Seismic Data Reconstruction via Shearlet-Regularized Directional Inpainting

Transcription

1 Seismic Data Reconstruction via Shearlet-Regularized Directional Inpainting Sören Häuser and Jianwei Ma May 15, 2012 We propose a new method for seismic data reconstruction by directional weighted shearlet-regularized image inpainting. The main idea is the application of a directional and scale dependent thresholding (sdt) to penalize the vertical artifact edges. The alternating direction method of multipliers (ADMM) is used to solve the resulting sparsity-promoting l 1 -norm regularization. Numerical experiments show a very good performance of the proposed method. 1 Introduction In applied geophysics various methods were developed to investigate the subsurface. Due to its ability to produce images down to depths of thousands of meters with a good resolution, the seismic method has become the most commonly used geophysical method in the oil and gas industry. The simplest possible seismic measurement would be a 1D pint measurement with a single source (often referred to as a shot) and receiver, both located in the same place. The results could be displayed as a seismic trace, which is a graph of the signal amplitude against travel time. Much more useful is a 2D measurement, with sources and receivers positioned along a line on the surface. In practice, many receivers regularly spaced along the line are used to record the signal from a series of source points. To avoid information losses, the seismic data should be sampled according to the Shannon-Nyquist criterion. However, seismic data are often sparsely or incompletely sampled along the spatial coordinates, partly caused by surface obstacles, dead trace, no-permit areas and economic constraints. So the reconstruction and interpolation of seismic data is a very important problem in the seismic community. The quality of reconstruction will impact the subsequent seismic processing steps, e.g., denoising, AVO, imaging, etc. Many methods have been proposed for the reconstruction of seismic data. These methods are mainly divided into two categories: wave-equation methods and signal processing methods. The former methods use the simulation of wave propagation to reconstruct seismic data, so Department of Mathematics, University of Kaiserslautern, Kaiserslautern, Germany, haeuser@mathematik.uni-kl.de Institute of Applied Mathematics, Harbin Institute of Technology, Harbin, China, jma@hit.edu.cn 1

2 that they require subsurface information (e.g., velocity distribution in earth medium), see e.g. [21, 26]. Most methods in the signal-processing-based category use sparse transforms such as the Radon transform [13, 28], Fourier transform [22, 29, 17], and curvelet transform [11, 12, 27, 18]. In particular, Herrmann et al. [11, 12, 27] applied the sparsity-promoting l 1 -norm minimization of curvelet coefficients, that leads to much improved results for edge/lineupspreserving reconstruction. It is well known that the curvelet transform sparsely represents seismic data that mainly includes line-like features. Thanks to the theory of compressed sensing, the seismic data can be efficiently reconstructed after random down-sampling (or taking less measurements) using a sparsity-promoting l 1 -norm convex optimization approach (even the sampling ratio is much less than the Shannon-Nyquist sampling ratio). This indicates that one can use less measured data (with more missing traces) to reconstruct the unknown seismic data with high precision. It could reduce the cost of seismic data survey. The method in this paper also follows this line. Instead of the curvelet transform we use the shearlet transform introduced in [14] and described in its fully finite discrete form in [10] in the sparsity-promoting l 1 -norm term. To our best knowledge this is the first time shearlets are used in the context of seismic data. Another popular techniques is the use of prediction filters. For instance, the low-frequency nonaliased components or observed data are used to build antialiasing prediction-error filters, and then the filters are applied to interpolate high-frequency components or missing traces. The prediction-filter method can be implemented in t x domain, f x domain, even curvelet domain, e.g., [25, 20, 4, 15, 18, 18]. Recently, rank-reducing methods are used for the seismic data reconstruction. In [19], Oropeza and Sacchi reorganized the seismic data into a Hankel matrix, and then use multichannel singular spectrum analysis (MSSA) to solve the rank-reduction problem of the Hankel matrix. In [31], Yang, Ma and Osher applied this theory and advanced fast algorithm of matrix completion (MC) to seismic data reconstruction. Many existing fast algorithms of nuclear-norm minimization can be applied in this framework. Recently, the authors [30] extended the lowrank method to a joint l 1 -norm of curvelet coefficient and nuclear-norm based low-rank method for seismic data reconstruction. In this paper, our contribution is two-fold. We will describe the missing traces recovery as an image inpainting problem using shearlets in the l 1 -sparsity regularization term. The regularization parameter Λ is usually chosen only scale dependent (st). We propose the additional use of the directional information given by the shearlet coefficients (scale-directional, sdt). In particular we use different weights for the horizontal and for the vertical directions to reduce visible artifacts. This can be done efficiently using the simple data structure provided by the recently published Fast Finite Shearlet Transform (FFST) [10, 9]. The organization of the paper is as follows: In Section 2 we introduce the shearlet transform and the basic steps for discretization. For our computations we will use FFST, an implementation of a translation invariant finite discrete shearlet transform discretized on the full grid. This is different to other implementations, see, e.g., ShearLab [24]. In Section 3 we apply the convex inpainting model to the missing trace reconstruction problem and show how to incorporate the shearlet transform into this model. We explain how the alternating direction method of multipliers (ADMM) can be used to find the minimizer of the functional exploiting the 2

3 Parseval-frame property of the discrete shearlet transform. In the last Section 4 we present the numerical results for two different data sets. 2 Shearlets and Shearlet transform In this first section we sketch the mathematical background of the shearlet transform that will be used as part of our inpainting algorithm. Similar to curvelets [2] shearlets were developed to provide a directional multiscale decomposition of images. However, the rotation used for curvelets to obtain the directional selectivity is replaced by a shear matrix S s that reads ( ) 1 s S s =, s R. 0 1 Note that the shearing is well suited for the discrete setting compared to the rotation used for the curvelet transform. Additionally shearlets stem from a square-integrable group representation, the so called shearlet group [5]. Similar to curvelets a shearlet is oriented in time (or space) domain and supported in a rectangle of width 2 2j and length 2 j, i.e. it fulfills the parabolic scaling relation width length 2. The shearlet ψ j,k,m emerges by dilation j, shear k and translation m from the mother shearlet ψ as (( 2 2j k2 ψ j,k,m (x) = ψ j ) ) 0 2 j (x m) We define the mother shearlet ψ L 2 (R 2 ) in the Fourier domain by ( ) ˆψ(ω) = ˆψ 1 (ω 1 ) ˆψ ω2 2 ω 1 where ψ 1 is usually the Meyer wavelet and ˆψ 2 is a bump function, for details see [9]. The dilated, sheared and translated shearlets in the Fourier domain are ˆψ j,k,m (ω) = ˆψ ( 1 4 j ) ω 1 ˆψ2 (2 j ω ) 2 + k e 2πi ω,m. ω 1 Fig. 1 shows a dilated and sheared shearlet in Fourier domain (left) and in time domain (right). The shearlet transform SH(f) of a function f L 2 (R 2 ) for a certain scale, shear and translation is the inner product between the function f and the respective shearlet ψ j,k,m, i.e., SH(f)(j, k, m) := f, ψ j,k,m = f(x)ψ j,k,m (x)dx. R 2 To get a good directional selectivity both from the horizontal and vertical point of view shearlets on the cone were introduced in [14]. We define the horizontal and the vertical cone by C h := {(ω 1, ω 2 ) R 2 : ω 1 1 2, ω 2 < ω 1 }, C v := {(ω 1, ω 2 ) R 2 : ω 2 1 2, ω 2 > ω 1 }, 3

4 (a) Shearlet in Fourier domain for j = 1 and k = 1 (b) Same shearlet in time domain (zoomed) Figure 1: Shearlet in Fourier and time domain. respectively. In the following, we consider digital images as functions sampled on the grid 1 N I := 1 N {(m 1, m 2 ) : m i = 0,..., N 1, i = 1, 2}, where we assume quadratic images for simplicity. Further, let j 0 := 1 2 log 2 N be the number of considered scales. To obtain a finite discrete shearlet transform, we use as finite discrete scaling, shear and translation parameters j = 0,..., j 0 1, 2 j k 2 j, t m := m N, m I. and discretize the frequency plane on Ω := { (ω 1, ω 2 ) : ω i = N 2,..., N } 2 1, i = 1, 2. We define the finite discrete shearlet transform as f, φ m for κ = 0, SH(f)(κ, j, k, m) = c(κ, j, k, m) := f, ψj,k,m κ for κ {h, v}, (1) f, ψ h v j,k,m for κ =, k = 2j. where j = 0,..., j 0 1, 2 j + 1 k 2 j 1 and m I if not stated in another way. The shearlet transform can be efficiently realized by applying the fft2 and its inverse ifft2. In view of the inverse shearlet transform it was proven in [9] that these discrete shearlets constitute a Parseval frame of the finite Euclidean space L 2 (I). Having a Parseval frame the inverse shearlet transform reads f = j j 1 κ {h,v} j=0 k= 2 j +1 m I j 0 1 f, ψj,k,m κ ψκ j,k,m + j=0 k=±2 j m I f, ψ h v j,k,m ψh v j,k,m + m I f, φ m φ m. (2) By (1) and (2) we can write the shearlet transform and its inverse in matrix-vector form as c = Sf and f = S c (3) where S S = I but SS I. The actual computation of f from given coefficients c(κ, j, k, m) := f, ψj,k,m κ is done in the Fourier domain. For details on the discretization and on computation we refer to [9]. 4

5 3 Inpainting Let the seismic measurement data be given as an image f R N N. We assume that the columns represent the spatial positions of the receivers and each row is a measurement at a certain time. The measured data are corrupted, i.e., a certain amount of columns is missing. The task is to recover the missing data. This can be seen as a general inpainting problem: Given an image with (some) regions of degraded pixels one wants to find a good approximation of the missing pixels. For a simple matrix-vector notation we assume the image to be column-wise reshaped as a vector. Thus, having an image f R N N we obtain the vectorized image vec(f) R N 2 where we retain the notation f for both the original and the reshaped image. Let C := {i {1,..., N 2 } : f(i) 0} be the set of indices of nonzero elements of f. With C we can define the mask H R N 2 N 2 as the diagonal matrix with H(i, i) = 1 if and only if i C and 0 otherwise. Hf is the application of the mask H to the image f. We want to find the recovered signal u as the minimizer of the functional { } 1 u := argmin 2 Hu f ΛSu 1 u R N2 where S is the shearlet transform applied to the vectorized image u in the sense of (3) and Λ is a diagonal matrix of regularization parameters. In our algorithm we will benefit from the fact that S S = I. (4) Alternating Direction Method of Multipliers To solve the problem in (4) we want to use the Alternating Direction Method of Multipliers (ADMM) see [1, 7, 8, 23]. To apply this algorithm we rewrite the problem (4) as { } 1 argmin 2 Hu f Λv 1 u R N2,v R ηn2 subject to Su = v (5) where η is the number of scales and shears in the shearlet transform S. Then the ADMM algorithm reads as follows: ADMM Algorithm for (5): Initialization: v (0), b (0) R ηn 2 and γ > 0. For i = 0,... repeat until a convergence criterion is reached: { 1 u (i+1) = argmin 2 Hu f } 2γ b(i) + Su v (i) 2 2, (6) u R N2 { v (i+1) = argmin Λv } 2γ b(i) + Su (i+1) v 2 2, (7) v R ηn2 b (i+1) = b (i) + Su (i+1) v (i+1). (8) The last step (8) is a simple update of b. We have to comment the first two steps. Due to the differentiability of (6) its solution follows by setting the gradient of the functional on the right-hand side to zero. Thus, the minimizer is given by the solution of 0 = H T (Hu f) + 1 ( γ S T Su + b (i) v (i)) 5

6 Since H T H = H, H T f = Hf = f and S T S = I N 2 no linear system has to be solved and we get u (i+1) = (H + 1γ ) 1 I N (f 2 1γ (b S T (i) v (i))) which means that we have just to apply an inverse shearlet transform. The matrix (H + 1 γ I N 2) is a diagonal matrix such that the inverse is the reciprocal of the diagonal elements. The second step (7) can be minimized as follows { v (i+1) = argmin Λv v R ηn2 = argmin v R ηn2 2γ v (b(i) + Su (i+1) ) }{{} { γ Λv v g 2 2 } =:g = S γλ (g) (componentwise soft-shrinkage). 2 2 } The soft-shrinkage S λ with a threshold λ is defined as x λ for x > λ, S λ (x) := 0 for x [ λ, λ], x + λ for x < λ. 3.1 The choice of Λ The regularization parameter Λ is a diagonal matrix in R ηn 2. Recall that η is the number of scales and shears of the shearlet transform and that we are thresholding the shearlet coefficients. The diagonal consists of η blocks of size N 2. The values within each block should be constant, otherwise we would threshold different regions of the image differently. Consequently, we characterize Λ by η variables. Further, these η variables represent the scale and shear, i.e., directional, information contained in the shearlet coefficients. We write Λ = (λ 0, λ 1,1,..., λ 1,4, λ 2,1,..., λ 2,8,..., λ j0,1,..., λ }{{}}{{} j0,2 j 0 +1 ) }{{} coarsest scale second scale finest scale where λ j,k is the parameter for the scale j and the direction k and λ 0 is the parameter for the low-pass which is usually set to zero. The idea behind our approach is now two-fold. As usual we use the shearlet transform to enforce the solution to be sparse in the shearlet domain, i.e., to contain curved structures, especially lines. This is reasonable since seismic data consists mostly of these kind of structures. For this approach it would be sufficient to choose the parameters in Λ only scale dependent, i.e, Λ = (λ 0, λ 1,..., λ j0 ). We will refer to this approach as only scale based thresholding (st). But we want to go one step further. The shearlet transform also contains directional information. Due to the missing columns a lot of additional (but unwanted) vertical edges are contained in the corrupted image. The original image in contrast has almost no edges in vertical direction. This is why we propose not to use the same parameter λ for all directions within 6

7 one scale but choose larger parameters for the vertical-like directions and smaller parameters for the horizontal-like directions. This results in penalizing vertical edges. We call this method scale-directional thresholding (sdt). But let us first illustrate this idea in more detail. In Fig. 2 we show the original data (left) and the corrupted data (right) with 50% of the columns missing. (a) original data X 0 (b) corrupted data with 50% of the columns missing Figure 2: Original data X 0 and corrupted data with 50% columns missing. For the shearlet coefficients of both images we plot in Fig. 3(a) for every scale and shear the mean absolute value of the shearlet coefficients over all translations, i.e, for given coefficients c we compute for scale j and respective shear k the value m j,k = 1 N 2 N x=1 y=1 N c(j, k, (x, y)). The regions between the vertical dashed lines represent the different scales and the annotation of the horizontal axis illustrates the respective orientation of the shearlet. The solid line is the mean value of the original data and the dashed line is the mean value of the corrupted data. Note that for a better comparison the mean value of the corrupted data is doubled. Comparing both graphs the different directions contained in the data become visible. The mean value of the corrupted data shows a peak at the most left position within each scale, i.e. the vertical direction. Clearly, the corrupted data contain much more large shearlet coefficients in vertical-like directions than the original data. For the horizontal-like directions the shearlet coefficients of the original and the corrupted data are similar, i.e. both data contain the same information. The difference between both lines is shown in Fig. 3(b). Again, we see the peaks for the vertical direction, i.e., large differences, and small values for the horizontal directions, i.e., small differences. Our regularization parameter Λ is set to these differences. During the numerical experiments it turned out that we obtain the best results when multiplying Λ with a factor α. This factor can be easily chosen to obtain the best SNR, i.e. we use αλ as regularization parameter. 7

8 Since the original image is not given in real applications we compute a reference image in the following way. In each row we (iteratively) replace every missing pixel with the mean value of its horizontal neighbors. In the first run these neighbors may be zero but with increasing iterations these gaps get filled from both sides. The SNR of the image reconstructed in this simple way is worse than our results presented below. But the difference of the shearlet coefficients compared to the ones of the original image is marginal such that we use this fast reconstructed image as reference image for computing Λ \ / \ / \ / \ / (a) mean absolute value of the shearlet coefficients for different scales and orientation, solid: original data, dashed: corrupted data (for a better visualization the mean value of the low-pass is set to zero) 0 0 \ / \ / \ / \ / (b) difference between mean absolute values of shearlet coefficients of corrupted and original data Figure 3: Comparison of the mean absolute values of the shearlet coefficients for the original and the corrupted data. 4 Numerical Results We want to present the results of our approach with two different data sets. The first one, a rather simple data set, has been shown in Fig. 2. We will refer to this data as X 0. As a second data set we choose a more complex one that is shown in Fig. 5(a) and the corrupted version in Fig 5(b). This data set is referred to as X 1. For both data sets we randomly set 50% of the columns to zero. As a performance measure we compute the signal-to-noise ratio (SNR) as ( X 2 ) F SNR = 10 log 10 X X 2 F where the original signal is denoted with X and the recovered signal with X. As a second measure we also compare the recovery of a the central trace, i.e., the central column of the matrix X. Therefore we plot the central trace of the original signal (solid line) and the same trace for the reconstructed signal (dashed line). In Fig. 4(a) we show the recovered signal X 0 and in 4(b) we compare the central traces of both the original and the recovered signal. The ADMM parameter γ only influences the convergence but not the result. For the given seismic data we obtained the best performance for γ = 8. 8

9 (a) Recovered data X 0 using scaledirectional thresholding, α = 1 6, SNR: (b) Comparison of central traces of the original signal (solid) and the recovered signal (dashed) Figure 4: Numerical results for dataset X 0 for scale-directional thresholding The recovered signal shows visually no difference to the original image and no artifacts are visible. A careful analysis of the trace comparison shows that the trace is recovered very well, especially for the larger amplitudes in the top. For the small amplitudes in the bottom some differences remain. For the more complex data set X 1 we obtain a similar result in Fig. 5. The first row of the figure shows the original signal on the left (Fig. 5(a)) and on the right the corrupted signal (Fig. 5(b)). The second row presents on left hand the recovered image (Fig. 5(c)) and on the right hand the trace comparison (Fig. 5(d)). Although, the SNR is worse than in the first experiment (due to the more complex structures), the recovered images has only very few remaining artifacts and no obvious difference is visible. The trace comparison also proves the very good reconstruction. For comparison we also present in Fig. 6 the results for the regularization parameters chosen only scale dependent (st), i.e., Λ = (λ 0, λ 1,..., λ j0 ). The parameters were again tuned for the best SNR. Each SNR is about 2-3 db worse compared to the first approach but additionally a lot of artifacts remain in the reconstructed images. At this point we should give a short heuristic explanation for these artifacts: As described above the main difference in the shearlet coefficients between the original and the corrupted image lies in the coefficients for the vertical directions. To reconstruct the image we just have to manipulate these coefficients. Thus we choose large parameters here and small parameters elsewhere. However, if we only have one parameter per scale we have to somehow find an average between these values. If the value is to small we end up with vertical artifacts, if the value is to big we will have artifacts in all directions because we manipulated to many coefficients. 9

10 (a) original data X 1 (b) corrupted data with 50% of the columns missing (c) Recovered data X 1 using scaledirectional thresholding, α = 1 8, SNR: (d) Comparison of central traces of the original signal (solid) and the recovered signal (dashed) Figure 5: Numerical results for dataset X 1 for scale-directional thresholding 10

11 Alternatively one could use for the images without noise an iterative algorithm which preserves the known pixels, see [3]. (a) Recovered data X 0 using only scale based thresholding for Λ = 1 (0, 0.1, 0.2, 0.4, 2), SNR: (b) Recovered data X 1 using only scale based thresholding for Λ = 1 (0, 0.1, 0.2, 0.4, 2), SNR: Figure 6: Numerical results for datasets X 0 and X 1 for only scale based thresholding To show the performance of our approach we now add white Gaussian noise to the data before applying the mask. Let σ denote the standard deviation. We start with a small σ = Since the noise will mostly effect the fine scales of the shearlet transform we adjust Λ by adding 0.1 to the values at the finest scale. Due to the small noise level the visual difference in the image is marginal and consequently we omit presenting the noisy images. Fig. 7 shows the results for both data sets X 0 and X 1, respectively. The second row shows the recovered images (X 0 in Fig. 7(c) and X 1 in Fig. 7(d)) using scale-directional weighting and the third row shows the trace comparisons to the respective original signal. In comparison to the non noisy data the SNR decreases slightly but visually the reconstruction is still very good for both data sets. However, the trace comparison shows more differences than before. The comparison with only scale-based weighting in the first row (X 0 in Fig. 7(a) and X 1 in Fig. 7(b)) shows that the results are still better for the additional directional weighting. For further comparisons we increase the noise level to σ = 0.1. As before we compare the directional thresholding with the scale based thresholding in Fig. 8. The first row shows the corrupted images, X 0 in Fig. 8(a) on the left and X 1 in Fig. 8(b) on the right. In the second row we show the results for the only scale based weighting. The last line shows the reconstruction with scale-directional weighting. We see that the the SNR is again smaller and a lot more artifacts appear for the only scale based weighting. Due to the higher noise the SNR decreases significantly in all cases but the main structures remain intact. We obtain small artifacts for the scale-directional weighting but much more for the only scale based weighting 11

12 (a) Recovered data X 0 with noise level σ = 0.01 and only scale based thresholding for Λ = 1 (0, 0.1, 0.2, 0.4, 2), SNR: (b) Recovered data X 1 with noise level σ = 0.01 and only scale based thresholding for Λ = 1 (0, 0.1, 0.2, 0.4, 2), SNR: (c) Recovered data X 0 with noise level σ = 0.01 and scale-directional thresholding for α = 1, SNR: (d) Recovered data X 1 with noise level σ = 0.01 and scale-directional thresholding for α = 1, SNR: (e) Comparison of central traces of the original signal X 0(solid) and the recovered signal (dashed) 12 (f) Comparison of central traces of the original signal X 1(solid) and the recovered signal (dashed) Figure 7: Numerical results for datasets X 0 and X 1 with noise σ = 0.01

13 noise level - σ = 0.01 σ = 0.1 data shearlets sdt st wavelets X X X X X X Table 1: Comparison of the SNR for different types of thresholds due to the thresholding over the complete fine scale. All parameters were chosen for the best SNR. Wavelet transform Using the wavelet transform in the regularizer and basic iterative thresholding [6, 16] with the iteration u (0) = 0, u (i+1) = W 1 S Λ W(u (i) + H T (f Hu (i) )) the results are worse than for the shearlet regularizer (in both cases). Obviously, shearlets (and curvelets) are better suited for these kind of curved structures. As wavelets we used Daubechies 6. Curvelet transform Instead of the shearlet transform it is also possible to use the curvelet transform in the ADMM. For the computation we used the publicly available toolbox Curve- Lab 1. The visual results and the SNR are very similar to the results presented here for the shearlet case, thus we omit presenting them here. It is also possible to implement the directional weighting in the curvelet setting. However, due to the more complex data structure (cell arrays) it is not as straight-forward as for the shearlet transform and the computation using CurveLab is about 2.5 times slower than using the FFST implementation of the shearlet transform. Summary In Table 1 we provide a summary of the SNR values for the different thresholds and regularizers. Comparing the SNR of both the scale-directional threshold (sdt) and the only scale based threshold (st) we see that the SNR is in all cases better for the scale-directional thresholding. The gab becomes smaller for higher noise level but the visual comparison of the results show that with the scale-directional thresholding we obtain much less artifacts. The performance of the wavelet regularizer is significant worse. This is why we omit listing the results for the noisy case here

14 (a) noisy and corrupted data X 0, σ = 0.1 (b) noisy and corrupted data X 1, σ = 0.1 (c) Recovered data X 0 with noise level σ = 0.1 and only scale based thresholding for Λ = 1 (0, 0.1, 0.2, 0.4, 1), SNR: (d) Recovered data X 1 with noise level σ = 0.1 and only scale based thresholding for Λ = 1 (0, 0.1, 0.2, 0.4, 1), SNR: (e) Recovered data X 0 with noise level σ = 0.1 and scale-directional thresholding for α = 1, SNR: (f) Recovered data X 1 with noise level σ = 0.1 and scale-directional thresholding for α = 1, SNR: Figure 8: Numerical results for datasets X 0 and X 1 with noise σ =

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