# An Optimization Based Empirical Mode Decomposition Scheme for Images

Huang, Boqiang and Kunoth, Angela (2012) An Optimization Based Empirical Mode Decomposition Scheme for Images. (Submitted)

Bidimensional empirical mode decompositions (BEMD) have been developed to decompose any bivariate function or image additively into multiscale components, so-called intrinsic mode functions (IMFs), which are approximately orthogonal to each other with respect to the $\ell_2$ inner product. In this paper, a novel optimization problem is designed to achieve this decomposition which takes into account important features desired of the BEMD. Specifically, we propose a data-adapted iterative method which we call Opt-BEMD which minimizes in each iteration a smoothness functional subject to inequality constraints involving the strictly local extrema of the image. In this way, the method constructs a sparse data-adapted basis for the input function as well as an envelope in a mathematically stringent sense. Moreover, we propose an ensemble version of Opt-BEMD to strengthen its performance when applied to noise-contaminated images or images with only few extrema.