A New Method For Locating Jump Points With Applications

The numerical differentiation problem is intended for reconstructing the differ-entiation of a given function, by use of noisy function values at nodes. It is a typicalill-posed problem, and has wide applications in the fields of image processing, materialscience, chemical engineering, computational mechanics etc. There are three classesof methods for numerical differentiation namely, the interpolation method (includingthe Tikhonov regularization method), the mollification method, and the integral op-erator method. For the Tikhonov regularization method, one needs to transform theoriginal problem into a feasible variational problem, and the efficiency of the methoddepends heavily on reasonably choosing regularization terms and regularization pa-rameters, and the computational cost is expensive comparably.Based on the above discussion, we devise in this thesis a new method for nu-merical differentiation, which enjoys the advantage of easy implementation and beingable to locate discontinuous points of the differentiation of the exact function. Theidea is to reconstruct the differentiation by certain continuous piecewise linear func-tion directly, whose function values at nodes are equal to the corresponding ones ofthe differentiation of certain locally defined quadratic interpolation polynomials. Af-ter technical derivations, it is proved that if the exact solution lies in ??2,∞or ??3,∞,the method exhibits optimal error estimates; if the exact solution is continuous andpiecewise smooth, one can construct a fast algorithm from the method to detect thejump points of the exact function. Compared to the regularization method, the newmethod does not require to solve any linear systems, alleviating computational costsignificantly. Numerical tests are provided to show the efficiency of the method.Finally, the proposed method is applied for edge detection of digital images. Insome cases, this method performs better than the existing methods in this field includ- ing the Sobel operator, the Prewitt operator, and the Log operator.