REU 2000 Project: Wavelets for local tomographic medical imaging
This project investigates the use of wavelet analysis in the problem of local tomography.
Introduction to tomography
Tomography deals with reconstruction of a function from its line integrals.
In our project we deal specifically with 2-dimensional images of slices of human body.
By shining an x-ray through a human brain and measuring its attenuation in the path,
we in effect measure the line integral of the attenuation function, namely its radon transform.
The radon transform of a two-dimensional function f(x,y) is defined as
The goal of tomography is to reconstruct f(x,y) given its radon transform.
Reconstruction by backprojection
The following result is used in many common reconstruction methods:
By the fourier slice theorem, the fourier transform of the radon transform at a
particular angle is equal to the slice of the two dimensional fourier transform at the
same angle.
One simple reconstruction technique is to acquire a full 2-dimensional fourier
transform of the function by computing fourier transforms of radon transforms at many different
angles, and consequently, compute a 2-D inverse fourier transform to get the original function
back. However, this method requires interpolation in frequency, and does not yield accurate
reconstructions.
An alternative method which gives much better results is the
reconstruction by filtered back-projection. The reconstruction fornula can be rewritten as
,
where S_theta(w) is the projection data
at angle theta.
With this approach, the image can be reconstructed angle by angle, so it is not necessary to wait for
projections at all angles to be completed. Also, the backprojection approach has much higher accuracy,
because the interpolation of missing data is done in the space domain.
Local tomography
In medical imaging, the area of the interest may be much smaller than the whole
crossection. By shining x-rays only through the area of interest, it is
possible to reduce the patient's exposure to radiation. However, due to the
non-local nature of the radon transform, a perfect reconstruction is impossible
with only local projections data.
The low frequency components of a function get severely distorted
by local radon transform, the high frequencies go through the reconstruction with a
smaller distortion. When local projections data is reconstructed using global backprojection,
setting the projections in uncollected regions to zero, the resulting image still shows the
contours of the objects. The contours are regions of rapid change, and they require many
high frequency components, which can be reconstructed with small distortion. However, the low
frequency data inside the region of interest gets distorted severely.
Example: Original image, Global reconstruction, local reconstruction of the centered circular part.
,
Notice that in the local reconstruction the contours of the phantom remain intact, however the colors of the
larger parts (lower spatial frequency) are distorted.
One approach to local tomography, described by Tim Olson, succeds in reducing
patient's exposure to radiation, but still requires some global projections information. The coarse idea behind
the method is separating the local projections data into high and low frequency components, and keeping the high
frequencies. Low frequencies are then estimated globally, on a very sparse grid
(low frequency => slow variation). Then the low and high frequency components are combined.
The approach in this project was also based on separation into high and low frequencies using orthogonal
wavelet filters, but the low frequency data was not recollected globally. We deal in this project with
dental images, which possess several helpful properties, such as symmetry. This allows to interpolate the
local frequency data from the local projections only. (The approach is rather ad-hoc and does not have a
very sound theoretical justification, but it does give good results).
