Using NUFFT in nonuniform sampling Fourier transform spectrometer and the comparison with conventional interpolation FFT method

Resampling by interpolation is the traditional method to process sample in nonunform sampling Fourier transform spectrometer. Nonuniform discrete Fourier transform is an alternative to interpolation that has not been overlooked before. With the aid of experiment, we systematically compare the NUFFT method with resampling by interpolation FFT method in nonuniform sampling Fourier transform spectrometer. We found that NUFFT is comparable to interpolation in spectral profile and spectral noise levels and is better in spectral amplitudes. We also found that It has significant advantage in under-sampling and anti-aliasing property which is offered by the unique non-periodic nature of nonuniform sampling. It is faster and consumes less computer memory in our python implementation. Overall, we found that NUFFT is superior to traditional interpolation method with mostly better performances as well as additional capabilities

An efficient high-order Nystr\"om scheme for acoustic scattering by inhomogeneous penetrable media with discontinuous material interface

This text proposes a fast, rapidly convergent Nystr\"{o}m method for the solution of the Lippmann-Schwinger integral equation that mathematically models the scattering of time-harmonic acoustic waves by inhomogeneous obstacles, while allowing the material properties to jump across the interface. The method works with overlapping coordinate charts as a description of the given scatterer. In particular, it employs "partitions of unity" to simplify the implementation of high-order quadratures along with suitable changes of parametric variables to analytically resolve the singularities present in the integral operator to achieve desired accuracies in approximations. To deal with the discontinuous material interface in a high-order manner, a specialized quadrature is used in the boundary region. The approach further utilizes an FFT based strategy that uses equivalent source approximations to accelerate the evaluation of large number of interactions that arise in the approximation of the volumetric integral operator and thus achieves a reduced computational complexity of $O(N \log N)$ for an $N$-point discretization. A detailed discussion on the solution methodology along with a variety of numerical experiments to exemplify its performance in terms of both speed and accuracy are presented in this paper

An efficient high-order Nyström scheme for acoustic scattering by inhomogeneous penetrable media with discontinuous material interface

This text proposes a fast, rapidly convergent Nyström method for the solution of the Lippmann–Schwinger integral equation that mathematically models the scattering of time-harmonic acoustic waves by inhomogeneous obstacles, while allowing the material properties to jump across the interface. The method works with overlapping coordinate charts as a description of the given scatterer. In particular, it employs “partitions of unity” to simplify the implementation of high-order quadratures along with suitable changes of parametric variables to analytically resolve the singularities present in the integral operator to achieve desired accuracies in approximations. To deal with the discontinuous material interface in a high-order manner, a specialized quadrature is used in the boundary region. The approach further utilizes an FFT based strategy that uses equivalent source approximations to accelerate the evaluation of large number of interactions that arise in the approximation of the volumetric integral operator and thus achieves a reduced computational complexity of O(N log N) for an N-point discretization. A detailed discussion on the solution methodology along with a variety of numerical experiments to exemplify its performance are presented in this paper

Modeling and tracking of the cardiac left ventricular motion by a linear harmonic model in MRI sequence

In this article, we propose a new method for modeling the left ventricular motion of the heart from a magnetic resonance imaging (MRI) sequence. We propose to model the space-time trajectory of the points of the endocardial (respectively epicardial) contour of the left ventricle (LV) using a harmonic model of movement, which is linear and can describe the dynamics of the left ventricle throughout the cardiac cycle. This new model is based on the assumption of quasi-periodicity of the cardiac cycle and uses a Kalman filter as estimation tool. We first refer to the main works in the field, then describing our method. We give the way to get the space-time trajectories of the contour points of the LV. We present the model with the selected state equations and the Kalman filter based motion estimate. We propose two methods of calculation. The direct one provides a solution for a fixed rank of the harmonic model. The recursive one allows progressively go from rank n to rank n + 1 without prior choice. The model is validated on simulated data by direct comparison with the traditional Fourier decomposition approach. It is shown that it fits well the studied trajectories. The results obtained on real cardiac sequences are particularly interesting because they demonstrate the capability of our method to discriminate unambiguously normal cases from pathological cases.Dans cet article, nous proposons une nouvelle méthode de modélisation du mouvement ventriculaire gauche du coeur à partir d'une séquence d'images acquises en imagerie par résonance magnétique (IRM). Nous proposons de modéliser la trajectoire spatio-temporelle des points appartenant au contour endocardique (respectivement épicardique) du ventricule gauche (VG) à l'aide d'un modèle de mouvement harmonique, linéaire, capable de décrire la dynamique du VG sur l'ensemble du cycle cardiaque. Ce modèle s'appuie sur l'hypothèse de quasi-périodicité du rythme cardiaque. Il utilise un filtre de Kalman comme outil d'estimation. Après une analyse commentée des travaux récents dans le domaine, nous détaillons les différentes étapes de la méthode. L'obtention des trajectoires spatio-temporelles des points de contours est décrite. Nous présentons le vecteur d'état canonique retenu et les équations d'état correspondantes, suivies de l'estimation des paramètres du mouvement par filtrage de Kalman. Nous proposons deux méthodes de calcul du modèle d'état harmonique, l'une directe qui fournit une solution pour un ordre fixé du modèle harmonique, l'autre récursive qui permet le passage progressif d'un ordre n à l'ordre n + 1 du modèle. Ce modèle est validé sur des données simulées par comparaison directe avec l'approche classique utilisant la décomposition de Fourier. On montre qu'il est nettement plus robuste en présence du bruit sur les trajectoires étudiées. Les résultats obtenus sur des séquences d'images cardiaques réelles sont particulièrement intéressants car ils permettent déjà d'identifier sans ambiguïté les cas normaux des cas pathologiques