# 3D reconstruction of tensors and vectors

## Abstract

Here we have developed formulations for the reconstruction of 3D tensor fields from planar (Radon) and line-integral (X-ray) projections of 3D vector and tensor fields. Much of the motivation for this work is the potential application of MRI to perform diffusion tensor tomography. The goal is to develop a theory for the reconstruction of both Radon planar and X-ray or line-integral projections because of the flexibility of MRI to obtain both of these type of projections in 3D. The development presented here for the linear tensor tomography problem provides insight into the structure of the nonlinear MRI diffusion tensor inverse problem. A particular application of tensor imaging in MRI is the potential application of cardiac diffusion tensor tomography for determining in vivo cardiac fiber structure. One difficulty in the cardiac application is the motion of the heart. This presents a need for developing future theory for tensor tomography in a motion field. This means developing a better understanding of the MRI signal for diffusion processes in a deforming media. The techniques developed may allow the application of MRI tensor tomography for the study of structure of fiber tracts in the brain, atherosclerotic plaque, and spine in addition to fiber structuremore »

- Authors:

- Publication Date:

- Research Org.:
- Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States)

- Sponsoring Org.:
- USDOE. Office of Science. Office of Biological and Environmental Research. Medical Applications and Biophysical Research Division; National Institutes of Health Grant EB000121; Benning Trust Funds (US)

- OSTI Identifier:
- 838184

- Report Number(s):
- LBNL-54936

R&D Project: 864K2B; TRN: US0501263

- DOE Contract Number:
- AC03-76SF00098

- Resource Type:
- Technical Report

- Resource Relation:
- Other Information: PBD: 17 Feb 2005

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 62 RADIOLOGY AND NUCLEAR MEDICINE; BRAIN; CONES; DIFFRACTION; DIFFUSION; FIBERS; FLEXIBILITY; IN VIVO; RADON; TENSOR FIELDS; TOMOGRAPHY; VECTORS

### Citation Formats

```
Defrise, Michel, and Gullberg, Grant T.
```*3D reconstruction of tensors and vectors*. United States: N. p., 2005.
Web. doi:10.2172/838184.

```
Defrise, Michel, & Gullberg, Grant T.
```*3D reconstruction of tensors and vectors*. United States. doi:10.2172/838184.

```
Defrise, Michel, and Gullberg, Grant T. Thu .
"3D reconstruction of tensors and vectors". United States. doi:10.2172/838184. https://www.osti.gov/servlets/purl/838184.
```

```
@article{osti_838184,
```

title = {3D reconstruction of tensors and vectors},

author = {Defrise, Michel and Gullberg, Grant T.},

abstractNote = {Here we have developed formulations for the reconstruction of 3D tensor fields from planar (Radon) and line-integral (X-ray) projections of 3D vector and tensor fields. Much of the motivation for this work is the potential application of MRI to perform diffusion tensor tomography. The goal is to develop a theory for the reconstruction of both Radon planar and X-ray or line-integral projections because of the flexibility of MRI to obtain both of these type of projections in 3D. The development presented here for the linear tensor tomography problem provides insight into the structure of the nonlinear MRI diffusion tensor inverse problem. A particular application of tensor imaging in MRI is the potential application of cardiac diffusion tensor tomography for determining in vivo cardiac fiber structure. One difficulty in the cardiac application is the motion of the heart. This presents a need for developing future theory for tensor tomography in a motion field. This means developing a better understanding of the MRI signal for diffusion processes in a deforming media. The techniques developed may allow the application of MRI tensor tomography for the study of structure of fiber tracts in the brain, atherosclerotic plaque, and spine in addition to fiber structure in the heart. However, the relations presented are also applicable to other fields in medical imaging such as diffraction tomography using ultrasound. The mathematics presented can also be extended to exponential Radon transform of tensor fields and to other geometric acquisitions such as cone beam tomography of tensor fields.},

doi = {10.2172/838184},

journal = {},

number = ,

volume = ,

place = {United States},

year = {2005},

month = {2}

}