Real-Time Computation of Legendre-Sobolev Approximations

The present work is motivated by the problem of mathematical handwriting recognition where symbols are represented as plane curves, (X(λ), Y(λ)) parameterized by arc length λ ϵ [0, L]. Earlier work has shown that approximating the coordinate functions as certain truncated orthogonal polynomial series yields fast and effective recognition. It has been previously shown how to compute Legendre series representation in real time, as the curve is being traced out. In this article we show how to compute Legendre-Sobolev series representation in real time. The idea is to numerically integrate the moments of the coordinate functions as the curve is being traced. We show how the Legendre-Sobolev coefficients may be constructed either from the Legendre series coefficients or directly from the moments. Computing via Legendre series coefficients requires two matrix vector products, while the direct method requires only one.