GPR Migration Imaging Algorithm Based on NUFFT

Transcription

1 PIERS ONLINE, VOL. 6, NO. 1, GPR Migration Imaging Algorithm Based on NUFFT Hao Chen, Renbiao Wu, Jiaxue Liu, and Zhiyong Han Tianjin Key Laboratory for Advanced Signal Processing, Civil Aviation University of China Tianjin , China Abstract Stolt migration is one of the useful GPR imaging algorithms, which has the advantage of fast implementation speed. The fast Fourier transform of integral formula in image reconstruction is realized by coordinate transformation and done through interpolation. Interpolation determines the effectiveness of imaging directly. In the practical applications, zero-fill interpolation in time-domain is used to improve accuracy, but which will increase the computational complexity. In this paper, non-uniform fast Fourier transform (NUFFT) algorithm is introduced to realize the fast computing of integral formula in image reconstruction directly. The simulation results show that this method achieve good results. 1. INTRODUCTION Stolt migration is one of the useful GPR imaging algorithms which has the advantage of fast implementation speed. The date, which is uniformly distribute in frequency-wavenumber domain, is transformed from frequency-wavenumber domain to wavenumber-wavenumber domain. Then using interpolation method to make sure that the date is uniformly distribute in wavenumberwavenumber domain. So it will be possibility to use the FFT, which lead efficiencies, to calculate the date. That is the key method of Stolt migration method. Usually, the Linear interpolation and sinc interpolation [, 3] is used. Interpolation determines the effectiveness of imaging directly, in the practical applications, zero-fill interpolation in time domain is used to improve accuracy but which will increase the computational complexity. This paper analyzed the root causes of why could not use FFT to calculate the formula for image reconstruction before stolt interpolation. Then introduced the algorithms of computing Fourier transform with unequally spaced data in any of the domain (time or frequency) have been proposed. These algorithms are known as Non-Uniform FFT (NUFFT), NUFFT was first proposed by Dutt and Rokhlin in 1993 [4], Beylkin; Nguyen [5] and Liu; Fessler [6] and Sutton [7] are made great contributions to its development. In this paper, NUFFT is introduced to realize the fast computing of formula in image reconstruction directly. Both the stolt interpolation and the final range inverse FFT have been substituted by a single NUFFT. This paper describes the detail application of the NUFFT s to the stolt migration algorithm. Compared the new method with the old ones in calculation efficiency and image result. The performance is illustrated with numerical simulations. The simulation of B-scan data of a two-layered medium is generated by the Gprmax.0 software. The simulation results show that this method reduce the complexity of the interpolation while ensure the operational speed. at the same time, it use least-square to approximate the function so the energy of the image result is more concentrated. This method achieves good results.. STOLT MIGRATION ALGORITHMS Migration method back propagates the received wave front to the instant the target explodes and the image is then the scalar field at the instant before it begins to propagate. From the recording section extended to the imaging section is called migration that is show in Fig. 1. Consider the -D case, the raw data e(x, z = 0, t) is measured on the surface z = 0. The imaging data after migration is e(x, z, t = 0), in particular, the -D Fourier transform can be written as: E(k x, z = 0, ω) = e(x, z = 0, t)e i(kxx+ωt) dxdt (1) The wave field extrapolation formula in the ω-k domain is: E(k x, z, ω) = E(k x, z = 0, ω)e ikzz ()

2 PIERS ONLINE, VOL. 6, NO. 1, ω k z. uniform data raw data k x k x (a) (b) Figure 1: Migration schematic. Figure : Coordinate transformation and stolt interpolation. The -D inverse Fourier transform about k x, ω can be written as: e(x, z, t) = 1 (π) E(k x, z, ω)e i(kxx+ωt) dk x dω (3) According to Eq. () and make t = 0, the integral formula in image reconstruction is: e(x, z, t = 0) = 1 (π) E(k x, z = 0, ω)e i(kxx+kzz) dk x dω (4) We could Calculate Eq. (4) by direct summation and thus is an accurate evaluation of Eq. (4) but it takes o(n ) as many as operations. An interpolation approach is usually utilized to improve efficiency. consider the ω-k domain algorithm: ( ) ω kx + kz = (5) v/ It is easy to see form Eq. (5) that k z = [ω/(v/)] kx, so k z is unequally spaced, that is why Eq. (4) can not calculate though FFT. However, according to Eq. (5), Eq. (4) becomes e(x, z, t = 0) = 1 (π) E Ge ikxx e ikzz dk x dz (6) That ( E = E k x, z = 0, v ) k x + kz v G = 1 + kx/k z Then, though interpolation k z becomes uniform, so that Eq. (6) can be Calculate by regular FFT, Coordinate transformation and interpolation as show in Figure MIGRATION BASED ON NUFFT In the stolt migration a variable transform is made form ω to k z and then interpolation is peformed along k z. There are many of the traditional interpolation methods, such as linear interpolation, sinc interpolation, and their improved methods [, 14]. If the method selected inappropriate, the energy will not concentrate and will Cause the parasitic Component []. By zero-padding in the time domain can improve the accuracy of interpolation, thereby improving image quality, but this will increase computational complexity. This paper introduces the non-uniform fast Fourier transform (NUFFT) to achieve Eq. (4) NUFFT Algorithms The main idea of NUFFT is for a known point of non-uniform space and its close points in the uniform space, we can use the points Fourier transform basis functions to approximate the point which is in the non-uniform space.

3 PIERS ONLINE, VOL. 6, NO. 1, Our aim is to develop a fast algorithm to find the following summation: F (α) j = N/ k= N/ α k e i π N jtk j = N,..., N (7) in which t k is un-uniform, for most NUFFT algorithms the procedures is: 1. Use (q + 1) uniform points to interpolate one point s (t k ) Fourier transform basis function (exp(iπjt k /N)) s j e i π N jtk = q/ l= q/ x l (t k )e i π mn j([mtk]+l) (8) where the accuracy factors 0 < s j 1 are chosen to minimaize the approximation error. Use least-square to approximate this sequence by a small number of uniform points.. Calculate Fourier coefficients η n = α k x l (t k ) (9) l,k,[mt k]+l=n 3. Regular FFT: use uniform FFT to evaluate F j = s 1 j mn/ 1 n= mn/ The number of arithmetic operations is O(N(q + 1) + mn log mn). 3.. Stolt Migration Using NUFFT Is easy to see that the discrete form of Eq. (5) is η n e i π mn nj (10) So [ ] πnω /(v/) = dtn t [ nω [ ] [ ] πnkx πnkz + (11) dxn x dzn z n kz = N z dz ] [ ] nkx /(v/) (1) dtn t dxn x the discrete form of Fourier transform basis function exp(ik z z) is exp(iπn kz z/n), use the NUFFT method can realize the fast Computing Depth/Nt Depth/m Along track possition/nx Along track possition/m Figure 3: GPR B-scan of a two-layered medium. Figure 4: NUFFT. Result image of migration based on

4 PIERS ONLINE, VOL. 6, NO. 1, EXPERIMENTAL RESULTS In the simulation experiment. The case of a metal ball existing at a depth of 0.55 m. The transmitter is located at 0.05 m Above the surface. The simulation of B-scan data of a two-layered medium is generated by the Gprmax.0 software. An electromagnetic pulse of ricker form is assumed to be emitted. The simulated depth of medium air and concrete is 0.05 m and 0.45 m respectively. The relative dielectric constant of each layer is 1 and 0 respectively. The center frequency of transmit antenna is f = 900 MHz and sampling time is dt = 5.89 ps. After suppression of ground reflective wave and direct coupling wave interference, the B-Scan image is shown in Fig. 3. the migrate result with the NUFFT migration method as show in Fig. 4, where q = 6 and over sampling rate m =, accuracy factors s j = cos πj/(mn). For comparison, the direct-summation migration and interpolation migration methods are implemented. we use entropy to measure the effect of the migration image, the more greater the entropy, the more confusion the migration image is. For the matrix A = (a ij ) m n, define that the entropy is [10, 13]: 1 n m a 4 ij j=1 i=1 E(A) = [ ] (13) n m The result is reported in Tab. 1. a ij j=1 i=1 Table 1: The entropy of the images. Direct-sum interpolation NUFFT entropy CONCLUSIONS Through the analysis above and simulation results we can see that, the Stolt migraton based on NUFFT is efficient while reducing the complexity of the interpolation, and has made good imaging results. NUFFT algorithm and stolt interpolation both are adopted to achieve the fast computing of Eq. (4) by approximate, but NUFFT algorithm is to approximate the Fourier transform-based by least-squares method in the frequency domain which is more normative, and the energy of the result image is more concentrated. In addition, in practical applications, due to obstructions, restricted areas and underground media, the raw data is also non-uniformity, NUFFT algorithms will be more widely used in this field. REFERENCES 1. Gu, K., G. Wang, and J. Li, Migration based SAR imaging for ground penetrating radar systems, IEEE Radar Sonar and Naviqation, Vol. 151, No. 5, , Oct Zhang, C. and Z. Zhou, A modified ground penetrating radar 3-D synthetic aperture imaging method based on stolt migration, Signal Processing, Vol. 0, No. 6, Dec Zeng, H., T. Zeng, C. Hu, and Y. Zhu, Comparison of several interpolation methods in spotlight SAR imaging algorithms, Signal Processing, Vol. 1, No. 4, Aug Dutt, A. and V. Rokhlin, Fast Fourier transforms for nonequispaced data, SLAM-J. Sci. Comp., Vol. 14, No. 6, Beylkin, G., On the fast fourier transform of functions with singularities, Applied and Computational Harmonic Analysis, Vol., No. 4, , Liu, Q. H. and N. Nguyen, An accurate algorithm for nonuniform fast fourier transforms (NUFFT s), IEEE Microwave and Gueded Wave Lett., Vol. 8, No. 1, 18 0, Fessler, J. A. and B. P. Sutton, Nonuniform fast Fourier transforms using min-max interpolation, IEEE Trans. Signal Process., Vol. 51, No., , Feb Liu, Q. H. and N. Nguyen, The regular fourier matrices and nonuniform fast fourier transforms, SLAM J. Sci. Comp., Vol. 1, No. 1, 83 93, GprMax.0.

5 PIERS ONLINE, VOL. 6, NO. 1, Wang, G., Study of Imaging Methods in Ground Penetrating Radar, Civil Aviation University of China, Tianjin, Stolt, R., Migration by Fourier transform, Geophysics, Vol. 43, No. 1, 3 48, Subiza, B., E. Gimeno-Nieves, and J. M. Lopez-Sanchez, An approach to SAR image by means of non-uninform FFT s, 003 IEEE International Geoscience and Remote Sensing Symposium Proceedings, IGARSS 03, Vol. 6, , Jul Xiu, Z., J. Chen, G. Fang, and F. Li, Ground penetrating radar imaging based on F-K migration and minimum entropy method, Journal of Electronics & Information Technology, Vol. 9, No. 4, Apr Chang, W., Q. Song, and D. Liang, Implementation and application of resampling interpolation, Systems Engineering and Electronics, Vol., No. 5, 000.