Signal Recovery from Highly Incomplete Data

科学和技术中的一个基本问题是从不完整的测量中恢复物体-数字信号或图像。这种情况的例子很多，从信号处理中连续信号的采样到生物医学成像中图像的二维频谱的测量。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重建高分辨率图像，因为这将允许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，有更多的未知信号值比可用的观测，先验的情况下，似乎绝望绝望。在许多情况下，我们希望恢复的对象是已知的，在这个意义上，它是稀疏或可压缩的结构。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少数有效条目。 这个前提从根本上改变了问题，使寻找解决方案变得可行。该研究涉及系统的努力，以利用和扩展数学突破，这表明它是令人惊讶的可能性，以重建这样的信号准确，有时甚至是完全，从有限的测量数量。 有三个主要成果：发展一个连贯和全面的知识，什么可以和不可以从基于不完整信息的重建策略中预期;发展灵活方便的算法，能够处理大规模的问题;部署由此产生的新概念和工具到有针对性的应用。 最初的应用重点是磁共振血管造影领域，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.