Projection of 2D signal Paper describes that a 2D signal(say video) can be projected to 1D for time series analysis by using a transformation function fx=Ax+constant and fy=Ay+constant where Ax , Ay represents the (x,y) coordinates of a single trajectory A. Then,these fx vs t can be plotted and various analysis can be performed. Going by their justification,is it logical to think that a coordinate position of an object can only be described by a single coordinate when doing time series analysis and signal compression?

Also,they mention about taking cartesian product.Why is that?
Yes you can re-represent the 2d signal as a 1-d signal and use signal processing algorithms on it. In doing this you can lose some of benefits of representing the data this way. 

Eg. If you were looking at compression of a mostly continuous 2-d dataset you may get benefits by treating the data as 2-dimensional and compressing in each dimension. Representing the data as concatenated rows or columns introduces discontinuities that the compression won't handle as well.