Computing Average Cell Shapes

When computing the Gromov-Wasserstein distance between two cells \(X\) and \(Y\), the optimal transport algorithm returns two pieces of information:

1. A coupling matrix, which represents the optimal probabilistic mapping of \(X\) onto \(Y\) that minimizes the distortion.

2. The distortion induced by this optimal coupling matrix, which is the Gromov-Wasserstein distance.

We can utilize the coupling matrix to construct a morphological average of a group or cluster of cells. CAJAL implements an algorithm called avg_shape_spt() to construct this morphological average. In brief, the algorithm proceeds as follows:

• Identify the medoid cell of the cluster, which is the cell that has the lowest average distance to the other cells.

• Use the optimal coupling matrices to reorient every other cell with respect to the medoid, so that they can be directly compared.

• Rescale all intracellular distance matrices to unit step size to ensure that differences in overall size do not distort the comparison.

• Cap the distance between points within each cell at 2. This destroys information about the global structure of the geodesic distances, preventing very distant points from having an outsize effect.

• Compute the arithmetic mean of all distance matrices, where the distance between any two points in the averaged matrix is the average distance between the corresponding pairs of points in each cell in the cluster.

• For neurons, construct a shortest-path tree through the weighted graph encoded by the average distance matrix. This tree represents the average neuronal morphology of the cluster.