SUPERRESOLUTION VIA SPARSITY CONSTRAINTS
SUPERRESOLUTION VIA SPARSITY CONSTRAINTS
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DOI：
10.1137/0523074
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发表时间：
1992-09-01
影响因子：
2
通讯作者：
DONOHO, DL
中科院分区：
文献类型：
--
作者：
DONOHO, DL
考虑恢复跨度为DELTA的网格上支持的测量μ的问题，当测量仅在小于或等于Ω的频率\Ω\处关于傅立叶变换μ（Ω）可用时。如果Ω比奈奎斯特频率pi/DELTA小得多，并且测量有噪声，则通常不可能稳定地恢复mu。在本文中，它表明，如果，此外，我们知道，测量μ满足一定的稀疏性约束，那么稳定的恢复是可能的。假设一个集合的瑞利指数小于或等于R，如果在任何长度为4-π/Ω的区间内。最多有R个元素。实际上，如果μ的（未知）支持先验地已知具有至多R的瑞利指数，则稳定恢复是可能的，其中稳定系数随着Δ-> 0以至多类似于Δ-2R-1的方式增长。这一结果验证了某些实际的努力，在光谱学，地震勘探和天文学，提供超分辨率的重建施加支持限制。结果相当于不平等的插值指数型的整个函数的值在特殊的点集是不规则的，但内部平衡，一致离散，并一致的密度1。
Consider the problem of recovering a measure-mu supported on a lattice of span DELTA, when measurements are only available concerning the Fourier Transform mu(omega) at frequencies \omega\ less-than-or-equal-to OMEGA. If OMEGA is much smaller than the Nyquist frequency pi/DELTA and the measurements are noisy, then, in general, stable recovery of mu is impossible. In this paper it is shown that if, in addition, we know that the measure-mu satisfies certain sparsity constraints, then stable recovery is possible. Say that a set has Rayleigh index less than or equal to R if in any interval of length 4-pi/OMEGA . R there are at most R elements. Indeed, if the (unknown) support of mu is known, a priori, to have Rayleigh index at most R, then stable recovery is possible with a stability coefficient that grows at most like DELTA-2R-1 as DELTA --> 0. This result validates certain practical efforts, in spectroscopy, seismic prospecting, and astronomy, to provide superresolution by imposing support limitations in reconstruction. The results amount to inequalities for interpolation of entire functions of exponential type from values at special point sets which are irregular, yet internally balanced, uniformly discrete, and of uniform density 1.