Towards a Mathematical Theory of Super- resolution
Towards a Mathematical Theory of Super- resolution
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DOI：
10.1002/cpa.21455
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发表时间：
2014-06-01
影响因子：
3
通讯作者：
Fernandez-Granda, Carlos
中科院分区：
文献类型：
--
作者：
Candes, Emmanuel J.;Fernandez-Granda, Carlos
本文提出了一种超分辨率的数学理论。从广义上讲，超分辨率是从光谱低端的样本中恢复物体光谱高端的精细细节的问题。假设我们有许多点源在[0，1]中的未知位置，并且具有未知的复值振幅。我们只观察到这个物体的傅立叶样本，直到频率截止f（c）。我们表明，一个可以超级解决这些点源具有无限的精度，即，通过解决一个简单的凸优化问题来恢复精确的位置和幅度，该问题基本上可以重新表述为半定规划。如果源之间的距离至少为2/f（c），则这一点成立。这个结果扩展到更高的维度和其他模型。例如，在一维中，也可以通过以无限精度解析不连续点来恢复分段光滑函数。我们还表明，理论和方法是强大的噪声。特别是，在离散设置中，我们开发了一些理论结果，解释如何的超分辨信号的准确性预计会降低时，噪声水平和超分辨率因子的变化。(c)2014年威利期刊公司
This paper develops a mathematical theory of super-resolution. Broadly speaking, super-resolution is the problem of recovering the fine details of an objectthe high end of its spectrumfrom coarse scale information onlyfrom samples at the low end of the spectrum. Suppose we have many point sources at unknown locations in [0,1] and with unknown complex-valued amplitudes. We only observe Fourier samples of this object up to a frequency cutoff f(c). We show that one can super-resolve these point sources with infinite precisioni.e., recover the exact locations and amplitudesby solving a simple convex optimization problem, which can essentially be reformulated as a semidefinite program. This holds provided that the distance between sources is at least 2/f(c). This result extends to higher dimensions and other models. In one dimension, for instance, it is possible to recover a piecewise smooth function by resolving the discontinuity points with infinite precision as well. We also show that the theory and methods are robust to noise. In particular, in the discrete setting we develop some theoretical results explaining how the accuracy of the super-resolved signal is expected to degrade when both the noise level and the super-resolution factor vary. (c) 2014 Wiley Periodicals, Inc.