Bio-Inspired Orientation Analysis for Seismic Image

Transcription

1 Proceedings of International Joint Conference on Neural Networks, Dallas, Texas, USA, August 4-9, 2013 Bio-Inspired Orientation Analysis for Seismic Image Yingwei Yu, Cliff Kelley, and Irina Mardanova Abstract In this paper, we propose a novel method of orientation analysis for seismic image. Unlike conventional methods of orientation estimation with a context window, our method is inspired by the neural mechanism of visual perception in the biological brain. In the brain, the primary visual cortex contains orientation columns, which is composed of an array of simple cells as orientation detectors. A log-gabor filter with a specific orientation and scale configuration simulates the neuronal mechanism of the simple cell, while an array of such orientation detectors simulates the neuronal responses of the orientation columns. The resulting orientation is derived at the pixel level of the input seismic image: the neuronal response of each simple cell competes with other cells in the same orientation column for the same pixel, and the winning neuron with maximum response defines the perceived orientation. We applied the proposed method to seismic interpretation to analyze the orientation patterns, and it also can be used as a general image orientation analysis tool. I. INTRODUCTION Seismic interpretation is the science and art of deducing geologic history by delineating geological surfaces as represented in and by seismic data. Sand and mud (which over time harden and become rocks) are naturally deposited in nearlyhorizontal layers. Boundaries between layers of buried rock are used to reveal subsurface geometry. The boundaries between layers are known as geologic surfaces, and rocks between boundaries are commonly known as formations. Seismic data are acquired digitally on land, on sea, and on the sea floor. Data are transmitted to processing centers, where noise is filtered out and the data are otherwise conditioned for interpretation. From the processor, seismic data are loaded onto computer for interpretation. The orientation patterns in a seismic image, such as the dip and azimuth attributes, serve as very important tools for 3D seismic data interpretation [1]. They can highlight subtle faults and fractures [2] or provide guidance for horizon autopicking [3]. Moreover, many seismic attributes are calculated either based on or guided by the orientation structures in the seismic image. For example, the attributes of the volumetric curvature and reflector shape are derived from the dip and azimuth volumes. In this paper, we propose a method to analyze the orientation pattern in frequency domain, and inspired by the neuronal circuits in the biological brain. Our method simulates the robust mechanism how humans perceive orientation patterns from a natural image. The biological brain s keen ability to identify orientation patterns in vision motivated us to study the neuronal mechanisms of visual perception in the Yingwei Yu, Cliff Kelley, and Irina Mardanova are with IHS Global, Inc. Suite 400, 8584 Katy Freeway, Houston, TX ( {yingwei.yu}@ihs.com). brain, simulate the neuronal circuits for orientation detection, and apply them to seismic image analysis. (a) The Primary Visual Cortex (V1) in the Brain. (b) Orientation Columns Fig. 1. The Primary Visual Cortex (V1) and Orientation Columns in the Brain. Redrawn from [4] (a) and [5] (b). As shown in Fig. 1a, visual input to the brain goes from the eye to the primary visual cortex (V1), which is located in the posterior of the occipital lobe. Hubel and Wiesel proposed the ice-cube model of the primary visual cortex like the one in Figure 1b by observing the cat s striate cortex [5]. As illustrated, the V1 area is composed by a slab of simple cells, which are tuned in the reponses to different orientation patterns. Neurophysiologically, the perceived angle in the visual input pattern is then determined by the strength of the simple cells responses. In the following analysis, we will model the simple cell with an orientation-tuned filter, and use an array of such filters to simulate the orientation column to estimate the perceived orientation. The rest of this paper is organized as follows: the next section introduces our methods to derive the orientation vector field from the input seismic image, and experiment and result are exposed in the third section, followed by a conclusion. II. METHODS In this section, we first introduce the simulation of a simple cell with a log-gabor filter. Then we will build the orientation column as an array of such filters. According to the activation profile of an orientation column observed in the biological brain, we can derive the perceived orientation from the filter with the maximum response. The result of the orientation detection method is to generate an orientation vector field (OVF), which indicates both directions and magnitudes of the orientation patterns from the input image /13/$ IEEE 2589

2 (a) Radial Component (b) Angular Component Fig. 3. Log-Gabor Filter in Frequency Domain. (a) Radial component: 2D Gaussian in logarithmic scale. (b) Angular component: Gaussian along orientations. Fig. 2. The Power Spectrum of the Seismic Data Overlay with A Logarithimic Gaussian Curve. One trace from the salt dome seismic data is transformed into frequency domain by DFFT. The x-axis is frequency and y-axis represents the strength of frequency responses. It shows that the seismic data s spectrum (blue) follows the logarithmic Gaussian distribution (red) with σ f = and f 0 = 25. A. Simulating the Orientation Detector: Log-Gabor Filter In 1980, Daugman first modeled the orientation-selective simple cells in the cortex as 2D form Gabor wavelet [6], which is often used as an orientation detector in many applications in the realm of imaging processing (e.g., see [7]). Field then followed an alternative to the Gabor filter [8], the log-gabor filter, which is an improved version. The log-gabor filter has no DC component, and overcomes the bandwidth limitation in the conventional Gabor filter. As proposed by Field [8], natural image has a frequency distribution as Gaussian in logarithmic scale, and the log- Gabor filter has the required logarithmic Gaussian profile. The seismic data follows the rule of natural images. Figure 2 shows the frequency plot of one trace from the salt dome seismic data in Figure 6. The frequency distribution reflects the logarithmic Gaussian patterns. The 2-D log-gabor filter is defined in frequency domain as follows [8]: H( f,α) = H f H α, (1) where H f is a radial component (see Figure 3a), and H α is an angular component (see Figure 3b). The radial component is 2D Gaussian in logarithmic scale, and the angular component is Gaussian along orientations. More specifically, ( ) ) H( f,α) = exp ln2 ( f / f 0 ) (α θ)2 2ln 2 exp ( (σ f / f 0 ) 2σ 2, (2) α where f 0 is central frequency, and θ is the filter direction. B. Orientation Column: Activation Profile A single orientation detector can respond maximally to a specific angle, but cannot tell the exact angle in the input. In the brain, it is the orientation column, i.e., an array of orientation detectors, that perceive the orientation patterns. First we study how orientation columns in the visual cortex interact in response to orientation patterns and then build the model of it to derive the OVF. For an orientation pattern with a specific angle, there is a corresponding simple cell in an orientation column that responds maximally, and the orientation column responses can be approximated by Gaussian distributions [9]. In Figure 4, the activation of simple cells in an orientation column in response to two angles is shown. The dashed curve is the response of the cells in an orientation column (x-axis) to a zero-degree line, while the solid curve is that to a 30 degree line [10]. It shows that in the biological brain, all of the orientation detectors tuned to various angles give responses to an input, and the perceived angle is determined by the detector with the peak response in the orientation column. Fig. 4. Orientation Column Activation Profile. The activation of simple cells in response to two angles is shown. The dashed curve is the response of the cells in an orientation column (x-axis) to a zero-degree line, and the solid curve that of a 30 degree line. Adapted from [10]. C. Generating the Orientation Vector Field Daugman first employed a bank of Gabor filters centered at different orientations and scales in the frequency domain for image analysis [6]. Others used a set of Gabor filters with similar configurations for texture analysis [11], or image compression [12]. Recently, Fischer et al. proposed an array of multi-resolution log-gabor filters that centered on a number of orientations for image representation, and successfully applied it to image denoise and compression [13]. We further extended Fischer s model [13] to simulate the orientation column responsis, and generate an orientation vector field (OVF) for image orientation analysis. The resulting OVF can be derived as follows. For input image in the spatial domain I(x,y), first apply a Fourier transform into the frequency domain Î(u, v). The 2590

3 application of log-gabor filter in frequency domain is, Ŷ <w,θ> (u,v) = H <w,θ> (u,v)î(u,v), (3) where H is the log-gabor filter with orientation θ and central frequency w. Since it is an array of log-gabor filters, their central orientations θ can be in a series of orientations, such as { π 2, 5π 8, π 4,...,+π 4,+5π 8 }, and w can be in single or multiple scales as well. By inverse Fourier transform, we convert Ŷ <w,θ> (u,v) into a complex image in the spatial domain. The resulting convolved image has a real part, Y r <w,θ> (u, v), and an imaginary part, Y <w,θ> im (u,v). The norm of the complex image is called orientation energy in the specific orientation θ and scale w: Y <w,θ> (x,y) = [Y <w,θ> r (x,y)] 2 + [Y <w,θ> im (x,y)] 2. (4) As indicated by the observed activation profile in the biological brain, when we line up all the filter responses for an individual pixel at (x,y), we can derive the perceived orientation at this point by getting the location of peak responses in the orientation column. In the model, the most salient orientation pattern γ at pixel (x,y) is defined by the index of the peak value of the summed orientation responses in all scales: γ(x,y) = argmax θ Y <w,θ> (x,y). (5) w The γ(x,y) is also referred as the apparent dip [1]. It is one of the two components of the OVF in the output, the other is the orientation energy E(x,y), which is the sum of filter responses with the orientation γ(x, y) in all scales: E(x,y) = Y <w,γ(x,y)> (x,y). (6) w The orientation energy E reflects the strength of orientation features. The low values of orientation energy mean that there are fewer oriented patterns in the neighborhood, while the stronger ones mean the orientation feature is more salient in the context. D. Multi-pass In the primary visual cortex, there are significant amount of feedback from higher level (e.g., layer 4) to V1 on layer 2 and 3. In our algorithm, the filter convolution response, Orientation Energy, is an orientation filter-smoothed version of the input amplitude. It can be fed into the filter array for another pass to further improve the accuracy of OVF generation. In our model, the feedback loop with Orientation Energy to the log-gabor filter array is analogies to the feedback signals from layer 4 to V1 in brain. The adaptive workflow is illustrated in figure 5. E. Main algorithm The algorithm (single pass) to generate the Orientation Vector Field is then summarized as follows. Fig. 5. Multi-pass workflow. The feedback loop with Orientation Energy to the log-gabor filter array is analogies to the feedback signals from layer 4 to V1 in brain. Algorithm 1 OVF Generation 1: procedure GENERATEOVF(I, f 0,σ f,σ α,a) I: seismic image in 2D, f 0 : log-gabor central frequency, σ f : log- Gabor frequency standard deviation, σ α :log-gabor angle standard deviation, A: an array of angles. 2: Î FFT 2(I) Apply 2D Fourier transform 3: for each orientation θ in A do 4: Ĥ[θ] loggabor( f 0,σ f,σ α,θ) 2D log-gabor filter (Eq. 1 and 2) 5: ˆR[θ] Ĥ[θ] Î Convolution (Eq. 3) 6: R[θ] IFFT 2( ˆR[θ]) Apply inverse Fourier transform 7: Y [θ] R[θ] Get absolute value (Eq. 4) 8: end for 9: for each pixel (x,y) in I do 10: γ(x,y) argmax θ Y [θ](x,y) Orientation (Eq. 5) 11: E(x, y) Y [γ(x, y)](x, y) Orientation energy (Eq. 6) 12: end for 13: return < γ, E > Orientation Vector Field 14: end procedure III. EXPERIMENTS AND RESULTS In this section, we performed two experiments with the algorithm applied to real data. The first test was with seismic image as shown in Figure 6. In the second test, the orientation analysis is applied to a natrual image. A. Test with Seismic Image The described algorithm was tested on a 3D seismic survey containing a salt dome in the center (Figure 6). The OVF was generated using a filter array with eight log-gabor filters, whose orientations are from π 2 to π 2 with angle interval of π 8, and central frequencies are at 25 Hz. Figure 6b shows the instantaneous dip generated from a vertical slice of amplitude in Figure 6a. Figure 7 shows the OVF generated for a 2591

4 2D seismic slice near a salt dome in zoom-in view. The orientation vector can also be used to derive other seismic attributes, such as curvature calculations. straight forward, but it is unstable to noise, and suffers from uncertainty with high frequency data with high dip angle. For example, as shown in figure 8b, for point A in the left wavelet, there can be multiple locations (B, C, or D) in its neighbor trace with high correlation values. To determine the right dip angle in such a case, it cannot purely depends on only two neighboring traces, and we need to analysis it in a larger context area. The proposed optical filter-based method has the ability to mimic the human orientation perception mechanism and it looks for the orientation patterns in a larger area as an image. (a) Amplitude Slice from a Salt Dome Survey (a) Fig. 8. Uncertainty in Correlation-based Dip Estimation Method. (a) The dip is usually estimated by locating the maximum cross-correlation point in its neighbor seismic trace. (b) Due to the limitation of the correlation-based method, it is unclear which location yields the correct dip angle. (b) (b) Dip Calculated from the Seismic Image in (a) Fig. 6. Amplitude and Dip of a Salt Dome Seismic Image. Seismic survey data courtesy of FairfieldNodal. Fig. 7. Orientation Vector Field Near a Salt Dome. The orientation vectors (red) are plotted on top of the seismic image in a region near the salt dome. The magnitudes of the vectors are normalized. Unlike our optical filter-based approach, conventional dip estimation is usually correlation-based. Given a seismic trace, by locating the maximum cross-correlation point in its neighbor trace, the dip angle is estimated. As shown in figure 8a, two seismic wavelets are immediate neighbors. One can cross-correlate these two wavelet from point A to point B, and the orientation pointed from A to B is the apprent dip calculated. The correlation-method is simple and B. Test with Natrual Image - Lenna In the second experiment, we used the standard testing image LENNA as the input (Figure 9a). An array of log-gabor filters are tuned to 36 orientations that evenly distributed from π 2 to + π 2 to generate the orientation vector field. In figure 9b, the orientation (dip value) is color-coded. Figure 9c shows a zoom-in region of the input image around the right eye with orientation vectors (red arrows) displayed on it. IV. CONCLUSIONS This paper introduced a novel method of orientation analysis in a seismic image. One unique feature of our method is that it is inspired by the neuronal mechanisms of the primary visual cortex in the brain for orientation perception. An array of log-gabor filters is employed to simulate the neuronal responses of the orientation column, and an orientation vector field is generated to represent the orientation patterns in the seismic image. As demonstrated by the experiment, our proposed method is an effective way to perform the orientation pattern analysis for seismic data, and can be applied to natural images for orientation detection in general. ACKNOWLEDGMENTS The authors would like to thank Dr. Yoonsuck Choe for helpful discussions and comments, and thank FairfieldNodal 2592

5 for permission to use the seismic data in this paper. The seismic orienation analysis method proposed in this paper is protected by US patent 8,265,876. REFERENCES [1] S. Chopra and K. J. Marfurt, Seismic Attributes for Prospect identification and Reservoir Characterization. Societry of Exploration Geophysicists, [2] A. E. Barnes, Shaded relief seismic attribute, Geophysics, vol. 68, pp , [3] Y. Yu, C. Kelley, and I. Mardanova, Automatic horizon picking in 3d seismic data using optical filters and minimum spanning tree, in 81st Annual International Meeting of Society of Exploration Geophysicists, SEG Expanded Abstracts, vol. 30. Society of Exploration Geophysicists, 2011, pp [4] S. Polyak, The vertebrate visual system. Univ. of Chicago Press, [5] D. H. Hubel and T. N. Wiesel, Receptive fields of single neurones in the cat s striate cortex, Journal of Physiololgy, vol. 150, pp , [6] J. Daugman, Two-dimension spectral analysis of cortical receptive field profile, Vision Research, vol. 20, pp , [7] A. Jain, N. Ratha, and S. Lakshmanan, Object detection using gabor filters, Pattern Recognition, vol. 30, pp , [8] D. J. Field, Relations between the statistics of natural images and the response properties of cortical cells, J. Opt. Soc. Am. A, pp , [9] L. M. Martinez, j. Alonso, R. C. Reid, and J. A. Hirsch, Laminar processing of stimulus orientation in cat visual cortex, Journal of Physiology, vol. 540(1), pp , [10] Y. Yu and Y. Choe, Neural model of disinhibitory interactions in modified pogendorff illusion, Biological Cybernetics, vol. 98(1), pp , [11] A. Bovik, M. Clark, and W. Geisler, Multichannel texture anslysis using localized spatial filters, IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 12(1), pp , [12] T. Lee, Image representation using 2d gabor wavelets, IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 18(10), pp , [13] S. Fischer, F. Sroubek, L. Perrinet, R. Redondo, and G. Cristobal, Self-invertible 2d log-gabor wavelets, International Journal of Computer Vision, vol. 75(2), pp , (a) A natural image input, Lenna (b) Color-coded orientations derived from (a) (c) Orientation Vector Field Fig. 9. Orientation Analysis with LENNA. (a) The original image with resolution of (b) color-coded dip orientations from (a), and (c) A zoom-in region of the input image with orientation vectors (red arrows) displayed on it. 2593