Nonlinear Approximations for Inverse Problems

该项目旨在开发一种新的方法来解决不同形式的反问题，例如开发非破坏性评估的新方法，如X射线断层扫描，磁共振成像（MRI），合成孔径雷达，衍射断层扫描，电子显微镜等。在这些问题中，数据可以直接在傅立叶域中收集（如在MRI中），或者可以为了图像形成的目的而转换到傅立叶域中（如在X射线断层扫描或电子显微镜中）。在所有情况下，图像分辨率由测量数据中存在的最大波数控制。为了最小化由于吉布斯现象引起的失真，当前的成像方法要么在傅立叶域的足够大的区域中收集数据，要么对数据加窗以强制人工衰减。可以表明，当前的方法需要收集比必要的更多的数据，或者对恢复的感兴趣的函数的空间奇点附近的分辨率产生负面影响。这些奇点通常是最终图像中信息量最大的部分。相比之下，这个建议依赖于非线性的，接近最佳的近似指数外推可用的傅立叶数据，也产生了一个接近最佳的合理表示空间。这种方法不仅提高了奇异点附近的分辨率，获得了接近最优的性能，而且还检测了数据中的噪声水平，为信号/噪声分离提供了一种实用的技术。 这些新的算法已经产生了显着的改善现有的技术在一维的问题，这个项目打算扩展到二维或三维的问题的方法。这样的扩展是非常不平凡的，因为在一个维度上使用的数学工具只有有限的use.Inverse问题出现在各种各样的科学和工程学科作为一种方法来分析生物或无机标本，分析制造设备的缺陷，执行遥感或地球物理勘探等许多其他应用。在所有这些模态中，从处理多个雷达数据到诸如MRI的生物医学成像，所收集的数据通过基于特定数学模型实现逆问题的解决方案的算法来处理。研究人员提出了一种开发和算法实现新的数学模型的方法，适用于许多（如果不是全部）这些问题。这种广泛适用性的原因在于，在新的数学模型中，感兴趣的函数用接近最佳数量的参数表示，从而显著改善了从测量数据中恢复信息的能力。该项目的数学和实际意义以及其挑战在于将研究人员在一维中开发的方法扩展到多维。由于非破坏性评估技术在自然科学、工程、医学诊断以及机场安全等可见应用中起着至关重要的作用，我们预计这些基于非线性近似的新数学模型将产生广泛的影响。 在这个项目中开发的数值方法应该为科学家提供计算工具，以有效地解决超出当今许多算法能力的问题。

This project aims to develop a new class of methods for solving inverse problems in their different modalities, for example to develop new methods for non-destructive evaluation, such as X-ray Tomography, Magnetic Resonance Imaging (MRI), Synthetic Aperture Radar, Diffraction Tomography, Electron Microscopy, and many others. In these problems, data may be collected directly in the Fourier domain (as in MRI), or may be transformed into the Fourier domain for the purpose of image formation (as in X-ray Tomography or Electron Microscopy). In all cases, the image resolution is controlled by the largest wave-number present in the measured data. In order to minimize the distortion due to Gibbs phenomenon, current imaging methods either collect data in a large enough area of the Fourier domain or window the data to force an artificial decay. It can be shown that current methods require collecting more data than necessary or negatively impact the resolution near spatial singularities of recovered functions of interest. These singularities are, typically, the most informative part of the final image. In contrast, this proposal relies on nonlinear, near optimal approximation by exponentials to extrapolate the available Fourier data, also yielding a near optimal rational representation in space. This approach not only improves the resolution near singularities and achieves a near optimal performance, but also detects the level of noise in data and provides a practical technique for signal/noise separation. These new algorithms already yield a significant improvement over existing techniques in one-dimensional problems and this project intends to extend the approach to problems in two or three dimensions. Such an extension is highly nontrivial since the mathematical tools used in one dimension are only of limited use.Inverse problems arise in a wide variety of scientific and engineering disciplines as a way to analyze biological or inorganic specimens, analyze manufactured devices for defects, perform remote sensing or geophysical exploration among many other applications. In all of these modalities, from processing multiple radar data to biomedical imaging such as MRI, the collected data is processed by algorithms implementing the solution of an inverse problem based on a specific mathematical model. The investigators propose a method of developing and algorithmically implementing new mathematical models applicable to many, if not all, of these problems. The reason for such wide applicability is the fact that in the new mathematical models the functions of interest are represented with a near optimal number of parameters, thus significantly improving the recovery of information from the measured data. The mathematical and practical significance of this project as well as its challenge lies in extending the methods developed by investigators in one dimension to multiple dimensions. Since techniques of non-destructive evaluation play a vital role in natural sciences, engineering, medical diagnostics as well as such visible applications as airport security, we expect a wide impact of these new mathematical models based on nonlinear approximations. The numerical methods developed within this project should provide scientists with computational tools to efficiently solve problems beyond the capabilities of many of today's algorithms.