) in Sect. 3, whereas (1) is neee in (16). Methos for unequally space fast Laplace transforms (USFLT) utilize convolution style operations with FFT to achieve approximations of the sums (1,) with arbitrary, but fixe, accuracy. We make a heuristic escription of how USFLT works, for a more precise formulation along with error estimates, see [3]. Let F enote the Fourier transform an introuce the Gaussian ϕ(x) = e α( x1 ) +x an moulate Gaussian ϕ ξ (x) = e μ ξ 1 j x 1+μ ξ j x ϕ(x), where α is a parameter which j we iscuss how to choose below. ote that in the frequency omain it hols that ϕ ξ j (v 1,v ) = π α e π α ( v 1 i μ ξ 1 j ) +( v i μ ξ ) j, i.e. ϕ ξ is a frequency moulate Gaussian. ext, by choosing g = f j ϕ one can erive the relation ( ) f ˆ(μ ξ j iξ j ) = f (x)e (μ ξ j iξ j ) x f x = F (ξ j ) =ĝ ϕ ξ (ξ j ) j = R ϕ ϕ ξ j ĝ(v 1,v ) ϕ ξ (v 1 ξ 1 j,v ξ j )v 1 v, (3) j The integral in (3) can be approximate by the trapezoial rule because of the fact that ϕ ξ j are smooth functions that ecay rapily to zero (the amount of oscillation is limite by the parameter μ). It turns out that accurate results can be obtaine by

18 448 J Fourier Anal Appl (018) 4: (v 1,v ) on an equally space lattice with some oversampling parameter ν. Crucial for the construction of fast algorithms is that the function ϕ ξ has small (numerical) support, since this will keep own the computational times for convolutions with ϕ ξ. It turns out that the (numerical) with of ϕ ξ is proportional to log ɛ, where ɛ is the esire precision. More precisely, values of ϕ ξ that are smaller than a certain j threshol level T ɛ = 1 π α log ɛ + α 4 + μ 4, can be omitte. In practice we thus replace ϕ ξ j by ϕ ɛ ξ j (v 1,v ) = { ϕ ξ (v 1,v ), j if v 1 T ɛ an v T ɛ. 0, otherwise, ow, if we assume that the iscrete Fourier transform (realize by FFT) provies an accurate approximation of ĝ at this oversample gri, the evaluation of () can be approximate in three steps: 1. Division in the space omain. FFT 3. Convolution-type operation in the frequency omain We refer to [3] for proofs an more etails, an remark that μ = log A in the notation of that paper. The approximations to prescribe precision ɛ can be achieve by a suitable choice of α in combination with proper oversampling ν. It then hols that the ifference between the sum of (1) an that escribe by the above algorithm can be boune for instance by Cɛ f l, where C is a constant that is inepenent of f. A similar result can also be obtaine for the approximation of (). Following [3], the parameter α shoul be chosen as α = ν 1 ν(ν 1) μ + 1 ( log ɛ), ν(ν 1) where ν is an oversampling factor for Fourier transform (usually ν = ). In practice this ( means that the integral (3) is replace by a summation over k 1, k Z using ϕ ɛ ξ k1 ν ξ 1 j, k ν ξ j ). ote that for each j = 1...J, these values will be nonzero only for values of k 1, k such that k 1 νξ 1 j νt ɛ an k νξ j νt ɛ. j By introucing M = νt ɛ we fin that the contributing values of k 1, k satisfy [νξ 1 j ] M k 1 [νξ 1 j ]+M an [νξ j ] M k [νξ j ]+M. The notations [x], x enote the nearest integer to x an the smallest integer less or equal to x, respectively.

19 J Fourier Anal Appl (018) 4: Finally, we have an approximation formula for () with accuracy ɛ: F j = f n1,n e (μ ξ 1 j iξ 1 j ) n 1 +( ξ j iξ j ) n n 1,n 1 ν ϕ ɛ ξ j ν 1 k 1 = ν ν 1 k = ν ( n 1,n ( k1 ν ξ 1 j, k ν ξ j ), f n1,n ϕ( n 1, n )e i ( k1 n 1 ν + k n ) ) ν where the sums over k 1, k variables consist each of (M +1) non-zero terms, respectively. The constant M is typically equal to 8-10 for single precision of computations, epening on the choice of μ an ν. Algorithm L Z C( f ) escribe these computational steps, with the assumption ξ j are insie a circle with raius slightly smaller than. The first operation (ivision) has O( ) computation cost, the secon Algorithm L C Z. 1. G k1,k = J j=1 F j ϕ ɛ ξ j. h n1,n = 1 ν 1 ν k 1 = ν 3. f n1,n = h n 1,n ϕ 0 ( n 1, n ( k1 ν ξ 1 j, k ) ν ξ j, ( ν k 1, k < ν ) ν 1 k = ν ), ( n 1, n < ). ( G k1,k e i k1 n 1 ν + k n ) ν, ( ν n 1, n < ν ) operation is evaluate by FFT in O(ν log()) time, an the thir operation cost O((M + 1) θ ) operations, since for each j there are only (M + 1) non-zero contributions. The total time complexity is thus O(ν log() + (M + 1) θ ). In terms of time complexity, this is ominate by FFT part, while in practice most time is typically spent on the thir step. The algorithm for L C Z( f ) (1) can be also constructe by applying the above operations in reverse orer, since L C Z ( f ) is ajoint to L Z C( f ). These steps are gathere in Algorithm L Z C below. Algorithm L Z C. 1. g n1,n = f n 1,n ϕ 0 ( n 1, n ), ( n 1, n < ). G k1,k = 1 ν 1 n 1 = 1 n = 3. F j = [νξ 1 j ]+M k 1 =[νξ 1 j ] M [νξ j ]+M k =[νξ j ] M G k 1,k ϕ ɛ ξ j ( g n1,n e i k1 n 1 ν + k n ν ), ( ν k 1, k < ν ) ( k1 ν ξ 1 j, k ) ν ξ j, (1 j J).

20 450 J Fourier Anal Appl (018) 4: References 1. Aguilar, V., Ehrenpreis, L., Kuchment, P.: Range conitions for the exponential Raon transform. J. Analyse Math. 68(1), 1 13 (1996). Al-Omari, S.K., Kılıçman, A.: On the exponential Raon transform an its extension to certain functions spaces. In: Abstract an Applie Analysis, vol Hinawi Publishing Corporation (014) 3. Anersson, F.: Algorithms for unequally space fast Laplace transforms. Appl. Comput. Harmon. Anal. 35(3), (013) 4. Arbuzov, E.V., Bukhgeim, A.L., Kazantsev, S.G.: Two-imensional tomography problems an the theory of a-analytic functions. Sib. Av. Math. 8(4), 1 0 (1998) 5. Beylkin, G.: The inversion problem an applications of the generalize Raon transform. Commun. Pure Appl. Math. 37(5), (1984) 6. Beylkin, G.: On the fast Fourier transform of functions with singularities. Appl. Comput. Harmon. Anal. (4), (1995) 7. Boman, J., Strömberg, J.-O.: ovikovs inversion formula for the attenuate Raon transforma new approach. J. Geom. Anal. 14(), (004) 8. Dutt, A., Rokhlin, V.: Fast Fourier transforms for nonequispace ata. SIAM J. Sci. Comput. 14(6), (1993) 9. Greengar, L., Lee, J.-Y.: Accelerating the nonuniform fast Fourier transform. SIAM Rev. 46, (004) 10. Hawkins, W.G., Leichner, P.K., Yang,.-C.: The circular harmonic transform for SPECT reconstruction an bounary conitions on the Fourier transform of the sinogram. IEEE Trans. Me. Imaging 7(), (1988) 11. Huang, Q., Zeng, G.L., Gullberg, G.T.: An analytical inversion of the 180 exponential Raon transform with a numerically generate kernel. Int. J. Image Graph. 7(01), (007) 1. Metz, C.E., Pan, X.: A unifie analysis of exact methos of inverting the - exponential Raon transform, with implications for noise control in SPECT. IEEE Trans. Me. Imaging 14(4), (1995) 13. atterer, F.: On the inversion of the attenuate Raon transform. umer. Math. 3(4), (1979) 14. atterer, F.: Inversion of the attenuate Raon transform. Inverse Probl. 17(1), 113 (001) 15. ovikov, R.G.: An inversion formula for the attenuate X-ray transformation. Arkiv för matematik, 40(1), (00) 16. Puro, A.: Cormack-type inversion of exponential Raon transform. Inverse Prob. 17(1), 179 (001) 17. Puro, A., Garin, A.: Cormack-type inversion of attenuate Raon transform. Inverse Prob. 9(6), (013) 18. Rullgår, H.: An explicit inversion formula for the exponential Raon transform using ata from 180. Arkiv för matematik 4(), (004) 19. Shepp, L.A., Logan, B.F.: The Fourier reconstruction of a hea section. IEEE Trans. ucl. Sci 1(3), 1 43 (1974) 0. Tretiak, O., Delaney, P.: The exponential convolution algorithm for emission compute axial tomography. A Review of Information Processing in Meical Imaging. Technical report. Oak Rige ational Laboratory ORL/BCTIC (1978) 1. Tretiak, O., Metz, C.: The exponential Raon transform. SIAM J. Appl. Math. 39(), (1980). Yarman, C.E., et al.: An inversion metho for the exponential Raon transform base on the harmonic analysis of the Eucliean motion group. In: Meical Imaging, pp. 6144A 6144A. International Society for Optics an Photonics (006) 3. Yarman, C.E., et al.: A new exact inversion metho for exponential Raon transform using the harmonic analysis of the Eucliean motion group. Inverse Probl. Imaging 1(3), (007)