Optimal density compensation factors for the reconstruction of the Fourier transform of bandlimited functions

An inverse nonequispaced fast Fourier transform (iNFFT) is a fast algorithm to compute the Fourier coefficients of a trigonometric polynomial from nonequispaced sampling data. However, various applications such as magnetic resonance imaging (MRI) are concerned with the analogous problem for bandlimited functions, i.e., the reconstruction of point evaluations of the Fourier transform from given measurements of the bandlimited function. In this paper, we review an approach yielding exact reconstruction for trigonometric polynomials up to a certain degree, and extend this technique to the setting of bandlimited functions. Here we especially focus on methods computing a diagonal matrix of weights needed for sampling density compensation

Fast and direct inversion methods for the multivariate nonequispaced fast Fourier transform

The well-known discrete Fourier transform (DFT) can easily be generalized to arbitrary nodes in the spatial domain. The fast procedure for this generalization is referred to as nonequispaced fast Fourier transform (NFFT). Various applications such as MRI, solution of PDEs, etc., are interested in the inverse problem, i.e., computing Fourier coefficients from given nonequispaced data. In this paper we survey different kinds of approaches to tackle this problem. In contrast to iterative procedures, where multiple iteration steps are needed for computing a solution, we focus especially on so-called direct inversion methods. We review density compensation techniques and introduce a new scheme that leads to an exact reconstruction for trigonometric polynomials. In addition, we consider a matrix optimization approach using Frobenius norm minimization to obtain an inverse NFFT

Nonuniform fast Fourier transforms with nonequispaced spatial and frequency data and fast sinc transforms

In this paper we study the nonuniform fast Fourier transform with nonequispaced spatial and frequency data (NNFFT) and the fast sinc transform as its application. The computation of NNFFT is mainly based on the nonuniform fast Fourier transform with nonequispaced spatial nodes and equispaced frequencies (NFFT). The NNFFT employs two compactly supported, continuous window functions. For fixed nonharmonic bandwidth, it is shown that the error of the NNFFT with two sinh-type window functions has an exponential decay with respect to the truncation parameters of the used window functions. As an important application of the NNFFT, we present the fast sinc transform. The error of the fast sinc transform is estimated, too

Fast and direct inversion methods for the multivariate nonequispaced fast Fourier transform

The well-known discrete Fourier transform (DFT) can easily be generalized to arbitrary nodes in the spatial domain. The fast procedure for this generalization is referred to as nonequispaced fast Fourier transform (NFFT). Various applications such as MRI and solution of PDEs are interested in the inverse problem, i.e., computing Fourier coefficients from given nonequispaced data. In this article, we survey different kinds of approaches to tackle this problem. In contrast to iterative procedures, where multiple iteration steps are needed for computing a solution, we focus especially on so-called direct inversion methods. We review density compensation techniques and introduce a new scheme that leads to an exact reconstruction for trigonometric polynomials. In addition, we consider a matrix optimization approach using Frobenius norm minimization to obtain an inverse NFFT

Saccadic latency in hepatic encephalopathy: a pilot study

Hepatic encephalopathy is a common complication of cirrhosis. The degree of neuro-psychiatric impairment is highly variable and its clinical staging subjective. We investigated whether eye movement response times—saccadic latencies—could serve as an indicator of encephalopathy. We studied the association between saccadic latency, liver function and paper- and pencil tests in 70 patients with cirrhosis and 31 patients after liver transplantation. The tests included the porto-systemic encephalopathy (PSE-) test, critical flicker frequency, MELD score and ammonia concentration. A normal range for saccades was established in 31 control subjects. Clinical and biochemical parameters of liver, blood, and kidney function were also determined. Median saccadic latencies were significantly longer in patients with liver cirrhosis when compared to patients after liver transplantation (244 ms vs. 278 ms p < 0.001). Both patient groups had prolonged saccadic latency when compared to an age matched control group (175 ms). The reciprocal of median saccadic latency (μ) correlated with PSE tests, MELD score and critical flicker frequency. A significant correlation between the saccadic latency parameter early slope (σE) that represents the prevalence of early saccades and partial pressure of ammonia was also noted. Psychometric test performance, but not saccadic latency, correlated with blood urea and sodium concentrations. Saccadic latency represents an objective and quantitative parameter of hepatic encephalopathy. Unlike psychometric test performance, these ocular responses were unaffected by renal function and can be obtained clinically within a matter of minutes by non-trained personnel