A New Look at Optimal Dual Problem Related to Fusion Frames

TL;DR

This paper explores the structure of optimal dual fusion frames, introduces a partial optimal dual concept for efficient erasure handling, and compares different duality definitions to improve reconstruction robustness.

Contribution

It introduces the partial optimal dual concept, offering a less computationally intensive method for identifying optimal duals in fusion frames.

Findings

Partial optimal duals reduce computation time.

Optimal duals improve reconstruction accuracy.

Relationship between local and global duals established.

Abstract

The purpose of this work is to examine the structure of optimal dual fusion frames and get more exibility in the use of dual fusion frames for erasures of subspaces. We deal with optimal dual fusion frames with respect to different definitions of duality and compare the advantages of these approaches. In addition, we introduce a new concept so called partial optimal dual which involves less time and computation for detecting optimal dual for erasures in known locations. Then we study the relationship between local and global optimal duals by partial optimal duals which leads to some applicable results. In the sense that, we obtain an overcomplete frame and a family of associated optimal duals by a given Riesz fusion basis. We present some examples to exhibit the effect of error rate when dual fusion frames are applied in reconstruction.