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Approaches to the recovery of three-dimensional information on a
biological object, which are often formulated or implemented
initially in an intuitive way, are concisely described here based
on physical models of the object and the image-formation process.
Both three-dimensional electron microscopy and X-ray tomography can
be captured in the same mathematical framework, leading to
closely-related computational approaches, but the methodologies
differ in detail and hence pose different challenges. The editors
of this volume, Gabor T. Herman and Joachim Frank, are experts in
the respective methodologies and present research at the forefront
of biological imaging and structural biology. Computational Methods
for Three-Dimensional Microscopy Reconstruction will serve as a
useful resource for scholars interested in the development of
computational methods for structural biology and cell biology,
particularly in the area of 3D imaging and modeling.
Goals of the Book Overthelast thirty yearsthere has been
arevolutionindiagnostic radiology as a result
oftheemergenceofcomputerized tomography (CT), which is the process
of obtaining the density distribution within the human body from
multiple x-ray projections. Since an enormous variety of possible
density values may occur in the body, a large number of projections
are necessary to ensure the accurate reconstruction oftheir
distribution. There are other situations in which we desire to
reconstruct an object from its projections, but in which we know
that the object to be recon structed has only a small number of
possible values. For example, a large fraction of objects scanned
in industrial CT (for the purpose of nonde structive testing or
reverse engineering) are made of a single material and so the ideal
reconstruction should contain only two values: zero for air and the
value associated with the material composing the object. Similar as
sumptions may even be made for some specific medical applications;
for example, in angiography ofthe heart chambers the value is
either zero (in dicating the absence of dye) or the value
associated with the dye in the chamber. Another example arises in
the electron microscopy of biological macromolecules, where we may
assume that the object to be reconstructed is composed of ice,
protein, and RNA. One can also apply electron mi croscopy to
determine the presenceor absence ofatoms in crystallinestruc tures,
which is again a two-valued situation."
This revised and updated second edition - now with two new
chapters - is the only book to give a comprehensive overview of
computer algorithms for image reconstruction. It covers the
fundamentals of computerized tomography, including all the
computational and mathematical procedures underlying data
collection, image reconstruction and image display. Among the new
topics covered are: spiral CT, fully 3D positron emission
tomography, the linogram mode of backprojection, and state of the
art 3D imaging results. It also includes two new chapters on
comparative statistical evaluation of the 2D reconstruction
algorithms and alternative approaches to image reconstruction.
Once we have accepted a precise replacement of the concept of algo
rithm, it becomes possible to attempt the problem whether there
exist well-defined collections of problems which cannot be handled
by algo rithms, and if that is the case, to give concrete cases of
this kind. Many such investigations were carried out during the
last few decades. The undecidability of arithmetic and other
mathematical theories was shown, further the unsolvability of the
word problem of group theory. Many mathematicians consider these
results and the theory on which they are based to be the most
characteristic achievements of mathe matics in the first half of
the twentieth century. If we grant the legitimacy of the suggested
precise replacements of the concept of algorithm and related
concepts, then we can say that the mathematicians have shown by
strictly mathematical methods that there exist mathematical
problems which cannot be dealt with by the methods of calculating
mathematics. In view of the important role which mathematics plays
today in our conception of the world this fact is of great
philosophical interest. Post speaks of a natural law about the
"limitations of the mathematicizing power of Homo Sapiens." Here we
also find a starting point for the discussion of the question, what
the actual creative activity of the mathematician consists in. In
this book we shall give an introduction to the theory of
algorithms."
"La narraci6n literaria es la evocaci6n de las nostalgias. "
("Literary narration is the evocation of nostalgia. ") G. G.
Marquez, interview in Puerta del Sol, VII, 4, 1996. A Personal
Prehistory In 1972 I started cooperating with members of the
Biodynamics Research Unit at the Mayo Clinic in Rochester,
Minnesota, which was under the direction of Earl H. Wood. At that
time, their ambitious (and eventually realized) dream was to build
the Dynamic Spatial Reconstructor (DSR), a device capable of
collecting data regarding the attenuation of X-rays through the
human body fast enough for stop-action imaging the full extent of
the beating heart inside the thorax. Such a device can be applied
to study the dynamic processes of cardiopulmonary physiology, in a
manner similar to the application of an ordinary cr (computerized
tomography) scanner to observing stationary anatomy. The standard
method of displaying the information produced by a cr scanner
consists of showing two-dimensional images, corresponding to maps
of the X-ray attenuation coefficient in slices through the body.
(Since different tissue types attenuate X-rays differently, such
maps provide a good visualization of what is in the body in those
slices; bone - which attenuates X-rays a lot - appears white, air
appears black, tumors typically appear less dark than the
surrounding healthy tissue, etc. ) However, it seemed to me that
this display mode would not be appropriate for the DSR.
This revised and updated second edition - now with two new
chapters - is the only book to give a comprehensive overview of
computer algorithms for image reconstruction. It covers the
fundamentals of computerized tomography, including all the
computational and mathematical procedures underlying data
collection, image reconstruction and image display. Among the new
topics covered are: spiral CT, fully 3D positron emission
tomography, the linogram mode of backprojection, and state of the
art 3D imaging results. It also includes two new chapters on
comparative statistical evaluation of the 2D reconstruction
algorithms and alternative approaches to image reconstruction.
Goals of the Book Overthelast thirty yearsthere has been
arevolutionindiagnostic radiology as a result
oftheemergenceofcomputerized tomography (CT), which is the process
of obtaining the density distribution within the human body from
multiple x-ray projections. Since an enormous variety of possible
density values may occur in the body, a large number of projections
are necessary to ensure the accurate reconstruction oftheir
distribution. There are other situations in which we desire to
reconstruct an object from its projections, but in which we know
that the object to be recon structed has only a small number of
possible values. For example, a large fraction of objects scanned
in industrial CT (for the purpose of nonde structive testing or
reverse engineering) are made of a single material and so the ideal
reconstruction should contain only two values: zero for air and the
value associated with the material composing the object. Similar as
sumptions may even be made for some specific medical applications;
for example, in angiography ofthe heart chambers the value is
either zero (in dicating the absence of dye) or the value
associated with the dye in the chamber. Another example arises in
the electron microscopy of biological macromolecules, where we may
assume that the object to be reconstructed is composed of ice,
protein, and RNA. One can also apply electron mi croscopy to
determine the presenceor absence ofatoms in crystallinestruc tures,
which is again a two-valued situation."
"La narraci6n literaria es la evocaci6n de las nostalgias. "
("Literary narration is the evocation of nostalgia. ") G. G.
Marquez, interview in Puerta del Sol, VII, 4, 1996. A Personal
Prehistory In 1972 I started cooperating with members of the
Biodynamics Research Unit at the Mayo Clinic in Rochester,
Minnesota, which was under the direction of Earl H. Wood. At that
time, their ambitious (and eventually realized) dream was to build
the Dynamic Spatial Reconstructor (DSR), a device capable of
collecting data regarding the attenuation of X-rays through the
human body fast enough for stop-action imaging the full extent of
the beating heart inside the thorax. Such a device can be applied
to study the dynamic processes of cardiopulmonary physiology, in a
manner similar to the application of an ordinary cr (computerized
tomography) scanner to observing stationary anatomy. The standard
method of displaying the information produced by a cr scanner
consists of showing two-dimensional images, corresponding to maps
of the X-ray attenuation coefficient in slices through the body.
(Since different tissue types attenuate X-rays differently, such
maps provide a good visualization of what is in the body in those
slices; bone - which attenuates X-rays a lot - appears white, air
appears black, tumors typically appear less dark than the
surrounding healthy tissue, etc. ) However, it seemed to me that
this display mode would not be appropriate for the DSR.
The conference was devoted to the discussion of present and future
techniques in medical imaging, including 3D x-ray CT, ultrasound
and diffraction tomography, and biomagnetic ima- ging. The
mathematical models, their theoretical aspects and the development
of algorithms were treated. The proceedings contains surveys on
reconstruction in inverse obstacle scat- tering, inversion in 3D,
and constrained least squares pro- blems.Research papers include
besides the mentioned imaging techniques presentations on image
reconstruction in Hilbert spaces, singular value decompositions, 3D
cone beam recon- struction, diffuse tomography, regularization of
ill-posed problems, evaluation reconstruction algorithms and
applica- tions in non-medical fields. Contents: Theoretical
Aspects: J.Boman: Helgason' s support theorem for Radon
transforms-a newproof and a generalization -P.Maass: Singular value
de- compositions for Radon transforms- W.R.Madych: Image recon-
struction in Hilbert space -R.G.Mukhometov: A problem of in- tegral
geometry for a family of rays with multiple reflec- tions
-V.P.Palamodov: Inversion formulas for the three-di- mensional ray
transform - Medical Imaging Techniques: V.Friedrich: Backscattered
Photons - are they useful for a surface - near tomography -
P.Grangeat: Mathematical frame- work of cone beam 3D reconstruction
via the first derivative of the Radon transform -P.Grassin,
B.Duchene, W.Tabbara: Dif- fraction tomography: some applications
and extension to 3D ultrasound imaging -F.A.Gr}nbaum: Diffuse
tomography: a re- fined model -R.Kress, A.Zinn: Three dimensional
reconstruc- tions in inverse obstacle scattering -A.K.Louis:
Mathemati- cal questions of a biomagnetic imaging problem - Inverse
Problems and Optimization: Y.Censor: On variable block algebraic
reconstruction techniques -P.P.Eggermont: On Volterra-Lotka
differential equations and multiplicative algorithms for monotone
complementary problems
Advances in Discrete Tomography and its Applications is a unified
presentation of new methods, algorithms, and select applications
that are the foundations of multidimensional image construction and
reconstruction. The self-contained survey chapters, written by
leading mathematicians, engineers, and computer scientists, present
cutting-edge research and results in the field. Three main areas
are covered: theoretical results, algorithms, and practical
applications. Following an historical and introductory overview of
the field, the book explores various mathematical and computational
problems of discrete tomography with an emphasis on new
applications. Topics and features include: historical overview and
summary chapter; uniqueness and complexity in discrete tomography;
3-D tomographic reconstruction from radiographic data; symbolic
projections; and, applications to neutron tomography.
Professionals, researchers, practitioners, and students in
mathematics, computer imaging, biomedical imaging, computer
engineering, and image processing will find the book to be a useful
guide and reference to state-of-the-art research, methods, and
applications.
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