Efficient Signal Reconstruction Research Articles

Consistent Sampling and Efficient Signal Reconstruction

We propose efficient signal reconstruction methods such that the reconstructed signal has the same measurements as the underlying original signal if the former was observed by the same system as that for the latter. The reconstructed signal is a linear combination of reconstruction functions, and the main task is to compute the coefficients from the measurements. This generally involves an inversion of the cross-correlation matrix between the sampling functions and the reconstruction functions. We show that the coefficients are computed efficiently using the discrete Fourier transform (DFT) when sampling is shift-invariant and when the reconstruction functions are shift-invariant functions, Fourier basis functions, or Fourier cosine basis functions. Because of the DFT, the proposed methods do not require matrix inversion computation; rather these methods make it possible to obtain the coefficients with an O(N log N) computation, where N stands for the number of samples. We show the effectiveness of the proposed method by experiments using an ECG signal.

Exact and efficient signal reconstruction in frequency-domain optical-coherence tomography

We address the problem of tomogram reconstruction in frequency-domain optical-coherence tomography. We propose a new technique for suppressing the autocorrelation artifacts that are commonly encountered with the conventional Fourier-transform-based approach. The technique is based on the assumptions that the scattering function is causal and that the intensity of the light reflected from the object is smaller than that of the reference. The technique is noniterative, nonlinear, and yields an exact solution in the absence of noise. Results on synthesized data and experimental measurements show that the technique offers superior quality reconstruction and is computationally more efficient than the iterative technique reported in the literature.

Best Fourier Approximation and Application in Efficient Blurred Signal Reconstruction

We expand upon the known results on sharp linear Fourier methods of approximation where the approximation is the best in terms of both rate and constant among all polynomial procedures of approximation. So far these results have been studied due to their mathematical beauty rather than their practical importance. In this paper we show that they are the core mathematics underlying best statistical methods of solving noisy ill-posed problems. In particular, we suggest a procedure for recovery of noisy blurred signals based on samples of small sizes where a traditional statistics concludes that the complexity of such a setting makes the problem not worthy of a further study. Thus, we present a problem where a combination of the classical approximation theory and statistics leads to interesting practical results.