Wavelet footprints: Theory, algorithms, and applications
Wavelet footprints: Theory, algorithms, and applications
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DOI：
10.1109/tsp.2003.810296
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发表时间：
2003-05-01
影响因子：
5.4
通讯作者：
Vetterli, M
中科院分区：
文献类型：
--
作者：
Dragotti, PL;Vetterli, M
近年来，基于小波的算法在不同的信号处理中都取得了成功。任务。小波变换是一个强大的工具，因为它能够用很少的变换系数来表示信号的暂态和平稳行为。不连续性通常带有相关的信号信息，因此，它们是需要分析的关键部分。在本文中，我们研究了由不连续产生的小波系数的跨尺度相关性。我们首先证明，任何分段平滑的信号都可以表示为一个和。分段多项式信号和一致光滑残差的关系(见第二节中的定理1)。然后我们引入了足迹的概念，它是尺度空间向量，准确地模拟分段多项式信号中的不连续性。我们证明了足迹形成了一个过完备的字典，并开发了高效和健壮的算法来寻找关于足迹的分段多项式函数的精确表示。这也导致了分段光滑函数的有效逼近。最后，我们将重点放在应用上，并证明了基于足迹的算法在去噪、压缩和(非盲)反卷积等不同应用中的性能优于标准小波方法。在压缩的情况下，我们还证明了在高速率下，基于足迹的算法获得了最佳性能(见第五节中的定理3)。
In recent years', wavelet-based algorithms have been successful in different signal processing. tasks. The wavelet transform is a powerful tool because it manages to represent both transient and stationary behaviors of a signal with few transform coefficients. Discontinuities often carry relevant signal information, and therefore, they represent a critical part to analyze. In this paper, we study the dependency across scales of the wavelet coefficients generated by discontinuities. We start by showing that any piecewise smooth signal can be expressed as a sum. of a piecewise polynomial signal and a uniformly smooth residual (see Theorem 1 in Section II). We then introduce the notion of footprints, which are scale space vectors that model discontinuities in piecewise polynomial signals exactly. We show that footprints form an overcomplete dictionary and develop efficient and robust algorithms to find the exact representation of a piecewise polynomial function in terms of footprints. This also leads to efficient approximation of piecewise smooth functions. Finally, we focus on applications and show that algorithms based on footprints outperform standard wavelet methods in different applications such as denoising, compression, and (nonblind) deconvolution. In the case of compression, we also prove that at high rates, footprint-based algorithms attain optimal performance (see Theorem 3 in Section V).