Recall that we are trying to reduce the variation in illumination across MR images. We can modify the image model to show this variation explicitly:
where is the desired constant illumination, is the undesired varying illumination, and is the reflectance. The goal now is to remove .
This can be accomplished in the log domain as follows:
Since varies slowly and varies rapidly, low pass filtering isolates :
Since is constant and equal to the average intensity of , is also constant. It is easily accounted for in computing the enhanced image:
Again, exponentiating produces the corrected image:
We approximate the low pass filter in Equation 6.12 by a neighborhood average:
which can be processed one dimension at a time. Taking into account that:
the average can be computed efficiently, independent of the size of the neighborhood (except for boundary voxels). This method provides a fast approximate homomorphic filter that can be used for RF correction.