Compressive sensing: Theory, algorithms and applications
Compressive sensing: Theory, algorithms and applications
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DOI：
10.1109/meco.2015.7181858
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发表时间：
2015-06
期刊：
影响因子：
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通讯作者：
S. Stankovic
中科院分区：
文献类型：
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作者：
S. Stankovic
在技术膨胀的时代，数字设备被用来实现高分辨率的信号采集，产生大量的数字数据。这是处理雷达信号、多媒体信号、医学和生物医学数据等的传感系统中的常见问题。采集过程根据Shanon-Nyquist定理进行，采样率通常至少是最高信号频率的两倍。因此，为了应对存储、传输和计算的挑战，这些获取的数据被压缩到可接受的质量。显然，通过使用压缩，我们能够显著减少数字数据量，保持高水平的解码信号质量。这种简化形式的数据在硬件和基础设施要求方面产生了积极影响，这打开了无数的应用程序。然而，数据采集过程在资源（例如传感器技术）和采集时间方面仍然要求很高。近年来，压缩感知方法已经得到了广泛的发展，其思想是克服传统采样理论的限制，并在感知过程中应用压缩的概念。在这个意义上，已经做出了大量努力来开发允许使用少得多的样本以压缩形式对数据进行采样的方法。在本教程中，将考虑压缩感知的基本概念。压缩感知理论指出，如果信号在某个变换域中具有稀疏（简洁）表示，则可以仅使用一小组随机获取的样本来重建信号。换句话说，由于大多数现实生活中的信号具有仅具有少量非零系数的可压缩表示，因此可以使用比传统采样定理所需的少得多的样本来重构信号。除了信号稀疏性之外，另一个重要条件是测量过程和稀疏基之间的不相干性。更多的不一致意味着更少的测量。为此，这些重要的性质：信号稀疏性，限制等距性质和非相干性将进行讨论。全信号重构问题是一个利用稀疏约束求解待定线性方程组的问题。存在可用于此目的的若干标准算法。例如，约束l1-最小化已被用作寻找稀疏解的第一种方法之一，它被称为基追踪。替代方法被称为贪婪算法，其中最流行的是迭代正交匹配追踪（具有各种修改）。另一方面，最新的CS重建方法是基于噪声模型，描述了影响稀疏域的缺失样本效应。该算法在一次或几次迭代过程中从各种非信号分量中区分和重构信号。本文综述的另一种方法是从稳健估计理论的假设出发的。它引入了一个新的最小化度量称为广义偏差，它被定义为总误差的p次范数，其中p来自噪声分布。因此，该方法有效地降低了噪声对重建性能的影响。在近似稀疏的条件下，定义了基于梯度的算法，以适应各种类型的信号和变换域。该方法对于1D和2D数据都是有效的，包括要求很高的自然图像。最后，整个概念的推广可以扩展到其他一些领域，在CS文献中不是那么典型，但在真实的应用中对信号非常重要，如多项式傅立叶和Hermite变换域。
In the era of technology expansion, the digital devices are made to achieve high resolution signal acquisition, producing a large amount of digital data. This is a common issue in sensing systems dealing with radar signals, multimedia signals, medical and biomedical data, etc. The acquisition process is done according to the Shanon-Nyquist theorem with the sampling rate which is usually at least twice the highest signal frequency. Consequently, in order to respond to the storage, transmission and computational challenges, such acquired data are compressed up to the acceptable quality. Obviously by using compression, we are able to significantly reduce the amount of digital data keeping a high level of decoded signal quality. This reduced form of data has a positive impact in the sense of hardware and infrastructure requirements, which opened an uncountable number of applications. However, the data acquisition process is still demanding in terms of resources (e.g. sensors technology) and acquisition time. In recent years, the compressive sensing approaches have been intensively developed with the idea to overcome the limits of traditional sampling theory and to apply a concept of compression during the sensing procedure. In that sense, significant efforts have been done toward the development of methods that would allow to sample data in the compressed form using much lower number of samples. In this tutorial the basic concept of compressive sensing will be considered. The compressive sensing theory states that the signal can be reconstructed using just a small set of randomly acquired samples if it has a sparse (concise) representation in certain transform domain. In other words, since most of the real-life signals have compressible representation with just a small number of non-zero coefficients, signals can be reconstructed using much fewer samples than required by the traditional sampling theorem. Another important condition, apart from the signal sparsity, is incoherence between the measurement process and the sparsity basis. More incoherence means fewer measurements. To that end, these important properties: signal sparsity, restricted isometric property and incoherence will be discussed. The full signal reconstruction is formulated as a problem of solving undetermined system of linear equations using sparseness constraints. There are several standard algorithms that could be employed for this purpose. For instance, the constrained l1-minimization has been used as one among first approaches for finding the sparse solutions and it is known as basis pursuit. Alternative approaches are called greedy algorithms and among them the most popular is the iterative Orthogonal Matching Pursuit (with a variety of modifications). On the other side, the most recent method for CS reconstruction is based on the noise model describing the missing samples effect influencing sparsity domain. The algorithm distinguishes and reconstructs signal among various non-signal components in a single or few-iteration process. Another method reviewed here, is derived from the postulates of robust estimation theory. It introduces a new minimization metrics called generalized deviations, which is defined as the p-th norm of the total error, where p arises from the noise distribution. Therefore, this approach effectively reduces the influence of noise to the reconstruction performance. In the conditions of approximate sparsity, the gradient based algorithm is defined to comply with various types of signals and transform domains. This method is efficient for both 1D and 2D data, including the highly demanding natural images. Finally, the generalization of the entire concept can be extended for some other domains, not so typical in CS literature but quite important for signals in real applications, such as polynomial Fourier and Hermite transform domain.