Total Variation as a Multiplicative Constraint for Solving Inverse Problems

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1 1384 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 10, NO. 9, SEPTEMBER 2001 Total Variation as a Multiplicative Constraint for Solving Inverse Problems Aria Abubakar and Peter M. van den Berg Abstract The total variation minimization method for deblurring noise is shown to be effective in increasing the resolution in a contrast source inversion approach to index reconstruction from measured scattered field data. Main drawback is the presence of an artificial weighting parameter in the cost functional, which can only be determined through considerable experimentation. Therefore, we introduce the total variation as a multiplicative constraint. Numerical examples demonstrate that the algorithm based on this multiplicative regularization seems to be robust and handling noisy data very well without the necessity of the weighting parameter. Index Terms Deblurring, inverse scattering, total variation. I. INTRODUCTION THE problem of determining the shape, location, and index of refraction of an inhomogeneity imbedded in a homogeneous background medium from measurement of the field scattered by the object when illuminated by known incident electromagnetic waves is one of the fundamental problems in inverse scattering. A number of methods of solution have been proposed and numerically implemented, see Colton and Kress [8], Gutman and Klibanov [12], Kleinman and van den Berg [24], [26] and the references therein. Almost all of the techniques involve iterative minimization of a cost functional. Some, such as those employing a Newton-type method, require the solution of a direct scattering problem at each iterative step (see [7], [13], [22], [23], [29]), while others avoid the repeated use of a direct solver [9], [20], [24], [26]. Of the latter approaches, one, a contrast source inversion method [1], [2], [26], has proven to be effective in reconstructing a variety of inhomogeneous and lossy refractive indices in two-dimensional (2-D) and three-dimensional (3-D) configurations. Concurrent with this progress in the inverse scattering problem there has been a significant breakthrough in image enhancement or deblurring noisy images by minimization of the total variation of the image, subject to constraints (see [4], [6], [11], [16], [19], [21], and [28]). In the original approach, the problem of nondifferentiability of the total variation is overcome by introducing nonlinear diffusion equation for minimizing the total variation. In [4] and [21], the problem is changed by introducing a small parameter to overcome nondifferentiability and adding the constraint as a penalty term. A similar approach was used in [11] to recover constant images Manuscript received January 18, 2000; revised April 26, The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Helen Na. The authors are with the Centre for Technical Geoscience, Delft University of Technology, 2600 GA, Delft, The Netherlands ( abubakar@its.tudelft.nl). Publisher Item Identifier S (01) and in [10] to enhance the reconstruction of conductivity in a problem in electrical impedance tomography. This last paper inspired the work of Van den Berg and Kleinman [25] which marries the minimization of the total variation as suggested by Rudin et al., in the modified form suggested by Acar and Vogel [4], with a nonlinear constraint as in Dobson and Santosa [10]. The addition of the total variation to the cost functional has a very positive effect on the quality of the reconstruction for both blocky and smooth profiles [25], but a drawback is the presence of an artificial weighting parameter in the cost functional, which can only be determined through considerable numerical experimentations and a priori information of the desired reconstruction. Van den Berg et al. [27] has suggested to include the total variation regularizer as multiplicative constraint with the result that the original cost functional is the weighting parameter, i.e., determined by the optimization process itself. This eliminates the choice of the artificial regularization parameters completely. In this paper, as in [27] we suggest to include the total variation as a multiplicative constraint and show the extension and the applicability of this technique to different applications. We present the inversion results from various measurements setup (static, induction, and electromagnetic) in the 2-D and 3-D configurations (single-well and cross-well configurations in geophysics). Numerical results show that the multiplicative type of regularization is able to handle noisy data in a robust way without the usual necessary a priori information. II. NOTATION AND PROBLEM STATEMENT We assume a time harmonic dependence i, where i, is angular frequency, and is time. The position vector in is denoted by. Consider to be a bounded, not necessarily connected scattering object (or objects) whose location and index of refraction or contrast is unknown but which is known to lie within another, larger, bounded simply connected domain. The sources and receivers are located in a domain outside. A. The 2-D Electromagnetic Problem If denotes an incident electric field with line source located at, then for 2-D electromagnetic configurations which is invariant in the direction the total electric field in is known to satisfy the integral equation i (1) /01$ IEEE

2 ABUBAKAR AND VAN DEN BERG: TOTAL VARIATION AS A MULTIPLICATIVE CONSTRAINT 1385 where the zero-order Hankel function of the first kind. The embedding is chosen to be homogeneous and lossless with permittivity of vacuum. Therefore, the wavenumber and the contrast function is given by i (2) where and are the unknown relative permittivity and conductivity. Observe that if is not in the vanishes, but if the location of is unknown then it is not known a priori where vanishes. However, with the assumption that it is known that vanishes for outside. In fact denoting by a domain or curve, or a discrete collection of points outside, for each excitation the scattered electric field (data quantity) in can be represented by i B. The 3-D Electromagnetic/Induction Problem For the electromagnetic problem the total vector electric field in for each excitation satisfies i where i with the permeability of vacuum and the complex conductivity of the background medium.in (4), denotes the spatial differentiation with respect to the position vector. The contrast is given by (3) (4) with i (5) where is the electrical conductivity, and is the dielectric permittivity. Note that for the low-frequency electromagnetic measurements (induction logging), the second term in (5) can be neglected. Then, is real and positive. In this case, the vector scattered magnetic field is taken as the data quantities. The data is represented in by C. The 3-D Static Problem For the static problem (electrode logging) the total vector electric field in for each excitation satisfies where is given in (5) with is now the real-valued conductivity. In this case, the scattered electric potential is taken as the data quantities. The data is represented in by i (6) (7) (8) D. Operator Notation In order to discuss our solution of the inverse scattering problem we write our equations in an operator form and we denote the electric field component or electric field vector by the symbol and the data quantities either the scattered electric field component, magnetic field vector or scalar electric potential field by. We assume that the unknown object or formation is irradiated successively by a number of known incident fields,. For each incident field, the total field will be denoted by and the measured scattered field data are denoted by. Then, the data equations in (3), (6), and (8) are written as while, the object equations in (1), (4), and (7) written as (9) (10) The profile reconstruction problem is that of finding of the object domain for given at the data domain, or solving the data equation in (9) for, subject to the additional condition that and satisfy (10) in for each excitation. III. CONTRAST SOURCE INVERSION A major observation is that the integral representations (data equations) contain both the unknown field and the unknown contrast in the form of a product; it can be written as a single quantity, viz. the contrast source (11) which can be considered as an equivalent source that produces the measured scattered field. The data equation (9) becomes while the object equation (10) becomes (12) (13) Substituting (13) into (11), we obtain an object equation for the contrast source rather than for the field, viz. (14) Although the data equation (9) is linear in the contrast source, it is a classic ill-posed equation. Therefore, Van den Berg and Kleinman [26] recasted the problem as an optimization problem in which not only the contrast sources were sought but also the contrast itself to minimize a cost functional consisting of two terms. The first term contains the errors in the data equation while the second term contains the errors in the object equation, rewritten in terms of the contrast and the contrast source rather than the field. Inspired by the form of the cost functional of the modified gradient method [24], the method consists of an

3 1386 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 10, NO. 9, SEPTEMBER 2001 algorithm to construct sequences and which iteratively reduce the value of the cost functional in which is the updated contrast sources obtained from (16). We observe that the numerator of (18) is minimized by taking (20) (15) where and denote the norms on and, respectively. The normalizations are chosen in such a way that both terms are equal to one if. This is a quadratic functional in, but highly nonlinear in. The algorithm involves the construction of sequences and, for in the following manner. A. Updating of the Contrast Sources Now suppose and are known. We update by (16) However, because of the presence of the contrast in the denominator of the second term of the right-hand side of (15), the updating scheme may not be error-reducing. To remedy this problem, the error reduction can be enforced by updating the contrast also in the conjugate gradient direction, defining the updating scheme for as (21) where is a constant parameter and the update direction are functions of position. The update directions are again the Polak Ribière conjugate gradient directions, making the updating scheme consistent with the updating of the contrast sources. These update directions are obtained as where is a constant parameter and the update directions are functions of position. The update directions are chosen to be the Polak Ribière conjugate gradient directions, which search for improved directions when a change with respect to the directions of the last iteration occurs and restart the optimization when practically no changes are made in the subsequent gradients. These updaète directions are obtained as where Re (22) Re (17) (23) where is the gradient (Frèchet derivative) of the cost functional with respect to evaluated at and. Explicitly, the gradient for the updating of the contrast source is found in terms of adjoint operators and, respectively. With the update directions completely specified, the parameter in (15) and is found explicitly by minimizing cost functional in (15) (see [26]). B. Updating of the Contrasts First we observe that the contrast is only present in the second term of (15) where (24) From the last expression we observe that the update added to in (21) is in fact a preconditioned gradient of the numerator of (18) with respect to variations of. To find the constant parameter in (21), we minimize the second term of the cost functional (18) (25) where (19) When we take real-valued, the minimization of (25) can be carried out explicitly (see [3]).

4 ABUBAKAR AND VAN DEN BERG: TOTAL VARIATION AS A MULTIPLICATIVE CONSTRAINT 1387 C. Starting Values Observe that we cannot start with and, since then the cost functional (15) is undefined. Therefore we start with finding the contrast sources that minimize the data error Using gradient method, we arrive at (26) (27) is the back propagation of the data from the data domain into the object domain. With this initial estimates, the initial field and contrast estimates are obtained by object equations when the TV-factor has reached a nearly constant value. If noise is present in the data, the data error term will remain at a large value during the optimization and therefore, the weight of the TV-factor will be more significant. Hence, the noise will, at all times, be suppressed in the reconstruction process and we automatically fulfill the need of a larger when the data contains noise as suggested by Chan and Wong [6] and Rudin et al. [19]. The factor in (30) is introduced for restoring differentiability to the TV-factor. We have chosen the value of to be large in the beginning of the optimization and small toward the end. In this way, the optimization will reconstruct the contrast in the first iterations in the normal way, before it will apply the minimization of variation to shape the image further. In particular, we have chosen (32) and (28) IV. TOTAL VARIATION MINIMIZATION According to [4], [11] and [25] the total variation (TV) can be incorporated as a penalty factor by defining where the TV-factor is defined as (29) (30) The choice of the weighting parameter is of significant importance for the final results of the inversion procedure. In the past years, there has been some experience with this weighting parameter and normally a constant value of around 10 or 10 is used (see, e.g., Dobson and Santosa [10], Kohn and McKenney [14], and Van den Berg and Kleinman [26]). The weighting parameter depends on the amount of variation in the scatterer and therefore must be modified in every problem. In order to overcome the problem of the artificial parameter, we introduce a new cost functional, in which the optimization process itself determines the weight of the TV-factor. We define a cost functional as a product of two factors, viz. (31) The new cost functional (31) is based on two things: the objective of minimizing the error in the data equations and object equations and the observation that the TV-factor, when minimized, converges to a constant factor. The structure of the new cost functional is such that it will minimize the TV-factor with a large weighting parameter in the beginning of the optimization process, because the value of is still large, and that it will gradually minimize more and more the error in the data and in which is the normalized error in the object equations, see (18). For a small number of iterations is large, while it decreases for an increasing number of iterations. By introducing this cost functional, the TV-factor actually influences the updating of the contrast source, because changes in each iteration. But, because the TV-factor does not actually change itself and therefore is only a multiplication factor when searching for a new update of,we do not change the updating scheme for the contrast sources; the contrast sources are determined by minimizing in each iteration the cost functional of (15), being the first factor of (31). The updating scheme for is given in (21), where the update directions are defined as Polak Ribière conjugate gradient directions of the cost functional (31). Using (22) for the update directions, the gradient is determined as where is the numerator of the second factor in (23) and is the gradient of the TV-factor, given by (33) (34) The weighting of the gradients clearly depends on the errors in the cost functional and the TV-factor. The real-valued constant in (21) is now found to minimize (35)

5 1388 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 10, NO. 9, SEPTEMBER 2001 This minimization can not be calculated analytically and is therefore determined by a numerical line minimization using the Fletcher Reeves Polak Ribière conjugate gradient algorithm of numerical recipes (Press et al. [17]). In this procedure, we take as initial value for the analytical expression by minimizing (31), obtained in absence of the TV-factor (see [3]). V. NUMERICAL EXAMPLE In this section, we will demonstrate the improvement of incorporating the TV-factor in our inversion method using electromagnetic, induction and static data in 2-D and 3-D configurations. A. The 2-D Inversion from Electromagnetic Data As the first example, we consider inversion of an object which is invariant in the direction illuminated by electric line sources. The test domain consists of a square with sides of length, while the measurement curve is a circle of radius, where is specified in terms of the wavelength. The homogeneous embedding is chosen to be lossless and therefore the wavenumber. The discrete form of the algorithm is obtained by dividing into rectangular subdomains, assuming the contrast, sources and fields to be piecewise constant and the integrals over subdomains were approximated by integrals over circles of equal area which were calculated analytically (see Richmond [18]). The incident fields were chosen to be excited by line sources parallel to the axis of the scatterer. The line sources were taken to be equally spaced on the measurement circle, and the source locations were also chosen as discretization points on the circle. The measured data were simulated by solving the direct scattering problem with a conjugate gradient fast Fourier transform (CGFFT) method. The circle was subdivided into 29 equally spaced arcs, each mid-point serving as the location of a line source as well as a receiver. The number of data is then equal to the number of unknown contrast values in the domain, viz., In Fig. 1(a), a scattering object is defined that consists of concentric square cylinders, an inner cylinder of dimension by, with complex contrast, surrounded by an outer cylinder, by, with contrast. The test domain is a square of dimension by. In the inversion we have used a priori information that the real part and the imaginary part of the contrast is positive, we remark that this positivity constraint is easily implemented by enforcing a negative value to zero after each update of the contrast. Numerical experiments have shown that this simple adjustment leads to reconstruction results, which are very similar to the ones of the original CSI method with positivity constraint [26]. The reconstruction results after 128 iterations are shown in Fig. 1(b) and (c). In the latter one, we have used the minimization with TV-factor. After 128 iterations the normalized errors in the data equation [the first term in (16)] are reduced to 0.39% for the algorithm without using TV-factor and to 0.32% for the algorithm with using TV-factor. The contrast error of minimization with TV-factor is 10% smaller than the contrast Fig. 1. Example of 2-D electromagnetic tomography experiment. (a) Original profile and reconstruction results using CSI method (b) without and (c) with TV-factor; 29 line sources and line receivers located on the measurement circle S were used. Fig. 2. Reconstruction results using CSI method (a) without and (b) with TV-factor; 10% noise. error without TV-factor. Further, the value of the original cost functional of CSI method with TV-factor is less than the one without TV-factor. It is anticipated that the multiplicative TV method amplifies the minimization of total variation when noise is present in the data, because noisy data will make the value of the cost functional larger. Therefore, we again corrupted the data with additive white noise of 10% of the maximum value of all the measured data. For these noisy data, the reconstructed profiles without TV-factor is given in Fig. 2(a). The reconstructed profiles with TV-factor as shown in Fig. 2(b) show a remarkable improvement. The influence of the TV-factor has become greater and has damped out all the oscillations. Note that now the normalized errors in the data equation are reduced to 5.44% for the algorithm without using TV-factor and to 3.04% for the algorithm with using TV-factor. Finally, we tested our multiplicative TV method with substantially less data, still with presence of 10% noise. Using ten instead of 29 source/receiver stations, Fig. 3(a) and (b) show a

6 ABUBAKAR AND VAN DEN BERG: TOTAL VARIATION AS A MULTIPLICATIVE CONSTRAINT 1389 Fig. 3. Same as Fig. 2, but now with ten stations instead of 29. Fig. 4. Example of 3-D cross-well induction tomography experiment. This configuration has two-blocks of m with =5 S/m in the test domain D with conductivity = 0:1 S/m. The dimensions of the object domain D are m. The lateral positions of the boreholes are (x ;x)=(050;0); (50;0); (0;050);(0;50). comparison of reconstructions using our CSI method without and with TV-factor. With this very limited and noisy data set (29 29 unknown contrast values and data points), the reconstruction is remarkable; numerical experiments demonstrated that, for even smaller number of source/receiver locations, our multiplicative TV algorithm still attempts to reconstruct the original image, while original CSI method clearly lacks data. B. The 3-D Inversion Cross-Well Induction Logging Data We consider the 3-D model shown in Fig. 4. A dual-block model with conductivity S/m located in a background medium with conductivity S/m. Each block has dimensions of m. We discretize the test domain in a rectangular mesh. The mesh is uniformly spaced the,, and directions with widths of,, and. In each subdomain, we assume the contrast to be constant. The details of the discretization procedure can be found in [1]. Thus, the m object domain is divided into subdomains of m. Hence, the total number of the rectangular subdomains is equal to The incident fields were chosen to be excited by point magnetic dipoles directed in the vertical direction. The synthetic data are generated by solving the forward scattering problem using the CGFFT method with twice the number of grid points we use in our inversion scheme. In this example 24 point sources provide the source of excitations. These sources are located in the four borehole at. Each source excitation located on a particular borehole has 27 multicomponents receivers located in the other three boreholes. Thus, in total we have 648 data points. In the inversion we have used a priori information that the electrical conductivity is positive. The reconstruction results after 1024 iterations from induction logging data at 20 khz are given in Fig. 5(a) and (b). Although the total number of the iteration is large, note that we do not solve a full forward problem in each iteration of the inversion procedure. One iteration of our scheme takes approximately 27 s. The latter figure gives the reconstruction results using CSI method with TV-factor. Note that after 128 iterations the normalized errors in the data equation [the first term in (16)] are reduced to 0.40% for the algorithm without using TV-factor and to 0.31% for the algorithm with using TV-factor. C. The 3-D Inversion from Single-well Induction Logging Data We consider a formation consisting four beds (two with invasion), and a borehole deviated with respect to the intersecting layers as shown in Fig. 6. The figure is a cross section of a 3-D conductivity distribution in the plane at. The conductivity from top to bottom are 1, 0.45, 0.01, and 0.45 S/m. The dimension of each layer in logging direction is 0.3 m. The conductivity of the invasion zone of the first (from the top) layer is 0.6 S/m, and of the third layer is 0.2 S/m. The radius of the borehole is 0.1 m. This formation is assumed to be embedded in an unbounded homogeneous domain with conductivity S/m. This single-well logging problem has been formulated in term of an integral representation in an oblique (nonorthogonal) coordinate system (see [2]). The incident fields were chosen to be excited by point magnetic dipoles directed in the borehole direction. The synthetic data are generated by solving the forward scattering problem using the CGFFT method with twice the number of grid points we use in our inversion scheme. In the inversion we use 15 different source and receiver positions at the borehole axis with spacing of 0.2 m. Thus, for each experiment we have 225 data points. In the reconstruction it was assumed that the unknown configuration was located entirely within a test domain of m. This test domain was partitioned into equal-sized subdomains with side lengths 0.1 m in all three-directions. Thus, the total grid points amount to The a priori information that the conductivity is positive and symmetric around the borehole have been used. A plot of the inverted conductivity distribution after 1024 iterations from the induction logging data at 20 khz using CSI method is given in Fig. 7(a), while the results using CSI method

7 1390 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 10, NO. 9, SEPTEMBER 2001 Fig. 6. Example of a 3-D single-well induction tomography experiment. This figure is a cross-section of 3-D conductivity distribution in the (x ;x) plane at x =0. This configuration contains four layers (two with invasion zones) and a borehole deviated 45 with respect to the intersecting layers. In borehole we have 15 sources and 14 receivers. Fig. 7. Plots of the conductivity distribution inverted from induction logging data (a) without and (b) with TV-factor. Fig. 5. Plots of the inverted conductivity distribution from the induction logging data at 20 KHz using CSI method (a) without and (b) with TV-factor. These plots give four slices of the 3-D conductivity distributions at the planes, x = 010 m, x =10m, x = 012:5 m, and x =12:5 m. with TV-factor is given in Fig. 7(b). After 1024 iterations the normalized errors in the data equation [the first term in (16)] are reduced to 0.26% for the algorithm without using TV-factor and to 0.24% for the algorithm with using TV-factor. Here, we observe again the improvement of the reconstruction results using CSI method with TV-factor. As seen in these figure, the reconstruction is quite reasonable despite the large variation of the conductivity contrast distribution in the unknown formation. So far, the results are represented for noiseless synthetic data. In order to simulate a more realistic field experiment we have added 10% random additive white noise to the synthetic data. The reconstruction results using the CSI method is given in Fig. 8(a) and the one with TV-factor is given in Fig. 8(b). We observe that in the reconstruction results using CSI method with TV-factor we still able to see the different between the high and low conductivity region (the first and the third from the top layer). Note that now the normalized errors in the data equation are reduced to 4.63% for the algorithm without using TV-factor and to 4.29% for the algorithm with using TV-factor. D. The 3-D Inversion from Single-Well Electrode Logging Data Next, to test whether the present approach also work in a more nonlinear problem, we carry the same experiments as for the 3-D single-well induction logging problem. The incident fields are excited by point electrodes. We measure the scalar scattered electric potential fields. The reconstructed profiles from the electrode logging data using CSI method without and with TV-factor are given in Fig. 9(a) and (b). After 1024 iterations the normalized errors in the data equation [the first term in (16)] are reduced to 0.27% for the algorithm without using TV-factor and to 0.15% for the algorithm with using TV-factor. We observe again the improvement of using TV-factor

8 ABUBAKAR AND VAN DEN BERG: TOTAL VARIATION AS A MULTIPLICATIVE CONSTRAINT 1391 Fig. 8. Same as Fig. 7, but now with 10% noise. Fig. 10. Same as Fig. 9, but now with 10% noise. seems to be robust, handling noisy as well as limited data very well without the necessity of choosing weighting parameters and without a significant increase of computation time. Fig. 9. Plots of the conductivity distribution inverted from electrode logging data (a) without and (b) with TV-factor. especially in the first layer from the top. The reconstructed profiles from the electrode logging data with 10% additive random white noise are given in Fig. 10(a) and (b). Now the normalized errors in the data equation are reduced to 3.55% for the algorithm without using TV-factor and to 1.99% for the algorithm with using TV-factor. VI. CONCLUSION A drawback of adding a TV-factor, is the use of regularization parameters, which must be determined by carrying out many numerical experiments. To overcome these artificial parameters, we have defined a new cost functional, which is the product of the original cost functional, consisting of the normalized errors in the data and object equations, and the TV-factor. This method makes use of a conjugate gradient method to update the contrast sources, followed by a conjugate gradient step updating the contrast. Numerical experiments showed that the algorithm REFERENCES [1] A. Abubakar and P. M. van den Berg, Three dimensional nonlinear inversion in cross-well electrode logging, Radio Sci., vol. 33, no. 4, pp , [2], Nonlinear inversion in electrode logging in a highly deviated formation with invasion using an oblique coordinate system, IEEE Trans. Geosci. Remote Sensing, vol. 38, pp , Jan [3], Non-linear three-dimensional inversion of cross-well electrical measurements, Geophys. Prospect., vol. 48, no. 1, pp , [4] R. Acar and C. R. Vogel, Analysis of bounded variation penalty methods for ill-posed problems, Inv. Probl., vol. 10, no. 6, pp , [5] D. L. Alumbaugh and H. F. Morrison, Theoretical and practical considerations for crosswell electromagnetic tomography assuming a cylindrical geometry, Geophysics, vol. 60, pp , [6] T. F. Chan and C. K. Wong, Total variation blind deconvolution, IEEE Trans. Image Processing, vol. 7, pp , Mar [7] W. C. Chew and Y. M. Wang, Reconstruction of two-dimensional permittivity distribution using the distorted Born iterative method, IEEE Trans. Med. Imag., vol. 9, pp , [8] D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory. Berlin, Germany: Springer, [9] D. Colton and P. Monk, The numerical solution of an inverse scattering problem for acoustic waves, J. Appl. Math., no. 49, pp , [10] D. C. Dobson and F. Santosa, An image-enhancement technique for electrical impedance tomography, Inv. Probl., vol. 10, no. 2, pp , [11] D. C. Dobson and F. Santosa, Recovery of blocky images from noisy and blurred data, SIAM J. Appl. Math., no. 56, pp , [12] S. Gutman and M. Klibanov, Three-dimensional inhomogeneous media imaging, Inv. Probl., vol. 10, no. 6, pp. L36 L46, [13] T. M. Habashy, M. L. Oristaglio, and A. T. de Hoop, Simultaneous nonlinear reconstruction of two-dimensional permittivity and conductivity, Radio Sci., vol. 29, no. 4, pp , [14] R. V. Kohn and A. McKenny, Numerical implementation of a variational method of a variational method for electrical impedance tomography, Inv. Probl., vol. 6, pp , [15] G. Newman, Crosswell electromagnetic inversion using integral and differential equations, Geophys., vol. 60, pp , [16] S. Osher and L. Rudin, Feature-oriented image enhancement using shock filters, SIAM J. Appl. Math., no. 29, pp , [17] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in Fortran: The Art of Scientific Computing. New York: Cambridge Univ. Press, 1992, pp

9 1392 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 10, NO. 9, SEPTEMBER 2001 [18] J. H. Richmond, Scattering by a dielectric cylinder of arbitrary cross section shape, IEEE Trans. Antennas Propagat., vol. AP-13, pp , [19] L. Rudin, S. Osher, and C. Fatemi, Nonlinear total variation based noise removal algorithm, Phys. D, vol. 60, pp , [20] H. A. Sabbagh and R. G. Lautzenheiser, Inverse problem in electromagnetic nondestructive evaluation, Int. J. Appl. Electromagn. Mater., no. 3, pp , [21] F. Santosa and W. Symes, Reconstruction of blocky impedance profile from normal-incidence reflection seismograms which are band-limited and miscalibrated, Wave Motion, no. 10, pp , [22] W. Tabbara, B. Duchene, C. Pichot, D. Lesselier, L. Chommeloux, and N. Joachimowicz, Diffraction tomography: Contribution to the analysis of applications in microwaves and ultrasonics, Inv. Probl., vol. 4, no. 2, pp , [23] C. Torres-Verdin and T. M. Habashy, Rapid 2.5-dimensional forward modeling and inversion via a new nonlinear scattering approximation, Radio Sci., vol. 29, no. 4, pp , [24] P. M. van den Berg and R. E. Kleinman, A modified gradient method for two-dimensional problems in tomography, J. Comput. Appl. Math., vol. 42, pp , [25] P. M. van den Berg and R. E. Kleinman, A total variation enhanced modified gradient algorithm for profile reconstruction, Inv. Probl., vol. 11, pp. L5 L10, [26] P. M. van den Berg and R. E. Kleinman, A contrast source inversion method, Inv. Probl., vol. 13, no. 6, pp , [27] P. M. van den Berg, A. L. van Broekhoven, and A. Abubakar, Extended contrast source inversion, Inv. Probl., vol. 15, pp , [28] C. R. Vogel and M. E. Oman, Iterative methods for total variation denoising, SIAM J. Sci. Comput., vol. 17, pp , [29] Q. Zhou, A. Becker, and H. F. Morrison, Audio-frequency electromagnetic tomography in 2-D, Geophysics, vol. 58, pp , Peter M. van den Berg was born in Rotterdam, The Netherlands, in He received the degree in electrical engineering from the Polytechnical School of Rotterdam in 1964, the B.Sc. and M.Sc. degrees in electrical engineering, and the Ph.D. degree in technical sciences, all from the Delft University of Technology, Delft, The Netherlands, in 1966, 1968, and 1971, respectively. From 1967 to 1968, he was a Research Engineer with the Dutch Patent Office. Since 1968, he has been a Member of the Scientific Staff of the Electromagnetic Research Group, Delft University of Technology. During these years, he carried out research and taught classes in the area of wave propagation and scattering problems. During the academic year , he was Visiting Lecturer in the Department of Mathematics, University of Dundee, Scotland, financed by an award from the Niels Stensen Stichting, The Netherlands. During , he was a Visiting Scientist at the Institute of Theoretical Physics, Goteborg, Sweden. He was appointed Professor at the Delft University of Technology in From 1988 to 1994, he carried out research at the Center of Mathematics of Waves, University of Delaware, Newark. During the summers of , he was a Visiting Scientist at Shell Research B.V., Rijswijk, The Netherlands. Since 1994, he has beem a Professor in the Delft Research School Centre of Technical Geoscience. His main research interest is the efficient computation of field problems using iterative techniques based on error minimization, the computation of fields in strongly inhomogeneous media, and the use of wave phenomena in seismic data processing. Major interest is in an efficient solution of the nonlinear inverse scattering problem. Dr. van den Berg is the receipient of a NATO Award. Aria Abubakar was born in Bandung, Indonesia, in He received the M.Sc. degree (cum laude) in electrical engineering and the Ph.D. degree (cum laude) in technical sciences, both from the Delft University of Technology, Delft, The Netherlands, in 1997 and 2000, respectively. In 1996, he was a Research Student with Shell Research B.V., Rijswijk, The Netherlands. He was a Summer Intern with Schlumberger-doll Research, Ridgefield, CT, in He is currently with the Laboratory of Electromagnetic Research and Section of Applied Geophysics, Delft University of Technology. His research includes solving forward and inverse scattering problems in acoustic, electromagnetic and elastodynamic. Dr. Abubakar is the recipient of the Best 1997 Master s Thesis Award in electrical engineering given by Delft University of Technology.