Ranadhir Roy and Eva M. Sevick-Muraca, "Active constrained truncated Newton method for simple-bound optical
tomography," J. Opt. Soc. Am. A 17, 1627-1641 (2000)

In the past, nonlinear unconstrained optimization of the optical imaging problem has
focused on Newton–Raphson techniques. Besides requiring expensive computation
of the Jacobian, the unconstrained minimization with Tikhonov regularization can pose
significant storage problems for large-scale reconstructions, involving a large
number of unknowns necessary for realization of optical imaging. We formulate the
inverse optical imaging problem as both simple-bound constrained and unconstrained
minimization problems in order to illustrate the reduction in computational time and
storage associated with constrained image reconstructions. The forward simulator of
excitation and generated fluorescence, consisting of the Galerkin finite-element
formulation, is used in an inverse algorithm to find the spatial distribution of
absorption and lifetime that minimizes the difference between predicted and synthetic
frequency-domain measurements. The inverse approach employs the truncated Newton
method with trust region and a modification of automatic reverse differentiation to
speed the computation of the optimization problem. The reconstruction results confirm
that the physically based, constrained minimization with efficient optimization
schemes may offer a more logical approach to the large-scale optical imaging problem
than unconstrained minimization with regularization.

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Optical Parameters Used for the Optimization Problems [See Eqs. (1 ) and (2 )]

Case

Unknown Variables

Background

Target 1

Target 2

Target 3

${\mu}_{{a}_{\mathit{xi}}}$ (cm^{-1} )

${\mu}_{{a}_{\mathit{xf}}}$ (cm^{-1} )

${\mu}_{{S}_{x}}{\mu}_{{S}_{m}}$ (cm^{-1} )

${\mu}_{{a}_{m}}$ (cm^{-1} )

τ (ns)

ϕ

${\mu}_{{a}_{\mathit{xf}}}$ (cm^{-1} )

τ (ns)

${\mu}_{{a}_{\mathit{xf}}}$ (cm^{-1} )

τ (ns)

${\mu}_{{a}_{\mathit{xf}}}$ (cm^{-1} )

τ (ns)

Problem 1: ${\mu}_{{a}_{\mathit{xf}}}$

0.0

0.02

10.0

0.02

10

0.034

0.2

10

0.1

8

0.05

5

Problem 2: τ

0.0

0.02

10.0

0.02

1

0.034

0.2

10

0.1

8

0.05

5

Problem 3: τ

0.0

0.02

10.0

0.02

10

0.034

0.2

1

0.1

8

0.05

5

Table 2

Computation Time Required for Reconstruction of Absorption Coefficients
${\mu}_{{a}_{\mathit{xf}}}$ (Problem 1) in the Two-Dimensional Domain with the
Unconstrained and Simple-Bound Constrained Optimization Methods

N/A, not applicable.
Lower bounds are less than the background (background value is 0.02
cm^{-1} ).
Upper bounds are higher than the target values (maximum target value is 0.2
cm^{-1} ).
Upper bounds are less than the maximum target values 0.2 cm^{-1} .
Problem is solved as a constrained optimization problem.
Problem is solved as an unconstrained optimization problem.
These results are similar but not identical to those in the figure listed.
Error ${\mathrm{\Sigma}}_{i=1}^{n=289}[({\mu}_{a}{)}_{m}-({\mu}_{a}{)}_{c}{]}^{2},$ where $m=\mathrm{measured}\mathrm{data},$$c=\mathrm{calculated}\mathrm{value},$ and $n=\mathrm{number}\mathrm{of}\mathrm{unknowns}.$

Table 3

Active and Free Variables of Absorption Coefficients after Each Iteration (Problem
1) with the Constrained Optimization Method

Iteration Number

Active Variables

Free Variables

1

0

289

2

114

175

3

137

152

4

137

152

5

203

86

6

218

71

7

235

54

8

243

46

9

251

38

10

254

35

11

258

31

12

266

23

13

270

19

14

276

13

15

277

12

Table 4

Computation Time for Reconstruction of Lifetime τ
(Background 1 ns, Problem 2) in the Two-Dimensional Domain with Longer
Fluorescence Lifetime in Three Heterogeneities Having Tenfold Uptake of
Fluorescent Dye with the Unconstrained and Simple-Bound Constrained Optimization
Methods

N/A, not applicable.
Lower bounds are less than the background (background value is 1 ns).
Upper bounds are higher than the target values (maximum target value is 10
ns).
Problem is solved as a constrained optimization problem.
Problem is solved as an unconstrained optimization problem.
These results are similar but not identical to those in the figure listed.
Error ${\mathrm{\Sigma}}_{i=1}^{n=289}[({\mu}_{a}{)}_{m}-({\mu}_{a}{)}_{c}{]}^{2}\times {10}^{10},$ where $m=\mathrm{measured}\mathrm{data},$$c=\mathrm{calculated}\mathrm{value},$ and $n=\mathrm{number}\mathrm{of}\mathrm{unknowns}.$

Table 5

Active and Free Variables of Lifetimes after Each Iteration (Background 1 ns,
Problem 2) with the Constrained Optimization Method

Iteration Number

Active Variable

Free Variable

1

0

289

2

0

289

3

126

163

4

153

136

5

185

104

6

210

79

7

233

56

8

246

43

9

249

40

10

255

34

11

259

30

12

266

23

13

270

19

14

277

12

Table 6

Computation Time Required for Reconstruction of Lifetime
τ (Background 10 ns, Problem 3) in the Two-Dimensional
Domain with Fluorescence Quenching in Three Heterogeneities Having Tenfold Uptake
of Fluorescent Dye with the Unconstrained and Simple-Bound Constrained
Optimization Methods

N/A, not applicable.
Lower bounds are less than the minimum target value (minimum target value is 1
ns).
Upper bounds are higher than the background (background is 10 ns).
Problem is solved as a constrained optimization problem.
Problem is solved as an unconstrained optimization problem.
These results are similar but not identical to those in the figure listed.
Error ${\mathrm{\Sigma}}_{i=1}^{n=289}[({\mu}_{a}{)}_{m}-({\mu}_{a}{)}_{c}{]}^{2}\times {10}^{10},$ where $m=\mathrm{measured}\mathrm{data},$$c=\mathrm{calculated}\mathrm{value},$ and $n=\mathrm{number}\mathrm{of}\mathrm{unknowns}.$

Table 7

Active and Free Variables of Lifetimes after Each Iteration (Background 10 ns,
Problem 3) with the Constrained Optimization Method

Iteration Number

Active Variable

Free Variable

1

0

289

2

0

289

3

131

158

4

131

158

5

176

113

6

224

65

7

236

53

8

254

35

9

258

31

10

267

22

11

271

18

12

277

12

13

277

12

Table 8

Computation Time Required for Finding the Background Absorption Coefficient
${\mu}_{{a}_{\mathit{xf}}}$ in the Two-Dimensional Domain with the
Unconstrained Optimization Methods

Absorption coefficient is constant at each nodal point.
Absorption coefficient is represented by a linear polynomial at each nodal
point.
Absorption coefficient is represented by a quadratic polynomial at each nodal
point.
Absorption coefficient is represented by a cubic polynomial at each nodal
point.

Tables (8)

Table 1

Optical Parameters Used for the Optimization Problems [See Eqs. (1 ) and (2 )]

Case

Unknown Variables

Background

Target 1

Target 2

Target 3

${\mu}_{{a}_{\mathit{xi}}}$ (cm^{-1} )

${\mu}_{{a}_{\mathit{xf}}}$ (cm^{-1} )

${\mu}_{{S}_{x}}{\mu}_{{S}_{m}}$ (cm^{-1} )

${\mu}_{{a}_{m}}$ (cm^{-1} )

τ (ns)

ϕ

${\mu}_{{a}_{\mathit{xf}}}$ (cm^{-1} )

τ (ns)

${\mu}_{{a}_{\mathit{xf}}}$ (cm^{-1} )

τ (ns)

${\mu}_{{a}_{\mathit{xf}}}$ (cm^{-1} )

τ (ns)

Problem 1: ${\mu}_{{a}_{\mathit{xf}}}$

0.0

0.02

10.0

0.02

10

0.034

0.2

10

0.1

8

0.05

5

Problem 2: τ

0.0

0.02

10.0

0.02

1

0.034

0.2

10

0.1

8

0.05

5

Problem 3: τ

0.0

0.02

10.0

0.02

10

0.034

0.2

1

0.1

8

0.05

5

Table 2

Computation Time Required for Reconstruction of Absorption Coefficients
${\mu}_{{a}_{\mathit{xf}}}$ (Problem 1) in the Two-Dimensional Domain with the
Unconstrained and Simple-Bound Constrained Optimization Methods

N/A, not applicable.
Lower bounds are less than the background (background value is 0.02
cm^{-1} ).
Upper bounds are higher than the target values (maximum target value is 0.2
cm^{-1} ).
Upper bounds are less than the maximum target values 0.2 cm^{-1} .
Problem is solved as a constrained optimization problem.
Problem is solved as an unconstrained optimization problem.
These results are similar but not identical to those in the figure listed.
Error ${\mathrm{\Sigma}}_{i=1}^{n=289}[({\mu}_{a}{)}_{m}-({\mu}_{a}{)}_{c}{]}^{2},$ where $m=\mathrm{measured}\mathrm{data},$$c=\mathrm{calculated}\mathrm{value},$ and $n=\mathrm{number}\mathrm{of}\mathrm{unknowns}.$

Table 3

Active and Free Variables of Absorption Coefficients after Each Iteration (Problem
1) with the Constrained Optimization Method

Iteration Number

Active Variables

Free Variables

1

0

289

2

114

175

3

137

152

4

137

152

5

203

86

6

218

71

7

235

54

8

243

46

9

251

38

10

254

35

11

258

31

12

266

23

13

270

19

14

276

13

15

277

12

Table 4

Computation Time for Reconstruction of Lifetime τ
(Background 1 ns, Problem 2) in the Two-Dimensional Domain with Longer
Fluorescence Lifetime in Three Heterogeneities Having Tenfold Uptake of
Fluorescent Dye with the Unconstrained and Simple-Bound Constrained Optimization
Methods

N/A, not applicable.
Lower bounds are less than the background (background value is 1 ns).
Upper bounds are higher than the target values (maximum target value is 10
ns).
Problem is solved as a constrained optimization problem.
Problem is solved as an unconstrained optimization problem.
These results are similar but not identical to those in the figure listed.
Error ${\mathrm{\Sigma}}_{i=1}^{n=289}[({\mu}_{a}{)}_{m}-({\mu}_{a}{)}_{c}{]}^{2}\times {10}^{10},$ where $m=\mathrm{measured}\mathrm{data},$$c=\mathrm{calculated}\mathrm{value},$ and $n=\mathrm{number}\mathrm{of}\mathrm{unknowns}.$

Table 5

Active and Free Variables of Lifetimes after Each Iteration (Background 1 ns,
Problem 2) with the Constrained Optimization Method

Iteration Number

Active Variable

Free Variable

1

0

289

2

0

289

3

126

163

4

153

136

5

185

104

6

210

79

7

233

56

8

246

43

9

249

40

10

255

34

11

259

30

12

266

23

13

270

19

14

277

12

Table 6

Computation Time Required for Reconstruction of Lifetime
τ (Background 10 ns, Problem 3) in the Two-Dimensional
Domain with Fluorescence Quenching in Three Heterogeneities Having Tenfold Uptake
of Fluorescent Dye with the Unconstrained and Simple-Bound Constrained
Optimization Methods

N/A, not applicable.
Lower bounds are less than the minimum target value (minimum target value is 1
ns).
Upper bounds are higher than the background (background is 10 ns).
Problem is solved as a constrained optimization problem.
Problem is solved as an unconstrained optimization problem.
These results are similar but not identical to those in the figure listed.
Error ${\mathrm{\Sigma}}_{i=1}^{n=289}[({\mu}_{a}{)}_{m}-({\mu}_{a}{)}_{c}{]}^{2}\times {10}^{10},$ where $m=\mathrm{measured}\mathrm{data},$$c=\mathrm{calculated}\mathrm{value},$ and $n=\mathrm{number}\mathrm{of}\mathrm{unknowns}.$

Table 7

Active and Free Variables of Lifetimes after Each Iteration (Background 10 ns,
Problem 3) with the Constrained Optimization Method

Iteration Number

Active Variable

Free Variable

1

0

289

2

0

289

3

131

158

4

131

158

5

176

113

6

224

65

7

236

53

8

254

35

9

258

31

10

267

22

11

271

18

12

277

12

13

277

12

Table 8

Computation Time Required for Finding the Background Absorption Coefficient
${\mu}_{{a}_{\mathit{xf}}}$ in the Two-Dimensional Domain with the
Unconstrained Optimization Methods

Absorption coefficient is constant at each nodal point.
Absorption coefficient is represented by a linear polynomial at each nodal
point.
Absorption coefficient is represented by a quadratic polynomial at each nodal
point.
Absorption coefficient is represented by a cubic polynomial at each nodal
point.