Equichordal Tight Fusion Frames
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DOI: 10.1007/s13226-020-0439-z
EQUICHORDAL TIGHT FUSION FRAMES Mozhgan Mohammadpour∗ , Rajab Ali Kamyabi-Gol∗∗ and Ghosheh Abed Hodtani∗ ∗ Department
of Pure Mathematics, Ferdowsi University of Mashhad, Iran
∗∗ Department
of Pure Mathematics, Ferdowsi University of Mashhad,
Center of Excellence in Analysis on Algebraic Structures (CEAAS), Mashhad, Iran e-mails: [email protected]; [email protected]; [email protected] (Received 12 June 2017; after final revision 15 April 2019; accepted 7 May 2019) A Grassmannian fusion frame is an optimal configuration of subspaces of a given vector space, that are useful in some applications related to representing data in signal processing. Grassmannian fusion frames are robust against noise and erasures when the signal is reconstructed. In this paper, we present an approach to construct optimal Grassmannian fusion frames based on a given Grassmannian frame. We also analyse an algorithm for sparse fusion frames which was introduced by Calderbank et al. and present necessary and sufficient conditions for the output of that algorithm to be an optimal Grassmannian fusion frame. Key words : Grassmannian frame; fusion frame; chordal distance; optimal Grassmannian fusion frame. 2010 Mathematics Subject Classification : 42C15, 41A58.
1. I NTRODUCTION When data is transmitted over the communication channel, it might be corrupted by noise or be lost. Grassmannian frames provide a representation which is robust against noise and multiple erasures [16, 17]. They are characterized by a property that the maximal cross correlation of the frame elements have minimum value among a given class of frames [16, 20]. The definition of Grassmannian frame can be generalized to the Grassmannian fusion frame which is a set of subspaces that the minimal chordal distance between subspaces has the maximum value among a given class of fusion frames [16,
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17]. Similar to Grasmanian frame, Grassmannian fusion frame is robust against noise and multiple erasures. In more details, Calderbank et al. [17] show that a fusion frame is optimally robust against noise if the fusion frame is tight and a tight fusion frame is optimally robust against one subspace erasure if the dimension of the subspaces are equal. They also proved that a tight fusion frame is optimally robust against multiple erasures if the subspaces are equidistant. Sparse recovery from combined fusion frame measurements is an interesting problem for scientist during the last decade. Grassmannian fusion frames has such a property since the mutual coherence of subspaces meets the lower bound. Hence, these fusion frames minimize the coherence and use a very few projection measurements to recover the signal [5]. The notion of sparse fusion frame is a fusion frame whose subspaces are generated by orthonormal basis vectors that are sparse in a uniform basis over all subspaces, thereby enabling low complexity fusion frame decompositions [8]. Moreover, an algorithmic construction to co
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