%0 Journal Article
%T About a Partial Differential Equation-Based Interpolator for Signal Envelope Computing: Existence Results and Applications
%A Oumar Niang
%A Abdoulaye Thioune
%A ¨¦ric Del¨¦chelle
%A Mary Teuw Niane
%A Jacques Lemoine
%J ISRN Signal Processing
%D 2013
%R 10.1155/2013/605035
%X This paper models and solves the mathematical problem of interpolating characteristic points of signals by a partial differential Equation-(PDE-) based approach. The existence and uniqueness results are established in an appropriate space whose regularity is similar to cubic spline one. We show how this space is suitable for the empirical mode decomposition (EMD) sifting process. Numerical schemes and computing applications are also presented for signal envelopes calculation. The test results show the usefulness of the new PDE interpolator in some pathological cases like input class functions that are not so regular as in the cubic splines case. Some image filtering tests strengthen the demonstration of PDE interpolator performance. 1. Introduction Interpolators are widely used in signal and processing or data analysis. In particular for the empirical mode decomposition (EMD) algorithm [1, 2], the iterative estimation of the signal trend is based on the computing of the envelopes obtained by the cubic spline interpolation of local extrema. The spline interpolation has been recognized as being very effective for EMD. But for signals that have no local extremum, the cubic spline interpolation fails. We proposed a PDE-based model which overcomes this limit of classical EMD implementation, in the computing of the envelopes for signals that have no local extremum [2¨C4]. Recently, the Spectral Intrinsic Decomposition (SID) method [5], based on the spectral decomposition of the PDE interpolator, provides a new application of our model. This PDE interpolator contribute, to the mathematical modeling of the EMD and has provided various applications in signal and image processing [4, 6]. In this paper, we describe the mathematical modeling of the new PDE interpolator by variational methods. The resolution of the variational problem leads to existence and uniqueness results in appropriate spaces. The paper is organized as follows. In Sections 2 and 2.1 recalls some PDE models in signal and image processing, and in Section 2.2 some mathematical preliminary notions are set out. In Section 3, the mathematical modeling is exposed. In Sections 4 and 5, the resolution of the variational problem is dealt. Subsequently, numerical implementation and applications are presented in Section 6. At last, we finish by conclusions. 2. Some Preliminaries 2.1. PDE Models in Signal and Image Processing and EMD Principle 2.1.1. Some Diffusion Equations This part consists of a brief and nonexhaustive presentation of classical nonlinear diffusion filters for 1D and 2D signal processing,
%U http://www.hindawi.com/journals/isrn.signal.processing/2013/605035/