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
(cm-1 )
(cm-1 )
(cm-1 )
(cm-1 )
τ (ns)
ϕ
(cm-1 )
τ (ns)
(cm-1 )
τ (ns)
(cm-1 )
τ (ns)
Problem 1:
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
(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 where and
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 where and
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 where and
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
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
(cm-1 )
(cm-1 )
(cm-1 )
(cm-1 )
τ (ns)
ϕ
(cm-1 )
τ (ns)
(cm-1 )
τ (ns)
(cm-1 )
τ (ns)
Problem 1:
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
(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 where and
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 where and
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 where and
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
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.