Explore possibilities using non-negative matrix factorization (NNMF or NMF) on the V × Q matrix that has one row for each voxel (e.g. ~1M rows for a single image or ~1G rows for a data set of many images) and one column for each q-vector (e.g., one column for each of ~90 distinct (direction, magnitude) magnetic gradients in dMRI).
TL;DR: We consider each voxel to be a single point in Q-dimensional space. This cloud of V points may fall within a simplex (e.g., triangle, tetrahedron, pentatope, ... higher-dimensional analogs up to Q dimensions), and the simplex's vertices can be discovered by NNMF. The location of any point in the interior of the simplex can be computed as a weighted sum of the locations of the simplex vertices, where (plus or minus errors in the measurements) the weights are all non-negative and sum to 1.0. That is, each point in the interior has a diffusion signature that is a convex combination of the simplex vertex diffusion signatures. This argues that the simplex vertices are signatures for pure microstructures (e.g., only unimpeded isotropic diffusion, or only all sticks of a given orientation), and the voxels in the interior of the simplex have the specified convex combination of these pure microstructures.
Instead of traditional approaches -- starting with proposed microstructures (and with the hope that the proposed microstructures are important) and mathematically coming up with diffusion signatures for them -- we are starting with the discovered locations of the simplex vertices, which by their location on the simplex are thus evidently important, and trying to figure out which microstructures they represent. These vertices will necessarily be well separated as signatures; which is not necessarily the case when one starts with hopeful proposed microstructures.
Depending upon the NNMF algorithm that we use, we may have to perform outlier rejection before factorization.
If two images are not too dissimilar in intensity normalization, the simplex vertices that they separately produce may be in an easily discoverable one-to-one correspondence. We can use the weights for a convex combination in one image to produce the corresponding signature in the other image, thus normalizing the diffusion signature of one image to that of another image. Once intensity is normalized across images (or if the input data arrive sufficiently normalized) we can more confidently look at voxels across all images as a single cloud, which could aid in image registration, as described next.
If the set of voxels with a signature near that of a pure microstructure is small enough in number and distinct enough, we may be able to discover the corresponding voxels in a second image. Alternatively, we would use more than one microstructure at a time. Alternatively, we would look at a small, distinct set of voxels near some simplex interior point, in each image. Regardless, this may aid in registering physical voxel locations between the two images.
If the cloud of points does not define a simplex very well, such as when the cloud does not get close enough to the simplex vertices, it can be difficult and/or slow to find the simplex vertices with a NNMF algorithm. The quickest algorithms (e.g., Newberg et al. 2018) work only when there is at least one cloud point near each simplex vertex.
We might be able to use more than Q+1 vertices for the convex hull; this would be more of a high-dimensional convex polyhedron than a high-dimensional tetrahedron in nature. As such there will be more than one set of convex weights that give the location of any interior point and we might use Lasso or other regularization techniques to choose among the competing sets of weights. The advantage of using more than Q+1 vertices is that it allows the set of vertices to represent a wide coverage of magnetic gradient orientations for each of several key pure microstructures.
Explore possibilities using non-negative matrix factorization (NNMF or NMF) on the V × Q matrix that has one row for each voxel (e.g. ~1M rows for a single image or ~1G rows for a data set of many images) and one column for each q-vector (e.g., one column for each of ~90 distinct (direction, magnitude) magnetic gradients in dMRI).
TL;DR: We consider each voxel to be a single point in Q-dimensional space. This cloud of V points may fall within a simplex (e.g., triangle, tetrahedron, pentatope, ... higher-dimensional analogs up to Q dimensions), and the simplex's vertices can be discovered by NNMF. The location of any point in the interior of the simplex can be computed as a weighted sum of the locations of the simplex vertices, where (plus or minus errors in the measurements) the weights are all non-negative and sum to 1.0. That is, each point in the interior has a diffusion signature that is a convex combination of the simplex vertex diffusion signatures. This argues that the simplex vertices are signatures for pure microstructures (e.g., only unimpeded isotropic diffusion, or only all sticks of a given orientation), and the voxels in the interior of the simplex have the specified convex combination of these pure microstructures.
Instead of traditional approaches -- starting with proposed microstructures (and with the hope that the proposed microstructures are important) and mathematically coming up with diffusion signatures for them -- we are starting with the discovered locations of the simplex vertices, which by their location on the simplex are thus evidently important, and trying to figure out which microstructures they represent. These vertices will necessarily be well separated as signatures; which is not necessarily the case when one starts with hopeful proposed microstructures.
Depending upon the NNMF algorithm that we use, we may have to perform outlier rejection before factorization.
If two images are not too dissimilar in intensity normalization, the simplex vertices that they separately produce may be in an easily discoverable one-to-one correspondence. We can use the weights for a convex combination in one image to produce the corresponding signature in the other image, thus normalizing the diffusion signature of one image to that of another image. Once intensity is normalized across images (or if the input data arrive sufficiently normalized) we can more confidently look at voxels across all images as a single cloud, which could aid in image registration, as described next.
If the set of voxels with a signature near that of a pure microstructure is small enough in number and distinct enough, we may be able to discover the corresponding voxels in a second image. Alternatively, we would use more than one microstructure at a time. Alternatively, we would look at a small, distinct set of voxels near some simplex interior point, in each image. Regardless, this may aid in registering physical voxel locations between the two images.
If the cloud of points does not define a simplex very well, such as when the cloud does not get close enough to the simplex vertices, it can be difficult and/or slow to find the simplex vertices with a NNMF algorithm. The quickest algorithms (e.g., Newberg et al. 2018) work only when there is at least one cloud point near each simplex vertex.
We might be able to use more than Q+1 vertices for the convex hull; this would be more of a high-dimensional convex polyhedron than a high-dimensional tetrahedron in nature. As such there will be more than one set of convex weights that give the location of any interior point and we might use Lasso or other regularization techniques to choose among the competing sets of weights. The advantage of using more than Q+1 vertices is that it allows the set of vertices to represent a wide coverage of magnetic gradient orientations for each of several key pure microstructures.