Intrinsic nonlinear multiscale image decomposition: A 2D empirical mode decomposition-like tool

Many works have been achieved for analyzing images with a multiscale approach. In this paper, an intrinsic and nonlinear multiscale image decomposition is proposed, based on partial differential equations (PDEs) and the image frequency contents. Our model is inspired from the 2D empirical mode decomposition (EMD) for which a theoretical study is quite nonexistent, mainly because the algorithm is based on heuristic and ad hoc elements making its mathematical study hard. This work has three main advantages. Firstly, we prove that the 2D sifting process iterations are consistent with the resolution of a nonlinear PDE, by considering continuous morphological operators to build local upper and lower envelopes of the image extrema. In addition to the fact that now differential calculus can be performed on envelopes, the introduction of such morphological filters eliminates the interpolation dependency that also terribly suffers the method. Also, contrary to former 2D empirical modes, precise mathematical definition for a class of functions are now introduced thanks to the nonlinear PDE derived from the consistency result, and their characterization on the basis of Meyer spaces. Secondly, an intrinsic multiscale image decomposition is introduced based on the image frequency contents; the proposed approach almost captures the essence and philosophy of the 2D EMD and is linked to the well known Absolutely Minimizing Lipschitz Extension model. Lastly, the proposed multiscale decomposition allows a reconstruction of images. The filterbank capability of the new multiscale decomposition algorithm is shown both on synthetic and real images, and results show that our proposed approach improves a lot on the 2D EMD. Moreover, the complexity of the proposed multiscale decomposition is very reduced compared to the 2D EMD by avoiding the surface interpolation approach, which is the core of all 2D EMD algorithms and is very time consuming. For that purpose also, our work will then be a great benefit; especially, in higher dimension spaces.