Hermite Interpolation Based Interval Shannon-Cosine Wavelet and Its Application in Sparse Representation of Curve

Abstract

Using the wavelet transform defined in the infinite domain to process the signal defined in finite interval, the wavelet transform coefficients at the boundary are usually very large. It will bring severe boundary effect, which reduces the calculation accuracy. The construction of interval wavelet is the most common method to reduce the boundary effect. By studying the properties of Shannon-Cosine interpolation wavelet, an improved version of the wavelet function is proposed, and the corresponding interval interpolation wavelet based on Hermite interpolation extension and variational principle is designed, which possesses almost all of the excellent properties such as interpolation, smoothness, compact support and normalization. Then, the multi-scale interpolation operator is constructed, which can be applied to select the sparse feature points and reconstruct signal based on these sparse points adaptively. To validate the effectiveness of the proposed method, we compare the proposed method with Shannon-Cosine interpolation wavelet method, Akima method, Bezier method and cubic spline method by taking infinitesimal derivable function cos(x) and irregular piecewise function as an example. In the reconstruction of cos(x) and piecewise function, the proposed method reduces the boundary effect at the endpoints. When the interpolation points are the same, the maximum error, average absolute error, mean square error and running time are 1.20 × 10−4, 2.52 × 10−3, 2.76 × 10−5, 1.68 × 10−2 and 4.02 × 10−3, 4.94 × 10−4, 1.11 × 10−3, 9.27 × 10−3, respectively. The four indicators mentioned above are all lower than the other three methods. When reconstructing an infinitely derivable function, the curve reconstructed by our method is smoother, and it satisfies C2 and G2 continuity. Therefore, the proposed method can better realize the reconstruction of smooth curves, improve the reconstruction efficiency and provide new ideas to the curve reconstruction method.