On optimal wavelet reconstructions from Fourier samples: linearity and universality of the stable sampling rate
On optimal wavelet reconstructions from Fourier samples: linearity and universality of the stable sampling rate
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DOI：
10.1016/j.acha.2013.07.001
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发表时间：
2012-08
期刊：
影响因子：
--
通讯作者：
B. Adcock;A. Hansen;C. Poon
中科院分区：
文献类型：
--
作者：
B. Adcock;A. Hansen;C. Poon
在本文中，我们研究了从傅里叶样本计算紧支持函数的小波系数的问题。为此，我们使用最近引入的广义抽样框架。我们的第一个结果表明，只要傅里叶样本的数量随着恢复的小波系数的数量线性增长，使用广义采样就可以获得稳定且准确的重建。对于 Daubechies 小波类，我们导出了精确的比例常数。我们的第二个结果涉及该问题的广义采样的最优性。在一些温和的假设下，我们表明广义抽样在近似质量方面的表现不能超过一个常数因子。此外，对于一类所谓的完美方法，任何将采样率降低到某个临界阈值以下的尝试都必然会导致指数病态。因此，广义抽样为这个问题提供了近乎最优的解决方案。
In this paper we study the problem of computing wavelet coefficients of compactly supported functions from their Fourier samples. For this, we use the recently introduced framework of generalized sampling. Our first result demonstrates that using generalized sampling one obtains a stable and accurate reconstruction, provided the number of Fourier samples grows linearly in the number of wavelet coefficients recovered. For the class of Daubechies wavelets we derive the exact constant of proportionality.Our second result concerns the optimality of generalized sampling for this problem. Under some mild assumptions we show that generalized sampling cannot be outperformed in terms of approximation quality by more than a constant factor. Moreover, for the class of so-called perfect methods, any attempt to lower the sampling ratio below a certain critical threshold necessarily results in exponential ill-conditioning. Thus generalized sampling provides a nearly-optimal solution to this problem.