Median Root Prior

Alenius et a/.45'46 have proposed a penalized-likelihood method based on a median filter. The algorithm can be written in the following form:

where Mj is the value of voxel j, obtained by median filtering the image fjo/d. Following the OSL interpretation discussed above, the difference between a voxel and the median of its neighbourhood is used as the gradient of some energy function. The derivation is empirical, and in fact, the corresponding energy function does not exist. Hsiao47 recently proposed a very similar

Figure 4. PET image obtained with MAP-reconstruction without (left) and with (right) the use of anatomical information from a registered MRI image (center), using a tissue composition model.43 The 3D sinogram was first rebinned with FORE,44 then the MAP-reconstructions were computed.12

algorithm, which does minimize a well defined objective function. Albeit empirical, the algorithm is very effective and has some interesting features. In contrast to most priors, it does not strive towards a completely flat image (which is not a very realistic prior assumption). Instead, it only requires that the image be locally monotonic. Locally monotonic images do not have small hot spots, but they can have sharp and/or smooth edges. In addition, similar to the relative difference prior, it penalizes relative differences, making b a "unit-less", easily tuned parameter. Stated intuitively, the MRP algorithm essentially suppresses all hot or cold spots that are small compared to the size of its median filter. It follows that, when applied in hot spot detection applications (such as PET whole body imaging in oncology), the mask should be chosen sufficiently small and b not too large.

Figure 5 illustrates the behaviour of ML-EM with moderate post-smoothing, MRP and the relative difference prior. Large homogenous regions (such as the liver and the mediastinum) are much better visualized in the MAP-reconstruction images, in particular with MRP. Smaller structures, such as the blood vessels, are somewhat attenuated by both priors. A very hot and small spot, such as the lesion in the neck, is strongly suppressed by the MRP-penalty. It is better preserved by the relative difference prior, because of its tolerance for large voxel differences. But it is best recovered by the noisier ML-EM image,

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