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，例如开发无损评估的新方法，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，数据可以直接在傅里叶域中收集（如在MRI中），或者可以转换到傅里叶域中以形成图像（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。为了尽量减少吉布斯现象造成的失真，目前的成像方法要么在足够大的傅里叶域中收集数据，要么对数据进行窗口处理以强制人工衰减。可以看出，当前的方法需要收集比必要的更多的数据，或者对感兴趣的恢复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，该建议依赖于非线性，近最优近似的指数来推断可用的傅里叶数据，也在空间中产生近最优的有理表示。该方法不仅提高了奇异点附近的分辨率，达到了接近最优的性能，而且还可以检测到数据中的噪声水平，为信噪分离提供了一种实用的技术。这些新算法已经对一维问题的现有技术产生了重大改进，本项目打算将该方法扩展到二维或三维问题。这样的扩展是非平凡的，因为在一维中使用的数学工具只有有限的用途。逆问题出现在各种各样的科学和工程学科中，作为分析生物或无机标本，分析制造设备的缺陷，在许多其他应用中执行遥感或地球物理勘探的一种方法。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），收集到的数据通过基于特定数学模型的反问题求解算法进行处理。研究人员提出了一种方法，开发和算法实现新的数学模型适用于许多，如果不是全部，这些问题。如此广泛适用性的原因是，在新的数学模型中，感兴趣的函数用接近最优数量的参数表示，从而显著提高了从测量数据中恢复信息的能力。该项目的数学意义和实际意义及其挑战在于将研究人员在一维上开发的方法扩展到多维。由于无损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发挥着至关重要的作用，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.