Hilbert-Schmidt Operators and Frames - Classification, Best Approximation by Multipliers and Algorithms!

22/10/2007

• P. Balazs, Hilbert-Schmidt Operators and Frames - Classification, Approximation by Multipliers and Algorithms, , International Journal of Wavelets, Multiresolution and Information Processing, (2007, accepted), preprint

Codes (MATLAB):

• Find the best Approximation of a matrix by a frame multiplier: ApprFramMult (T,D,Ds).
• Build the matrix for a frame multiplier: FramMultMat(s,D,Ds)
• Two simple examples for the best approximation by frame multipliers: testapprframmult
• Find the best approximation of a frame multiplier by a frame multiplier: testapprframmult2
• Test the approximation of matrices by filters, Gabor and wavelet mulitpliers: TestBestApprMult
• Approximation of Matrices by irregular Gabor multiplier: GMAPPir
• Creates the irregular Gabor frame n the lattice xpo: Gabbaspir
• Create a synthesis matrix containing mexican hat wavelets: waveletmat
• Find the best approximation by circulant matrices: bestapprcircl
  
• Calculation of cross-Gram Matrix for Gabor systems : HSGramMatrXXL
• Fourier Transformation of a matrix FMFxxl 
  
• Inverse Fourier Transformation of a matrix: iFMFxxl
• Creation of a translation matrix: transmatxxl
  
• All codes collected in one ZIP-file.

Pictures:

1. Using the algorithm for approximation with frame multipliers in the time invariant filter, Gabor and wavelet case.
   
   Top Left: original system, Top Right: approximation by circulant matrices, i.e. time-invariant filters, Bottom Left: Gabor case, Bottom Right:wavelet case
   

   
   
   
   
2. The time frequency spread of the best approximations in the last figure.
   Top Left: original system, Top Right: approximation by circulant matrices, i.e. time-invariant filters, Bottom Left: Gabor case, Bottom Right:wavelet case