Nonlinear Approximations for Inverse Problems

该项目旨在开发一类新的方法来解决不同形式的逆问题，例如开发新的无损评估方法，如X射线层析成像、磁共振成像(MRI)、合成孔径雷达、衍射层析成像、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(如在磁共振成像中)，或者可以转换到傅立叶域以用于成像(如在X射线断层摄影术或电子显微镜中)。在所有情况下，图像分辨率都由测量数据中存在的最大波数控制。为了最大限度地减少Gibbs现象引起的失真，目前的成像方法要么在足够大的傅立叶域内采集数据，要么对数据进行加窗以强制进行人工衰减。可以看出，当前的方法需要收集更多的数据，或者对恢复的感兴趣函数的空间奇异性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。相反，这一建议依赖于非线性的、近乎最优的指数逼近来推断可用的傅立叶数据，也产生了近乎最优的空间有理表示。这种方法不仅提高了奇点附近的分辨率，获得了接近最优的性能，而且还检测了数据中的噪声水平，为信噪分离提供了一种实用的技术。这些新的算法已经在一维问题中产生了对现有技术的重大改进，该项目打算将这种方法扩展到二维或三维问题。这种扩展是非常重要的，因为在一维中使用的数学工具只有有限的使用。在广泛的科学和工程学科中出现反向问题，作为分析生物或无机样本、分析制造的设备的缺陷、执行遥感或地球物理勘探等许多其他应用的方法。在所有这些模式中，从处理多个雷达数据到生物医学成像(如MRI)，收集的数据由算法处理，该算法基于特定的数学模型实现反问题的解决。研究人员提出了一种方法，开发并通过算法实现适用于许多(如果不是全部)这些问题的新数学模型。之所以具有如此广泛的适用性，是因为在新的数学模型中，感兴趣的函数用接近最佳数量的参数来表示，从而显著提高了从测量数据中恢复信息的能力。这个项目的数学和实际意义以及它的挑战在于将研究人员在一个维度上开发的方法扩展到多个维度。由于无损评估技术在自然科学、工程学、医学诊断学以及机场安全等可视化应用中发挥着重要作用，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，以有效地解决当今许多算法无法解决的问题。

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.