Mathematics in Biomedical Imaging › journals › ijbi › 2007 › 064954.pdf2 International...

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

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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