Constructions of Dual Frames Compensating for Erasures with Implementation

Let \(I\subseteq \mathbb N\) be a finite or infinite set of indices and let \((x_n)_{n\in I}\) be a frame for a separable Hilbert space \(\mathcal {H}\) . Consider the transmission of a signal \(h\in \mathcal {H}\) where a finite subset \(({\langle }h,x_n{\rangle })_{n\in E}\) of the frame coefficients \(({\langle }h,x_n{\rangle })_{n\in I}\) is lost. There are several approaches in the literature that aim at recovering h. In this paper, we focus on the approach based on the construction of a dual frame of the reduced frame \((x_n)_{n\in I\setminus E}\) which is then used for a perfect reconstruction of h from the preserved frame coefficients \(({\langle }h,x_n{\rangle })_{n\in I\setminus E}\) . There are several methods for such construction, starting from the canonical dual frame or any other dual frame of \((x_n)_{n\in I}\) . We implemented algorithms for these methods and performed tests to compare their computational efficiency.