Continuous-Domain Fractional Derivative Model for Signal Decomposition

Abstract We present a continuous-domain variational model for the morphological decomposition of 1D noisy signals into a sum of meaningful constituent components. The model relies on sparsifying fractional-order derivatives of the sought components to capture intricate signal structures. An in-depth analysis of the model leads to a representer theorem , establishing the equivalence between the infinite-dimensional problem and a finite-dimensional counterpart, which serves as its exact discretization. To efficiently solve the resulting discrete and convex optimization problem, an alternating direction method of multipliers-based algorithm is presented. Furthermore, we introduce a bilevel optimization framework for the automatic selection of all free model parameters, including the fractional derivative orders, based on the generalized whiteness principle. Numerical results validate the effectiveness of our approach, which can provide accurate decompositions even in demanding scenarios characterized by high noise levels and abrupt signal discontinuities.