Accurate discontinuity detection using limited resolution information

作者：Wright R K^*

来源：Journal of Computational and Applied Mathematics, 2010, 234(4): 1249-1257.

DOI：10.1016/j.cam.2009.08.112

• Discontinuity detection
• Splines
• Spline wavelets
• Fourier coefficients

Let low resolution spline wavelet or Fourier coefficient information be available for a function f = g + epsilon where g is a piecewise polynomial with jump discontinuities of itself and its derivatives and epsilon is the noise. We construct a function r such that the convolution r * g is a polynomial in the neighborhood of the jump and has the jump location as root, and such that the convolution can be calculated using only the available information and a rectangle rule quadrature. Applying this calculation to f = g + epsilon yields a polynomial which is perturbed from r * g by an amount proportional to the L(2)-norm of epsilon. Some methods lose accuracy when large derivative jumps coincide with function jumps and resolution is limited, especially in the presence of noise. The present method maintains reasonable accuracy even with large derivative jumps and noise parallel to epsilon parallel to(2) approximate to .02 parallel to f parallel to(2). The present method is a local method, and requires some other strategy to locate the proper polynomial regions. We present a simple method which produces approximate jump locations close enough to actual ones to locate the desired polynomial regions.

• 出版日期2010-6-15