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Tomography, which enables imaging of cross sections of the body without dissection,
is among the most important
tools in diagnostic medicine. Tomographic reconstruction algorithms, which reconstruct a
function from linear views or line integrals, are the basis of tomography.
Moreover, the same underlying
mathematical principles can be used to study the structure of objects of
all sizes, ranging from huge supernova remnants to tiny viruses.
Ideally, to obtain an accurate reconstruction, data should be collected
along as many views as possible and distributed in a full angular range.
However, due to physical constraints or to minimize costs, either
a limited angular range of views is imaged or only
a few views are available, resulting in notorious errors in the reconstructed images.
IMA postdoc Hstau Y. Liao created an algorithm which produces
images with high accuracy even when the projection data are not finely
sampled in a full angular range. One application of his work
is to dental surgery, where
limited range tomographic reconstructions can assist with the
restoration of the jaw bone for the subsequent tooth implantation, as
well as enhance the root canal therapy. Lesions or concavities in the
supporting bone can be identified in the reconstructed images, in order to
place grafts and artificial titanium roots.
Liao presented his work at the 2007 International Symposium on
Biomedical Imaging.
Mathematics like Liao's can help put the smile back on a patient's face.
Tomographic reconstruction of a third mandibular molar from
only 23 views. Left,
with Liao's algorithm. Right, with an Algebraic Reconstruction Technique,
widely used in commercial scanners. Data from Maaria Rantala of PaloDEx Group.
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