A simple shearlet-based 2D Radon inversion with an application to computed tomography

Low RCS targets detection is a challenging problem for marine radar in sea clutter. In this paper, a new detection algorithm is developed based on Radon and inverse Radon transform which accounts for the linear feature of the targets in sea clutter. After Radon transform the SCR is increased and clutter can be suppressed more easily in Radon domain. Then the data in Radon domain can be inverse Radon transformed into time-range domain, and the target track can be detected. The simulation results show that the proposed algorithm makes the detection of very small targets possible.

Seismic waves split in an anisotropic medium, instead of rotating horizontal component to principal direction, Radon-Fourier is derived to observe the signature of shear wave reflection. Synthetic model with fracture is built and discretized using finite difference scheme for spatial and time domain. Common depth point (CDP) with single shot gives traces and automatic gain is preprocessed before Radon Transform (RT), a filtering technique gives radon domain. It makes easier to observe fractures at specific incidence and improves its quality in some way by removing the noise. A comparison of synthetic data and BF-data is performed on the basis of root means square error (RMS) values. The RMS error is minimum at the 10 th trace in radon domain.

Summary The Radon transform has been widely used in seismic data regularization. Since the standard Radon transform is expressed as the summation of seismic data along the trajectories of different curvature radius, the AVO characteristics cannot be taken into account during the reconstructing processing. In this paper, we present a new 3D multiple frequency high-order parabolic Radon transform that combines parabolic Radon transform with orthogonal polynomial fitting techniques. The proposed method expresses the new transform as a problem of solving a multiple frequency matrix equations in frequency domain. We use SVD to solve the matrix equations for high resolution. The full 3D synthetic examples demonstrate that, compared to the traditional Radon transform, our method performs better in restoring missed traces in time-space domain, and has abilities in reconstructing seismic data with true amplitude and restoring missed frequency-space slices.

X-ray computed tomography (CT) reconstruction with sparse projection views was proposed to reduce both the radiation dose and scan time. However, lacking of sufficient projection views may lead to severe artifacts for analytical reconstruction method such as the filtered back projection (FBP). Although the projection data is incomplete, we can generate the full-sampling system matrices according to the sufficient-sampling conditions [5]. Thus, we propose a novel iterative reconstruction model to fit the target images and the corresponding high resolution measurements in Radon domain by the full-sampling system matrices. Our proposed model is solved by the learned alternating minimization method, which accounts for a forward operator in deep neural network by the unrolling strategy. Numerical results demonstrate that the proposed approach outperforms some latest learning based reconstruction methods for the sparse-view CT problems.

Multiple attenuation based on connected-component analysis and high-resolution parabolic Radon transform

The time-invariant Radon transform (RT) is commonly used to regularize and interpolate sparsely sampled or irregularly acquired prestack seismic data. The sparseness of the Radon model significantly influences the results of regularization. We have developed an effective and efficient method for the regularization and interpolation of 2D as well as 3D prestack seismic data. We used an accelerated sparse time-invariant RT in the mixed frequency-time domain to improve the performance of RT-based seismic data regularization. This 2D sparse RT incorporated the iterative 2D model shrinkage algorithm instead of the traditional iteratively reweighted least-squares (IRLS) algorithm in the time domain, and we computed the forward and inverse RTs in the frequency domain to solve the sparse inverse problem, which dramatically reduced the computational cost while obtaining a high-resolution result. The 2D synthetic and real data examples revealed that our 2D approach can better interpolate beyond aliasing a 2D prestack seismic record that contains a large gap, compared with the least-squares-based RT and the frequency-domain sparse RT methods. To extend the 2D technique to 3D more efficiently, we first formulate the 3D RT as a problem of solving a special matrix equation. Next, we use the iterative 3D model shrinkage algorithm to obtain a high-resolution 3D Radon model. The proposed 3D sparse RT method can be applied in the regularization of 3D prestack gathers, such as in the cable interpolation in a 3D marine survey. We achieved robustness and effectiveness with our 3D approach with successful applications to 3D synthetic and real data.

For parallel beam geometry the Fourier reconstruction works via the Fourier slice theorem (or central slice theorem, projection slice theorem). For fan beam situation, Fourier slice can be extended to a generalized Fourier slice theorem (GFST) for fan-beam image reconstruction. We have briefly introduced this method in a conference. This paper reintroduces the GFST method for fan beam geometry in details. The GFST method can be described as following: the Fourier plane is filled by adding up the contributions from all fanbeam projections individually; thereby the values in the Fourier plane are directly calculated for Cartesian coordinates such avoiding the interpolation from polar to Cartesian coordinates in the Fourier domain; inverse fast Fourier transform is applied to the image in Fourier plane and leads to a reconstructed image in spacial domain. The reconstructed image is compared between the result of the GFST method and the result from the filtered backprojection (FBP) method. The major differences of the GFST and the FBP methods are: (1) The interpolation process are at different data sets. The interpolation of the GFST method is at projection data. The interpolation of the FBP method is at filtered projection data. (2) The filtering process are done in different places. The filtering process of the GFST is at Fourier domain. The filtering process of the FBP method is the ramp filter which is done at projections. The resolution of ramp filter is variable with different location but the filter in the Fourier domain lead to resolution invariable with location. One advantage of the GFST method over the FBP method is in short scan situation, an exact solution can be obtained with the GFST method, but it can not be obtained with the FBP method. The calculation of both the GFST and the FBP methods are at O(N<formula>^3</formula>), where N is the number of pixel in one dimension.

A generalized Peano Kernel Theorem for distributions of exponential decay

Des que estava cursant els primers anys de la Llicenciatura en Matemtiques he tingut la illusi de fer una tesi doctoral en aquesta disciplina, com deia Gauss, la reina de les cincies. Segurament aquesta illusi ha sorgit de la curiositat que les Matemtiques i la cincia

We consider a model problem of recovering a function $f(x_1,x_2)$ from noisy Radon data. The function $f$ to be recovered is assumed smooth apart from a discontinuity along a $C^2$ curve, that is, an edge. We use the continuum white-noise model, with noise level $\\varepsilon$.\n¶ Traditional linear methods for solving such inverse problems behave poorly in the presence of edges. Qualitatively, the reconstructions are blurred near the edges; quantitatively, they give in our model mean squared errors (MSEs) that tend to zero with noise level $\\varepsilon$ only as $O(\\varepsilon^{1/2})$ as $\\varepsilon\\to 0$. A recent innovation--nonlinear shrinkage in the wavelet domain--visually improves edge sharpness and improves MSE convergence to $O(\\varepsilon^{2/3})$. However, as we show here, this rate is not optimal.\n¶ In fact, essentially optimal performance is obtained by deploying the recently-introduced tight frames of curvelets in this setting. Curvelets are smooth, highly anisotropic elements ideally suited for detecting and synthesizing curved edges. To deploy them in the Radon setting, we construct a curvelet-based biorthogonal decomposition of the Radon operator and build "curvelet shrinkage" estimators based on thresholding of the noisy curvelet coefficients. In effect, the estimator detects edges at certain locations and orientations in the Radon domain and automatically synthesizes edges at corresponding locations and directions in the original domain.\n¶ We prove that the curvelet shrinkage can be tuned so that the estimator will attain, within logarithmic factors, the MSE $O(\\varepsilon^{4/5})$ as noise level $\\varepsilon\\to 0$. This rate of convergence holds uniformly over a class of functions which are $C^2$ except for discontinuities along $C^2$ curves, and (except for log terms) is the minimax rate for that class. Our approach is an instance of a general strategy which should apply in other inverse problems; we sketch a deconvolution example.

Sampling theorem for bandlimited Hardy space functions generated by Regge problem

Computationally efficient filtered-backprojection algorithm for tomographic image reconstruction using Walsh transform

The redatuming approach, often referred to as up-down deconvolution, is well-known and is applied to remove water-layer and source-signature effects in seabed seismic surveys. The upgoing wavefield can be expressed as the multidimensional convolution of the downgoing wavefield with the earth’s reflectivity. Consequently, deconvolving the downgoing wavefield from the upgoing wavefield gives us the earth’s reflectivity response. The deconvolution process requires solving a multidimensional integral equation but, in a laterally invariant medium, after that wavefields are decomposed into plane-wave components, deconvolution can be enormously simplified if performed as a spectral division in the Fourier or Radon domain. It has been experimentally observed that deconvolution carried out one plane-wave component at a time gives good results, even in the presence of complex subsurface structures, provided that the seabed is relatively flat. When this geological condition is not satisfied, the same problem can be formulated in terms of interferometric redatuming using multidimensional deconvolution (MDD), in which the integral equation solution is achieved by introducing the point-spread function concept. We have developed a methodology based on numerical simulations to determine when the integral equations associated with the problem of up-down deconvolution can be solved under the assumption of shift-invariant wavefields and when it requires MDD. In the latter case, we have developed a regularized inverse procedure that mitigates the numerical problems due to the typically ill-posed nature of the inversion and that, combined with an interpolation strategy for the downgoing, enables the application of MDD within the range of sampling scenarios considered so far. We apply this methodology to synthetic data, and we discuss the potential to extend up-down deconvolution to a broader range of geological conditions.

This chapter constructs a functional analytic machinery that is adapted to the analysis of the hypoelliptic Laplacian ℒA,bX. The analysis of the hypoelliptic Laplacian essentially consists in the construction of Sobolev spaces on which the operators ℒA,bX act as unbounded operators, and in the proof of regularizing properties of their resolvents and of their heat operators. The heat operators are shown to be given by smooth kernels. To make the analysis easier, the chapter first works with a scalar hypoelliptic operator AbX. This operator does not contain a quartic term. The results on AbX then easily extend to a scalar operator AbX acting over X with circumflex, which also does not contain the quartic term. A scalar operator AbX on ̂X containing the quartic term is introduced. Finally, the chapter extends the analysis to the operator ℒA,bX.

Methods for reconstructing tomographic images based on the inversion of the Radon transform are used in problems arising in medicine, biology, astronomy, and many other fields. In the presence of noise in the projection data, as a rule, it is necessary to apply regularization methods. Recently, methods of the threshold processing of wavelet expansion coefficients have become popular. The analysis of errors of these methods is an important practical task, since it makes it possible to assess the quality of both the methods themselves and the equipment used. When using threshold processing, it is usually assumed that the number of expansion coefficients is fixed, and the noise distribution is Gaussian. This model is well studied in the literature, and the optimal values of the threshold processing parameters are calculated for different classes of functions. However, in some situations, the amount of data is not known in advance and has to be modeled with a certain random variable. In this paper, we consider a model with a random amount of data containing Gaussian noise, and estimate the order of the mean-square risk with an increasing number of decomposition coefficients.