In comparing two datasets, the purpose is to estimate a smoothly varying warping function, , required to align one dataset, , to a reference dataset, ,
We can represent the warping function with the shifts, , as follows:
where the denotes the original independent axis and are the shifts required to match the datasets as defined in Equation A-1. The correlation coefficient can be used to quantify the quality of the match between datasets (Hampson-Russell, 1999). The LSIM method begins with the observation that the correlation coefficient () only provides one number to describe the datasets; however, we am interested in understanding the local changes in the datasets' similarity. Therefore, the LSIM method computes local similarity , which is a function of time, . The square of can be split into a product of two factors (Fomel, 2007a):
where and are the solutions to the following regularized least-squares problems, respectively
The regularization operator, , is implemented using shaping regularization (Fomel, 2007b) and designed to enforce smoothness. To estimate the solution, LSIM is calculated for a series of shifts. The results of this calculation are accumulated and displayed on a `similarity scan.'