Numerical Realizations of X-ray Computerized
Tomography by Cauchy-type Boundary Integration
Hiroshi Fujiwara (Kyoto University)
Alexandru Tamasan (University of Central Florida)
We study numerical feasibility and properties of a Cauchy-type boundary integration
formula in X-ray Computerized Tomography (CT). The integration formula has beendeveloped by Bukhgeim et. al. [1, 2, 3] based on mainly the theory of A-analytic functions
by converting the mathematical model of X-ray CT into the inverse source problem forthe transport equation. However, to the best of the authors’ knowledge, its numericalrealization have not been realized so far. It is also worthy of notice that the method
has a possibility in application to the radiative transport equation [4], while the inverse
Radon transform is unavailable due to its crucial dependence on straightness of X-ray
propagation.
Let D be a convex domain in R2 , and µ(x) be the absorption coeﬃcient at x ∈ D.
The standard mathematical model of the X-ray CT [7] is to ﬁnd µ(x) satisfying theintegral equation∫µ(x) dℓ = Rµ(ω, s),x·ω=sfrom measurement of Rµ(ω, s) for all ω ∈ S 1 and s ∈ R, which is called the Radontransform of µ. The ﬁltered back projection (FBP) with the inverse Radon transform [8]has been well established in reconstruction of µ(x), which is given as∫ π1Pω ∗ h(x · ω) dΩ,x ∈ D,µ(x) ≈2π 0
where Ω = arg ω, Pω (s) = Rµ(ω, s), and h(s) is the ﬁlter function whose Fourier transform approximates |r|.
On the other hand, Arbuzov, Bukhgeim and Kazantsev have proposed alternative
inversion algorithm based on A-analytic theory [1, 2]. They considered the equivalent
inverse source problem of the transport equation
ξ · ∇u(x, ξ) = µ(x),u(x, ξ) = 0,(x, ξ) ∈ D × S 1 ,(x, ξ) ∈ Γ− ,u(x, ξ) = Rµ(ξ ⊥ , x · ξ ⊥ ), (x, ξ) ∈ Γ+ ,where Γ± = {(x, ξ) ; x ∈ ∂D, ξ∑∈ S 1 , n(x) · ξ ≷ 0}. It is shown that introducing the()Fourier expansion u z, ξ(θ) =un (z)e−inθ , the attenuation µ is reconstructed asn∈Zµ(z) = 2 Re

∂u1(z),∂zz ∈ D,(1)

FBP

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1(a) Filtered Back Projection with Inverse Radon Transform

R=1.1, FD

10.50Ellipse, P1

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00.51(b) Proposed Method, D : Circle with Radius
R = 1.1, ∂/∂z : Finite Diﬀerence [5]

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00.51(c) Proposed Method, D : Ellipse, ∂/∂z :Numerical Diﬀerentiation by Least SquareMethod [6]Figure 1: Numerical Reconstructions for Modiﬁed Shepp-Logan Phantom [9].

where x = (x1 , x2 ) ∈ R2 is identiﬁed with z = x( 1 + ix2 ∈ C, and u1 (z),) z ∈ D is obtainedby the by the boundary integral of uodd (ζ) = u1 (ζ), u3 (ζ), u5 (ζ), . . . , ζ ∈ ∂D as()∫1dζ − S ∗ dζζ −zuodd (z) =Ruodd (ζ), z ∈ D,(2)2πi ∂Dζ −z
ζ −z
where R(λ) = (λI − S ∗ )−1 and S ∗ (u1 , u3 , u5 , . . . ) = (u3 , u5 , u7 , . . . ). In the formula,
the resolvent R might cause numerical instability. We show a regularization schemeas truncation of high-frequency modes based on the expression given by Finch [3]. A
strategy of optimal choice of regularization parameters is also exhibited by calculating
the norm of u0 which does not explicitly appear in the reconstruction procedure.
Figures 1 show the numerical realization by (1) and (2). The results by the proposed
method, (b) and (c), exhibit similar accuracy as FBP shown in (a). Details are discussed
in [5] and [6].

References[1] E. V. Arbuzov, A. L. Bukhgeı̆m, and S. G. Kazantsev, Two-dimensional
tomography problems and the theory of A-analytic functions [translation of algebra,

geometry, analysis and mathematical physics (russian) (novosibirsk, 1996), 6–20,
189, Izdat. Ross. Akad. Nauk Sibirsk. Otdel. Inst. Mat., Novosibirsk, 1997], SiberianAdv. Math., 8 (1998), pp. 1–20.
[2] A. L. Bukhgeim, Inversion formulas in inverse problems, in Linear Operators andIll-Posed Problems by M. M. Lavrentiev and L. Ya. Savalev, Plenum, New York,(1995), pp. 323–378.
[3] D. V. Finch, The attenuated X-ray transform: recent developments, in Inside out:
inverse problems and applications, vol. 47 of Math. Sci. Res. Inst. Publ., Cambridge
Univ. Press, Cambridge, 2003, pp. 47–66.
[4] H. Fujiwara, K. Sadiq, and A. Tamasan, A Fourier approach to the inverse
source problem in an absorbing and non-weakly scattering medium, Inverse Problems,
36 (2020), pp. 015005, 33.
[5] H. Fujiwara and A. Tamasan, Meshless X-ray computerized tomography by theCauchy-type integral formula, JASCOME, 19 (2019), pp. 1–6. (in Japanese).
[6], Numerical realization of a new generation tomography algorithm based on theCauchy-type integral formula, Adv. Math. Sci. Appl., 28 (2019), pp. 413–424.

[7] F. Natterer and F. Wübbeling, Mathematical methods in image reconstruction, SIAM Monographs on Mathematical Modeling and Computation, Society forIndustrial and Applied Mathematics (SIAM), Philadelphia, PA, 2001.
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gewisser mannigfaltigkeiten, Berichte Sächsische Akademie der Wissenschaften zu
Leipzig, Math.-Phys. Kl., 69 (1917), pp. 262–277. (translated : On the determination of functions from their integral values along certain maniforlds, in IEEE Trans.
Med. Imaging, MI-5 (1986), pp. 170–176.).
[9] P. A. Toft, The Radon Transform — Theory and Implementation, PhD thesis,
Technical University of Denmark, 1996.