从光子统计到锥形束CT：计算机断层扫描原理与算法

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"本书《Computed Tomography》由Thorsten M. Buzug撰写，全面介绍了X射线技术、CT系统的发展历程以及计算体层成像的主要重建方法。书中强调了数学在重建技术中的基础作用，同时也深入探讨了X射线生成、物质与光子的相互作用、X射线检测、光子统计以及信号处理基本概念和专门的测量系统。内容涵盖了从光子统计到现代锥形束CT系统的基础知识，并重点解析了二维和三维锥形束CT的重建算法。书中还分析了CT成像中的伪影，并讨论了剂量考虑等实际问题，旨在帮助研究生和从业者深入理解现代CT系统。" 《Computed Tomography》一书结构清晰，从基础的CT理论开始，包括光子统计，逐步深入到现代锥形束系统。作者着重讲解了重建算法的详细推导，不仅限于2D，还包括了3D锥形束CT系统。这部分内容对于理解和应用这些技术至关重要，因为CT图像的质量和准确性很大程度上取决于这些算法。此外，书中对CT成像中的艺术性和误差进行了详尽的分析，如各种可能产生的伪影，这些伪影可能由于设备性能、扫描参数选择或数据采集过程中的问题而出现。通过对这些问题的探讨，读者可以了解到如何优化CT扫描以获得更高质量的图像，并理解如何减少潜在的副作用，比如辐射剂量的问题。这本书不仅适合生物医学工程、医学工程科学、医学物理、放射学、数学、电气工程和物理学的研究生阅读，同时，对于在这些领域有经验的实践者来说，也是一本有价值的参考书。通过阅读，读者不仅能掌握CT的基本原理，还能了解到最新的发展和技术挑战。《Computed Tomography》是一部综合性的CT技术指南，结合理论与实践，旨在提供一个全面的视角来理解和运用CT技术。它深入浅出地介绍了CT技术的核心要素，包括物理基础、重建算法、伪影分析和剂量管理，对于想要在CT领域深化研究或实践的人来说，是一本不可或缺的教材。

 1Introduction

1.3

Historical Perspective

A particular kind of mathematical problem in CT became popular in the s

when the astrophysicist Bracewell proved that the resolution of telescopes could

be signiﬁcantly improved if spatially distributed telescopes are appropriately syn-

chronized. However, in , similar problems with the same mathematical basis

had already been discussed (cf. Cormack [] and the references therein; many

examples are collected by Deans []).

One example can be found in the ﬁeld of statistics: Given all marginal distri-

butions of a probability distribution density, is it possible to deduce the probability

density itself? In this example, the marginal distributions represent an equivalent

example for the measured projections in computed tomography. Another example

can be found in astrophysics: Looking from earth into a particular direction of the

universe, only the radial velocity component of the stars can be obtained via the

spectral Doppler red shi. is again represents the same inverse problem that has

to be solved if the distribution of the actual three-dimensional velocity vector is to

be reconstructed from the radial velocity components acquired from all available

directions.

In computed tomography, the meaning of the mathematical term inverse prob-

lem is immediately apparent. In contrast to the situation shown in Fig. ., the spa-

tial distribution of the attenuating objects that produce the projection shadow is

not known a priori. is, actually, is the reason for acquiring the projections along

the rotating detector coordinate ξ over a projection angle interval of at least 



Figure . illustrates this situation. It is an inversion of integral transforms. From

a sequence of measured projection shadows p



(ξ), p



(ξ), p



(ξ),...,thespa-

Fig. .. Schematic illustration of the inverse problem posed by CT. Attenuation pro-

ﬁles, p

(ξ), have been measured for a set of projection angulations, γ



and γ



. e unknown

geometry, or the object with its associated spatial distribution of attenuation coeﬃcients, has

to be calculated from a complete set of attenuation proﬁles p



(ξ), p



(ξ), p



(ξ),...

 1Introduction

tial distribution of the objects, or more precisely, the spatial distribution of the at-

tenuation coeﬃcients within a chosen section through the patient, must be esti-

mated.

In , the solution to this problem was applied for the ﬁrst time to a sequence

of X-ray projections for which an anatomical object had been measured from diﬀer-

ent directions. Allen MacLeod Cormack (–) and Sir Godfrey Hounsﬁeld

(–) are pioneers of medical computed tomography and in  received

the Nobel Prize for Medicine for their epochal work during the s and s.

Figure . shows a picture of the two Nobel Prize winners.

Table .. Summary of historical CT milestones

Year Milestone

 Röntgen discovers a new kind of radiation, which he named X-ray

 Röntgen receives the Nobel Prize for physics

 Bockwinkel employs the Lorentz’s solution in the reconstruction of three-

dimensional functions from two-dimensional area integrals

 Radon publishes his epochal work on the solution of the inverse problem of

reconstruction

 Ehrenfest extends the solution of Lorentz to n dimensions using the Fourier

transform

 Cramer and Wold solve the reconstruction problem in statistics in which the

probability distribution is obtained from a complete set of marginal probability

distributions

 Eddington solves the reconstruction problem in the ﬁeld of astrophysics to cal-

culate the distribution of star velocities from the distribution of their measured

radial components

 Bracewell applies Fourier techniques for the solution of the inverse problem in

radio astronomy

 e Ukrainian scientist Korenblyum developsan X-ray scanner and tries to meas-

ure thin slices through the patient with analogue reconstruction principles

 Cormack contributes the ﬁrst mathematical implementationsfor tomographic re-

construction in South Africa

 Hounsﬁeld shows proof of the principle with the ﬁrst CT scanner based on a ra-

dioactive source at the EMI research laboratories

 Hounsﬁeldand Ambrose publish the ﬁrst clinical scans with an EMI head scanner

 Set-up of the ﬁrst whole body scanner with a fan-beam system

 Hounsﬁeld and Cormack receive the Nobel Prize for Medicine

 Demonstration of electron beam CT (EBCT)

 Kalender publishes the ﬁrst clinical spiral-CT

 Demonstration of multi-slice CT (MSCT)

1.4 Some Examples 

Cormack pointed out that previously the Dutch physicist H.A. Lorentz had

found a solution to the three-dimensional problem in which the desired function

had to be reconstructed from two-dimensional surface integrals (Cormack ).

Lorentz himself did not publish the results and so, unfortunately, the context of

his work is still unknown today. e result, however, is associated with Lorentz by

H. Bockwinkel, who mentioned the work in a  publication on light propagation

in crystals.

A detailed, mathematical basis to the solution of the inverse problem in com-

puted tomography was published by the Bohemian mathematician Johann Radon

in  (cf. Fig. ., le) (Radon ). Due to the complexity and depth of the mathe-

matical publication, however, the consequences of his ground-breaking results were

revealed very late in the mid-th century. Additionally, the paper was published in

German, which hindered a wide distribution of the work. In Chap. , an excerpt

of his original work is reprinted.

In Table ., some of the historical milestones and development steps of com-

puted tomography are summarized. e list undoubtedly has to start with Wilhelm

Conrad Röntgen (–), who received the Nobel Prize for physics in  (cf.

Fig. ., right). Before , a signiﬁcant number of mathematical contributions in

the ﬁeld of inverse problems were developed independently and are summarized

here only retrospectively.

1.4

Some Examples

Figures . to . show several examples of computer tomographic images that illus-

trate diﬀerent anatomical regions oen used in clinical practice. Modern CT scan-

ners yield images with an excellent so tissue contrast. In Fig. ., the slices are

annotated with the relevant scan protocol parameter. e most important parame-

ters are the acceleration voltage (which determines the energy of the X-ray quanta),

the tube current (which determines the intensity of the radiation), the slice thick-

ness (which is the axial thickness of the X-ray fan beam), and the gantry tilt (which

is the angulation of the CT frame with respect to the axial axis). In spiral-CT, the

pitch is an additional parameter that deﬁnes the table feed in units of slice thick-

ness.

In clinical practice, besides choosing an appropriate set of scan parameters

(cf. Figs. . and .), it is necessary to have a planning step for accurate anatom-

ical scanning before CT slice sequence acquisition. In this planning step, the slices

must be adapted to the anatomical situation, and furthermore, the dose for sensi-

tive organs must be minimized. e planning is accomplished on the basis of an

overview scan that looks similar to simple projection radiography (cf. Chap. ).

Here, the exact position and orientation of the slice can be interactively deﬁned.

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