Signal Recovery from Highly Incomplete Data

科学和技术中的一个基本问题涉及从未完成的测量中恢复对象-数字信号或图像--。这种情况的例子很多，从信号处理中的连续信号采样，到生物医学成像中的图像二维频谱的测量。例如，在磁共振成像(MRI)中，人们希望从严重欠采样的频率数据中重建高分辨率图像，因为这将使图像采集速度远远超过当前技术所提供的速度。在所有这些应用中，未知信号值比可用观测值多得多，这是先验似乎是绝望的情况。在许多情况下，我们希望恢复的对象是已知的结构，其意义是稀疏或可压缩的。这意味着未知对象依赖于较少数量的未知参数，在某些固定表示中只有几个重要条目。这一前提从根本上改变了问题，使寻找解决方案变得可行。这项研究涉及一项系统的努力，以利用和扩展一项数学突破，该突破表明，从有限数量的测量中准确地、有时甚至精确地重建这种信号是令人惊讶的可能。有三个主要成果：形成一个连贯和全面的知识，了解根据不完全信息制定的重建战略能做什么，不能做什么；开发能够处理大规模问题的灵活方便的算法；将由此产生的新概念和工具部署到有针对性的应用程序中。最初的应用重点是磁共振血管成像领域，以及全新一代编码方案的设计。

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.