Dictionary redundancy elimination

Two criteria for dictionary redundancy elimination are discussed. One of them operates by disregarding linearly dependent atoms, whilst the other selects linearly independent atoms. The latter is implemented by the modified Gram-Schmidt orthogonalisation with pivoting technique, and is suitable for handling the effect of 'quasi-linear dependence', most likely to be present in a redundant dictionary. The corresponding reciprocal waveforms are easily obtained within the workings of the selection process. Such waveforms are biorthogonal to the selected atoms and allow computation of the respective coefficients of the linear combination approximating an arbitrary signal at best in a minimum distance sense.     

Signal representation for compression and noise reduction throughframe-based wavelets

A mathematical framework for data representation and for noise reduction is presented in this paper. The basis of the approach lies in the use of wavelets derived from the general theory of frames to construct a subspace capable of representing the original signal excluding the noise. The representation subspace is shown to be efficient in signal modeling and noise reduction, but it may be accompanied by an ill-conditioned inverse problem. This is further examined, and a more adequate orthonormal representation for the generated subspace is proposed with an improvement in compression performance     

Frame-theory-based approach for determining the time-frequency distribution characterizing a dense group of reflecting objects

An inverse-scattering problem concerning the determination of a time-frequency spreading function is addressed. Such a function characterizes a dense group of reflecting objects at different ranges and moving with different velocities. The problem, arising in radar and other remote-sensing techniques, is a classical inverse problem. The aim is to reconstruct a function of two variables by means of signals (of one variable) reflected from the environment being observed. The proposed approach is developed by recourse to the frame theory in order to provide a reconstruction formula that asymptotically converges to a unique spreading function. The realistic situation with respect to the transmission of a finite number of signals is further considered. In this case the reconstruction formula is shown to yield the orthogonal projection of the spreading function onto a subspace generated by the outgoing signals.     

The continuous wavelet transform as a maximum entropy solution ofthe corresponding inverse problem

The continuous wavelet transform is obtained as a maximum entropy solution of the corresponding inverse problem. It is well known that although a signal can be reconstructed from its wavelet transform, the expansion is not unique due to the redundancy of continuous wavelets. Hence, the inverse problem has no unique solution. If we want to recognize one solution as “optimal”, then an appropriate decision criterion has to be adopted. We show here that the continuous wavelet transform is an “optimal” solution in a maximum entropy sense