Spectral Intrinsic Decomposition Method for Adaptive Signal Representation

We propose a new method called spectral intrinsic decomposition (SID) for the representation of nonlinear signals. This approach is based on the spectral decomposition of partial differential equation- (PDE-) based operators which interpolate the characteristic points of a signal. The SID’s components which are the eigenvectors of these PDE interpolation operators underlie the new signal decomposition-reconstruction method. The usefulness and the efficiency of this method is illustrated, in signal reconstruction or denoising aim, in some examples using artificial and pathological signals. 1. Introduction The signal decomposition into atoms is an popular approach in signal analysis. The Fourier representation technics and other based on wavelets and time-frequency, or time scales analysis methods [1], and recently the Empirical Mode Decomposition [2] are extensively used in signal and image processing. The objective is to understand the contents of the signal by analyzing its components. It is sometimes desirable to have these components well suited to the separation of the noise or data in some scale analysis. Sparse representations of signals have like pursuit methods [3], the Poper Orthogonal Decomposition (POD) [4], or Singular Value Decomposition (SVD) received a great deal of attentions in last recent years. The problem solved by the sparse representation is the most compact representation of a signal in terms of combination of atoms in an overcomplete dictionary. The Empirical Mode Decomposition [2] is a self adaptive decomposition method which is essentially algorithmic and can decompose a nonlinear signal into Amplitude Modulation-Frequency Modulation (AM-FM) component plus an residue. The characteristic points of a signal like local extrema are very useful in signal analysis as its shown in EMD algorithm. The interpolation of the characteristic points provides a low frequency component of a signal whose iterative extraction is the basis of the EMD sifting process. To overcome the lack of a solid theoretical framework of EMD, we have proposed an analytical approach for sifting process based on partial differential equation (PDE) in [5–8]. We give in particular a noniterative scheme to solve the coupled PDEs system for upper and lower envelopes estimation with an adequate definition of the characteristic points of the signal to be decomposed, see [6, 8]. In this following work, we use all the eigenvectors of upper and lower PDE-envelope operators and propose a new Spectral Intrinsic Decomposition (SID) method for nonlinear signal representation.