An Inner-Product Calculus for Periodic Functions and Curves

Our motivation is the design of efficient algorithms to process closed curves represented by basis functions or wavelets. To that end, we introduce an inner-product calculus to evaluate correlations and $L _{ 2 }$ distances between such curves. In particular, we present formulas for the direct and exact evaluation of correlation matrices in the case of closed (i.e., periodic) parametric curves and periodic signals. We give simplifications for practical cases that involve B-splines. To illustrate this approach, we also propose a least-squares approximation scheme that is able to resample curves while minimizing aliasing artifacts. Another application is the exact calculation of the enclosed area.