Fourier rebinning algorithm for inverse geometry CT
Samuel R. Mazina?
Department of Radiology, Stanford University, Stanford, California 94305
Norbert J. Pelc
Department of Radiology, Stanford University, Stanford, California 94305
and Department of Bioengineering, Stanford University, Stanford, California 94305
?Received 1 June 2008; revised 26 August 2008; accepted for publication 27 August 2008;
published 13 October 2008?
Inverse geometry computed tomography ?IGCT? is a new type of volumetric CT geometry that
employs a large array of x-ray sources opposite a smaller detector array. Volumetric coverage and
high isotropic resolution produce very large data sets and therefore require a computationally
efficient three-dimensional reconstruction algorithm. The purpose of this work was to adapt and
evaluate a fast algorithm based on Defrise’s Fourier rebinning ?FORE?, originally developed for
positron emission tomography. The results were compared with the average of FDK reconstructions
from each source row. The FORE algorithm is an order of magnitude faster than the FDK-type
method for the case of 11 source rows. In the center of the field-of-view both algorithms exhibited
the same resolution and noise performance. FORE exhibited some resolution loss ?and less noise?
in the periphery of the field-of-view. FORE appears to be a fast and reasonably accurate recon-
struction method for IGCT. © 2008 American Association of Physicists in Medicine.
Key words: volumetric CT, inverse geometry CT, reconstruction algorithm, Fourier rebinning
Inverse geometry computed tomography ?IGCT? is a new
investigation.1–5Instead of using a single x-ray focal spot to
acquire a projection, IGCT employs a large array of focal
spots opposite a detector array that has the same axial extent
as the source array but is much smaller in the transverse
direction ?Fig. 1?.
It is important to find a reconstruction algorithm for this
geometry that is accurate, fast, and signal-to-noise ratio
?SNR? efficient. Computational efficiency is important due to
the need to process a very large data set that is inherent to
IGCT due to the presence of rays at multiple angles with
respect to the axis-of-rotation.
Previous work focused on a three-dimensional filtered-
backprojection ?3DFBP? approach.2Although 3DFBP is ac-
curate and SNR efficient, it is computationally expensive.
Positron emission tomography ?PET? has a similar sam-
pling distribution to IGCT ?after a rotation?. The 3D re-
projection algorithm6?3DRP? is a PET algorithm that per-
forms a two-pass reconstruction. The 3DRP algorithm is
accurate, however, it is also slow due to a costly reprojection
The purpose of this work is to evaluate a much faster
algorithm based on a Fourier rebinning ?FORE? algorithm
developed by Defrise et al. for PET,7and also applied to
multiple orbit SPECT.8
II.A. FORE algorithm description
FORE was designed for a PET geometry and we will
attempt to use the same notation as described in Ref. 7. Each
line-of-response ?LOR? can be characterized by four param-
eters: two in-plane parameters ?s,?? and two longitudinal
parameters ?z,?? ?Fig. 2?. s and ? are the radial distance and
angle, respectively, of the LOR projected onto a transverse
plane. z is the midpoint of the LOR and ? is the tangent of
the angle the LOR makes with a transverse plane.
Any LOR can then be described as the projection
f?s cos ? − l sin ?,s sin ? + l cos ?,z
For each z and ??0, we call the two-dimensional ?2D? data
as a function of s and ? an oblique sinogram. The ?=0
subset data are the in-plane sinograms. One can take a 2D
Fourier transform of p with respect to s and ? to get
P??,k,z,??, where ? and k are the frequency variables as-
sociated with s and ?, respectively. One of the main results
of Ref. 7 is the approximation
P??,k,z,0? ? P??,k,z + ?k/???,??,
which suggests that one can rebin all of the oblique sino-
grams into in-plane sinograms. After the rebinning process,
the data is in the form of a stack of in-plane 2D sinograms
that can then be reconstructed using any 2D algorithm. The
data set is reduced in size, resulting in a very computation-
ally efficient algorithm. The basic steps of the FORE algo-
rithm are as described in Ref. 7:
?1? Initialize a stack of 2D Fourier-transformed in-plane si-
nograms P??,k,z,0? to zero.
?2? For each ?z,?? sinogram:
48574857 Med. Phys. 35 „11…, November 20080094-2405/2008/35„11…/4857/6/$23.00 © 2008 Am. Assoc. Phys. Med.
III.B. Physical experiments
Figure 9?a? shows a 2D FBP reconstruction of the central
slice of the anthropomorphic torso phantom using only in-
plane rays. Figures 9?b? and 9?c? are FDK and FORE recon-
structions of the same slice which show the advantage of
utilizing all ray measurements in reducing noise.
IV. DISCUSSION AND CONCLUSIONS
FORE’s superior speed is primarily due to the rebinning
step. It reduces the 3D reconstruction problem into a series
of 2D reconstruction problems. Also, no interpolation is nec-
essary with respect to the in-plane parameters s, ?, w, or k
?assuming the data are parallel and uniform in s and ??.
However, FORE loses some axial resolution and has an
asymmetric point spread function in the periphery. This is
compensated by lower noise in these regions.
Like FDK, FORE suffers from cone-beam effects. Unlike
FDK, FORE did not suffer from streak artifacts in the De-
frise phantom reconstruction. FORE is an approximate algo-
rithm, and we believe that the residual cone-beam effects are
due to the approximate nature of FORE. An exact version of
the FORE algorithm for PET, FORE-J,13is currently being
investigated for application to IGCT.
In conclusion, FORE is a reasonably accurate algorithm
for IGCT data with significantly lower reconstruction times
compared to a FDK-like algorithm based on 3D backprojec-
The authors would like to thank GE Healthcare, the NIH
?R01 EB0006837?, the American Heart Association, and the
Lucas Foundation for their financial support. They would
also like to thank NovaRay, Inc. for the use of the SBDX
system. They are grateful to N. Robert Bennett and Josh
Star-Lack for help with experiments.
a?Electronic mail: email@example.com
1T. G. Schmidt, R. Fahrig, E. G. Solomon, and N. J. Pelc, “An inverse-
geometry volumetric CT system with a large-area scanned source: A fea-
sibility study,” Med. Phys. 31, 2623–2627 ?2004?.
2T. G. Schmidt, R. Fahrig, and N. J. Pelc, “Athree-dimensional reconstruc-
tion algorithm for an inverse-geometry volumetric CT system,” Med.
Phys. 32, 3234–3245 ?2005?.
3T. G. Schmidt, J. Star-Lack, N. R. Bennett, S. R. Mazin, E. Solomon, R.
Fahrig, and N. J. Pelc, “A prototype table-top inverse-geometry volumet-
ric CT system,” Med. Phys. 33, 1867–1878 ?2006?.
4S. R. Mazin, J. Star-Lack, N. R. Bennett, and N. J. Pelc, “Inverse-
geometry volumetric CT system with multiple detector arrays for wide
field-of-view imaging,” Med. Phys. 34, 2133–2142 ?2007?.
5B. De Man, S. Basu, D. Beque, B. Claus, P. Edic, M. Iatrou, J. LeBlanc,
B. Senzig, R. Thompson, M. Vermilyea, C. Wilson, Z. Yin, and N. Pelc,
“Multi-source inverse geometry CT: A new system concept for x-ray
computed tomography,” Proc. SPIE 6510, 65100H1–65100H8 ?2007?.
6P. E. Kinahan and J. G. Rogers, “Analytic three-dimensional image re-
construction using all detected events,” IEEE Trans. Nucl. Sci. 36, 964–
7M. Defrise, P. E. Kinahan, D. W. Townsend, C. Michel, M. Sibomana,
and D. F. Newport, “Exact and approximate rebinning algorithms for 3-D
PET data,” IEEE Trans. Med. Imaging 16, 145–158 ?1997?.
8D. S. Lalush, “Fourier rebinning applied to multiplanar circular-orbit
cone-beam SPECT,” IEEE Trans. Med. Imaging 18, 1076–1084 ?1999?.
9J. O’Sullivan, “A fast sinc function gridding algorithm for Fourier inver-
sion in computer tomography,” IEEE Trans. Med. Imaging 4, 200–207
10J. Jackson, C. Meyer, D. Nishimura, and A. Macovski, “Selection of a
convolution function for Fourier inversion using gridding,” IEEE Trans.
Med. Imaging 10, 473–478 ?1991?.
11L. A. Feldkamp, L. C. Davis, and J. W. Kress, “Practical cone-beam
algorithm,” J. Opt. Soc. Am. A 1, 612–619 ?1984?.
12E. Solomon, B. Wilfley, M. Van Lysel, A. Joseph, and J. Heanue,
“Scanning-beam digital x-ray ?SBDX? system for cardiac angiography,”
Proc. SPIE 3659, 246–257 ?1999?.
13M. Defrise and X. Liu, “A fast rebinning algorithm for 3D positron emis-
sion tomography using John’s equation,” Inverse Problems 15, 1047–
4862 S. R. Mazin and N. J. Pelc: Fourier rebinning algorithm for inverse geometry CT4862
Medical Physics, Vol. 35, No. 11, November 2008