Signal Recovery from Highly Incomplete Data

从高度不完整的数据中恢复信号

• 批准号: 0515362
• 负责人: Emmanuel Candes
• 金额: $ 30万
• 依托单位: California Institute of Technology
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-05-01 至 2008-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0515362&HistoricalAwards=false
• 关键词: Signal; Recovery; Highly; Incomplete; Data

A fundamental problem in science and technology concerns the recovery of an object---a digital signal or image---from incomplete measurements. The examples of such situations are numerous, ranging from the sampling of continuous signals in signal processing, to the measurement of the two-dimensional frequency spectrum of an image as in biomedical imaging. In Magnetic Resonance Imaging (MRI) for instance, one would like to reconstruct high-resolution images from heavily undersampled frequency data as this would allow image acquisition speeds far beyond those offered by current technologies. In all these applications, there are many more unknown signal values than available observations, a situation which a priori seems desperately hopeless.In many instances, the object we wish to recover is known to be structured in the sense that it is sparse or compressible. This means that the unknown object depends upon a smaller number of unknown parameters, with only a few significant entries in some fixed representation. This premise radically changes the problem, making the search for solutions feasible. The research involves a systematic effort to exploit and extend a mathematical breakthrough which shows that it is surprisingly possible to reconstruct such signalsaccurately, and sometimes even exactly, from a limited number of measurements. There are three main outcomes: the development of a coherent and comprehensive knowledge of what can and cannot beexpected from reconstruction strategies based upon incomplete information; the development of flexible and convenient algorithms able to handle large scale problems; the deployment of the resulting new concepts and tools into targeted applications. The initial applicative focus is in the field of Magnetic Resonance angiography, and on the design of a brand new generation of encoding schemes.

科学技术中的一个基本问题涉及对象的恢复 - 数字信号或图像 - 从不完整的测量中。这种情况的示例很多，从信号处理中的连续信号采样到图像的二维频率频谱的测量，如生物医学成像中。例如，在磁共振成像（MRI）中，人们希望重建来自重采样频率数据的高分辨率图像，因为这将允许图像采集速度远远超出了当前技术提供的速度。 在所有这些应用中，与可用的观察值相比，信号值更多，这种情况似乎是绝望的。在许多情况下，我们希望恢复的对象在稀疏或可压缩的意义上是构造的。这意味着未知对象取决于较小数量的未知参数，在某些固定表示中只有少数重要条目。 这个前提从根本上改变了问题，从而搜索解决方案可行。这项研究涉及系统的努力来利用和扩展数学突破，这表明可以从有限数量的测量值中重建此类信号，有时甚至完全是确切的。 有三个主要结果：开发基于不完整信息的重建策略可以证明什么和不能指出的一致而全面的知识；能够处理大规模问题的灵活和方便算法的发展；将产生的新概念和工具部署到目标应用程序中。 最初的应用重点是磁共振血管造影领域以及全新的编码方案的设计。