Minimizing total distance traveled by points in points cloud transformation
If the trajectories must be lines and $\epsilon$ is small enough, the problem can be solved with min-cost matching. If all coordinates are integers with absolute value bounded by $H$, then two disjoint intervals with integer endpoints have distance at least $\frac{1}{4H}$ from each other. Hence if $\epsilon$ is less than this value, it might as well be 0.

Take any four points $A, B, C, D$ and assume that there exists a line segment connecting $A$ and $B$, one connecting $C$ and $D$, and that these line segments intersect at point $X$. Then $|A - B| + |C - D| = |A - X| + |B - X| + |C - X| + |D - X|$. Triangle inequality is strict when the three points do not lie on the same line, so if not all of $A,B,C,D$ are colinear, we can exchange the startpoints of the segments to decrease their total length. Thus, if no four points lie on a line, the minimum total length solution doesn't have intersecting line segments.

To find the minimum total length solution, we use min-cost matching, with cost $|A - B|$ to match points $A$ and $B$ (assuming one of $A$ and $B$ is a starting point, and the other is an endpoint).

To handle the case where some four points lie on a line, we just disallow matching $A$ and $B$ if some other point lies between them. Matching in this case is not always possible, for example if we have two startpoints, then two endpoints on a line.