Efficient Nearly Piecewise Continuous Signal Basis Expansion by Precision Calculation

Many one-dimensional (1D) signals, including ECG signals, economic data, particle distributions, seismic waveforms, image rows, and traffic statistics, are nearly piecewise continuous. These signals vary gradually in most regions but vary rapidly in certain narrow regions. Due to the non-uniform distribution of gradients, it is very challenging to well expand these nearly piecewise continuous signals. In this paper, we propose a novel algorithm, called “precise calculation,” to adaptively expand such signals. It categorizes signals into three types and then segments each type into smooth and large-gradient parts by using gradient-based analysis, adaptive thresholding, and morphology refinement. After decomposition, adaptive expansion is applied in each segment. Experiments on 1D ECG signals, electron-density distributions, seismic waveforms, image-row data, and economic time-series demonstrate that the proposed methods accurately expand nearly piecewise continuous signals. Under a fixed number of basis functions, our approach achieves significantly lower approximation errors than existing methods, highlighting its effectiveness for high-fidelity signal reconstruction. While applying only 5% of the expansion functions, experiments show that, for ECG signals, the expansion errors obtained by the proposed method range from 0.0003 to 0.0094—significantly lower than those achieved using only the DCT (0.0023–0.1506) and the Legendre method (0.00318–0.1677).