X-Ray Microtomography Using Cone-Beam Geometry
S. J. Pan1, W. S. Liou1, A. Shih1, W. Chang1, M. S. Park1, G. Wang2, S. P. Newberry3,
H. Kim4, D. M. Shinozaki5, P. C. Cheng1
1
Advanced Microscopy and Imaging Laboratory,
Department of Electrical and Computer Engineering,
State University of New York, Buffalo, NY 14260, USA
2
Mallinckrodt Institute of Radiology,
Washington University, School of Medicine, St. Louis, MO 63110, USA
3
CBI Labs, Box 11, S. Wescott Rd., Schenectady, NY 12306, USA
4
Department of Material Sciences and Engineering,
Kwangju Institute of Science and Technology, Kwangju, Republic of Korea
5
Department of Mechanical and Materials Engineering,
University of Western Ontario, London, Ontario, Canada, N6A 5B7
Abstract. An X-ray microtomographic system utilizing cone-beam geometry and
generalized Feldkamp cone-beam algorithm has been developed in our laboratory.
This system is capable of handling spherical, rod-shaped and plate-like specimens.
1 Introduction
Recent developments in confocal microscopy and two-photon fluorescent microscopy
have made the optical study of three-dimensional microstructures feasible in
transparent and semi-transparent materials [1,2]. However, the examination of threedimensional microstructures in opaque materials remains a very difficult task, since Xrays must be employed. For practical usefulness, the spatial resolution of any
microscopy technique must be at least similar to that obtained with optical
microscopy. Specimens used in microscopic investigations are often in geometric
shapes which are preparation dependent. The present work describes the most recent
attempts to develop an experimentally useful high resolution microtomographic
system which can rapidly produce accurate three dimensional images from
cylindrically symmetric, and flat plate-shaped specimens.
2 Tomographic Imaging System
Single two dimensional images are acquired using a cone beam of X-rays emanating
from a small source. A succession of images is taken at a variety of specimen
orientations which depend on the specimen geometry and three dimensional spatial
resolution required. The quality of the reconstructed image depends on the quality of
the two dimensional images and the number of such images, and on the reconstruction
algorithm. Tomographic systems can be separated into high and low resolution
instruments, with somewhat different kinds of end-use applications. In the present
work the results of a relatively low resolution system are shown, and it is shown that
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the various components of the system work together to produce rapid accurate three
dimensional images.
The macrotomographic system, for which the most recent results are described
here, consists of a conventional X-ray source, a three-dimensional translation and
rotation specimen stage, a high resolution phosphor screen and a slow scan cooled
CCD camera. The resolution is limited by the large source size.
The
microtomographic system is distinguished from the macrotomographic system only by
the relatively small source size, which is needed to decrease the size of the smallest
resolvable object. The most common X-ray sources employ an electron beam stopped
in a metal target, and the source size depends on the electron beam crossectional size,
and on the spreading of the beam in the metal target by scattering. A convenient,
stable small beam size can be obtained in a conventional scanning electron
microscope (electron beam diameters approach 10 nm) [3], which in addition offers
very precise beam positioning capability. Beam spread in the target can be limited by
reducing the thickness of the target, although intensity is reduced. Fig. 1 shows a
schematic diagram of the X-ray projection imaging system. The sample was attached
to the axis of a rotational stage. The projection images were formed on a high
resolution phosphor screen. The scintillated visible light image was then captured by
a slow scan cooled CCD camera (Kodak KAF 1400).
Fig. 1. Schematic representation of a cone-beam x-ray microtomographic imaging system.
3 Generalized Feldkamp Algorithm
To perform tomographic reconstruction from cone-beam projection data, a
generalized Feldkamp algorithm suitable for various hardware configurations were
developed [4,5,6,7]. The Generalized Feldkamp algorithm is formulated as following:
ρ(β)t
ρ(β)
1 2π ρ2 (β) ∞
g(x, y, z) = ∫
Rβ ( p,ς) f (
− p)
dpdβ
∫
2
ρ(β) − s
2 0 (ρ(β) − s) −∞
ρ2 (β) + p2 +ς 2
t = x cos β + y sin β, s = −x sin β + y cos β
X-Ray Microtomography Using Cone-Beam Geometry
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where g ( x , y , z ) is the X-ray absorption coefficient at voxel ( x , y , z ) , ρ ( β )
describes the horizontal distance between the source and the origin of the coordinate
system, β is the rotation angle of the specimen, Rβ ( p, ς ) is the absorbance of
equispatial cone-beam projection, and f (⋅) is the reconstruction filter. In addition to
the single cone-beam reconstruction, this generalized algorithm is also capable of
handling rod and plate-like specimen [8].
4 Results
A small fresh water snail was used as our test object. Fifty X-ray projections were
taken along a circular locus of 63.1 mm in radius with a specimen-to-detector distance
of 6.9mm. Each image was obtained with an integration time of 0.1 second. The
resulting projections were normalized against background to remove the contribution
from phosphor screen imperfection. Each frame of equispatial cone-beam projection
covers a rectangular region of 20.2x15.8 mm2 (at rotation axis position) and was
digitized into 1277x1004 pixels at 8-bit resolution. For ease of reconstruction
calculations, the original projection images were cropped and converted into 256x256
pixels. Fig. 2 (a, and b) shows a stereo-pair of the original X-ray projection. Fig. 3
shows four different projection views of the snail (X-ray absorbance). Fig. 4 shows
two slices (a and b) obtained from the reconstructed volumetric data of 14.5
x14.5x14.5mm3 (at 256x256x256 voxels). Fig. 5a shows the outer surface of the snail
and Fig. 5b shows the internal spiral of the snail by digitally removing a portion of the
shell.
Fig. 2. An X-ray stereo-pair of the shell of a fresh water snail.
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Fig. 3. Four projection views (absorbance) of the shell of a fresh water snail
at 0°, 90°, 180° and 270° respectively.
Fig. 4. Two sections cutting through the reconstructed volumetric data set showing cross
sections of the snail. Note the outer circle defines the reliable reconstruction region of the conebeam algorithm. The radiant rays in the image are the result of tomographic reconstruction
from relative low number of projections (50 in this experiment).
Fig. 5. (a) Surface shaded images of the snail. (b) Internal spiral of the snail shell by digitally
remove a portion of the shell.
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5 Conclusions
Satisfactory X-ray cone-beam microtomographic reconstruction is shown to be
practically feasible using the instrumentation and reconstruction algorithms developed
for this work. Further efforts are needed to make it a practical tool in biomedical and
material applications. The effects of source size and its intensity distribution on the
reconstruction algorithm should be studied. The data acquisition process, including
the electronic read-out speed of the CCD and the scintillation efficiency of the
phosphor screen, can be further improved. Future implementation of reconstruction
algorithm on a specialized computational hardware should allow near-real time
performance essential for medical and industrial applications.
Acknowledgments
This project was supported in part by the Academic Development Fund of SUNY to
PCC. We thanks the wonderful machining job of Mr. Willi Schulze. This work is part
of the Ph.D. thesis of SJP.
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