Latest views of the sparse Radon transform
Latest views of the sparse Radon transform
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DOI：
10.1190/1.1543224
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发表时间：
2003
期刊：
影响因子：
3.3
通讯作者：
D. Trad;T. Ulrych;M. Sacchi
中科院分区：
文献类型：
--
作者：
D. Trad;T. Ulrych;M. Sacchi
Radon变换（RT）遭受的典型问题的分辨率损失和混叠所产生的不完整的信息，包括有限的孔径和离散化的结果。Radon域中的稀疏性是提供这种缺失信息的有效且有用的标准，在某种程度上相当于假设已知和未知（缺失）数据之间的过渡中的平滑幅度变化。在尊重数据的同时应用该约束可能成为常规地震处理的严重挑战，因为通常每个公共中点可用的处理时间非常有限。为了开发鲁棒、易于使用且灵活地适应不同问题的方法，我们必须注意各种算法、运算符设计以及负责解的正则化的超参数估计。在本文中，我们讨论了快速实现几个品种的RT在时域和频域。一个迭代共轭梯度算法与快速傅立叶变换乘法在所有情况下使用。为了保持迭代子空间方法通过迭代次数正则化解的重要性质，将模型权重并入到算子中。事实证明，这一点特别重要，并且可以根据加权变换的奇异向量来理解。根据子空间的一般交叉验证准则停止迭代算法。我们将这个想法应用到几个已知的实现和比较结果，以便更好地了解这些算法之间的差异和优点。
The Radon transform (RT) suffers from the typical problems of loss of resolution and aliasing that arise as a consequence of incomplete information, including limited aperture and discretization. Sparseness in the Radon domain is a valid and useful criterion for supplying this missing information, equivalent somehow to assuming smooth amplitude variation in the transition between known and unknown (missing) data. Applying this constraint while honoring the data can become a serious challenge for routine seismic processing because of the very limited processing time available, in general, per common midpoint. To develop methods that are robust, easy to use and flexible to adapt to different problems we have to pay attention to a variety of algorithms, operator design, and estimation of the hyperparameters that are responsible for the regularization of the solution. In this paper, we discuss fast implementations for several varieties of RT in the time and frequency domains. An iterative conjugate gradient algorithm with fast Fourier transform multiplication is used in all cases. To preserve the important property of iterative subspace methods of regularizing the solution by the number of iterations, the model weights are incorporated into the operators. This turns out to be of particular importance, and it can be understood in terms of the singular vectors of the weighted transform. The iterative algorithm is stopped according to a general cross validation criterion for subspaces. We apply this idea to several known implementations and compare results in order to better understand differences between, and merits of, these algorithms.