A wavelet phase filter for emission tomography Page: 3 of 12
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photon density of order 10 photons per pixel (nearer 100 per pixel in the numerical experiments
described here) and hence is inherently noisy.
One way to reduce noise sensitivity is to model the emission process as a random process
and to use reconstruction algorithms based on the statistical model. In 1982, Shepp and Vardi
[SHEP82] introduced a Poisson process model for emission tomography which seems to be an
excellent model. Maximum likelihood (ML) and maximum a posteriori (MAP) methods based
on this model give what we consider to be the best possible reconstruction from noisy sinogram
The problem with the ML and MAP techniques is that they are very expensive in comparison
with Fourier based techniques. The DFR method requires O(N2log N) operations, and CBP
requires O(N3) operations, where N is the number of detectors. On the other hand, an iterative
ML algorithm (called the EM algorithm) requires 0(N4) operations for each iteration and it
usually takes 50 iterations to achieve sufficient precision, [VARD85]. Even with acceleration
techniques such as Lewitt and Muehllehner's relaxation method, [LEWI86], the EM algorithm
still remains computationally very expensive.
Another way to reduce noise sensitivity is to filter the sinogram data. Several filtering tech-
niques for emission tomography are described in the literature. Linear low pass filters are easy
to implement in connection with Fourier based methods, and are the current method of choice.
A special lowpass filter called the parabola filter was developed at Washington University for
the purpose of PET noise reduction. The filter is called a "parabola" filter since the window
function is a quadratic function in the frequency domain (the filter is nonetheless a linear filter).
The window function is the product of a triangle function
and a ramp function
u(w) = Sp~)w
where Q is the cutoff frequency. Yang, [YANG91], implemented a noise filtering scheme using
projection onto convex sets. Kuan et al., [KUAN85], developed an adaptive filter for PET
image restoration. According to Yang's simulation results, the parabola filter is among the
most effective for use in connection with the CBP method, from the standpoint of subjective
evaluation of the resulting reconstructed image. The underlying assumption for using a lowpass
filter such as the parabola filter is that the energy of a typical image is primarily concentrated in
its low-frequency components and that the energy of a random noise is more spread out over the
whole frequency domain. While this assumption seems to be a reasonable one even for the noise
in emission tomography, it is not clear to us that this assumption alone captures the significant
features of the noise present in emission tomography applications. In these applications, the
noise is signal-dependent Poisson type noise. Furthermore, while the gross structure of an image
appears as low-frequency components, sharp edges give rise to high-frequency components, and
small details also give rise to high-frequency components. Therefore, the linear lowpass filtering
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Olsen, E.T. & Lin, B. A wavelet phase filter for emission tomography, report, July 1, 1995; Illinois. (digital.library.unt.edu/ark:/67531/metadc794263/m1/3/: accessed February 21, 2019), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.