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Center for Nonlinear Studies |
A sparse representation is an adaptive signal decomposition consisting of a linear combination of atoms from an overcomplete dictionary, where the coefficients of the linear combination are optimized according to some sparsity criterion. Denoising can be perform by allowing a misfit equal to the desired noise variance (between the original and reconstructed signals) when calculating the signal's sparse reprensentation. On the other hand, Total Variation (TV) regularization is a widely used method to directly calculate a signal's denoised version. TV regularization is known to be exceptionally effective when applied over blocky signals. Although both techniques solve the same problem, their philosophy is fundamentally different. It have been shown in the related literature that BP (specifically BP denoising) and TV are equivalent in the 1-dimensional case. The 2-D case is non-trivial and the equivalence has not been proven. We aim to show that Total Variation can be used as an adaptive dictionary for sparse representation (i.e. Basis Pursuit), showing an equivalence between the two methods in the case of 2-D signals. Software routines will be written to test our hypothesis and increase the functionality of the NUMIPAD (numerical methods for inverse problems and adaptive decomposition) software being developed at T-7.
Host: Paul Rodriguez