Counting Lock-Screen Patterns

An Android-style unlock pattern connects some of $9$ points arranged in a $3\times 3$ grid, subject to these rules: choose a starting point, then move the finger through a sequence of points (at least $4$ points in all); no point may be visited more than once; and the finger may not skip over an unvisited point (if the straight segment between two consecutive chosen points passes through another grid point, that point must already have been visited). Find the total number of valid patterns.