Backward adaptive biorthogonalization

Abstract

A backward biorthogonalization approach is proposed, which modifies biorthogonal functions so as to generate orthogonal projections onto a reduced subspace. The technique is relevant to problems amenable to be represented by a general linear model. In particular, problems of data compression, noise reduction, and sparse representations may be tackled by the proposed approach.

Cardinal B-spline dictionaries on a compact interval

“…Hence, in order to account for the inclusion (or respective elimination) of one term in (1), all the elements of the dual sequence need to be modified for the coefficients of the new approximation to minimize the distance to the signal f . Such modifications can be performed in an effective manner by means of adaptive biorthogonalization techniques [14,15]. However, the problem of choosing the M elements of a non-orthogonal basis best representing the signal is a highly nonlinear one [8,9,16].…”

Cooperative greedy pursuit strategies for sparse signal representation by partitioning

“…We extend here the Backward Optimized Orthogonal Matching Pursuit (BOOMP) strategy [24] to select also the blocks from which the atoms are to be removed for downgrading the approximation of a whole signal. The BOOMP strategy is stepwise optimal because it minimizes, at each step, can be quickly obtained through the adaptive backward equations [24,25] b{q} n ← b{q} n − b{q} j b{q} n , b{q} j b{q} j 2 , n = 1, . .…”

Measurements design and phenomena discrimination

“…The graph on the right depicts the result obtained by applying the proposed technique to the signal on the left (it coincides with the line representing the true signal). On the contrary, although the spaces (27) and span{x n (t), t ∈ [0, 1]} 405 n=1 are 'theoretically' complementary, since the construction of the corresponding oblique projector is very ill posed, the projection fails to correctly separate the signals. Figure 1: The left graph depicts the register of the motion of a system consisting of 100 damped harmonic oscillators, whose frequencies are integer numbers randomly taken from the interval [1,405], corrupted by 200 pulses randomly taken from the subspace W ⊥ given in (27).…”