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An Exact Method for Computing the Area Moments of Wavelet and Spline Curves

M. Jacob, T. Blu, M. Unser

IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 23, no. 6, pp. 633–642, June 2001.

We present a method for the exact computation of the moments of a region bounded by a curve represented by a scaling function or wavelet basis. Using Green's Theorem, we show that the computation of the area moments is equivalent to applying a suitable multidimensional filter on the coefficients of the curve and thereafter computing a scalar product. The multidimensional filter coefficients are precomputed exactly as the solution of a two-scale relation. To demonstrate the performance improvement of the new method, we compare it with existing methods such as pixel-based approaches and approximation of the region by a polygon. We also propose an alternate scheme when the scaling function is sinc(x). 

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