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Research topics

Reconstruction of three-dimensional images from projections.

Non-invasive imaging techniques reconstruct the distribution in space, f(x,y,z),

of some physical caracteristics of an organ or object by probing that object using

electromagnetic or acoustic radiation. After reconstruction the distribution can be

displayed as selected two-dimensional (2D) slices or using volume rendering.

We study tomographic techniques where the measured data are well modeled by

the integrals of f(x,y,z) along a set of straight lines. The set of line integrals which

is measured by a scanner determines the properties of the reconstruction problem:

existence and unicity of the solution, stability in the presence of noise, symmetry



Standard tomographic techniques decompose the 3D object or organ in a stack of

parallel 2D slices, and each slice is measured and reconstructed independently. This

simplifies the reconstruction but makes a poor utilization of the available radiation

since only those photons that are flying within the plane of a slice can be used.

Our research focusses on "truly" 3D tomography, where line integrals are measured

with arbitrary orientations and the factorisation of the organ or object in 2D slices

is no longer possible. Truly 3D tomography is becoming the standard approach and

is the object of intensive research (see the 1999 International Meeting on Fully 3D

Image reconstruction in radiology and nuclear medicine).




Two main types of geometries:

In 3D transmission tomography (CT, "Computerized Tomography") the object is

probed by a cone-beam of X-rays. The attenuation of the beam along each line

connecting the source of X-rays to a point on the detector yields the integral of the

attenuation coefficient f(x,y,z) along that line. The set of all line integrals diverging

from the source is a cone-beam projection. To reconstruct f(x,y,z) a set of cone-beam

projections are measured for various positions of the X-ray source around

the object. In medical applications (see for example the new LightSpeed scanner of GE),

the source usually moves along a spiral path, but other types of paths are used for industrial


Reconstruction from cone-beam projections is also important in single-photon emission

tomography (SPECT).

Most recent developements:

Exact reconstruction algorithms for truncated (incompletely measured) cone-beam

projections acquired with a spiral path covering only a section of a long object [13].

Fast approximate algorithms for the same geometry, based on rebinning the measured

cone-beam projections into ordinary 2D tomographic data for a stack of 2D slices [14].


In positron emission tomography (PET), the unknown function f(x,y,z) is the concentration

in the point (x,y,z) of an organ of a tracer molecule which has been labelled with a

positron emitting radionuclide. Line integrals of f(x,y,z) are measured by detecting pairs

of 511 keV photons emitted when the positron annihilates with an atomic electron.

Fully 3D PET scanner have a cylindrical geometry, and the integral of f(x,y,z) is measured

along all lines connecting two detectors on the surface of the cylinder [7]. Usually these lines

are grouped in 2D parallel projections, i.e. in sets of lines parallel to a given direction in space.

Collaboration with the PET Facility of the University of Pittsburgh and the Laboratoire de

Tomographie Positon de l'Université Catholique de Louvain




Most recent developments:

Exact and approximate reconstruction algorithms based on rebinning the measured data into

ordinary 2D tomographic data for a stack of 2D slices. Fourier rebinning algorithm.[10, 15]

Combination of rebinning algorithms with 2D iterative reconstruction methods based on

a statistical model of the data. FORE+OSEM [11].


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Selected recent publications

1. M.D. and R. Clack, "Cone-beam reconstruction using shift-variant filtering and cone-beam backprojection", IEEE Trans. Med. Imag., MI-13, 186-195, 1994.

2. R. Clack and M.D., "Cone-beam reconstruction using Radon Transform intermediate functions", J. Opt. Soc. Am. A, 11, 580-585 (1994).

3. D. W. Townsend, L.Byars, M.D., A.Geissbuhler, R.Nutt, "Rotating positron tomographs revisited", Phys. Med. Biol. 39, 401-410, 1994.

4. Q. Chen, M.D., F. Deconinck, "Symmetric phase-only matched filtering of Fourier-Mellin transforms for image registration and recognition", IEEE Trans. Pattern Analysis and Machine Intelligence, 16, 1156-1168, 1994.

5. M.D., D.W Townsend, R. Clack, "Image reconstruction from truncated two-dimensional projections", topical review, Inverse Problems, 11, 287-313 (1995).

6. M.D., "A factorisation method for the 3D X-ray transform", Inverse Problems, 11, 983-994 (1995).

7. M.D. and P. Kinahan, "Data acquisition and image reconstruction for 3D PET", in "The theory and practice of 3D PET", eds. B. Bendriem and D. Townsend, Kluwer 1998, 11-53.

8. F. Noo, M.D., R. Clack, "FBP reconstruction of cone-beam data acquired with a vertex path containing a circle", IEEE Trans. Image Proc. 7, 854-867 (1998)

9. F. Noo, R. Clack, M. D., "Cone-beam reconstruction from general discrete vertex sets using Radon rebinning algorithms", IEEE Trans. Nucl. Sc., NS-44, 1309-1316 (1997)

10. M.D., P. E. Kinahan, D. W. Townsend, C. Michel, M. Sibomana, D. F. Newport, "Exact and approximate rebinning algorithms for 3D PET data", IEEE Trans. Med. Imag. , MI-16, 145-158 (1997).

11. C. Comtat, P E Kinahan, M.D., C. Michel, D W Townsend, "Fast reconstruction of 3D PET data with accurate statistical modeling", IEEE Trans. Nucl. Sc. NS-45, 1083-1089 (1998).

12. R. Clackdoyle, M.D., F. Noo, "Early results on general vertex sets and truncated projections in cone-beam tomography", in "Computational Radiology and Imaging: therapy and diagnostics", eds C. Börgers and F. Natterer, IMA Volumes in Mathematics and its Applications, 110, 113-136, Springer-Verlag, 1999.

13. H. Kudo, F. Noo, M.D., "Cone-beam filtered backprojection algorithm for truncated helical data", Phys. Med. Biol., 43, 2885-2909 (1998).

14. F. Noo, M. D., R. Clackdoyle, "Single-slice rebinning for helical cone-beam CT", Phys. Med. Biol., 44, 561-570 (1999).

15. M. D, X. Liu, "Fast rebinning algorithm for 3D PET using John’s equation", Inverse Problems, 15, 1047-1065 (1999)


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Contact Information

Michel Defrise

Department of Nuclear Medicine

University Hospital AZ-VUB

Laarbeeklaan 101

B-1090 Brussels, BELGIUM


phone 32-2-4774611, 32-2-6505513 (ULB)

fax 32-2-4774611

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