## Algorithms in Tomography (1997)

Venue: | The State of the Art in Numerical Analysis |

Citations: | 3 - 0 self |

### BibTeX

@INPROCEEDINGS{Natterer97algorithmsin,

author = {Frank Natterer},

title = {Algorithms in Tomography},

booktitle = {The State of the Art in Numerical Analysis},

year = {1997},

publisher = {Clarendon Press}

}

### OpenURL

### Abstract

this paper is as follows. In section 2 we survey the mathematical models used in tomography. In section 3 we give a fairly detailed survey on 2D reconstruction algorithm which still are the work horse of tomography. In section 4 we describe recent developments in 3D reconstruction. In section 5 we make a few remarks on the beginning development of algorithms for nonstraight -line tomography. 2 Mathematical Models in Tomography

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Citation Context ...so-called backprojection and K = 1 2 (2��) 1\Gamman ( (\Gamma1) (n\Gamma2)=2 H @ n\Gamma1 @s n\Gamma1 ; n even (\Gamma1) (n\Gamma1)=2 @ n\Gamma1 @s n\Gamma1 ; n odd (2.6) with H the Hilbert transf=-=orm [39]-=-. In fact the numerical implementation of (2.4) leads to the filtered backprojection algorithm which is the standard algorithm in commercial CT scanners, see section 3. For n = 3, the relevant integra... |

215 |
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Citation Context ...ompletely undetermined [47]. This is reminiscent of vector tomography. 3 Basic Algorithms in 2D Tomography The numerical implementation of Radon's inversion formula (2.6) is now well understood [39], =-=[25], [2-=-8]. We consider only the simplest case of parallel scanning. This means that g(`; s) = (Ra)(`; s) is sampled at ` = ` j = ` cos ' j sin ' j ' ; ' j = �� j =p ; j = 0; : : : ; p \Gamma 1 ; (3.1) s ... |

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Citation Context ...d attempt to maximize "posterior likelihood". Thus f is assumed to have a prior probability distribution, called a Gibbs-Markov random field ��(f), which gives preference to certain func=-=tions f [46], [21], [18]. -=-Most prior �� simply add a penalty term to the likelihood function to account for correction between neighboring pixels and do not use biological information. However if �� is carefully chosen... |

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Citation Context ...owing. Assume that g(x; `) = u(x) ; ` 2 S n\Gamma1 (2.12b) is known outside\Omega\Gamma Determine f inside\Omega\Gamma Uniqueness and stability of the inverse problem (2.12) has recently been settled =-=[36]-=-. However, stability is only logarithmic [2], i.e. a data error of size ffi results in a reconstruction error 1= log(1=ffi). Numerical algorithms did not emerge from this work. Numerical methods for (... |

77 |
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Citation Context ...ecompose a in its solenoidal and potential part, i.e. a = a 1 +ra 2 div a 1 = 0 a 2 = 0 on @\Omega : It can be shown that a 1 can be recovered uniquely from (2.28), but a 2 is completely undetermined =-=[47]-=-. This is reminiscent of vector tomography. 3 Basic Algorithms in 2D Tomography The numerical implementation of Radon's inversion formula (2.6) is now well understood [39], [25], [28]. We consider onl... |

57 |
An inversion formula for cone-beam reconstruction
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Citation Context ...lane. In view of (4.5) this means that g is available for a neighbourhood of the fan in that plane converging to the source v. This is Grangeat's completeness condition. The inversion formulas of Tuy =-=[53], B. Smith [50] -=-and Gelfand and Goncharov [17] can be obtained by puttingsh(oe) = (2��) 1=2 (joej \Gamma oe),sk(oe) = (2��) \Gamma1=2 oe andsh(oe) = (2��) 1=2 joej,sk(oe) = (2��) \Gamma1=2 joej, respe... |

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Citation Context ...rstood. Direct algebraic algorithms [39] compute a minimal norm solution in L 2 for finitely many data by FFT techniques. Iterative methods work on discrete versions of linear reconstruction problems =-=[10]-=-, [24]. ART (= algebraic reconstruction technique [24], [25]) is based on the Kaczmarz iteration for linear systems and can be viewed as an SOR method [14]. The EM iteration is described in section 2(... |

42 |
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Citation Context ...4.4) holds. The simplest choice for h, k issh = \Gamma(\Gamma2��) \Gamma1=2 ioe,sk = (2��) 1=2 ioe. In this case, both h = \Gammaffi 0 , k = 2��ffi 0 are local. We obtain Grangeat's invers=-=ion formula [19] G(`; v) -=-= Z S 2 "` ? @ @` g(v; !)d! ; (4.5) a(x) = 1 8�� 2 Z S 2 F 0 (!; ! \Delta x)d! (4.6) where @ @` is the derivative in direction ` with respect to the second argument and F 0 is the partial der... |

36 |
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Citation Context ...thm is based on (3.6). Theoretically the complexity of this algorithm isO(\Omega 2 log \Omega\Gammag but the accurate and efficient implementation is in no way easy. It seems that the gridding method =-=[45]-=- works satisfactorily, but the theory behind this method is not well understood. Direct algebraic algorithms [39] compute a minimal norm solution in L 2 for finitely many data by FFT techniques. Itera... |

31 |
A filtered backpropagation algorithm for diffraction tomography, Ultrasonic Imaging 4
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Citation Context ... in a reconstruction error 1= log(1=ffi). Numerical algorithms did not emerge from this work. Numerical methods for (2.12) are mostly based on linearizations, such as the Born and Rytov approximation =-=[13]-=-. In order to derive the Born approximation, one rewrites (2.12a) as u(x) = u ` (x) \Gamma k 2 Z \Omega G(x \Gamma y)f(y)u(y)dy (2.13) Frank Natterer 7 where G is an appropriate Green's function. For ... |

28 |
Image reconstruction from cone-beam projections necessary and sufficient conditions and reconstruction methods
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Citation Context ...f (4.5) this means that g is available for a neighbourhood of the fan in that plane converging to the source v. This is Grangeat's completeness condition. The inversion formulas of Tuy [53], B. Smith =-=[50] and Gelfand and-=- Goncharov [17] can be obtained by puttingsh(oe) = (2��) 1=2 (joej \Gamma oe),sk(oe) = (2��) \Gamma1=2 oe andsh(oe) = (2��) 1=2 joej,sk(oe) = (2��) \Gamma1=2 joej, respectively [12]. T... |

27 |
The exponential Radon transform
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Citation Context ...(2.10) for f . Apart from the exponential factor, (2.10) is identical - up to notation - to the integral equation in transmission CT. Except for very special cases - e.g. a constant in a known domain =-=[52]-=-, [42] no explicit inversion formulas are available. Numerical techniques have been developed but are considered to be slow. Again the situation becomes worse if scatter is taken into account. This ca... |

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Citation Context ...gnetic field H space dependent in a controled way the local magnetization M 0 (x) (together with the relaxation times T 1 (x), T 2 (x)) can be imaged. In the following we derive the imaging equations =-=[27]-=-. The magnetization M (x; t) caused by a magnetic field H(x; t) satisfies the Bloch equation @M @t = flM \Theta H \Gamma 1 T 2 (M 1 e 1 + M 2 e 2 ) \Gamma 1 T 1 (M 3 \Gamma M 0 )e 3 : (2.20) Here, M i... |

21 |
A Cone-Beam Reconstruction Algorithm Using Shift-Variant Filtering and Cone-Beam Backprojection
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Citation Context ...th [50] and Gelfand and Goncharov [17] can be obtained by puttingsh(oe) = (2��) 1=2 (joej \Gamma oe),sk(oe) = (2��) \Gamma1=2 oe andsh(oe) = (2��) 1=2 joej,sk(oe) = (2��) \Gamma1=2 joe=-=j, respectively [12]-=-. These formulas are not as useful as Grangeat's formula since h is no longer local. In practice g(v; \Delta) is measured on a detector plane D v = p v \Gamma v + (p v \Gamma v) ? where p v is the ort... |

18 |
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Citation Context .... Direct algebraic algorithms [39] compute a minimal norm solution in L 2 for finitely many data by FFT techniques. Iterative methods work on discrete versions of linear reconstruction problems [10], =-=[24]-=-. ART (= algebraic reconstruction technique [24], [25]) is based on the Kaczmarz iteration for linear systems and can be viewed as an SOR method [14]. The EM iteration is described in section 2(b). Th... |

17 |
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Citation Context ...ation to the transport equation by a parabolic differential equation. Since inverse problems for parabolic equations are severely ill-posed, this approach is questionable. Higher order approximations =-=[29]-=-, [20] are hyperbolic, making the inverse problem much more stable. As an alternative to the transport equation one can also model light tomography by a discrete stochastical model [22]. In the 2D cas... |

17 |
Sampling in fan-beam tomography
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Citation Context ...orithm can also be used for fan beam geometry, in which the X-ray tube sits on a circle surrounding the patient, with a detector array on the opposite side. The sampling requirements are discussed in =-=[38]-=-. 4 Formulas for 3D reconstruction In 3D CT one reconstructes a from the values of g(`; v) = Z a(v + t`)dt where v runs through the source curve V outside supp(a) and ` 2 S 2 . Most reconstruction for... |

16 |
Determination of tissue attenuation in emission tomography of optically dense media. Inverse Problems 9
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Citation Context ...ideal mathematical solution would be a method for determining f and a in (2.9) simultaneously. Under strong assumption on a (e.g. a constant in a known region [26], a affine distortion of a prototype =-=[37]-=-, a close to a known distribution [8]) encouraging results have been obtained. Theoretical results based on the transport formulation have been obtained, even for models including scatter [43]. But a ... |

15 |
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Citation Context ...y the determination of a in such an arrangement is, though uniquely possible, highly unstable. The condition of stability is the following: Each plane meeting supp(a) must contain at least one source =-=[16]-=-. This condition is obviously violated for sources on a circle. Cases in which the condition is satisfied include the helix and a pair of orthogonal circles. A variety of inversion formulae has been d... |

14 |
A modified gradient method for two dimensional problems in tomography
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Citation Context ...t know how to use Gelfand-Levitan, nor do we know anything about stability. Of course one can always solve the nonlinear problem (2.17) by a Newton type method. Such methods have been developed [23], =-=[30]-=-. They suffer from excessive computing time and from their apparent inability to handle large wave numbers k. (d) Optical tomography Here one uses NIR (= near infra-red) lasers for the illumination of... |

11 |
Theory of three-dimensional reconstruction: Conditions of a complete set of projections. Soviet Physics Crystallography
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Citation Context ... 1 4�� 2 \Delta Z G Z ` ? Pf(`; x \Gamma y) jyjL(`; y) dyd` (4.8) where G ` S 2 is a spherical zone around the equator and L(`; y) is the length of the intersection of G and the plane spanned by `=-=, y [41]-=-. With (4.8) one still has problems near the openings of the cylinder. More satisfactory reconstruction formulas based on the principle of the stationary phase have been given in [11]. 5 Algorithm for... |

10 |
The identi problem for the constantly attenuated Radon transform
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Citation Context ...catter. For the attenuation problem, the ideal mathematical solution would be a method for determining f and a in (2.9) simultaneously. Under strong assumption on a (e.g. a constant in a known region =-=[26]-=-, a affine distortion of a prototype [37], a close to a known distribution [8]) encouraging results have been obtained. Theoretical results based on the transport formulation have been obtained, even ... |

9 |
Integrated Photoelasticity
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Citation Context ...ding on a. The problem is to recover a in\Omega from the knowledge of U (x 0 ; x 1 ) for x 0 , x 1 2 @ For n = 1 we regain (2.1). Applications of (2.25) for n ? 1 have become known in photoelasticity =-=[1], bu-=-t applications to medicine are not totally out of question. Frank Natterer 13 In a further extension we let a depend on the direction �� = (x 1 \Gamma x 0 )=jx 1 \Gamma x 0 j. Such problems occur ... |

9 |
Reconstructing from efficiently sampled data in parallel-beam computed tomography
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Citation Context ... suffice. Without loosing resolution we can drop g(` j ; s ` ) for j + ` = p mod 2. In that case the interpolation step for (3.7) has to be done with care, e.g. with a stepsize much smaller than ae=q =-=[15]. 2. The con-=-dition ae q ��\Omega has to be satisfied strictly. If it is violated, (3.5) is not valid even approximately, making the reconstruction completely unacceptable. 3. If �� ps�� \Omega ae is n... |

9 |
Principles of Computerized Tomography Imaging
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Citation Context ...ely undetermined [47]. This is reminiscent of vector tomography. 3 Basic Algorithms in 2D Tomography The numerical implementation of Radon's inversion formula (2.6) is now well understood [39], [25], =-=[28]. We-=- consider only the simplest case of parallel scanning. This means that g(`; s) = (Ra)(`; s) is sampled at ` = ` j = ` cos ' j sin ' j ' ; ' j = �� j =p ; j = 0; : : : ; p \Gamma 1 ; (3.1) s = s ` ... |

5 |
A performance study of 3D reconstruction algorithms for positron emission tomography. Phys Med Biol
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Citation Context ... spanned by `, y [41]. With (4.8) one still has problems near the openings of the cylinder. More satisfactory reconstruction formulas based on the principle of the stationary phase have been given in =-=[11]-=-. 5 Algorithm for more general inverse problems The problem of CT is probably the most simple inverse problem. In general one has to solve the inverse problem for a partial differential equation. The ... |

5 |
An inversion method for an attenuated x-ray transform, Inverse Problems 12
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Citation Context ... for f . Apart from the exponential factor, (2.10) is identical - up to notation - to the integral equation in transmission CT. Except for very special cases - e.g. a constant in a known domain [52], =-=[42]-=- no explicit inversion formulas are available. Numerical techniques have been developed but are considered to be slow. Again the situation becomes worse if scatter is taken into account. This can be d... |

5 |
Maximum Liklihood Reconstruction for Emission Tomography
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Citation Context ... a from the emission data has not yet been found. Noise and scatter are stochastic phenomena. Thus, besides models using integral equations, stochastic models have been set up for emission tomography =-=[48]-=-. These models are completely discrete. We subdivide the reconstruction region into m pixels or voxels. The number of events in pixel/voxel j is a Poisson random variable ' j whose mathematical expect... |

4 |
Recovery of a compactly supported function starting from its integrals over lines intersecting a given set of points
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Citation Context ...ailable for a neighbourhood of the fan in that plane converging to the source v. This is Grangeat's completeness condition. The inversion formulas of Tuy [53], B. Smith [50] and Gelfand and Goncharov =-=[17] can be obtained-=- by puttingsh(oe) = (2��) 1=2 (joej \Gamma oe),sk(oe) = (2��) \Gamma1=2 oe andsh(oe) = (2��) 1=2 joej,sk(oe) = (2��) \Gamma1=2 joej, respectively [12]. These formulas are not as useful... |

4 |
Regularized quasi-Newton method for inverse scattering problems
- Gutman, Klibanov
- 1993
(Show Context)
Citation Context ... do not know how to use Gelfand-Levitan, nor do we know anything about stability. Of course one can always solve the nonlinear problem (2.17) by a Newton type method. Such methods have been developed =-=[23]-=-, [30]. They suffer from excessive computing time and from their apparent inability to handle large wave numbers k. (d) Optical tomography Here one uses NIR (= near infra-red) lasers for the illuminat... |

2 |
Stable Determination of Conductivity by Boundary Mesurements
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(Show Context)
Citation Context ...amma1 (2.12b) is known outside\Omega\Gamma Determine f inside\Omega\Gamma Uniqueness and stability of the inverse problem (2.12) has recently been settled [36]. However, stability is only logarithmic =-=[2]-=-, i.e. a data error of size ffi results in a reconstruction error 1= log(1=ffi). Numerical algorithms did not emerge from this work. Numerical methods for (2.12) are mostly based on linearizations, su... |

2 |
E.E.: Investigation of Scattering and Absorbing Media by the Methods of X-ray Tomography
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Citation Context ...overing a from (2.8) in much harder. An explicit formula for a such as (2.4) has not become known and is unlikely to exist. Nevertheless numerical methods have been developed for special choices of j =-=[3]-=-, [7]. The situation gets even more difficult if one takes into account that, strictly speaking, j is object dependent and hence not known in advance. (2.8) is a typical example of an inverse problem ... |

2 |
Uniqueness of Simultaneous Determination of two Coefficients of the Transport
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(Show Context)
Citation Context ...o be quite different from what we have seen in other types of tomography. The mathematical theory of the inverse problem (2.18) is in a deplorable state. There exist some Russian papers on uniqueness =-=[4]-=-. General methods have been developed, too, but apparently they have been applied to 1D problems only [44]. Nothing seems to be known about stability. The numerical methods which have become known are... |

2 |
Degradation transform in tomography
- Bronnikov
- 1994
(Show Context)
Citation Context ... method for determining f and a in (2.9) simultaneously. Under strong assumption on a (e.g. a constant in a known region [26], a affine distortion of a prototype [37], a close to a known distribution =-=[8]-=-) encouraging results have been obtained. Theoretical results based on the transport formulation have been obtained, even for models including scatter [43]. But a clinically useful way of determining ... |

2 |
D.: Statistical Methods for Tomographic Image
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(Show Context)
Citation Context ...mpt to maximize "posterior likelihood". Thus f is assumed to have a prior probability distribution, called a Gibbs-Markov random field ��(f), which gives preference to certain functions =-=f [46], [21], [18]. Most p-=-rior �� simply add a penalty term to the likelihood function to account for correction between neighboring pixels and do not use biological information. However if �� is carefully chosen so th... |

2 |
Medical Imaging: State of the Art
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Citation Context ...cal Models in Tomography In the description of the mathematical models we restrict ourselves to those features which are important for the mathematical scientist. For physical and medical aspects see =-=[35]-=-, [6], [32]. (a) Transmission CT. This is the original and simplest case of CT. In transmission tomography one probes an object with non diffracting radiation, e.g. x-rays for the human body. If I 0 i... |

1 |
V.: X-Ray Tomography
- Bondarenko
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(Show Context)
Citation Context ...ng a from (2.8) in much harder. An explicit formula for a such as (2.4) has not become known and is unlikely to exist. Nevertheless numerical methods have been developed for special choices of j [3], =-=[7]-=-. The situation gets even more difficult if one takes into account that, strictly speaking, j is object dependent and hence not known in advance. (2.8) is a typical example of an inverse problem for a... |

1 |
Block-iterative methods for consistent and inconsistent linear equations
- Elving
- 1980
(Show Context)
Citation Context ...e versions of linear reconstruction problems [10], [24]. ART (= algebraic reconstruction technique [24], [25]) is based on the Kaczmarz iteration for linear systems and can be viewed as an SOR method =-=[14]-=-. The EM iteration is described in section 2(b). The filtered backprojection algorithm can also be used for fan beam geometry, in which the X-ray tube sits on a circle surrounding the patient, with a ... |

1 |
et al.: A novel approach to laser tomography
- Gratton
- 1993
(Show Context)
Citation Context ...to the transport equation by a parabolic differential equation. Since inverse problems for parabolic equations are severely ill-posed, this approach is questionable. Higher order approximations [29], =-=[20]-=- are hyperbolic, making the inverse problem much more stable. As an alternative to the transport equation one can also model light tomography by a discrete stochastical model [22]. In the 2D case, bre... |

1 |
F.: A propagation-backpropapagation method for ultrasound tomography
- Natterer
- 1995
(Show Context)
Citation Context ...ton method for solving the inverse problem simply as a nonlinear problem [31]. A method for solving the inverse problem (2.12) of ultrasound CT by an ART type method (compare section 3) is as follows =-=[40]-=-. For each direction ` j , j = 1; : : : ; p, define the nonlinear (Radon-type) operator R j = L 2(\Omega\Gamma ! L 2 (@ R j (f) = uj @\Omega j where u is the solution to (2.12a) for ` = ` j with bound... |

1 |
Conditional Stability Estimates for the Problem of Recovering of Absorption Coefficients and Right Hand Side of Transport Equations
- Romanov
(Show Context)
Citation Context ...ototype [37], a close to a known distribution [8]) encouraging results have been obtained. Theoretical results based on the transport formulation have been obtained, even for models including scatter =-=[43]-=-. But a clinically useful way of determining a from the emission data has not yet been found. Noise and scatter are stochastic phenomena. Thus, besides models using integral equations, stochastic mode... |

1 |
N.J.: General Solutions to Inverse Transport Problems
- Sanchez
- 1981
(Show Context)
Citation Context ...inverse problem (2.18) is in a deplorable state. There exist some Russian papers on uniqueness [4]. General methods have been developed, too, but apparently they have been applied to 1D problems only =-=[44]-=-. Nothing seems to be known about stability. The numerical methods which have become known are of the Newton type, either applied directly to the transport equation or to the so-called diffusion appro... |

1 |
ESNM: Ein rauschunterdruckendes EM - Verfahren fur die Emissionstomographie. Thesis, Fachbereich Mathematik der Universitat
- Setzepfandt
- 1992
(Show Context)
Citation Context ...on" and attempt to maximize "posterior likelihood". Thus f is assumed to have a prior probability distribution, called a Gibbs-Markov random field ��(f), which gives preference to c=-=ertain functions f [46], [21], -=-[18]. Most prior �� simply add a penalty term to the likelihood function to account for correction between neighboring pixels and do not use biological information. However if �� is carefully ... |

1 |
W.: Doppler tomography for vector fields
- Strakl'en
- 1995
(Show Context)
Citation Context ... (2.25) is similar to the Radon transform. One can show that curl u can be computed from Ru, and an inversion formula similar to the Radon inversion formula exists. Numerical simulations are given in =-=[51]-=-. (h) Tensor Tomography As an immediate extension of transmission CT to non-isotropic media we consider a matrix valued attenuation a(x) = (a ij (x)), i, j = 1; : : : ; n. We solve the vector differen... |