## A numerical study of gradient-based nonlinear optimization methods for contrast enhanced optical tomography

Optics Express, Vol. 9, Issue 1, pp. 49-65 (2001)

http://dx.doi.org/10.1364/OE.9.000049

Acrobat PDF (509 KB)

### Abstract

Numerical performance of two gradient-based methods, a truncated-Newton method with trust region (TN) and a nonlinear conjugate gradient (NCG), is studied and compared for a given data set and conditions specific for the contrast enhanced optical tomography problem. Our results suggest that the relative performance of the two methods depends upon the error functions, specific to the problem to be solved. The TN outperforms the NCG when maps of fluorescence lifetime are reconstructed while both methods performed well when the absorption coefficient constitutes the parameter set that is to be recovered.

© Optical Society of America

## 1. Introduction

1. Y. Q. Yao, Y. Wang, Y. L. Pei, W. W. Zhu, and R. L. Barbour, “Frequency-domain optical imaging of absorption and scattering distributions by Born iterative method,” J. Opt. Soc. Am A , **14**, .325–342, (1997). [CrossRef]

2. D. Y. Paithankar, A. U. Chen, B. W. Pogue, M. S. Patterson, and E. M. Sevick-Muraca, “Imaging of fluorescent yield and lifetime from multiply scattered light re-emitted from tissues and other random media,” Appl. Opt , **36**, 2260–2272, (1997). [CrossRef] [PubMed]

13. R. Roy and E. M. Sevick-Muraca, “Truncated Newton’s optimization scheme for absorption and fluorescence optical tomography: Part I theory and formulation,” Opt. Express , **4**, 353–371, (1999). http://www.opticsexpress.org/oearchive/source/9268.htm [CrossRef] [PubMed]

15. R. Roy and E. M. Sevick-Muraca, “Active constrained truncated Newton method for simple-bound optical tomography,” J. Opt. Soc. Am. A **17**, 1627–1641, (2000). [CrossRef]

16. S. R. Arridge, “Optical tomography in medical imaging,” Inverse Problems , **15**, R41–R93, (1999). [CrossRef]

*µ*

_{a}(

*r̄*), and lifetime,

*τ*(

*r̄*), owing to a fluorescent agent within a given region.

17. R. S. Dembo and T. Steihaug, “Truncated Newton algorithms for large-scale unconstrained optimisation,” Math. Programming , **26**, 190–212, (1983). [CrossRef]

18. L. C. W. Dixon and R. C. Price, “Numerical experience with the truncated Newton method for unconstrained optimization,” J. optimization theory and applications. **56**, 245–255, (1988). [CrossRef]

19. T. Steihaug, “The conjugate gradient method and trust region in large scale optimisation,” SIAM J. Numerical analysis , **20**, 626–637, (1983). [CrossRef]

20. M. R. Hestenes and E. Stiefel, ‘Methods of conjugate gradients for solving linear systems,” J. Res. Nat. Bur. Stand. **49**, 409–436 (1952). [CrossRef]

21. R. Fletcher and C. M. Reeves, “Function minimization by conjugate gradients,” Computer J. , **7**, 149–154 (1964). [CrossRef]

24. M. J. D. Powell, “Restart procedures for the conjugate gradient method,” Math. Programming , **12**, 241–254, (1977). [CrossRef]

13. R. Roy and E. M. Sevick-Muraca, “Truncated Newton’s optimization scheme for absorption and fluorescence optical tomography: Part I theory and formulation,” Opt. Express , **4**, 353–371, (1999). http://www.opticsexpress.org/oearchive/source/9268.htm [CrossRef] [PubMed]

15. R. Roy and E. M. Sevick-Muraca, “Active constrained truncated Newton method for simple-bound optical tomography,” J. Opt. Soc. Am. A **17**, 1627–1641, (2000). [CrossRef]

27. A. J. Davies, B. Christianson, L. C. W. Dixon, R. Roy, and P. van der Zee, “Reverse differentiation and the inverse diffusion problem,” Adv. In Eng. Software , **28**, 217–221, (1997). [CrossRef]

## 2. Governing equations for intensity-modulated excitation and emission light propagation

13. R. Roy and E. M. Sevick-Muraca, “Truncated Newton’s optimization scheme for absorption and fluorescence optical tomography: Part I theory and formulation,” Opt. Express , **4**, 353–371, (1999). http://www.opticsexpress.org/oearchive/source/9268.htm [CrossRef] [PubMed]

_{x}and Φ

_{m}are the AC components of the excitation and emission photon fluence (photon/cm

^{2}s) at position

**r**, respectively;

*ω*is the angular modulation frequency (

*rad*/

*s*);

*c*is the speed of light within the medium (cm/s);

*cm*

^{-1}); and

*cm*

^{-1}).

*D*

_{x,m}is the optical diffusion coefficient equivalent to

^{-1}) and

*ϕ*represents the quantum efficiency of the fluorescence process, which is defined as the probability that an excited fluorophore will decay radiatively, and

*τ*is the fluorophore lifetime (ns). Note that the source term requires coupling with the solution of excitation fluence described by equation (1). For the purposes of this study, we assume that the absorption coefficient at the excitation wavelength is only due to fluorophores, i.e.,

*S*is the complex flux density associated with a point source of excitation light located at position

**r**

_{s}on

*d*Ω and

**n**denotes the unit outward vector normal to the surface. The boundary condition for emission fluence is identical to equation (3) with the exception that

*S*=0 on the surface.

## 2.1 Galerkin finite element formulation of excitation and emission equations

_{x,m},

*τ*are assumed linear. The formulation of the Galerkin finite element method of the diffusion equations is described in the references [13

**4**, 353–371, (1999). http://www.opticsexpress.org/oearchive/source/9268.htm [CrossRef] [PubMed]

15. R. Roy and E. M. Sevick-Muraca, “Active constrained truncated Newton method for simple-bound optical tomography,” J. Opt. Soc. Am. A **17**, 1627–1641, (2000). [CrossRef]

**K**, and the solution is obtained by solving the complex equations:

## 2.2 Synthetic data sets for reconstruction

_{x,m}in a two dimensional, square domain of 4×4 cm

^{2}. For the forward simulation, the mesh contained 2209 internal and 192 boundary nodes with 4608 triangular elements. Four point sources of excitation light intensity-modulated at frequency

*ω*were simulated midpoint on each side of the domain with a total of 60 point detectors simulated equidistant along the domain periphery, excluding positions occupied by one of the four sources and at each corner of the domain. Predictions of Φ

_{x,m}were made at every detector for each of the four sources, each providing intensitymodulation with unit depth ofmodulation and zero phase. For this simulated measurement configuration there were 4×59=236 simulated observations. The main objective of this paper is to investigate the performance of TN and NCG methods using identical conditions. Hence, the same number of sources and detectors were used for both these methods. (For this focused study we choose not to optimize the number of sources/detectors required for the inverse problem). The simulator was coded in Fortran 77 and took 6 seconds to run on a SUN Ultrasparc 10 Workstation.

**4**, 353–371, (1999). http://www.opticsexpress.org/oearchive/source/9268.htm [CrossRef] [PubMed]

## 3. Inverse problemof recovering optical properties

*τ*are to be determined from synthetic fluorescence measurements. Let

*l*=1,…,

*N*

_{q}denote the number of sources q on the boundary and

*j*=1,…,

*N*

_{B}denote the number of detectors

*B*on the boundary. The error function for optimization is taken as:

*c*denotes the values calculated by the forward problem and the subscript

*me*denotes the experimentally measured values, or in this case, the synthetic data sets. The superscript ∗ denotes the complex conjugate of the complex function Φ

_{m}. For simplicity, we will use

*µ*

_{a}to denote

## 4 Gradient based methods

### 4.1 The truncated Newton method (TN)

*E*((

*µ*

_{a})

_{k}+

**d**) by the quadratic model [13

**4**, 353–371, (1999). http://www.opticsexpress.org/oearchive/source/9268.htm [CrossRef] [PubMed]

**17**, 1627–1641, (2000). [CrossRef]

17. R. S. Dembo and T. Steihaug, “Truncated Newton algorithms for large-scale unconstrained optimisation,” Math. Programming , **26**, 190–212, (1983). [CrossRef]

19. T. Steihaug, “The conjugate gradient method and trust region in large scale optimisation,” SIAM J. Numerical analysis , **20**, 626–637, (1983). [CrossRef]

**g**

_{k}and

**G**

_{k}are the gradient and Hessian of the error function

*E*at the

*k*

^{th}iteration, respectively. The Newton direction is obtained from an exact minimum of equation (6); i.e. the search direction

**d**at iteration

*k*is defined by the equation

17. R. S. Dembo and T. Steihaug, “Truncated Newton algorithms for large-scale unconstrained optimisation,” Math. Programming , **26**, 190–212, (1983). [CrossRef]

**d**

_{i}1≤

*i*≤

*n*, where

*n*is the number of unknowns to the solution

**d**. A sequence of approximations of the solution is generated until a vector

**d**

_{i}is obtained for which the following condition on the relative residual is satisfied.

**r**

_{i}=

**G**

_{k}

**d**

_{i}+

**g**

_{k}.

**d**

_{i}is used as the search direction for the minimisation. The relative residual is used as a measure of accuracy of

**d**

_{i}and the required level of accuracy improves as the minimisation proceeds and the minimum is approached.

**26**, 190–212, (1983). [CrossRef]

**G**, as only the product

**Gd**was required. The product is calculated using the finite difference formula given below:

*k*, the cost of each outer iteration increases.

## 4.2 Nonlinear conjugate gradient method (NCG)

*E*: ℜ

^{n}→ℜ is to generate a sequence of iterates (

*µ*

_{a})

_{k}according to

**d**

_{k}is the search direction and

*α*

_{k}is the step length, which minimises

*E*along

**d**

_{k}from the point (

*µ*

_{a})

_{k}. For

*k*=1, the search direction

**d**

_{1}=-∇̄

*E*((

*µ*

_{a})

_{1})=-

**g**

_{1}can be used, and for subsequent iterations, given

**x**

_{k}with

**g**

_{k}=∇̄

*E*((

*µ*

_{a})

_{k})≠0

*k*≥1, we use

*β*

_{k}used to construct the search direction given by equation (11). The Polak and Ribiere choice (PR) for

*β*

_{k}is given by

## 4.3 Line search for both TN and NCG

**d**

_{k}, the Armijo line search [22, 30] is used in both TN and NCG methods to find an appropriate step size,

*α*

_{k}, which minimises the function

*E*along the line (

*µ*

_{a})

_{k}+

*α*

_{k}

**d**

_{k}. Gilbert and Nocedal [31] and Dembo and Steihaug [17

**26**, 190–212, (1983). [CrossRef]

32. P. Wolfe, “Convergence condition for ascent methods,” SIAM Rev , **11**, 226–253, (1969). [CrossRef]

**d**

_{k}is acceptable if

*ε*

_{1}is a small positive constant, typical

*ε*

_{1}=0.001. This ensures that the angle between

**d**and the gradient vector is negative and bounded away from zero.

*ε*

_{2}is a constant, such that 0.5>

*ε*

_{2}>0 (typically (

*ε*

_{2}=0.1). This condition ensures that the step taken produces a non-trivial reduction of the objective function.

*ε*

_{3}is a constant such that 0.5>

*ε*

_{3}>0. This condition requires that the rate of decrease of

*E*in the direction

**d**at (

*µ*

_{a})

_{k+1}must be larger than some prescribed fraction of the rate of decrease in the direction

**d**at (

*µ*

_{a})

_{k}. This condition prevents the steps from being too small, relative to the initial rate of decrease of

*E*.

*E*to be bounded away from the linear predicted reduction.

## 5. Approach and Methods

*µ*

_{a}, and fluorescence lifetime,

*τ*, were reconstructed from synthetic fluorescence measurements. We assume that all parameters other than the property to be mapped are known and constant for both absorption and fluorescence lifetime reconstructions. In problem 1, absorption imaging with up to tenfold increase in absorption within heterogeneities was investigated. In problem 2, fluorescence lifetime imaging with up to tenfold decrease in lifetimes within heterogeneities (fluorescence quenching) was investigated. Problem 3 investigated fluorescence lifetime imaging with up to tenfold increase in lifetime within heterogeneities. Thus we considered lifetime imaging cases in which (i) the background possesses smaller fluorescent lifetime than the heterogeneities and, (ii) the background possesses larger lifetime than the heterogeneities.

*µ*

_{a})

_{actual}-(

*µ*

_{a})

_{c}]

^{2}where c is the calculated value and there are 289 unknowns.

**g**‖ was less than 10

^{-8}(chosen by trial and error) or a maximum of 2000 iterations was reached.

## 6. Results and Discussion

### 6.1 Absorption coefficient imaging from synthetic fluorescence measurements

## 6.2 luorescence imaging from synthetic fluorescence measurements with increased decay kinetics

_{h}=10, 8, and 5 ns) within a background that possessed a shorter lifetime (τ

_{b}=1 ns). Figure 3(b) shows the reconstructed map of lifetime from synthetic fluence values at 150 MHz modulation frequency using the TN method while the NCG method was unable to reconstruct the heterogeneities. Indeed, while the TN method was capable of recovering lifetime maps at all modulation frequencies, the NCG method could recover lifetimes when the modulation frequency,

*ω*, was greater than 250 MHz (data not shown for brevity). When the optimisation problem was redefined to map the parameter

*ωτ*instead of

*τ*, recovery of lifetime maps was still not possible with the NCG method at

*ω*<250 MHz.

2. D. Y. Paithankar, A. U. Chen, B. W. Pogue, M. S. Patterson, and E. M. Sevick-Muraca, “Imaging of fluorescent yield and lifetime from multiply scattered light re-emitted from tissues and other random media,” Appl. Opt , **36**, 2260–2272, (1997). [CrossRef] [PubMed]

33. H. Jiang, “Frequency-domain fluorescent diffusion tomography: a finite-element-based algorithm and simulations,” Appl. Opt. **37**, 5337–5343, (1998). [CrossRef]

*E*is small. As discussed in Section 7, the TN method gives the good results due to the fact that this method uses the gradients as well as the Hessian while the NCG method uses only the gradients.

## 6.3 Fluorescence lifetime imaging from synthetic fluorescence measurements with quenching

_{h}=8, 5, and 1 ns) within a background which possessed a longer lifetime (τ

_{b}=10 ns). Figure 4(b) shows the reconstructed map of lifetime from synthetic fluence values at modulation frequency of 150 MHz using the TN method while Figure 4(c) shows the lifetime map obtained from the NCG method. Reconstruction using the NCG method resulted in only one of three heterogeneities distinguished from the background while reconstruction using the TN method resulted in all three heterogeneities distinguished from the background. The sum of square errors given by the TN method is shown in Table 5. Time taken by the TN method was 9417 seconds. As in problem 2, the TN method was capable of recovering lifetime maps at all modulation frequencies while the NCG method could recover lifetimes when the modulation frequency,

*ω*, was greater than 250MHz (data not shown for brevity). When the optimization was redefined tomap parameter

*ωτ*instead of

*τ*, the frequency dependence on TN and NCG performance remain unchanged.

## 7 Relative performance of TN and NCG

## 7.1 Sensitivity Analysis

*E*for a small change in the parameter under study. Two cases were considered: (i) influence of the absorption coefficients on the target regions, and (ii) influence of the lifetimes on the target regions.

## (a) Sensitivity of error function to absorption coefficient

_{m}, were calculated using a absorption coefficient of 0.02

*cm*

^{-1}throughout the square except on the three target regions, as shown in Figure 1, where the absorption coefficient was 0.2

*cm*

^{-1}. Each heterogeneity consisted of four nodal points. The calculated values, Φ

_{c}, was obtained using different absorption coefficients on the target regions that varied from 0.02 to 0.2 cm

^{-1}while all the remaining parameters were kept constant. The variation of the error function at modulation frequency of 150 MHz due to variation of the absorption coefficients is shown in Figure 5. The error function dependence of absorption varied little with modulation frequency.

## (b) Sensitivity of error function to lifetime

_{m}data at modulation frequencies of 15, 150, and 500 MHz, were obtained using a lifetime of 10 ns throughout the square except on the three target regions, as shown in Figure 1, where the lifetime was 1 ns. The calculated values, Φ

_{c}, were obtained using different lifetimes on the target region which varied between 10 to 1 ns while all other parameters remained constant. The influence of modulation frequency upon the sensitivity of the error function to lifetime is illustrated in Figure 6. Similar sensitivities were obtained when the lifetime of the background was 1 ns and of the target regions was 10 ns. While the TN method successfully recovers absorption and lifetime maps at all modulation frequencies, the NCG method fails to recover lifetime maps at modulation frequencies less than 250MHz, where the variation of the error function with target lifetime is small. Regardless of whether the reconstruction involved lifetime or the product of lifetime and modulation frequency, the variation of the error function with lifetime is small when

*ω*<250 MHz. Since the NCG method uses only the gradient of the error function while the TN method employs both the gradient and the Hessian of the error function, it may not be surprising that TN method performance is independent of modulation frequency.

## 7.2 Mathematical formulation

*E*: ℜ

^{n}→ℜ is twice continuously differentiable and if there exists 0<m<M<∞ such that, for all x, d∊ℜ

^{n},

*E*(·) is either a function of the absorption coefficients,

*µ*

_{a}(

*r̄*), or the lifetimes,

*τ*(

*r̄*). However x represents spatial distribution of either the absorption coefficients or the lifetimes.

*k*∊N

*E*(·).

*E*(·).

**d**

^{T}

**G**(

**x**)

**d**

*E*(

**x**)‖

**d**‖.

**d**

^{T}

**Gd**is decreasing and positive, indicatry that a solution should exist. Since the function decreases along the direction

**d**and the curvature direction is positive (and hence the Hessian is positive definite), the error function in problem 1 must also be convex. Thus the assumption of the NCG method applies as given in Equation (16). It is not surprising that, the NCG method as well as the TN method enable recovery of absorption.

_{k}‖

*k*=1,2,3,…} of problems 2 and 3 are decreasing, bounded away from zero, but are large. The values of

*α*calculated by the Armijo line search for both methods are given in Tables 8 and 9, respectively. The values of

*α*used in the NCG method are zeroes in problem 2 at 150 MHz.

*E*(

*τ*

_{k}+

*α*

**d**

_{k})<

*E*(

*τ*

_{k}), for small positive values of

*α*. For this property we assume that the step length satisfies

*α*>0 (as shown in Table 9). If the lifetimes (

*τ*

_{k}) are greater than zero, then some of the Newton directions

**d**

_{k}should be negative. Since the directions of all the gradient vectors of problem 2 are negative at modulation frequencies less than 250 MHz (data at 150 and 500 MHz are listed in Table 10), problem 2 does not satisfy equation (18) or (10). It should be noted that the Newton direction is accepted as a search direction if it satisfies the Wolfe condition 1 to ensure that it has a significant component in the negative gradient direction.

*α*=1 satisfies the

*Wolfe conditions*. If it does not, then we try

*α*with a very small number taken for instance from Table 9 as

*α*=0.39×10

^{-18}. We take the typical parameters,

*ε*

_{1}=0.001 and

*ε*

_{2}=

*ε*

_{3}=0.1 for this analysis. Table 11 presents values of

*E*(

*τ*),

*E*(

*τ*+

*α*

**d**), and

*E*(

*τ*+2

*α*

**d**) for the minimum and maximum values of step size and

*τ*equivalent to 1 ns. These error values are computed to assess the validity of Wolfe criteria 1, 2, and 3. Evaluation was performed at 200 MHz where the NCG failed to provide

*τ*recovery, and at 500 MHz, where NCG recovery was successful. When the step size is maximum (i.e.

*α*=1), problem 2 at 200 and 500 MHz modulation frequencies does not satisfyWolfe condition 2 because the reduction of error function is much higher than

**d**

^{T}

**g**(even when the parameter

*ε*

_{2}=0.5.). When the step size is minimum, taken to be

*α*=0.39×10

^{-18}, problem 2 at 200 MHz does not satisfy Wolfe condition 3, i.e. that the rate of decrease of error function

*E*in the direction

**d**at (

*τ*)

_{k+1}is not larger than some prescribed fraction of the rate of decrease in the direction

**d**at (

*τ*)

_{k}. Whether the reconstruction parameter is

*ωτ*or

*τ*, Wolfe criteria 2 and 3 for the NCG method are not met.

*E*is smaller at 200 MHz than that at 500 MHz (c.f. Table 11). Problem 2 does not converge at 200 MHz due to small variations of the error function, but does converge when the modulation frequency is greater than 200 MHz as discussed in Section 6.3. Powell and others have shown NCG may not converge for non-convex optimization problems and will fail due to round off for ill-conditioned problems [35].

**d**

_{k}‖

*k*=1, 2, 3,…} using the linear conjugate gradient method to approximately minimize the quadratic model of the error function given by equation (6) within a trust region. That is, the linear conjugate method calculates the Newton direction by solving the equation (7) with the gradient on the right hand side. Whenever the linear conjugate gradient method detects a nonascent negative curvature direction (a direction

**d**

_{k}such that g(

**x**

_{k})

^{T}

**d**

_{k}<0 and

**G**(

**x**

_{k})

**d**

_{k}<0), the minimization of the model is stopped and the new point is obtained by performing the trust region method along this direction (for detailed discussion see reference [18

18. L. C. W. Dixon and R. C. Price, “Numerical experience with the truncated Newton method for unconstrained optimization,” J. optimization theory and applications. **56**, 245–255, (1988). [CrossRef]

36. S. G. Nash, “A survey of truncated-Newton methods,” J. of Computational and Applied Math. **124**, 45–59, (2000). [CrossRef]

## 8 Conclusions

## Acknowledgements

## References

1. | Y. Q. Yao, Y. Wang, Y. L. Pei, W. W. Zhu, and R. L. Barbour, “Frequency-domain optical imaging of absorption and scattering distributions by Born iterative method,” J. Opt. Soc. Am A , |

2. | D. Y. Paithankar, A. U. Chen, B. W. Pogue, M. S. Patterson, and E. M. Sevick-Muraca, “Imaging of fluorescent yield and lifetime from multiply scattered light re-emitted from tissues and other random media,” Appl. Opt , |

3. | K. D. Paulsen and H. Jiang, “Enhanced frequency domain optical image reconstruction in tissues through total variation minimization,” Appl. Opt. |

4. | H. B. Jiang, K. D. Paulsen, U. L. Osterberg, B. W. Pogue, and M. S. Patterson, “Optical image reconstruction using frequency domain data: Simulations and experiments,” J. Opt. Soc. Am. A |

5. | M. A. O’Leary, D. A. Boas, B. Chance, and A. G. Yodh, “Experimental images of heterogeneous turbid media by frequency-domain diffusion-photon tomography,” Opt. Lett. |

6. | S. R. Arridge, M. Schweiger, and D. T. Delpy, “Iterative reconstruction of near-infrared absorption images,” in |

7. | R. Roy, “Image reconstruction from light measurement on biological tissue,’ Ph. D thesis, Hatfield, England, (1996). |

8. | S. R. Arridge and M. Schweiger, “A gradient-based optimization scheme for optical tomography”, Opt. Express ; |

9. | A. H. Hielscher, A. D Klose, and K. M. Hanson, “Gradient-based iterative image reconstruction scheme for time-resolved optical tomography,” IEEE Trans Med. Imag , |

10. | A. D. Klose and A.H. Hielscher, “Iterative reconstruction scheme for optical tomography based on the equation of radiative transfer,” Med. Phy. |

11. | M. V. Klibanov, T. R. Lucas, and R. M. Frank, “A fast and accurate imaging algorithm in optical diffusion tomography,” Inverse Problems , |

12. | S. S. Saquib, K. M. Hanson, and G. S. Cunningham, “Model-based image reconstruction from time-resolved diffusion data,” in Medical Imaging: Image Processing, Proc. of the SPIE-The International Society for Optical Engineering , |

13. | R. Roy and E. M. Sevick-Muraca, “Truncated Newton’s optimization scheme for absorption and fluorescence optical tomography: Part I theory and formulation,” Opt. Express , |

14. | R. Roy and E. M. Sevick-Muraca, “Truncated Newton’s optimization scheme for absorption and fluorescence optical tomography: Part II Reconstruction from synthetic measurements,” Opt. Express , |

15. | R. Roy and E. M. Sevick-Muraca, “Active constrained truncated Newton method for simple-bound optical tomography,” J. Opt. Soc. Am. A |

16. | S. R. Arridge, “Optical tomography in medical imaging,” Inverse Problems , |

17. | R. S. Dembo and T. Steihaug, “Truncated Newton algorithms for large-scale unconstrained optimisation,” Math. Programming , |

18. | L. C. W. Dixon and R. C. Price, “Numerical experience with the truncated Newton method for unconstrained optimization,” J. optimization theory and applications. |

19. | T. Steihaug, “The conjugate gradient method and trust region in large scale optimisation,” SIAM J. Numerical analysis , |

20. | M. R. Hestenes and E. Stiefel, ‘Methods of conjugate gradients for solving linear systems,” J. Res. Nat. Bur. Stand. |

21. | R. Fletcher and C. M. Reeves, “Function minimization by conjugate gradients,” Computer J. , |

22. | S. Bazaraa, H. D. Sherali, and C. M. Shetty, “Nonlinear programming theory and algorithms,” John Wiley & Sons Inc, New York, (1993). |

23. | R. Fletcher, “Practical methods of optimisation,” Second edition, John Wiley & Sons, Chichester, (1996). |

24. | M. J. D. Powell, “Restart procedures for the conjugate gradient method,” Math. Programming , |

25. | A. Griewank, “On automatic differentiation,” edited M. Iri and K. Tanaka, |

26. | B. Christianson, A. J. Davies, L. C. W. Dixon, R. Roy, and P. van der Zee, “Giving reverse differentiation a helping hand,” Opt. Meth. And Software , |

27. | A. J. Davies, B. Christianson, L. C. W. Dixon, R. Roy, and P. van der Zee, “Reverse differentiation and the inverse diffusion problem,” Adv. In Eng. Software , |

28. | A. Ishimaru, |

29. | O. C. Zienkiewcz and R. L. Taylor, |

30. | L. Armijo, “Minimization of functions having Lipschitz continuous first partial derivatives,” Pacific J. Mathematics , |

31. | J. C. Gilbert and J Nocedal, “Global convergence properties of conjugate gradient methods for optimization,” Report no. 1268, INRIA, (1990) |

32. | P. Wolfe, “Convergence condition for ascent methods,” SIAM Rev , |

33. | H. Jiang, “Frequency-domain fluorescent diffusion tomography: a finite-element-based algorithm and simulations,” Appl. Opt. |

34. | E. Polak, |

35. | M. J. D. Powell, “Nonconvex minimization calculations and the conjugate gradient methods,” Lecture Notes in |

36. | S. G. Nash, “A survey of truncated-Newton methods,” J. of Computational and Applied Math. |

**OCIS Codes**

(100.3190) Image processing : Inverse problems

(110.6960) Imaging systems : Tomography

**ToC Category:**

Research Papers

**History**

Original Manuscript: May 1, 2001

Published: July 2, 2001

**Citation**

Ranadhir Roy and Eva Sevick-Muraca, "A numerical study of gradient-based nonlinear optimization methods for contrast enhanced optical tomography," Opt. Express **9**, 49-65 (2001)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-9-1-49

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### References

- Q. Yao, Y. Wang, Y. L. Pei, W. W. Zhu, R. L. Barbour, "Frequency-domain optical imaging of absorption and scattering distributions by Born iterative method," J. Opt. Soc. Am A, 14, .325-342, (1997). [CrossRef]
- D.Y. Paithankar, A.U. Chen, B.W. Pogue, M.S. Patterson,and E.M. Sevick-Muraca, "Imagingof fluorescent yield and lifetime from multiply scattered light re-emitted from tissues and other random media," Appl. Opt, 36, 2260-2272, (1997). [CrossRef] [PubMed]
- K. D. Paulsen and H. Jiang, "Enhanced frequency domain optical image reconstruction in tissues through total variation minimization," Appl. Opt. 35, 3447-3458, (1996). [CrossRef] [PubMed]
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