Here's the answer, from one of our software engineers:
Both the solidity and extent are ratios of the area of your object (which Matlab refers to as region sometimes) and some other surrounding container. For Extent, the container is the enclosing rectangle and for Solidity, the container is the convex hull.
Extent is the easier of the two concepts: imagine the smallest rectangle that fits around your object. The extent is the area of your object divided by the area of this rectangle. Your cells will have relatively large extents if they are circular or elliptical and have no protrusions.
Solidity is a little more difficult, but potentially a better differentiator of cells with protrusions or irregular shape versus generally round cells. The containing shape is the convex hull. Imagine putting a rubber band around your cell's shape. The rubber band will stretch in a straight line across indentations in the cell but will follow the rounded outside contours. The area within the rubber band is the area of the convex hull. The solidity should be one for something without any indentations and, for a cell with an indent, will have a value of one minus the proportion of the area within the rubber band not covered by the cell.
Here are a couple of links that may explain the convex hull in a more visual manner:
softsurfer.com/Archive/algorithm ... ex%20Hulls
cse.unsw.edu.au/~lambert/jav ... xHull.html