Learnable Empirical Mode Decomposition based on Mathematical Morphology

. Empirical mode decomposition (EMD) is a fully data driven method for multiscale decomposing 4 signals into a set of components known as intrinsic mode functions. EMD is based on lower and 5 upper envelopes of the signal in an iterated decomposition scheme. In this paper, we put forward a 6 simple yet effective method to learn EMD from data by means of morphological operators. We pro-7 pose an end-to-end framework by incorporating morphological EMD operators into deeply learned 8 representations, trained using standard backpropagation principle and gradient descent-based opti-9 mization algorithms. Three generalizations of morphological EMD are proposed: a) by varying the 10 family of structuring functions, b) by varying the pair of morphological operators used to calculate 11 the envelopes, and c) by considering a convex sum of envelopes instead of the mean point used 12 in classical EMD. We discuss in particular the invariances that are induced by the morphological 13 EMD representation. Experimental results on supervised classification of hyperspectral images by 14 1D convolutional networks demonstrate the interest of our method. 15