Representation of frame image pixels
The paper that you are quoting assumes that the motions along the different coordinates are independant functions, i.e., $$P(t) = \left(x(t),y(t)\right) = (f_1(t),f_2(t)) \neq f(x,y)(t),$$ where $P$ stand for position.
The various 1D functions are obtained in this way.

A bit of smoothing is generally helpful. 
However, when to comes to gait recognition, you will probably have to test different smoothing procedures: my intuition is that gait recognition relies on high frequency / low amplitude clues that can easily be cancelled out (but I may be wrong for this).