Hi Remi,
> authors says that \"two signals are
> maximally interdependent when they are related by a one to one mapping\".
> What does this mean exactly? is there a bijection between the two
> signals?
The existence of a bijection would be a case in which knowledge of either signal
would determine the other. The other situation of maximal interdependence can
happen when one signal predicts the other, but the other does not predict the
first [this would be a case where the second signal contains less information
and is wholly redundant with respect to the first].
In optimizing MI, the creation of a bijection between two signals is not usually
possible, but the goal of mutual information coregistration techniques is to
create something that is as close to a bijection as possible.
I can give an example that may be more concrete: coregistering a T1 and T2
image.
T1 T2
gray matter dark gray light gray
white matter light gray dark gray
CSF black white
If you want to coregister these images to one another, one method might be to
hold the first fixed, and slightly rotate and transpose the second to match the
first. If this is an iterative process, you would want to check at every step
how well the first image matches the (slightly rotated and shifted) second
image by comparing intensities in corresponding locations. If every pixel that
was dark gray in the first image was light gray in the second, you might
conclude that the gray matter voxels match up pretty well across the two
images. This would be a condition of high mutual information. If you found a
light gray voxel in the first image that was black in the second image, this
may be a spot that is not in register (as this pair of T1 and T2 intensities
does not correspond to any known tissue type). For any given T1 intensity
level, the less it predicts the T2 level of a corresponding voxel, the less
likely the image is in register.
More generally, the technique depends on the fact that each of the images is a
specific mapping from different tissue types to different intensities. This
would be an ideal case, but we know that is not true- due to noise in the
image, some white matter voxels will have a lower intensity in a T1 image than
some gray matter voxels. Or, more generally, that the peaks in a histogram of a
T1 or T2 image will blend into each other. The slightly inspecific mapping of
tissue->intensity is one reason why the mutual information function between the
two images would not reach its theoretical maximum. Another limitation might be
that two tissues that are distinguishable in one modality are not
distinguishable in the other. This would also prevent the formation of a
bijection.
Ken
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Ken Roberts
Woldorff Laboratory
Center for Cognitive Neuroscience, Duke University
(919) 668-1334
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