## A family of approximations spanning the Born and Rytov scattering series

Optics Express, Vol. 14, Issue 19, pp. 8837-8848 (2006)

http://dx.doi.org/10.1364/OE.14.008837

Acrobat PDF (184 KB)

### Abstract

A new hybrid scattering series is derived that incorporates as special cases both the Born and Rytov scattering series, and includes a parameter so that the behavior can be continuously varied between the two series. The parameter enables the error to be shifted between the Born and Rytov error terms to improve accuracy. The linearized hybrid approximation is derived as well as its condition of validity. Higher order terms of the hybrid series are also found. Also included is the integral equation that defines the exact solution to the forward scattering problem as well as its Fréchet derivative, which is used for the solution of inverse multiple scattering problems. Finally, the linearized hybrid approximation is demonstrated by simulations of inverse scattering off of uniform circular cylinders, where it is shown that the hybrid approximation achieves smaller error than either the Born or Rytov approximations alone.

© 2006 Optical Society of America

## 1. Introduction

3. G. Beylkin and M. L. Oristaglio, “Distorted-wave Born and distorted-wave Rytov approximations,” Opt. Commun. **53**, 213–216 (1985). [CrossRef]

4. S. D. Rajan and G. V. Frisk, “A comparison between the Born and Rytov approximations for the inverse backscattering problem,” Geophysics **54**, 864–871 (1989). [CrossRef]

5. M. J. Woodward, “Wave-equation tomography,” Geophysics **57**, 15–26 (1992). [CrossRef]

6. M. I. Sancer and A. D. Varvatsis, “A comparison of the Born and Rytov methods,” Proc. IEEE **58**, 140–141 (1970). [CrossRef]

7. M. Slaney, A. C. Kak, and L. E. Larsen, “Limitations of imaging with first-order diffraction tomography,” IEEE Trans. Microwave Theory Tech. **MTT-32**, 860–874 (1984). [CrossRef]

8. F. C. Lin and M. A. Fiddy, “The Born-Rytov controversy: I. Comparing the analytical and approximate expressions for the one-dimensional deterministic case,” J. Opt. Soc. Am. A **9**, 1102–1110 (1992). [CrossRef]

9. F. C. Lin and M. A. Fiddy, “Born-Rytov controversy: II Applications to nonlinear and stochastic scattering problems in one-dimensional half-space media,” J. Opt. Soc. Am. A **10**, 1971–1983 (1993). [CrossRef]

*e.g.*refractive index or acoustic impedance). Therefore, the reconstructed contrast is usually estimated as linear in the detected field. The Born approximation is accurate when the product of the index contrast and object size is less than one-quarter wavelength. On the other hand, the Rytov approximation is typically used when the field is detected in the forward scattering direction, as is employed in diffraction tomography, many geophysical problems, and X-ray computed tomography [10

10. G. Gbur and E. Wolf, “Relation between computed tomography and diffraction tomography,” J. Opt. Soc. Am. A **18**, 2132–2137 (2001). [CrossRef]

*n*becomes a parameter that can be varied from one to infinity. If

*n*=1, then

*u*=

*u*

_{0}+

*u*

_{0}

*ϕ*, which is the additive relationship assumed in the Born approximation, if we identify

*u*

_{s}=

*u*

_{0}

*ϕ*. This suggests that by using Eq. (2) as a definition of a generalized complex phase where

*n*can take any value from one to infinity, a range of models are available lying between the Born or Rytov approximations by varying the parameter

*n*. It is possible that intermediate values of

*n*may provide better accuracy in situations where neither the Born and Rytov approximations apply. Lu also considered generalized transformations [11

11. Z. Q. Lu, “Multidimensional structure diffraction tomography for varying object orientation through generalised scattered waves,” Inv. Prob. **1**, 339–356 (1985). [CrossRef]

*n*can be used to incorporate both the Born and Rytov models in the search for a solution to inverse scattering problems. This way, a solution to the multiple scattering problem can be consistently defined that uses both the Born and Rytov models. This will add a degree of flexibility that may enable inverse scattering problems to be solved where the scatterer has high contrast and large extent. For this reason, results pertaining to the application of the hybrid approximation to multiple scattering methods such as higher order terms of the hybrid series are included in this work.

## 2. Derivation of a Hybrid Approximation

*k*(

**r**) is the spatial frequency of the wave in the medium at position

**r**, and

*k*

_{0}is the freespace spatial frequency of the scalar field. We posit that the solution to this equation has the form

*u*

_{0}(

**r**) is an unperturbed field that satisfies the homogeneous wave equation ∇

^{2}

*u*

_{0}+

*u*

_{0}=0, and

*ϕ*(

*r*) is a “complex phase” that will be determined to find the scattered field. The function

*ϕ*will be expanded as a perturbation series

*ϕ*

_{0}=0 because the field

*u*

_{0}is defined as satisfying the unperturbed reduced wave equation, so that the lowest order non-zero term of

*ϕ*is first order. The constant

*n*will be a free parameter [1,∞]. This form is chosen so that

*u*(

**r**)=

*u*

_{0}(

**r**)+

*u*

_{0}(

**r**)

*ϕ*(

**r**) when

*n*=1, which is the model used for Born scattering with

*u*

_{0}(

**r**)

*ϕ*(

**r**) being the scattered field. As

*n*→∞,

*u*(

**r**)=

*u*

_{0}(

**r**)exp(

*ϕ*(

**r**)), which is the form of the total field in the Rytov approximation. By adjusting

*n*, we can produce an approximation that is intermediate between the two approximations.

*k*

^{2}-

*κ*=

*k*

^{2}-

^{2}(

*u*

_{0}

*ϕ*)=

*u*

_{0}∇

^{2}

*ϕ*+2∇

*u*

_{0}·∇

*ϕ*+

*ϕ*∇

^{2}

*u*

_{0}, and that ∇

^{2}

*u*

_{0}=-

*u*

_{0}, which combined yield ∇

^{2}(

*u*

_{0}

*ϕ*)+

*u*

_{0}

*ϕ*)=

*u*

_{0}∇

^{2}

*ϕ*+2∇

*u*

_{0}·∇

*ϕ*. Substituting this into Eq. (7) and rearranging terms, one finds that

*G*(

**r**

^{′},

**r**) is the Green’s function of the homogeneous wave equation. Note that no approximations have been made yet. An interesting feature that follows from the derivation is that when

*n*=1, and the quantity

*u*

_{0}is added to both sides of this equation, then this equation is the Lippmann-Schwinger equation. To linearize this integral to form the first-order approximation, we explicitly list the quantities summed inside the integral on the right hand side of Eq. (9) separately to find their respective orders of ε:

*ε*-order of ϕ that is non-zero is first order. Equation (10) is the product of a term

*ε*

^{0}), and ∇

_{ϕ}·∇

*ϕ*of order ε

^{2}, so this term is ε

^{2}order. Equation (11) is of order ε

^{1}, because it does not contain a

*ϕ*, only an ε. Finally, Eq. (12) is of order

*ε*

^{2}, because it contains the product of

*ϕ*and ε. Therefore to first order only the term of Eq. (11) needs to be retained, so that to a first-order approximation Eq. (9) is

*n*, so it has the same form for any value of

*n*. This is why the first Born and Rytov approximations have the same form; only the definition of

*ϕ*is different. Therefore

*n*can be chosen as needed as long as the conditions of the approximation are satisfied. To derive these conditions, we require that the omitted terms produce a contribution to the complex phase much smaller than one:

*u*

_{0}(

**r**) as a constant, and assuming that

*G*(

**r**

^{′},

**r**)≈|

**r**-

**r**

^{′}|

^{-1}is a worst-possible case of constructive interference between the fields scattered inside the volume, then this can be simplified to

*ϕ*, denoted the average wave number perturbation by Δ

*k*(which is of order

*ε*), and the average distance between points in the object (or alternately, the radius of a sphere enclosing the scatterer) as

*R*. This inequality combines the constraints of both the Born and the Rytov approximations. The Rytov constraint, which is weighted by

*n*, one can trade off the amount of error in the reconstruction produced by either the Born or Rytov models, so that accuracy can be maintained when neither of the models separately applies.

## 3. Higher-Order Terms of the Hybrid Series

*ϕ*is inserted into the following equation, which is Eq. (7) after the equation is multiplied by

*ε*

^{p}for

*p*>1:

^{2}+

*u*

_{0}

*ϕ*

_{m})=2∇

*u*

_{0}·

*ϕ*

_{m}+

*u*

_{0}∇

^{2}

*ϕ*

_{m}, the following recurrence relation is obtained for (∇

^{2}+

*u*

_{0}

*ϕ*

_{p}):

*u*

_{0}

*ϕ*

_{p}can be transformed into integral solutions using the Green’s function in a manner identical to the transformation between Eq. (8) and Eq. (9).

6. M. I. Sancer and A. D. Varvatsis, “A comparison of the Born and Rytov methods,” Proc. IEEE **58**, 140–141 (1970). [CrossRef]

*ε*

^{p}by expanding the right hand side of the equation with the binomial theorem. If this is done, the relationship between the first three Born and the hybrid series terms are:

*u*

_{p}is always

*u*

_{0}

*ϕ*

_{p}, the Born terms up to order

*p*can be calculated, and then the term

*ϕ*

_{p}can be calculated from

*ϕ*

_{1}to

*ϕ*

_{p-1}and

*u*

_{p}. This may be a more rapid way to calculate

*ϕ*

_{p}. These higher-order forward-scattering terms can be used to compute an inverse-scattering series as given in [13

13. G. A. Tsihrintzis and A. J. Devaney, “Higher order (nonlinear) diffraction tomography: inversion of the Rytov series,” IEEE Trans. Inf. Theory **46**, 1748–1761 (2000). [CrossRef]

## 4. Evaluating the Fréchet derivative for Inverse Multiple Scattering

14. W. C. Chew and Y. M. Wang, “Reconstruction of two-dimensional permittivity distribution using distorted Born iterative method,” IEEE Trans. Med. Imaging **9**, 218–225 (1990). [CrossRef] [PubMed]

15. R. E. Kleinman and P. M. van der Berg, “A modified gradient method for two-dimensional problems in tomography,” J. Comput. Appl. Math **42**, 17–35 (1992). [CrossRef]

16. R. E. Kleinman and P. M. van der Berg, “An extended-range modified gradient technique for profile inversion,” Radio Sci. **28**, 877–884 (1993). [CrossRef]

13. G. A. Tsihrintzis and A. J. Devaney, “Higher order (nonlinear) diffraction tomography: inversion of the Rytov series,” IEEE Trans. Inf. Theory **46**, 1748–1761 (2000). [CrossRef]

17. K. Belkebir and A. G. Tijhuis, “Modified gradient method and modified Born method for solving a two-dimensional inverse scattering problem,” Inv. Prob. **17**, 1671–1688 (2001). [CrossRef]

*ϕ*and

*κ*, which is the analogue of the Lippmann-Schwinger equation for the hybrid complex phase. This method is akin to the Modified Gradient Method, but allows an arbitrary

*n*to be used rather than the Born approximation for which

*n*=1. The functional that will be minimized to estimate a solution to Eq. (9) is

*L*(

*ϕ*,

*κ*) and is given by

*γ*-norm of the total error in Eq. (9) integrated over the entire scatterer volume, where the exponent

*γ*determines the norm to be minimized. The quantity

*M*(

**r**′) is the error in Eq. (9) at point

**r**′ The equation has been further divided into

*M*

_{R}and

*M*

_{B}components which is the Rytov and Born contributions to the error

*M*. The function

*q*() is an optional regularization term constraining the reconstructed

*κ*and potentially its gradient. Minimizing

*L*finds the minimum-norm solution to Eq. (9) for

*ϕ*and

*κ*. To find this minimum, we find the Fréchet derivatives of

*L*with respect to these quantities so that a gradient descent method can be used to minimize

*L*. These derivatives can be computed using the Euler-Lagrange derivative formula

*M*

_{R}, which is weighted by

*M*

_{B}, which is weighted by

*ϕ*that conform to the assumptions of either approximation, and the derivative is a weighted sum of these. One should note that the second term of

*n*is not 1 or ∞, so it effectively couples the Born and Rytov approximations together in a way that would not be achieved if the Born and Rytov search directions were used separately. The complex phase, as defined in Eq. (4), may be advantageous over the conventional Born and Rytov definitions because it enables one to find a solution where both the Born and Rytov search directions can be used consistently to find a solution to Eq. (9).

*ϕ*(

**r**) and

*κ*(

**r**) that successively minimize

*L*. With both the Born and Rytov search directions available, there is greater flexibility in finding a solution that can solve the multiple scattering problem without stagnation. Unlike other methods such as the Born Iterative Method or Newton-Kantorovich, because Eq. (9) is nonlinear in

*ϕ*, a successive approximation approach may be more efficient than directly solving for

*κ*(

**r**) and

*ϕ*(

**r**) each iteration, and it can allow the regularization to be introduced in a consistent way.

## 5. Inverse Scattering using the Hybrid Series

*u*(

**r**) are made, and from these the complex phase can be computed by

*ϕ*(

**r**)=log(

*u*/

*u*

_{0}). Because the log function is multiple-valued for complex numbers, there is an ambiguity in

*ϕ*of a multiple of 2

*πi*. To account for this, typically the measured value of

*ϕ*is phase unwrapped as a function of position of the scattered field. The unwrapping is done such that the difference between the imaginary parts of two adjacent samples of

*ϕ*differs by less than

*π*. This complex phase is used to solve Eq. (13) for

*κ*(

**r**).

*ϕ*is given by

^{th}root of a complex number. This leads to an unwrapping problem similar to that which exists in the Rytov approximation, because there are

*n*roots of a given complex number if

*n*is an integer. To properly unwrap the n

^{th}root of a complex function

*f*(

*x*), one can write

*f*(

*x*)=|

*f*(

*x*)|exp[

*iθ*(

*x*)], where

*θ*(

*x*)=arg

*f*(

*x*), wrapped from [-

*π*,

*π*]. If

*θ*

^{′}(

*x*) is the unwrapped

*θ*(

*x*), then the unwrapped

*f*(

*x*)

^{1/n}=|

*f*(

*x*)|

^{1/n}exp[

*iθ*

^{′}(

*x*)/

*n*]. Alternatively, one can unwrap the logarithm of

18. G. A. Tsihrintzis and A. J. Devaney, “Higher-order (Nonlinear) Diffraction Tomography: Reconstruction Algorithms and Computer Simulation,” IEEE Trans. Imag. Proc. **9**, 1560–1572 (2000). [CrossRef]

13. G. A. Tsihrintzis and A. J. Devaney, “Higher order (nonlinear) diffraction tomography: inversion of the Rytov series,” IEEE Trans. Inf. Theory **46**, 1748–1761 (2000). [CrossRef]

19. G. A. Tsihrintzis and A. J. Devaney, “A Volterra series approach to nonlinear traveltime tomography,” IEEE Trans. Geo. Rem. Sens. **38**, 1733–1742 (2000). [CrossRef]

**G**

_{i}given by

^{th}Born-series term

*u*

_{0}

**G**

_{i}(

*κ*) of the field

*u*

_{0}scattered from an inhomogeneity κ. The integration can often be performed using a Fast Fourier Transform if the kernel

*G*(

**r**

^{′},

**r**) is shift-invariant. Next, operators are needed that calculate the terms of the hybrid series (based on Eqs. (20–22)):

*u*

_{0}corresponding to many illumination source locations for which the scattered phases

*ϕ*will be measured or computed. Braces around a phase {

*ϕ*} denote the set of all phase functions

*ϕ*that are computed or measured for all of the incident fields

*u*

_{0}.

**G**

_{1}(κ) is denoted by the operator

**B**({

*ϕ*}) which computes an estimate of

*κ*from all {

*ϕ*} such that

**B**[{

**G**

_{1}(

*κ*)}]≈

*κ*. This inverse corresponds to the linearized inverse diffraction tomography operator, implemented by the Fourier diffraction theorem [2] and the filtered backpropagation algorithm [20

20. A. J. Devaney, “A filtered back propagation algorithm for diffraction tomography,” Ultrason. Imaging **4**, 336–350 (1982). [CrossRef] [PubMed]

*ϕ*} is calculated from the measured fields {

*u*} using Eq. (27) including phase unwrapping. From this data function, the first two inverse terms of the hybrid series are calculated as follows:

*κ*=

*κ*

_{1}+

*κ*

_{2}. As a potentially simpler method, one may wish to explore applying the nonlinear inverse methods of [21

21. V. A. Markel, J. A. O’Sullivan, and J. C. Schotland, “Inverse problem in optical diffusion tomography. IV. Nonlinear inversion formulas,” J. Opt. Soc. Am. A **20**, 903–912 (2003). [CrossRef]

## 6. Simulation

22. M. Slaney, “Diffraction Tomography Algorithms from Malcolm Slaney PhD Dissertation,” obtained from http://rvl4.ecn.purdue.edu/~ malcolm/purdue/diffract.tar.Z.

7. M. Slaney, A. C. Kak, and L. E. Larsen, “Limitations of imaging with first-order diffraction tomography,” IEEE Trans. Microwave Theory Tech. **MTT-32**, 860–874 (1984). [CrossRef]

*n*for which the RMS error was minimized for the hybrid approximation.

*n*=2 may be a good starting point to find the optimal exponent. In addition to this figure, animations are available on-line that show the evolution of the computed reconstruction of three of these cylinders from the Born to the Rytov solutions. [Media 1, Media 2, Media 3]

*N*being the refractive index of the cylinder, and

*R*being the radius of the cylinder in wavelengths. The equation for

*n*

_{opt}suggests that perhaps the optimal exponent is proportional to the product of the index contrast and the area of the object. However, with a sufficiently large index difference and object area neither the first Born, Rytov, or the hybrid approximation can be expected to provide accurate results. In compiling the figures, hybrid reconstructions that produced optimal RMS errors greater than 40% were excluded from the table because it was decided that these points produced results too inaccurate to be of interest. Therefore it can not be expected that the empirical formula will be accurate when the RMS error is over 40%.

## Acknowledgements

## References and links

1. | W. C. Chew, |

2. | A. C. Kak and M. Slaney, |

3. | G. Beylkin and M. L. Oristaglio, “Distorted-wave Born and distorted-wave Rytov approximations,” Opt. Commun. |

4. | S. D. Rajan and G. V. Frisk, “A comparison between the Born and Rytov approximations for the inverse backscattering problem,” Geophysics |

5. | M. J. Woodward, “Wave-equation tomography,” Geophysics |

6. | M. I. Sancer and A. D. Varvatsis, “A comparison of the Born and Rytov methods,” Proc. IEEE |

7. | M. Slaney, A. C. Kak, and L. E. Larsen, “Limitations of imaging with first-order diffraction tomography,” IEEE Trans. Microwave Theory Tech. |

8. | F. C. Lin and M. A. Fiddy, “The Born-Rytov controversy: I. Comparing the analytical and approximate expressions for the one-dimensional deterministic case,” J. Opt. Soc. Am. A |

9. | F. C. Lin and M. A. Fiddy, “Born-Rytov controversy: II Applications to nonlinear and stochastic scattering problems in one-dimensional half-space media,” J. Opt. Soc. Am. A |

10. | G. Gbur and E. Wolf, “Relation between computed tomography and diffraction tomography,” J. Opt. Soc. Am. A |

11. | Z. Q. Lu, “Multidimensional structure diffraction tomography for varying object orientation through generalised scattered waves,” Inv. Prob. |

12. | Z.-Q. Lu, “JKM Perturbation Theory, Relaxation Perturbation Theory, and Their Applications to Inverse Scattering: Theory and Reconstruction Algorithms,” IEEE Trans. Ultra. Ferr. Freq. Cont. |

13. | G. A. Tsihrintzis and A. J. Devaney, “Higher order (nonlinear) diffraction tomography: inversion of the Rytov series,” IEEE Trans. Inf. Theory |

14. | W. C. Chew and Y. M. Wang, “Reconstruction of two-dimensional permittivity distribution using distorted Born iterative method,” IEEE Trans. Med. Imaging |

15. | R. E. Kleinman and P. M. van der Berg, “A modified gradient method for two-dimensional problems in tomography,” J. Comput. Appl. Math |

16. | R. E. Kleinman and P. M. van der Berg, “An extended-range modified gradient technique for profile inversion,” Radio Sci. |

17. | K. Belkebir and A. G. Tijhuis, “Modified gradient method and modified Born method for solving a two-dimensional inverse scattering problem,” Inv. Prob. |

18. | G. A. Tsihrintzis and A. J. Devaney, “Higher-order (Nonlinear) Diffraction Tomography: Reconstruction Algorithms and Computer Simulation,” IEEE Trans. Imag. Proc. |

19. | G. A. Tsihrintzis and A. J. Devaney, “A Volterra series approach to nonlinear traveltime tomography,” IEEE Trans. Geo. Rem. Sens. |

20. | A. J. Devaney, “A filtered back propagation algorithm for diffraction tomography,” Ultrason. Imaging |

21. | V. A. Markel, J. A. O’Sullivan, and J. C. Schotland, “Inverse problem in optical diffusion tomography. IV. Nonlinear inversion formulas,” J. Opt. Soc. Am. A |

22. | M. Slaney, “Diffraction Tomography Algorithms from Malcolm Slaney PhD Dissertation,” obtained from http://rvl4.ecn.purdue.edu/~ malcolm/purdue/diffract.tar.Z. |

**OCIS Codes**

(100.3190) Image processing : Inverse problems

(110.2990) Imaging systems : Image formation theory

(290.3200) Scattering : Inverse scattering

**ToC Category:**

Scattering

**History**

Original Manuscript: June 26, 2006

Revised Manuscript: August 18, 2006

Manuscript Accepted: August 19, 2006

Published: September 18, 2006

**Citation**

Daniel L. Marks, "A family of approximations spanning the Born and Rytov scattering series," Opt. Express **14**, 8837-8848 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-19-8837

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### References

- W. C. Chew, Waves and Fields in Inhomogeneous Media (IEEE Press, Piscataway, NJ, 1995).
- A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE Press, New York, 1988).
- G. Beylkin and M. L. Oristaglio, "Distorted-wave Born and distorted-wave Rytov approximations," Opt. Commun. 53, 213-216 (1985). [CrossRef]
- S. D. Rajan and G. V. Frisk, "A comparison between the Born and Rytov approximations for the inverse backscattering problem," Geophysics 54, 864-871 (1989). [CrossRef]
- M. J. Woodward, "Wave-equation tomography," Geophysics 57, 15-26 (1992). [CrossRef]
- M. I. Sancer and A. D. Varvatsis, "A comparison of the Born and Rytov methods," Proc. IEEE 58, 140-141 (1970). [CrossRef]
- M. Slaney, A. C. Kak, and L. E. Larsen, "Limitations of imaging with first-order diffraction tomography," IEEE Trans. Microwave Theory Tech. MTT-32, 860-874 (1984). [CrossRef]
- F. C. Lin and M. A. Fiddy, "The Born-Rytov controversy: I. Comparing the analytical and approximate expressions for the one-dimensional deterministic case," J. Opt. Soc. Am. A 9, 1102-1110 (1992). [CrossRef]
- F. C. Lin and M. A. Fiddy, "Born-Rytov controversy: II Applications to nonlinear and stochastic scattering problems in one-dimensional half-space media," J. Opt. Soc. Am. A 10, 1971-1983 (1993). [CrossRef]
- G. Gbur and E. Wolf, "Relation between computed tomography and diffraction tomography," J. Opt. Soc. Am. A 18, 2132-2137 (2001). [CrossRef]
- Z. Q. Lu, "Multidimensional structure diffraction tomography for varying object orientation through generalized scattered waves," Inv. Prob. 1, 339-356 (1985). [CrossRef]
- Z.-Q. Lu, "JKM Perturbation Theory, Relaxation Perturbation Theory, and their Applications to Inverse Scattering: Theory and Reconstruction Algorithms," IEEE Trans. Ultra. Ferroelectr. Freq. Control UFFC-32, 722-730 (1986).
- G. A. Tsihrintzis and A. J. Devaney, "Higher order (nonlinear) diffraction tomography: inversion of the Rytov series," IEEE Trans. Inf. Theory 46, 1748-1761 (2000). [CrossRef]
- W. C. Chew and Y. M. Wang, "Reconstruction of two-dimensional permittivity distribution using distorted Born iterative method," IEEE Trans. Med. Imaging 9, 218-225 (1990). [CrossRef] [PubMed]
- R. E. Kleinman and P. M. van der Berg, "A modified gradient method for two-dimensional problems in tomography," J. Comput. Appl. Math 42, 17-35 (1992). [CrossRef]
- R. E. Kleinman and P. M. van der Berg, "An extended-range modified gradient technique for profile inversion," Radio Sci. 28, 877-884 (1993). [CrossRef]
- K. Belkebir and A. G. Tijhuis, "Modified gradient method and modified Born method for solving a two dimensional inverse scattering problem," Inv. Prob. 17, 1671-1688 (2001). [CrossRef]
- G. A. Tsihrintzis and A. J. Devaney, "Higher-order (Nonlinear) Diffraction Tomography: Reconstruction Algorithms and Computer Simulation," IEEE Trans. Image Process. 9, 1560-1572 (2000). [CrossRef]
- G. A. Tsihrintzis and A. J. Devaney, "A Volterra series approach to nonlinear traveltime tomography," IEEE Trans. Geosco. Remote Sens. 38, 1733-1742 (2000). [CrossRef]
- A. J. Devaney, "A filtered back propagation algorithm for diffraction tomography," Ultrason. Imaging 4, 336-350 (1982). [CrossRef] [PubMed]
- V. A. Markel, J. A. O’Sullivan, and J. C. Schotland, "Inverse problem in optical diffusion tomography. IV. Nonlinear inversion formulas," J. Opt. Soc. Am. A 20, 903-912 (2003). [CrossRef]
- M. Slaney, "Diffraction Tomography Algorithms from Malcolm Slaney PhD Dissertation," obtained from http://rvl4.ecn.purdue.edu/˜malcolm/purdue/diffract.tar.Z.

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