Estimation of Tikhonov Regularization Parameter for Image Reconstruction in Electromagnetic Geotomography

Transcription

1 Rafał Zdunek Andrze Prałat Estimation of ikhonov Regularization Parameter for Image Reconstruction in Electromagnetic Geotomograph Abstract: he aim of this research was to devise an efficient wa of finding the optimal value of the parameter in the ikhonov regularization applied to image reconstruction in electromagnetic (EM) geotomograph. We emploed three basic approaches: a plot of norm l of the error vector versus the regularization parameter, the Ordinar Cross-Validation () and the L-curve. hese tools were applied to snthetic noise-free and nois data generated from an original image. In a simulation stud, the first tool was used as the background for comparisons with the other methods. he criterion has been found to estimate the regularization parameter most precisel. Its rather high computing cost can be reduced through a combination of powerful hardware with sophisticated implementation. Furthermore, this criterion combined with an regularization method constitutes a robust tool for solving ill-posed inverse problems. Kewords:, L-curve, ikhonov regularization, EM geotomograph 1. IRODUCIO In electromagnetic (EM) geotomograph [4], an electromagnetic wave is transmitted through the earth and an image of the electromagnetic wave attenuation coefficient distribution, representing the interior of a cross-borehole section, is reconstructed from signal attenuation measurements along multiple ras (proections). Each ra is described b a linear equation and a sstem of such equations is formed for the investigated area. Similarl as in man applications of tomograph, the problems of image reconstruction in EM geotomograph boil down to the solution of a sstem of linear equations. Unfortunatel, such a sstem has man disadvantages. Its coefficient matrix is large, sparse, unsmmetrical, ill-conditioned and ver often it has no full rank. Because of this, man eas-to-use methods of solving linear equations cannot be applied here. Wroclaw Universit of echnolog, Institute of elecommunications and Acoustics, Electronic Circuits Division, s: rafal@zue.ita.pwr.wroc.pl, pralat@zue.ita.pwr.wroc.pl

2 An measurement errors make this sstem inconsistent. Moreover, since the sstem is ill-conditioned, the reconstructed image is strongl affected b inaccurate data. his leads to the question of how to choose the right solution to this sstem. Regularization is a technique which can handle such a task. here are man regularization techniques [, 3], but the ikhonov regularization is the most popular. It makes an ill-posed problem well-posed b offsetting all the singular values in the coefficient matrix. he shifted spectrum is much easier to invert because the regularized matrix has a full rank and it is better conditioned. But the deformation of the spectrum introduces an error into the solution. he value of the offset depends on the so-called regularization parameter. he choice of an optimal regularization parameter is ver difficult and it depends on the features of a given problem. Some methods of optimal regularization parameter selection, i.e. the L-curve, the Ordinar Cross-Validation () and the General Cross-Validation (GCV), and a few extrapolation techniques have been proposed [1,, 3], but neither of them works ideall. In the paper, the ikhonov regularization is applied to solve inverse problems in EM geotomograph. Images are reconstructed from snthetic noise-free and nois data generated from an original image. he regularization parameter is estimated b the L-curve and. he most efficient method for the estimation of the regularization parameter is chosen on the basis of a comparison of the obtained results.. IKHOOV REGULARIZAIO In EM geotomograph a forward model can be formulated as the linear sstem: A x =, (1) M, where A R is a coefficient matrix, x R M is an image vector and R is a measurement vector. he least-squares solution is the one that minimizes the quadratic distance between real data and forward proected data A x. his can be expressed as: min{ A x }. () x Since matrix A has no full rank, man solutions satisf least-squares criterion (). Moreover, each of them is strongl affected b inaccurate data. o choose the desired solution, additional information (so-called a priori information) should be imposed on (). In the ikhonov regularization, this information is defined as the smoothest solution satisfing the condition: min{ x }. Hence criterion () can be improved as follows: x

3 min x { A x + x } α. (3) he balance between the least-squares solution and the smoothest one is controlled b regularization parameter α. Considering (3), the solution of (1) can be expressed in a closed form as: ( A 1 A + I ) A x = α (4), where I R is an identit matrix. he inverse matrix in (4) is well-conditioned, smmetrical, positive-defined and it has a full rank. evertheless, the second term in (3) shifts the spectrum of matrix A and the estimation of x is no longer the truth solution. he goal is to find the right value of regularization parameter α which minimizes errors introduced b both the shifted spectrum of matrix A and the ill-posed nature of the problem. 3. ESIMAIO OF REGULARIZAIO PARAMEER here are man techniques [1,, 3] which estimate the optimal value of the regularization parameter. Since the exact characteristics of the noise are unknown, our stud is limited to the L-curve and cross-validation methods L-CURVE he L-curve is a plot for all valid regularization parameters of norm l of the regularized solution, e.g. x, versus corresponding residual norm A x. An important feature of the L-curve is its L-shaped corner that appears for regularization parameters close to the optimal value. he idea behind the L-curve criterion for finding regularization parameter α is to choose a point on this curve as close as possible to the corner. 3.. ORDIARY CROSS-VALIDAIO () he cross-validation criteria can be seen as data prediction error criteria. For the optimal regularization parameter an error between predicted missing data values and real data values should be minimized. he Ordinar Cross-Validation () criterion is formulated as:

4 ( ( ) ) = M F = 1 ˆ, (5) ( ) where, ˆ stand for the -th component of measurement vector, and the -th component predicted from incomplete vector for each ( 1 M ) the computation of difference ( ) M 1 ( ) R, respectivel. Since ( ) ˆ involves the inversion of matrix A A + α I, where is matrix A with its -th A ( ) ( ) ( ) row deleted, the number of matrix operations can become ver large. o keep the computing cost at a reasonable level, criterion (5) ma be rewritten as follows: [ ( ) ] M I A B 1 A M F =, (6) = 1 1 s, where S = A B 1 A and s, are the diagonal elements of S. Matrix B denotes regularized normal matrix ( A A + α I ). 1 he computation of F requires onl one inversion, e.g. evaluation of B, for each value of α. In practice, criterion (6) should be evaluated for several regularization parameter samples covering the likel range of regularization parameter variation. he lowest value of F corresponds to the optimal value of parameter α. he accurac of the criterion and the overall computing cost depend on the sampling interval of the regularization parameter. In our experiment, two samples per decade were set. 4. RESULS he methods of estimating the regularized parameter were tested in the following different imaging situations: snthetic noise-free data, snthetic nois data. In computer simulations, snthetic data were generated from the original image presented in Fig. 1.

5 Fig.1. Original image he reconstruction results obtained from noise-free data are shown in Fig.. he plot of F versus parameter α is presented (in the log-log scale) in Fig..a. he L- curve is shown in Fig..b. o allow the comparison of the plots, the normalized error norm and the normalized residual norm versus α are plotted (in the log-log scale) in Fig..c and Fig..d., respectivel. he normalized error norm is computed as x xs, where x is a vector of the regularized solution and x s is a vector of the xs original image shown in Fig.1. he normalized residual norm is expressed as res, where res is a residual vector and is a measurement vector. Figs..e. and.f. show images reconstructed b the ikhonov regularization for the optimal value of α estimated b the criterion and for α = 1, respectivel. ois data are generated as follows: = ( 1 + ( μ, σ ) 0, for = 1,, M, (7) M where R is an accurate proection vector, and (, ) μ 0 σ stands for a Gaussian probabilit distribution with mean μ 0 and variance σ. We set μ 0 = 0 and σ = (0.0). he reconstruction results obtained from nois data are shown in Fig. 3 (the legend is the same as in Fig.).

6 (a) (b) (c) (d) (e) (f) Fig.. Reconstruction results obtained from noise-free data: (a), (b) L-curve, (b) norm l of error vector, (c) norm l of residual vector, (d) image reconstructed for α α = 10 6, (e) image reconstructed for α = 1. = optimal

7 (a) (b) (c) (d) (e) (f) Fig.3. Reconstruction results obtained from nois data: (a), (b) L-curve, (b) norm l of error vector, (c) norm l of residual vector, (d) image reconstructed for α α = 1, (e) image reconstructed for α = = optimal

8 5. COCLUSIOS he goal of this research was to devise an efficient technique of estimating the regularization parameter of ill-posed inverse problems. Such problems occur in tomograph and being ver difficult to solve, the usuall need to be regularized. here are man regularization techniques but all of them are parameter-dependent. he most difficult task is the right choice of regularization parameters since there are still no efficient tools for performing it. his paper represents an attempt at solving ill-posed inverse problems in EM geotomograph. Our stud is limited to the estimation of the parameter in the ikhonov regularization b means of the and L-curve criteria. o choose the most efficient criterion, error and residual norms versus the regularization parameter were plotted. he minimum values of the norms should correspond to the optimal value of this parameter. In the case of noise-free data, the error norm (Fig..c.) has low values in a wide range of the variable and thus the optimum of α can be attributed onl to the minimum of the residual norm (Fig..d.). his minimum matches better the lowest value of F (Fig..a.) than the corner in the L-curve. hus the criterion appears to be more efficient for noise-free data. For nois data onl the error norm (Fig.3.c.) can be used as the background for comparisons. In this case, the criterion (Fig. 3.a.) also performs better than the L-curve criterion (Fig. 3.b.). o conclude, the criterion seems to be more efficient than the L-curve criterion. Its rather high computing cost can be reduced through a combination of powerful hardware with sophisticated implementation. Furthermore, this criterion combined with an regularization method constitutes a robust tool for solving ill-posed inverse problems. REFERECES [1] COHE-BACRIE C., GOUSSARD Y., GUARDO R., Regularized Reconstruction in Electrical Impedance omograph Using a Variance Uniformization Constraint, IEEE rans. Medical Imaging, Vol. 16, o. 5, October 1997, , [] HASE Ch., Rank-Deficient and Discrete Ill-Posed Problems, Philadelphia, SIAM, 1998, [3] EUMAIER A., Solving Ill-Conditioned and Singular Linear Sstems: A utorial on Regularization, SIAM Review 40, 1998, , [4] PRAŁA A., ZDUEK R., he Electromagnetic Geotomograph the Comparison of Various Image Reconstruction Methods, Acta Montana IRSM AS CR, Series A, Vol. 118, o. 16, 000, ,