Signal Reconstruction from Frame and Sampling Erasures

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• 作者：David Larson ; Sam Scholze
• 关键词：Finite frame ; Omission ; Erasure ; Reconstruction ; Bridging ; 46G10 ; 46L07 ; 46L10 ; 46L51 ; 47A20 ; 42C15 ; 46B15 ; 46B25 ; 47B48
• 刊名：Journal of Fourier Analysis and Applications
• 出版年：2015
• 出版时间：October 2015
• 年：2015
• 卷：21
• 期：5
• 页码：1146-1167
• 全文大小：666 KB
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• 作者单位：David Larson (1)
  Sam Scholze (1)
  
  1. Department of Mathematics, Texas A&M University, College Station, USA
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Fourier Analysis
  Abstract Harmonic Analysis
  Approximations and Expansions
  Partial Differential Equations
  Applications of Mathematics
  Signal,Image and Speech Processing
  
• 出版者：Birkh盲user Boston
• ISSN：1531-5851

文摘

We give some new methods for perfect reconstruction from frame and sampling erasures in a small number of steps. By bridging an erasure set we mean replacing the erased Fourier coefficients of a function with respect to a frame by appropriate linear combinations of the non-erased coefficients. We prove that if a minimal redundancy condition is satisfied bridging can always be done to make the reduced error operator nilpotent of index 2 using a bridge set of indices no larger than the cardinality of the erasure set. This results in perfect reconstruction of the erased coefficients. We also obtain a new formula for the inverse of an invertible partial reconstruction operator. This leads to a second method of perfect reconstruction from frame and sampling erasures in a small number of steps. This gives an alternative to the bridging method for many (but not all) cases. The methods we use employ matrix techniques only of the order of the cardinality of the erasure set, and are applicable to rather large finite erasure sets for infinite frames and sampling schemes as well as for finite frame theory. These methods are usually more efficient than inverting the frame operator for the remaining coefficients because the size of the erasure set is usually much smaller than the dimension of the underlying Hilbert space. Some new classification theorems for frames are obtained and some new methods of measuring redundancy are introduced based on our bridging theory. Keywords Finite frame Omission Erasure Reconstruction Bridging