Ming Jiang

Alfred K. Louis

Didier Wolf

Hongkai Zhao

Christian Daul

Zhaotian Zhang

Tie Zhou

Mathematics in Biomedical Imaging

We propose a semi-automatic thalamus and thalamus nuclei segmentation algorithm from Diffusion Tensor Magnetic Resonance Imaging (DT-MRI) based on the mean-shift algorithm. Comparing with existing thalamus segmentation algorithms which are mainly based on K-means algorithm, our mean-shift based algorithm is more
flexible and adaptive. It does not assume a Gaussian distribution or a fixed number of clusters. Furthermore, the single parameter in the mean-shift based algorithm supports hierarchical clustering naturally.

Ye  Duan

Xiaoling  Li

Yongjian  Xi

Thalamus Segmentation from Diffusion Tensor Magnetic Resonance Imaging

Extravascular signal decay rate R2 or R<mml:math id="E1" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mn>2</mml:mn><mml:mo>&#x2217;</mml:mo></mml:msup></mml:mrow></mml:math> as a function of blood oxygenation, geometry, and field strength was calculated using a Monte Carlo (MC) algorithm for a wider parameter
 range than hitherto by others. The relaxation rates of gradient-recalled-echo (GRE) and Hahn-spin-echo
 (HSE) imaging in the presence of blood vessels (ranging from capillaries to veins) have been computed
 for a wide range of field strengths up to 9.4 T and 50&#x25; blood deoxygenation. The maximum HSE decay
 was found to be shifted to lower radii in higher compared to lower field strengths. For GRE, however, the
 relaxation rate was greatest for large vessels at any field strength. In addition, assessments of computational
 reliability have been carried out by investigating the influence of the time step, the Monte Carlo step procedure,
 boundary conditions, the number of angles between the vessel and the exterior field <mml:math id="E2" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mtext>B</mml:mtext><mml:mn>0</mml:mn></mml:msub></mml:mrow></mml:math>, the influence of
 neighboring vessels having the same orientation as the central vessel, and the number of proton spins.
 The results were compared with those obtained from a field distribution of the vessel computed by an analytic
 formula describing the field distribution of an ideal object (an infinitely long cylinder). It was found that the
 time step is not critical for values equal to or lower than 200 microseconds. The choice of the MC step procedure
 (three-dimensional Gaussian diffusion, constant one- or three-dimensional diffusion step) also failed to
 influence the results significantly; in contrast, the free boundary conditions, as well as taking too few angles
 into account, did introduce errors. Next neighbor vessels with the same orientation as the main vessel did
 not contribute significantly to signal decay. The total number of particles simulated was also found to play
 a minor role in computing R2/ R<mml:math id="E3" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mn>2</mml:mn><mml:mo>&#x2217;</mml:mo></mml:msup></mml:mrow></mml:math>.

Bernd Michael Mueller-Bierl

Kamil Uludag

Philippe L. Pereira

Fritz Schick

Magnetic Field Distribution and Signal Decay in Functional MRI in Very High Fields (up to 9.4 T) Using Monte Carlo Diffusion Modeling

In positron emission tomography (PET), a radioactive compound is injected into the body to promote a tissue-dependent emission rate.
Expectation maximization (EM) reconstruction algorithms are iterative techniques which estimate the concentration coefficients
that provide the best fitted solution, for example, a maximum likelihood estimate. In this paper, we combine the EM algorithm with a level set approach.
The level set method is used to capture the coarse scale information and the discontinuities of the concentration coefficients.
An intrinsic advantage of the level set formulation is that anatomical information can be efficiently incorporated and used in an easy and natural way.
We utilize a multiple level set formulation to represent the geometry of the objects in the scene. The proposed algorithm can be applied to any PET configuration, without major modifications.

Tony F. Chan

Hongwei Li

Marius Lysaker

Xue-Cheng Tai

Level Set Method for Positron Emission Tomography

Elasticity imaging is a reconstructive imaging technique where tissue motion in response to mechanical excitation is measured using modern imaging systems, and the estimated displacements are then used to reconstruct the spatial distribution of Young's modulus. Here we present an ultrasound elasticity imaging method that utilizes the model-based technique for Young's modulus reconstruction. Based on the geometry of the imaged object, only one axial component of the strain tensor is used. The numerical implementation of the method is highly efficient because the reconstruction is based on an analytic solution of the forward elastic problem. The model-based approach is illustrated using two potential clinical applications: differentiation of liver hemangioma and staging of deep venous thrombosis. Overall, these studies demonstrate that model-based reconstructive elasticity imaging can be used in applications where the geometry of the object and the surrounding tissue is somewhat known and certain assumptions about the pathology can be made.

Salavat R. Aglyamov

Andrei R. Skovoroda

Hua Xie

Kang Kim

Jonathan M. Rubin

Matthew O'Donnell

Thomas W. Wakefield

Daniel Myers

Stanislav Y. Emelianov

Model-Based Reconstructive Elasticity Imaging Using Ultrasound

We generalize the total variation restoration model, introduced by Rudin, Osher, and Fatemi in 1992, to matrix-valued data, in particular, to diffusion tensor images (DTIs). Our model is a natural extension of the color total variation model proposed by
Blomgren and Chan in 1998. We treat the diffusion matrix <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>D</mml:mi></mml:math> implicitly as the product <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>D</mml:mi><mml:mo>=</mml:mo><mml:mi>L</mml:mi><mml:msup><mml:mi>L</mml:mi><mml:mi>T</mml:mi></mml:msup></mml:mrow></mml:math>, and work with the elements of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>L</mml:mi></mml:math> as variables, instead of working directly on the elements of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>D</mml:mi></mml:math>. This ensures positive definiteness of the tensor during the regularization flow, which is essential when regularizing DTI. We perform numerical experiments on both synthetical data and 3D human brain DTI, and measure the quantitative behavior of the proposed model.

Oddvar  Christiansen

Tin-Man  Lee

Johan  Lie

Usha  Sinha

Total Variation Regularization of Matrix-Valued Images

We present numerical results for two reconstruction methods for a
new planar electrical impedance tomography device. This prototype
allows noninvasive medical imaging techniques if only one side of a patient
is accessible for electric measurements. The two reconstruction methods
have different properties: one is a linearization-type method that allows
quantitative reconstructions; the other one, that is, the factorization method, is a qualitative one,
and is designed to detect anomalies within the body.

Mustapha  Azzouz

Martin  Hanke

Chantal  Oesterlein

Karl  Schilcher

The Factorization Method for Electrical Impedance Tomography Data from a New Planar Device

The mathematical principles of tomographic imaging using detected (unscattered) X- or gamma-rays are based on the two-dimensional Radon transform and many of its variants. In this paper, we show that two new generalizations, called conical Radon transforms, are related to three-dimensional imaging processes based on detected Compton scattered radiation. The first class of conical Radon transform has been introduced recently to support imaging principles of collimated detector systems. The second class is new and is closely related to the Compton camera imaging principles and invertible under special conditions. As they are poised to play a major role in future designs of biomedical imaging systems, we present an account of their most important properties which may be relevant for active researchers in the field.

T. T. Truong

M. K. Nguyen

H. Zaidi

The Mathematical Foundations of 3D Compton Scatter Emission Imaging

The cone beam transform of a tensor field of order <mml:math id="E2" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>m</mml:mi></mml:math> in <mml:math id="E3" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>n</mml:mi><mml:mo>≥</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:math> dimensions is considered. We prove that the image of a tensor field under this transform is related to a derivative of the <mml:math id="E4" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>n</mml:mi></mml:math>-dimensional Radon transform applied to a projection of the tensor field. Actually the relation we show reduces for <mml:math id="E5" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>m</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math> and <mml:math id="E6" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>n</mml:mi><mml:mo>=</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:math> to the well-known formula of Grangeat. In that sense, the paper contains a generalization of Grangeat's formula to arbitrary tensor fields in any dimension. We further briefly explain the importance of that formula for the problem of tensor field tomography. Unfortunately, for <mml:math id="E7" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>m</mml:mi><mml:mo>&gt;</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math>, an inversion method cannot be derived immediately. Thus, we point out the possibility to calculate reconstruction kernels for the cone beam transform using Grangeat's formula.

T. Schuster

The Formula of Grangeat for Tensor Fields of Arbitrary Order in <svg style="vertical-align:-0.1638pt" height="7.9499998" version="1.1" viewBox="0 0 8.6625004 7.9499998" width="8.6625004" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,7.675)"><path id="x1D45B" d="M495 86q-46 -47 -87 -72.5t-63 -25.5q-43 0 -16 107l49 210q7 34 8 50.5t-3 21t-13 4.5q-35 0 -109.5 -72.5t-115.5 -140.5q-21 -75 -38 -159q-50 -10 -76 -21l-6 8l84 340q8 35 -4 35q-17 0 -67 -46l-15 26q44 44 85.5 70.5t64.5 26.5q35 0 10 -103l-24 -98h2&#xA;q42 56 97 103.5t96 71.5q46 26 74 26q9 0 16 -2.5t14 -11.5t9.5 -24.5t-1 -44t-13.5 -68.5q-30 -117 -47 -200q-4 -19 -3.5 -25t6.5 -6q21 0 70 48z" /></g></svg> Dimensions

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International Journal of Biomedical Imaging aims to promote research and development of biomedical imaging by publishing high-quality research articles and reviews in this rapidly growing interdisciplinary field.

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Papers are published upon acceptance, regardless of the Special Issue publication date.

International Journal of Biomedical Imaging

Mathematics in Biomedical Imaging

Mathematics in Biomedical Imaging

Articles

Mathematics in Biomedical Imaging

Thalamus Segmentation from Diffusion Tensor Magnetic Resonance Imaging

Magnetic Field Distribution and Signal Decay in Functional MRI in Very High Fields (up to 9.4 T) Using Monte Carlo Diffusion Modeling

Level Set Method for Positron Emission Tomography

Model-Based Reconstructive Elasticity Imaging Using Ultrasound

Total Variation Regularization of Matrix-Valued Images

The Factorization Method for Electrical Impedance Tomography Data from a New Planar Device

The Mathematical Foundations of 3D Compton Scatter Emission Imaging

The Formula of Grangeat for Tensor Fields of Arbitrary Order in Dimensions