Nonconvex optimization-based inverse spectral decomposition

Abstract

Abstract As a time–frequency analysis tool, inverse spectral decomposition (ISD) could be utilized to obtain a high-resolution time–frequency map via the inversion strategy. In the established inversion function, an analytic signal is disintegrated as a coefficient matrix whose elements represent the weights of the wavelet components with the different dominant frequencies and the time location in the complex wavelet library. By using a sparse constraint, a high-quality inverse decomposition result could be generated. In this paper, a modified ISD technique based on a nonconvex optimization algorithm is proposed to pursue a sparser coefficient solution found in the inversion processing by decreasing the redundant information. This new approach applies the lp(0 < p < 1) penalty term to build an accurate mapping relationship between the original signal and its time–frequency spectrum. This adequate regularization in reconstructed function serves as a better alternative to the l1 norm one in conventional ISD. Lower signal-to-noise ratio (SNR) and much weaker incoherence of wavelet library cannot impact on the accuracy of output with the ISD based on the lp norm (0 < p < 1) constraint. Synthetic data tests with the Gabor transform, the conventional ISD method and the nonconvex optimization-based ISD are applied to demonstrate the performance gaps. In the real application area, a blended image with instantaneous spectral attribute volumes produced by the new approach assists the identification of the geological anomalous body, which is verified via a physical model data and field data tests.