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Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 14, Iss. 19 — Sep. 18, 2006
  • pp: 8837–8848
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A family of approximations spanning the Born and Rytov scattering series

Daniel L. Marks  »View Author Affiliations


Optics Express, Vol. 14, Issue 19, pp. 8837-8848 (2006)
http://dx.doi.org/10.1364/OE.14.008837


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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

The mathematics of inverse scattering problems is important to optical imaging and microscopy, medical imaging, radar, acoustics, geophysics, and many other disciplines. Inverse scattering is the inference of properties of an inhomogeneous medium from detection of waves that are scattered by the medium. In general, scattering from inhomogeneous media [1

1. W. C. Chew, Waves and Fields in Inhomogeneous Media (IEEE Press, Piscataway, NJ, 1995).

] is a complex process where the scattered fields are nonlinear in the medium property. Because of the difficulty of modeling arbitrary inhomogeneous scatterers, approximations are frequently made that simplify the mathematics and modeling of scatterers in special cases. For various practical inverse scattering problems, specific tools have been developed that enable practical solution of specific problems. Over time, these tools have been synthesized into general methodologies that have enabled investigators in many disciplines to mutually improve the state of the art in inverse scattering. This work presents a new series that synthesizes two well-known scattering series into one. Here it is shown how two approximations typically used for linearized inverse scattering, the Born and Rytov approximations, are two extremes of a more generalized family of hybrid approximations, some of which can achieve better accuracy for particular inverse scattering problems than either the Born or Rytov approximations alone. In addition, it is shown how the hybrid approximations can be applied to more complicated multiple scattering problems.

The applicability and accuracy of the Born and Rytov scattering series has been well studied [2

2. A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE Press, New York, 1988).

, 3

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

, 4

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

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

, 6

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

, 7

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

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

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]

]. The Born approximation is frequently employed in geometries where the radiation is collected in the backscattering direction, such as monostatic radar, B-mode ultrasound, and optical coherence tomography. In the first Born approxmation, the scattered field is modeled as a linear function of the scattering contrast (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]

]. In the first Rytov approximation, the scattered field is modeled as an exponentially dependent function of the scattering contrast. The Rytov approximation is accurate when the square of the phase gradient is much less than the index contrast divided by the wavelength squared. The approximations are asymptotically equivalent for low-contrast, small objects. However, the areas of applicability of the two approximations is usually seen as mostly disjoint, because of the very different assumptions implicit in each about the relation of the field to the scatterer, either linear or exponential.

By using a common limiting form of the exponential function, it can be seen how the two are related. The Born series may be obtained by writing the field as u=u 0+u s , where u 0 is the incident field and us is the scattered field and a perturbative expansion in orders of the contrast is made for u 0. On the other hand, the Rytov approximation is derived by first expressing the total field u=u 0 expϕ, where one identifies ϕ as a complex phase of the scattered field. By an identity,

limn(1+xn)n=expx
(1)

one can rewrite the Rytov model as

u=u0limn(1+ϕn)n
(2)

Now consider removing the limit from Eq. (2), so that 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]

, 12

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. UFFC-32, 722–730 (1986).

] on the field that yield the Born and Rytov approximations, including an intermediate transform that can be varied between Born and Rytov. Here we explore the validity and utility of a specific field transformation that may be especially useful and adaptable to current inverse scattering methods.

2. Derivation of a Hybrid Approximation

To derive the hybrid approximation discussed above, we study scattering of a scalar wave in an inhomogeneous medium. The field satisfies the spatially inhomogeneous reduced wave equation:

2u+k2(r)u=0
(3)

where 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(r)=u0(r)(1+ϕ(r)n)n
(4)

where u 0(r) is an unperturbed field that satisfies the homogeneous wave equation ∇2 u 0+k02 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(r)=m=0εmϕm(r), where ε is the order parameter proportional to the magnitude of the index perturbation of the medium. Note that 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

1. W. C. Chew, Waves and Fields in Inhomogeneous Media (IEEE Press, Piscataway, NJ, 1995).

,∞]. 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.

Substituting Eq. (4) into Eq. (3), we find that

·[u0(1+ϕn)n]+k2[u0(1+ϕn)n]=0
(5)

Expanding out the differential operators, this becomes

(1+ϕn)n2u0+2(1+ϕn)n1u0·ϕ+u0(1+ϕn)n12ϕ+
u0n1n(1+ϕn)n2(ϕ·ϕ)+k2u0(1+ϕn)n=0
(6)

If we note that (1+ϕn)n2u0+(1+ϕn)nk02u0=0 and subtract this relation from Eq. (6), and then divide the resulting equation by (1+ϕn)n1, one finds that

2u0·ϕ+u02ϕ+u0n1n(1+ϕn)1(ϕ·ϕ)+εκu0(1+ϕn)=0
(7)

Note that we have inserted the order parameter ε to indicate that the index perturbation k 2-k02 is of first-order, and defined the contrast κ=k 2-k02 for brevity. Now we note that ∇2(u 0 ϕ)=u 02 ϕ+2∇u 0·∇ϕ+ϕ2 u 0, and that ∇2 u 0=-k02 u 0, which combined yield ∇2(u 0 ϕ)+k02(u 0 ϕ)=u 02 ϕ+2∇u 0·∇ϕ. Substituting this into Eq. (7) and rearranging terms, one finds that

2(u0ϕ)+k02(u0ϕ)=u0n1n(1+ϕn)1(ϕ·ϕ)εκu0(1+ϕn)
(8)

Equation (8) may be recast as an integral equation, with the following result obtained

u0(r')ϕ(r')=Vd3rG(r',r)u0(r)[n1n(1+ϕn)1(ϕ·ϕ)+εκ(r)(1+ϕn)]
(9)

where 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:

G(r',r)u0(r)[n1n(1+ϕn)1(ϕ·ϕ)]=𝒪(ε2)
(10)
G(r',r)u0(r)εκ(r)=(ε1)
(11)
G(r',r)u0(r)εκ(r)ϕn=𝒪(ε2)
(12)

As explained earlier, the lowest ε-order of ϕ that is non-zero is first order. Equation (10) is the product of a term (1+ϕn)1 which is of order zero (ε 0), and ∇ϕ·∇ϕ of order ε2, so this term is ε2 order. Equation (11) is of order ε1, 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

u0(r')ϕ(r')=εVd3rG(r',r)κ(r)u0(r)
(13)

Note that this linearization does not contain the parameter 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:

u0(r)1Vd3rG(r',r)u0(r)[n1n(1+ϕn)1(ϕ·ϕ)+εκ(r)ϕn]1
(14)

Approximating 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

1Rn1nVd3rϕ2+2k0ΔkR1nVd3rϕ1
(15)

We have discarded the third-order and higher terms in ϕ, denoted the average wave number perturbation by Δk (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 n1n, restricts the total integrated squared gradient of the complex phase over the scatterer volume. The Born constraint, which is weighted by 1n, restricts the total integrated complex phase magnitude over the volume. By adjusting 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

To derive the higher order terms of the hybrid series, we expand the complex phase as a power series, ϕ(r)=m=0εmϕm(r). The power series for ϕ is inserted into the following equation, which is Eq. (7) after the equation is multiplied by (1+ϕn)u0

2u0u0·ϕ(1+ϕn)+2ϕ(1+ϕn)+n1n(ϕ·ϕ)+εκ(1+ϕn)2=0
(16)

After expanding out the products of the power series, and collecting terms, the following equation found for the coefficients of εp for p>1:

2u0u0.(ϕp+1nm=0p1ϕpmϕm)+(2ϕp+1nm=0p1ϕpm2ϕm)+
n1n(m=1p1ϕpm·ϕm)+κ(2ϕp1n+1n2m=0p1ϕpm1ϕm)=0
(17)

By rearranging the terms, and noting that (∇2+k02)(u 0 ϕ m )=2∇u 0·ϕ m +u 02 ϕm , the following recurrence relation is obtained for (∇2+k02)(u 0 ϕp ):

(2+k02)(u0ϕp)=[1nm=0p1ϕpm((2+k02)(u0ϕm))+
n1nu0m=1p1ϕpm·ϕm+κn2u0m=0p1ϕpm1ϕm+2κnu0ϕp1]
(18)

Evaluating the recurrence relation for the first three terms of the series:

(2+k02)(u0ϕ1)=u0κ
(2+k02)(u0ϕ2)=u0[n1n(ϕ1·ϕ1)+1nϕ1κ]
(2+k02)(u0ϕ3)=u0[n1n(2ϕ1·ϕ21nϕ1(ϕ1·ϕ1))+1nϕ2κ]
(19)

These differential equations for u 0 ϕp can be transformed into integral solutions using the Green’s function in a manner identical to the transformation between Eq. (8) and Eq. (9).

In an alternative expansion [6

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

], we can expand the total field u=m=0εmum as a Born series, and the hybrid series as u=u0(1+ϕn)n where ϕ=m=0εmϕm and equate terms of order 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:

u1u0=ϕ1
(20)
u2u0=ϕ2+n12nϕ12
(21)
u3u0=ϕ3+n1nϕ1ϕ2+(n1)(n2)6n2ϕ13
(22)

Because the leading term of u p is always u 0 ϕp , the Born terms up to order p can be calculated, and then the term ϕp can be calculated from ϕ 1 to ϕ p-1 and u p . This may be a more rapid way to calculate ϕp . 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

L=Vd3r'[Mγ+q(κ,κ)]
whereM=u0(r')ϕ(r')+Vd3ru0[n1nMR+1nMB+G(r',r)ϕ(r)]
MR=G(r',r)(1+ϕn)1(ϕ·ϕ)
MB=G(r',r)κ(r)ϕ(r)
(23)

This functional represents the 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 δ(Mγ)δϕ=(Mγ)ϕr'.(Mγ)(ϕ):

δ(Mγ)δϕ=γMγ2M*δMδϕ
(24)

where

δMδϕ=u0+Vd3ru0[n1nδMRδϕ+1nδMBδϕ]
δMRδϕ=2[(1+ϕn)1(r'G(r',r)·rϕ(r))][1nG(r',r)(1+ϕn)2(ϕ·ϕ)]
δMBδϕ=κ(r)G(r',r)
(25)

With the gradient of Eq. (25) and the other functional gradient δ(Mγ+q)δκ given by

δ(Mγ+q)δκ=γMγ2M*[Vd3rG(r',r)u0(1+ϕn)]+qκr'·qκ
(26)

5. Inverse Scattering using the Hybrid Series

Implementing this method for inverse scattering is similar to implementing the first Rytov approximation in practice. For the Rytov approximation, measurements of the total field 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 2πi. 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).

In the hybrid approximation, the complex phase ϕ is given by

ϕ=n[(uu0)1n1]=n[exp(1nloguu0)1]
(27)

There is a multiple-valued inverse associated with taking the nth 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 nth root of a complex function f (x), one can write f(x)=|f(x)|exp[(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[ (x)/n]. Alternatively, one can unwrap the logarithm of uu0 as would be done in the Rytov case, and use the second form of Eq. (27).

With this change in unwrapping, the inverse scattering solution proceeds the same as it would in the Rytov case. The only change is the defintion of the complex phase and how it is unwrapped. Therefore this method should be easy to implement in cases where the Rytov approximation is already used.

As an example of implementing higher order diffraction tomography using the hybrid series, we adapt the method of inversion of nonlinear Volterra operators, which has been applied to the Born series [18

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]

], Rytov series [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]

], and the eikonal equation [19

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]

]. Below we outline the steps of estimating the inhomogeneity of an object with a nonlinear Volterra operator modified to use the hybrid series. For this, several operators will need to be defined. One is a repeated forward-scattering operator G i given by

[u0(r')Gi(κ)]=Vd3rG(r',r)κ(r)[u0(r)Gi1(κ)]
(28)
andG0(κ)=1
(29)

which computes the ith 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. (2022)):

Φ1(κ)=G1(κ)
(30)
Φ2(κ)=G2(κ)n12nΦ1(κ)2
(31)
Φ3(κ)=G3(κ)n1nΦ1(κ)Φ2(κ)(n1)(n2)6n2Φ1(κ)3
(32)

In general, there will be many incident fields 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.

The linearized inverse of 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

2. A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE Press, New York, 1988).

] 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]

] methods, among others. Regularization of this operator may be needed to ensure that the inverse is stable.

To implement the inverse, the complex phase {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:

κ1=B({ϕ})
(33)
κ2=B({Φ2(κ1)})
(34)

The total object estimate is then given as 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]

] to this series.

6. Simulation

To test the utility of the hybrid approximation, a two-dimensional simulation of a diffraction tomography inverse scattering experiment was performed using the first-order hybrid approximation. The error in the hybrid reconstruction is compared to the error of the first-order Born and Rytov approximations. Simulation computer codes [22

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

] were obtained and modified to implement the hybrid reconstruction, so that the details of the simulation are almost identical to that of [7

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]

] and [2

2. A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE Press, New York, 1988).

]. The simulated objects were circular cylinders of varying contrasts and sizes. The circular cylinder was a useful test object because there is an exact solution for plane wave scattering from a cylinder. The simulated diffracted field from the cylinder was computed on a receiver line located at a distance of 100 wavelengths from the center of the cylinder sampled every 1/4 wavelength, so that 400 points are sampled in the detector field. Only one projection needed to be computed because because of the cylindrical symmetry of the object, and the Fourier space was sampled as a 256×256 grid. Fourier reconstruction was implemented using the Fourier Diffraction Theorem [2

2. A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE Press, New York, 1988).

] including interpolation in the Fourier domain. The only change in how the inverse was implemented from the original code was to employ Eq. (27) to compute the complex phase of the scattered field, with the unwrapped phase included in the imaginary part of the complex logarithm of log uu0.

The results of the simulations are shown in Figs. 1 and 2. Each of the figure parts show the computed reconstructions of a cylinder, varying the radii and refractive indices. The reconstructed index of each cylinder as a function of radius is plotted, with the blue line corresponding to the first Born approximation, the magenta to the first Rytov approximation, and the black line to the first hybrid approximation. For each reconstruction, the root-mean-square (RMS) percentage error was computed between the computed reconstruction and the ideal cylinder profile, which is indicated in the inset of each graph. The inset of each graph also specifies the radius and refractive index of its respective cylinder, and the exponent n for which the RMS error was minimized for the hybrid approximation.

The tendency appears to be that the Born approximation reproduces the boundary well but not the interior, while the Rytov fills the interior but removes the boundary. By choosing the optimal hybrid exponent, these two tendencies can be balanced and the result it a more uniformly reconstructed cylinder. It appears that the hybrid reconstruction can achieve the best improvement over both the Born and Rytov reconstructions when the optimal exponent is close to two, so that 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]

Based on the reconstructions shown in these figures, empirical formulas for the optimal exponent and RMS error for that exponent are fitted:

nopt=1+0.205(N1)1.08R1.93
RMSopt=2.34(N1)0.93R0.46
(35)

with 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%.

We have derived a hybrid approximation that incorporates the advantages of both the Born and Rytov approximations. In cases where either of these approximations individually may not apply, this hybrid method helps obtain additional accuracy, but retains the convenient frame-work of linear inverse scattering. Furthermore, we show that the Born and the Rytov method are two endpoints of a family of linearized scattering approximations.

Fig. 1. Profile plots of computed reconstructions of cylinders with various refractive index and radii using the Born, Rytov, and hybrid approximations. The blue curve is the Born reconstruction of the refractive index contrast, the magenta curve the Rytov reconstruction, and the black curve the hybrid reconstruction. The text in the upper left corners of each subfigure (a) to (j) specifies the radius and index contrast of each cylinder, and the optimal exponent for which the error was minimized. The text in the upper right corners is the RMS error for the Born, Rytov, and hybrid reconstructions.
Fig. 2. More profile plots of computed reconstructions of cylinders with various refractive index and radii using the Born, Rytov, and hybrid approximations. The blue curve is the Born reconstruction of the refractive index contrast, the magenta curve the Rytov reconstruction, and the black curve the hybrid reconstruction. The text in the upper left corners of each subfigure (a) to (h) specifies the radius and index contrast of each cylinder, and the optimal exponent for which the error was minimized. The text in the upper right corners is the RMS error for the Born, Rytov, and hybrid reconstructions.

Acknowledgements

D.L.M. is grateful to Stephen A. Boppart for his helpful comments, suggestions, and support, to P. Scott Carney and Brynmor Davis for their help improving the manuscript, and to the Biophotonics Imaging Laboratory members for their suggestions. This research was supported in part by the National Science Foundation (BES 03-47747, S. A. Boppart), and the National Institutes of Health (1 R21 EB005321, S. A. Boppart). Additional information can be found at http://biophotonics.uiuc.edu/.

References and links

1.

W. C. Chew, Waves and Fields in Inhomogeneous Media (IEEE Press, Piscataway, NJ, 1995).

2.

A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE Press, New York, 1988).

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]

10.

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

11.

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

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. UFFC-32, 722–730 (1986).

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]

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]

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]

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]

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]

20.

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

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]

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

  1. W. C. Chew, Waves and Fields in Inhomogeneous Media (IEEE Press, Piscataway, NJ, 1995).
  2. A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE Press, New York, 1988).
  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]
  10. G. Gbur and E. Wolf, "Relation between computed tomography and diffraction tomography," J. Opt. Soc. Am. A 18, 2132-2137 (2001). [CrossRef]
  11. Z. Q. Lu, "Multidimensional structure diffraction tomography for varying object orientation through generalized scattered waves," Inv. Prob. 1, 339-356 (1985). [CrossRef]
  12. 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).
  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]
  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]
  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]
  18. 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]
  19. 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]
  20. A. J. Devaney, "A filtered back propagation algorithm for diffraction tomography," Ultrason. Imaging 4, 336-350 (1982). [CrossRef] [PubMed]
  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]
  22. 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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