Constructions of dual frames compensating for erasures with implementation

Abstract

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