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Hindawi Publishing CorporationInternational Journal of
Biomedical ImagingVolume 2007, Article ID 64954, 2
pagesdoi:10.1155/2007/64954
EditorialMathematics in Biomedical Imaging
Ming Jiang,1 Alfred K. Louis,2 Didier Wolf,3 Hongkai Zhao,4
Christian Daul,3
Zhaotian Zhang,5 and Tie Zhou1
1 School of Mathematical Sciences, Peking University, Beijing
100871, China2 Institute for Applied Mathematics, Saarland
University, Postfach 151150, 66041 Saarbrücken, Germany3 Institut
National Polytechnique de Lorraine, Centre de Recherche en
Automatique de Nancy, 54516 Vandoeuvre-Les-Nancy, France4
Department of Mathematics, University of California Irvine, Irvine,
CA 92697, USA5 Division of Electronics & Information System,
Department of Information Science, National Natural Science
Foundation of China,Beijing 100085, China
Received 18 September 2007; Accepted 18 September 2007
Copyright © 2007 Ming Jiang et al. This is an open access
article distributed under the Creative Commons Attribution
License,which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly
cited.
Biomedical imaging is critically important for life scienceand
health care. In this rapidly developing field, mathemat-ics is one
of the most powerful tools for developing imagereconstruction as
well as image processing theory and meth-ods. Many of the
innovations in biomedical imaging are fun-damentally related to the
mathematical sciences. With im-provements of traditional imaging
systems and emergence ofnovel imaging modalities such as molecular
imaging towardsmolecular medicine, imaging equations that link
measure-ments to original images become increasingly more com-plex
to reflect the reality upto an ever-improving accuracy.Mathematics
becomes increasingly useful and leads to a newarray of
interdisciplinary and challenging research opportu-nities.The
future biomedical imaging will include advancedmathematical methods
as major features.
It is a current trend that more mathematicians becomeengaged in
biomedical imaging at all levels, from image re-construction to
image processing, and upto image under-standing and various
high-level applications. This special is-sue addresses the role of
mathematics in biomedical imaging.The themes include theoretical
analysis, algorithm design,system modeling and assessment, as well
as various biomed-ical imaging applications. From 10 submissions, 7
papers arepublished in this special issue. Each paper was reviewed
byat least two reviewers and revised according to review com-ments.
The papers cover the following imaging modalities:X-ray computed
tomography (CT), positron emission to-mography (PET), magnetic
resonance imaging (MRI), diffu-sion tensor imaging (DTI),
electrical impedance tomography(EIT), and elasticity imaging using
ultrasound.
The field of X-ray imaging has been expanding rapidlysince
Röntgen’s historical discovery in 1895. X-ray CT, as
the first noninvasive tomographic method, has revolution-ized
imaging technologies in general, which was also the firstsuccessful
application of mathematics in biomedical imag-ing. The mathematics
is the theory of Radon transform in-vented by Radon in 1917.
Further research may rejuvenatethis classic topic to meet modern
imaging challenges such asscattering effects. In Truong et al.’s
paper, the authors pre-sented two further generalizations of the
Radon transform,namely, two classes of conical Radon transforms
which origi-nate from imaging processes using Compton scattered
radia-tion. The first class, called C1-conical Radon transform, is
re-lated to an imaging principle with a collimated gamma cam-era
whereas the second class, called C2-conical Radon trans-form,
contains a special subclass which models the Comptoncamera imaging
process. They demonstrated that the inver-sion of C2-conical Radon
transform can be achieved under aspecial condition.
PET is currently a major imaging modality for
clinicaldiagnostics and pharmacological research. The
expectationmaximization (EM) algorithm has been used in PET
foryears. In Chan et al.’s paper, the authors propose to com-bine
the level set method with the EM algorithm for PET. Thelevel set
method, which was originally developed for captur-ing moving
interfaces in multiphase physics, is used here tocapture geometric
information, for example, the anatomicalstructure. If another type
of information is available, for ex-ample, CT or MRI images, it can
be used as prior knowledgeand can be incorporated into the
formulation. The idea ofcombining geometric information with
statistic methods isquite interesting and promising.
In Mueller-Bierl et al.’s paper, the authors investigatedthe
magnetic field distribution and signal decay of high field
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2 International Journal of Biomedical Imaging
strength functional MRI imaging. The static dipole modelhas been
extended to a dynamic model to describe the sam-pling of phases of
the individual protons moving in the in-homogeneous magnetic field.
The dynamic Brownian mo-tion process is implemented using a Monte
Carlo methodwith different step parameters. Various factors for
signal de-cay and artifacts formation are investigated. Results
from dif-ferent methods were compared.
Earlier work on total variation (TV) regularization forcolor
(vector valued) images is naturally extended to DTI,which is
composed of a symmetric positive definite (SPD)matrix at each
pixel. In the last decade, a new magnetic res-onance modality, DTI,
has caught a lot of interest. DTI canreveal anatomical structure
information. In this special issue,there are two papers on this
imaging technique. In Chris-tiansen et al.’s paper, this type of
tensor-valued images isdenoised using TV regularization. Recently,
partial differen-tial equation- (PDE-) based image processing
methods havebeen very successful in many applications due to its
intrin-sic geometric nature. TV regularization, which can
effec-tively remove noise while keeping sharp features, is one
ofthe most important techniques for PDE-based image pro-cessing
methods. Although TV regularization is very naturalfor scalar
(gray) images, there is no easy and natural way toextend to vector
values (color) images. This paper proposesto use TV regularization
to denoise DTI based on previouswork on generalizing TV to vector
value images. To maintainthe SPD structure of the tensor, the
authors propose to workon the LU factorization of the tensor rather
than on the ten-sor itself. The results demonstrated the expected
strength ofTV regularization. Another paper on DTI is by Duan.
Theauthor proposes a semi-automatic thalamus and thalamusnuclei
segmentation algorithm based on the mean-shift al-gorithm. The main
advantages of the proposed method overmethods based on K-means are
its flexibility and adaptivity,since assumptions of Gaussian or a
fixed number of clustersare not needed.
The EIT and elasticity imaging in the following two
con-tributions are inverse problems of partial differential
equa-tions. In EIT, electric currents are applied to the boundary
ofan object and the induced surface voltages are measured.
Themeasured voltage data are then used to reconstruct the inter-nal
conductivity distribution of the object. In Azzouz et al.’spaper,
the authors establish two reconstruction methods fora new planar
electrical impedance tomography device. Thisprototype allows
noninvasive medical imaging techniques ifonly one side of a patient
is accessible for electric measure-ments. The two reconstruction
methods have different prop-erties: one is a linearization-type
method that allows quan-titative reconstructions; and the other
one, that is, the fac-torization method, is a qualitative one, and
it is designedto detect anomalies within the body. Numerical
results arealso presented. In elasticity imaging, tissue motion in
re-sponse to mechanical excitation is measured using modernimaging
systems, and the estimated displacements are thenused to
reconstruct the spatial distribution of Young’s mod-ulus. In
Aglyamov et al.’s paper, the authors propose a novelreconstruction
technique for elastic properties of biologi-cal tissues from
compressional ultrasound elastography. The
technique assumes spherical or cylindrical symmetry so
thatstrain equations can be simplified. The reconstruction is
con-ducted with inverse problem computations for partial
differ-ential equations. The proposed method is applied to
imageliver hemangioma (spherical symmetry) and rat DVT
(cylin-drical symmetry). The reconstruction results are
comparedwith traditional elastography images. This paper offers
someinteresting thoughts especially for some special clinical
caseswhere elasticity properties are spherical symmetric.
These papers represent an exciting, insightful observa-tion into
the state of the art, as well as emerging future topicsin this
important interdisciplinary field. We hope that thisspecial issue
would attract a major attention of the peers.
ACKNOWLEDGMENTS
We would like to express our appreciation to all the
authors,reviewers, and the Editor-in-Chief Dr. Ge Wang for
greatsupport that to make this special issue possible.
Ming JiangAlfred K. Louis
Didier WolfHongkai Zhao
Christian DaulZhaotian Zhang
Tie Zhou
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