A Computational Approach to Edge Detection
IEEE Transactions on Pattern Analysis and Machine Intelligence
Detection, classification, and measurement of discontinuities
SIAM Journal on Scientific and Statistical Computing
Accurate reconstructions of functions of finite regularity from truncated Fourier series expansions
Mathematics of Computation
Journal of Computational and Applied Mathematics - Special Issue: Proceedings of the 10th international congress on computational and applied mathematics (ICCAM-2002)
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Let low resolution spline wavelet or Fourier coefficient information be available for a function f=g+@e where g is a piecewise polynomial with jump discontinuities of itself and its derivatives and @e 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+@e yields a polynomial which is perturbed from r*g by an amount proportional to the L"2-norm of @e. 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 @?@e@?"2~.02@?f@?"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.