# An inversion formula for the exponential Radon transform in spatial domain with variable focal-length fan-beam collimation geometry

## Abstract

Inverting the exponential Radon transform has a potential use for SPECT (single photon emission computed tomography) imaging in cases where a uniform attenuation can be approximated, such as in brain and abdominal imaging. Tretiak and Metz derived in the frequency domain an explicit inversion formula for the exponential Radon transform in two dimensions for parallel-beam collimator geometry. Progress has been made to extend the inversion formula for fan-beam and varying focal-length fan-beam (VFF) collimator geometries. These previous fan-beam and VFF inversion formulas require a spatially variant filtering operation, which complicates the implementation and imposes a heavy computing burden. In this paper, we present an explicit inversion formula, in which a spatially invariant filter is involved. The formula is derived and implemented in the spatial domain for VFF geometry (where parallel-beam and fan-beam geometries are two special cases). Phantom simulations mimicking SPECT studies demonstrate its accuracy in reconstructing the phantom images and efficiency in computation for the considered collimator geometries.

- Authors:

- Department of Biomedical Engineering, Beijing Institute of Technology, Beijing, 100081 (China) and Department of Radiology, State University of New York, Stony Brook, New York 11794 (United States)
- (United States)

- Publication Date:

- OSTI Identifier:
- 20775098

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Medical Physics; Journal Volume: 33; Journal Issue: 3; Other Information: DOI: 10.1118/1.2170596; (c) 2006 American Association of Physicists in Medicine; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 62 RADIOLOGY AND NUCLEAR MEDICINE; ACCURACY; ATTENUATION; BRAIN; COLLIMATORS; EFFICIENCY; FILTERS; IMAGE PROCESSING; IMAGES; PHANTOMS; RADON; SINGLE PHOTON EMISSION COMPUTED TOMOGRAPHY

### Citation Formats

```
Wen Junhai, Liang Zhengrong, and Department of Radiology, State University of New York, Stony Brook, New York 11794.
```*An inversion formula for the exponential Radon transform in spatial domain with variable focal-length fan-beam collimation geometry*. United States: N. p., 2006.
Web. doi:10.1118/1.2170596.

```
Wen Junhai, Liang Zhengrong, & Department of Radiology, State University of New York, Stony Brook, New York 11794.
```*An inversion formula for the exponential Radon transform in spatial domain with variable focal-length fan-beam collimation geometry*. United States. doi:10.1118/1.2170596.

```
Wen Junhai, Liang Zhengrong, and Department of Radiology, State University of New York, Stony Brook, New York 11794. Wed .
"An inversion formula for the exponential Radon transform in spatial domain with variable focal-length fan-beam collimation geometry". United States.
doi:10.1118/1.2170596.
```

```
@article{osti_20775098,
```

title = {An inversion formula for the exponential Radon transform in spatial domain with variable focal-length fan-beam collimation geometry},

author = {Wen Junhai and Liang Zhengrong and Department of Radiology, State University of New York, Stony Brook, New York 11794},

abstractNote = {Inverting the exponential Radon transform has a potential use for SPECT (single photon emission computed tomography) imaging in cases where a uniform attenuation can be approximated, such as in brain and abdominal imaging. Tretiak and Metz derived in the frequency domain an explicit inversion formula for the exponential Radon transform in two dimensions for parallel-beam collimator geometry. Progress has been made to extend the inversion formula for fan-beam and varying focal-length fan-beam (VFF) collimator geometries. These previous fan-beam and VFF inversion formulas require a spatially variant filtering operation, which complicates the implementation and imposes a heavy computing burden. In this paper, we present an explicit inversion formula, in which a spatially invariant filter is involved. The formula is derived and implemented in the spatial domain for VFF geometry (where parallel-beam and fan-beam geometries are two special cases). Phantom simulations mimicking SPECT studies demonstrate its accuracy in reconstructing the phantom images and efficiency in computation for the considered collimator geometries.},

doi = {10.1118/1.2170596},

journal = {Medical Physics},

number = 3,

volume = 33,

place = {United States},

year = {Wed Mar 15 00:00:00 EST 2006},

month = {Wed Mar 15 00:00:00 EST 2006}

}