A PDE based and interpolation-free framework for modeling the sifting process in a continuous domain El Hadji S. Diop ⋆ , Radjesvarane Alexandre † and Valérie Perrier ‡ ⋆ Center of Mathematical Morphology, Department of Mathematics and Systems MINES ParisTech, 35 rue Saint-Honoré, 77305 Fontainebleau Cedex, France. † Department of Mathematics, Shanghai Jia Tong University, Shanghai 200240, PRC China. ‡ Laboratoire Jean-Kunztmann, Tour IRMA, BP 53, 38041 Grenoble Cedex 9, France. E-mails: el-hadji.diop@mines-paristech.fr, alexandreradja@gmail.com, Valerie.Perrier@imag.fr Abstract The empirical mode decomposition (EMD) is a powerful tool in signal processing. Despite its algorithmic origin making its theoretical analysis and formulation very difficult, a few recent works has contributed to its theoretical framework. Herein, the former local mean is formulated in a more convenient way by introducing op- erators to calculate local upper and lower envelopes. This enables the use of dif- ferential calculus and other classical calculations on the new local mean. Based on its more accurate formulation, a partial differential equation (PDE) consistency re- sult is provided to approximate the sifting process iterations, without any envelope interpolation. In addition, a new stopping criterion based on the introduced local mean is proposed. This new criterion is a local measure and resolves the null in- tegral conservative property of the previous derived PDE, which made any signal having a null integral be a PDE-based mode. Moreover, the δ inner model parame- ter is now linked to the signal intrinsic properties, providing to the latter a physical meaning and making the proposed model keep the auto-adaptive property of the EMD. New decomposition modes are now analytically and fully characterized, and also interpolation free. Finally, properties of the interpolation free PDE model are presented. Results obtained with our proposed approach by explicit computations thanks to the eigendecomposition of the Laplacian operator, and also by numerical resolution of the derived PDE, show noticeable improvements for both stationary and non stationary signals, in comparison to the former EMD algorithm. Keywords : Eigenfunctions, eigenvalues, empirical mode decomposition, intrinsic mode functions, partial differential equations. 1