Weighted frames of exponentials and stable recovery of multidimensional functions from nonuniform Fourier samples
Weighted frames of exponentials and stable recovery of multidimensional functions from nonuniform Fourier samples
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DOI：
10.1016/j.acha.2015.09.006
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发表时间：
2014-05
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影响因子：
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通讯作者：
B. Adcock;M. Gataric;A. Hansen
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文献类型：
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作者：
B. Adcock;M. Gataric;A. Hansen
在本文中，我们考虑了一个紧支持多元函数从其傅立叶变换的非一致点态样本集合中恢复的问题。我们通过加权傅里叶帧的概念来做到这一点。Beurling的一个开创性的结果表明，如果采样点相对分离且密度足够大，就会产生经典的傅立叶框架。然而，这个结果不允许采样点的任意聚类，这是在实践中经常出现的情况。在保持密度条件清晰和尺寸无关的同时，我们的第一个结果去掉了分离条件，并表明仅密度就足够了。然而，这个结果并不能得到帧边界的估计。Gröchenig的已知结果提供了明确的估计，但仅受密度随维度线性恶化的条件的约束。在我们的第二个结果中，我们通过减少维度依赖来改进这些边界。特别地，我们提供了明确的无量纲框架边界，对于包含在球面上的紧支持函数。接下来，我们展示了我们的两个主要结果如何为基于现有广义采样框架的重构算法提供了新的见解，该算法允许从有限的样本集合中在任何特定的基础上进行稳定和准最优重构。最后，我们构建了在实践中经常使用的足够密集的采样方案-抖动，径向和螺旋采样方案-并提供了几个例子来说明我们的方法在这些方案上进行测试时的有效性。
In this paper, we consider the problem of recovering a compactly supported multivariate function from a collection of pointwise samples of its Fourier transform taken nonuniformly. We do this by using the concept of weighted Fourier frames. A seminal result of Beurling shows that sampling points give rise to a classical Fourier frame provided they are relatively separated and of sufficient density. However, this result does not allow for arbitrary clustering of sampling points, as is often the case in practice. Whilst keeping the density condition sharp and dimension independent, our first result removes the separation condition and shows that density alone suffices. However, this result does not lead to estimates for the frame bounds. A known result of Gröchenig provides explicit estimates, but only subject to a density condition that deteriorates linearly with dimension. In our second result we improve these bounds by reducing the dimension dependence. In particular, we provide explicit frame bounds which are dimensionless for functions having compact support contained in a sphere. Next, we demonstrate how our two main results give new insight into a reconstruction algorithm—based on the existing generalized sampling framework—that allows for stable and quasi-optimal reconstruction in any particular basis from a finite collection of samples. Finally, we construct sufficiently dense sampling schemes that are often used in practice—jittered, radial and spiral sampling schemes—and provide several examples illustrating the effectiveness of our approach when tested on these schemes.