Latest views of the sparse Radon transform

Abstract

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.

Simply denoise: Wavefield reconstruction via jittered undersampling

“…The objective of this paper is to demonstrate that irregular/random undersampling is not a drawback for particular transform-based interpolation methods and for many other advanced processing algorithms as was already observed by other authors (Zhou and Schuster, 1995;Sun et al, 1997;Trad and Ulrych, 1999;Xu et al, 2005;Abma and Kabir, 2006;Zwartjes and Sacchi, 2007). We explain why random undersampling is an advantage and how it can be used to our benefit when designing coarse sampling schemes.…”

“…As mentioned before, CS relies on a sparsifying transform for the to-berecovered signal and uses this sparsity prior to compensate for the undersampling during the recovery process. For the reconstruction of wavefields in the Fourier (Sacchi et al, 1998;Xu et al, 2005;Zwartjes and Sacchi, 2007), Radon (Trad et al, 2003), and curvelet (Hennenfent and Herrmann, 2005; domains, sparsity promotion is a well-established technique documented in the geophysical literature. The main contribution of CS is the new light shed on the favorable recovery conditions.…”

“…The to-be-recovered signal f 0 is given by f 0 = S H x 0 and the observations y are obtained by undersampling f 0 regularly, randomly according to a discrete uniform distribution, or optimally-jittered, i.e., ξ = γ. Finally, the estimated spectrumx of f 0 is obtained by solving equation 3 with the Spectral Projected Gradient for 1 solver (SPGL1 -van den Berg and Friedlander, 2007). Keep in mind that the number k of nonzero entries of x 0 is not known a priori.…”

“…Wavefield-operator-based methods represent another type of interpolation approaches that explicitly include wave propagation (Canning and Gardner, 1996;Biondi et al, 1998;Stolt, 2002). Finally, transform-based methods also provide efficient algorithms for seismic data regularization (Sacchi et al, 1998;Trad et al, 2003;Zwartjes and Sacchi, 2007;. However, for irregularly-sampled data, e.g., binned data with some of the bins that are empty, or data that are continuous random undersampled, the performance of most of the aforementioned interpolation methods deteriorates.…”

Fourier reconstruction of marine-streamer data in four spatial coordinates

“…Transformation-based reconstruction requires estimation of the coefficients of a particular transform ͑Fourier, Radon, wavelet, etc.͒ that synthesize the signal on the nonuniform grid, either through inversion ͑Hugonnet and Canadas, 1997; Duijndam et al, 1999b;Trad et al, 2003͒ or through an iterative approach ͑Kabir and Verschuur, 1995; Xu et al, 2005͒. These methods are fast when the transform can be computed efficiently, require little user input, and can handle both nonuniform and uniform grids.…”