Huabei Jiang, Keith D. Paulsen, Ulf L. Osterberg, Brian W. Pogue, and Michael S. Patterson, "Optical image reconstruction using frequency-domain data: simulations and experiments," J. Opt. Soc. Am. A 13, 253-266 (1996)
Optical image reconstruction in a heterogeneous turbid medium with the use of frequency-domain measurements is investigated in detail. A finite-element reconstruction algorithm for optical data based on a diffusion equation approximation is presented and confirmed by a series of simulations and experiments using phantoms having optical properties in the range of those expected for tissues. Simultaneous reconstruction of absorption and scattering coefficients is achieved both theoretically and experimentally. Images with different target locations and contrast levels between target and background are also successfully recovered. All reconstructed images from both simulated and experimental data are derived directly from absolute optical data in which no differential measurement scheme is used. Results from the use of simulated and measured data suggest that quantitative images can be produced in terms of absorption and scattering coefficient values and location, size, and shape of heterogeneities within a circular background region over a range of contrast levels. Further, the effects of modulation frequency are found to be relatively modest, although boundary conditions appear to be important factors.
You do not have subscription access to this journal. Cited by links are available to subscribers only. You may subscribe either as an Optica member, or as an authorized user of your institution.
You do not have subscription access to this journal. Figure files are available to subscribers only. You may subscribe either as an Optica member, or as an authorized user of your institution.
You do not have subscription access to this journal. Article tables are available to subscribers only. You may subscribe either as an Optica member, or as an authorized user of your institution.
You do not have subscription access to this journal. Equations are available to subscribers only. You may subscribe either as an Optica member, or as an authorized user of your institution.
Image Errors (Absolute Difference between the True and the Reconstructed Values) for Images from Simulated Data with Different Noise Levelsa
Diffusion Coefficient (mm)
Absorption Coefficient (mm−1)
Noise Levels
Maximum
Average
Maximum
Average
No noise
0.1338
0.01926
0.004896
0.0006393
1% noise
0.2041
0.02633
0.007780
0.0008466
5% noise
0.2269
0.04984
0.01111
0.002215
The true values for the background are μa = 0.006 mm−1 and
(i.e., D = 0.56 mm); for the target region they are μa 0.012 mm−1 and
(i.e., D = 0.28 mm). The target is located at 3 o’clock (off-center 10 mm).
Table 2
Geometric Information for Reconstructed Images from Simulated Data with Different Noise Levelsa
Target Location
Target Size
Target Shape
D Image
μa Image
D Image
μa Image
D Image
μa Image
X
Y
X
Y
EF
GH
EF
GH
EF/GH
EF/GH
Exact
10.4
0.0
10.4
0.0
27.0
28.0
27.0
28.0
0.96
0.96
No noise
10.5
0.0
10.5
1.0
23.5
28.0
31.0
29.0
0.84
1.07
1% noise
10.6
1.0
6.3
−1.5
18.5
24.0
15.0
25.0
0.77
0.60
5% noise
10.4
1.6
8.2
0.8
18.0
19.7
22.4
22.4
0.91
1.0
X and Y refer to the x and y coordinates (in millimeters) of the target center, respectively. EF and GH are the transect length (in millimeters) of the target region along the x and y directions, respectively [see Fig. 1(a)]. This table is for the off-centered target case with 2:1 contrast level between the target and the background medium. Note that EF ≠ GH in the exact case, because the property profile is modeled as being linearly interpolated across the jump discontinuity assumed in the optical properties [see Figs. 4(a) and 4(e)].
Table 3
Reconstructed Contrast Levels between the Target and the Background and Optical Property Ratios of the Target between the Different Contrast Levels for Images from Simulated Data, Where the Target Is Located Off-centera
μa
μa
μa
D
D
D
μa(5:2)
μa(10:5)
D(5:2)
D(10:5)
Exact
2:1
5:1
10:1
1:2
1:5
1:10
2.5:1
2:1
1:2.5
1:2
Reconstructed
1.99:1
3.7:1
4.8:1
1:1.5
1:1.7
1:1.7
2.1:1
1.6:1
1:1.2
1:1.1
Average values for optical properties in the target and background regions have been used. Note that (5:2) and (10:5) in the 8th–11th columns express target comparisons of contrast pairs 5:1 and 2:1 and contrast pairs 10:1 and 5:1, respectively.
Table 4
Image Errors (Absolute Difference between the True and Reconstructed Values) for Images from Simulated Data with a Centered Target and Different Modulation Frequencya
Diffusion Coefficient (mm)
Absorption Coefficient (mm−1)
Modulation Frequency
Maximum
Average
Maximum
Average
50 MHz
0.1717
0.02521
0.03416
0.002561
200 MHz
0.1479
0.02501
0.03572
0.002646
300 MHz
0.1714
0.02819
0.02914
0.002319
The contrast level used for this table is 5:1 between the target and the background medium. The true values for the background are μa = 0.006 mm−1 and
(i.e., D = 0.56 mm); for the target region they are μa = 0.03 mm−1 and
(i.e., D = 0.112 mm).
Table 5
Image Errors (Absolute Difference between the True and Reconstructed Values) for Images from Experimental Data with Centered and Off-Centered Target Locationsa
Diffusion Coefficient (mm)
Absorption Coefficient (mm−1)
Target Location
Maximum
Average
Maximum
Average
Centered
0.1939
0.03456
0.02121
0.003671
Off-Centered
0.2278
0.03202
0.02365
0.003645
The true values for the background are μa = 0.006 mm−1 and
(i.e., D = 0.56 mm); for the target region they are μa = 0.012 mm−1 and
(i.e., D = 0.28 mm). The off-centered target is located at 3 o’clock (approximately 10 mm away from the center of the phantom).
Table 6
Geometric Information for Reconstructed Images from Experimental Data with Centered and Off-Centered Target Locationsa
Target Location
Target Size
Target Shape
D Image
μa Image
D Image
μa Image
D Image
μa Image
X
Y
X
Y
EF
GH
EF
GH
EF/GH
EF/GH
CenteredE
0.0
0.0
0.0
0.0
25.0
25.0
25.0
25.0
1.00
1.00
CenteredR
0.0
0.0
0.0
0.0
34.2
35.0
17.8
19.0
0.98
0.94
Off-centeredE
10.4
0.0
10.4
0.0
25.0
25.0
25.0
25.0
1.00
1.00
Off-centeredR
10.2
0.0
8.0
0.25
22.6
27.2
21.2
21.6
0.83
0.98
X and Y refer to the x and y coordinates (in millimeters) of the target center, respectively. EF and GH are the transect length (in millimeters) of the target region along the x and y directions, respectively [see Fig. 1(a)]. This table is for the case with 2:1 contrast level between the target and the background medium. CenteredE and CenteredR indicate the centered target for exact and reconstructed data, respectively, and Off-centeredE and Off-centeredR indicate the off-centered target for exact and reconstructed data, respectively.
Table 7
Reconstructed Contrast Levels between the Target and the Background and Optical Property Ratios of the Target between the Different Contrast Levels for Images Obtained from Experimental Data, Where the Target Is Located Off-Centera
μa
μa
μa
D
D
D
μa(5:2)
μa(10:5)
D(5:2)
D(10:5)
Exact
2:1
5:1
10:1
1:2
1:5
1:10
2.5:1
2:1
1:2.5
1:2
Reconstructed
2.5:1
3.8:1
4.2:1
1:1.1
1:1.2
1:1.2
1.6:1
1.2:1
1:1.02
1:1.02
Average values for optical properties in the target and background regions have been used. Note that (5:2) and (10:5) in the 8th–11th columns express target comparisons of contrast pairs 5:1 and 2:1 and contrast pairs 10:1 and 5:1, respectively.
Table 8
Image Errors (Absolute Difference between the True and Reconstructed Values) for Images from Experimental Data with Centered Target and Different Modulation Frequencya
Diffusion Coefficient (mm)
Absorption Coefficient (mm−1)
Modulation Frequency
Maximum
Average
Maximum
Average
50 MHz
0.3406
0.05257
0.04273
0.005152
200 MHz
0.3441
0.05543
0.03748
0.004831
300 MHz
0.3807
0.05236
0.04546
0.004703
The contrast level used for this table is 5:1 between the target and the background medium. The true values for the background are μa =0.006 mm−1 and
(i.e., D = 0.56 mm); for the target region they are μa = 0.03 mm−1 and
(i.e., D = 0.112 mm).
Tables (8)
Table 1
Image Errors (Absolute Difference between the True and the Reconstructed Values) for Images from Simulated Data with Different Noise Levelsa
Diffusion Coefficient (mm)
Absorption Coefficient (mm−1)
Noise Levels
Maximum
Average
Maximum
Average
No noise
0.1338
0.01926
0.004896
0.0006393
1% noise
0.2041
0.02633
0.007780
0.0008466
5% noise
0.2269
0.04984
0.01111
0.002215
The true values for the background are μa = 0.006 mm−1 and
(i.e., D = 0.56 mm); for the target region they are μa 0.012 mm−1 and
(i.e., D = 0.28 mm). The target is located at 3 o’clock (off-center 10 mm).
Table 2
Geometric Information for Reconstructed Images from Simulated Data with Different Noise Levelsa
Target Location
Target Size
Target Shape
D Image
μa Image
D Image
μa Image
D Image
μa Image
X
Y
X
Y
EF
GH
EF
GH
EF/GH
EF/GH
Exact
10.4
0.0
10.4
0.0
27.0
28.0
27.0
28.0
0.96
0.96
No noise
10.5
0.0
10.5
1.0
23.5
28.0
31.0
29.0
0.84
1.07
1% noise
10.6
1.0
6.3
−1.5
18.5
24.0
15.0
25.0
0.77
0.60
5% noise
10.4
1.6
8.2
0.8
18.0
19.7
22.4
22.4
0.91
1.0
X and Y refer to the x and y coordinates (in millimeters) of the target center, respectively. EF and GH are the transect length (in millimeters) of the target region along the x and y directions, respectively [see Fig. 1(a)]. This table is for the off-centered target case with 2:1 contrast level between the target and the background medium. Note that EF ≠ GH in the exact case, because the property profile is modeled as being linearly interpolated across the jump discontinuity assumed in the optical properties [see Figs. 4(a) and 4(e)].
Table 3
Reconstructed Contrast Levels between the Target and the Background and Optical Property Ratios of the Target between the Different Contrast Levels for Images from Simulated Data, Where the Target Is Located Off-centera
μa
μa
μa
D
D
D
μa(5:2)
μa(10:5)
D(5:2)
D(10:5)
Exact
2:1
5:1
10:1
1:2
1:5
1:10
2.5:1
2:1
1:2.5
1:2
Reconstructed
1.99:1
3.7:1
4.8:1
1:1.5
1:1.7
1:1.7
2.1:1
1.6:1
1:1.2
1:1.1
Average values for optical properties in the target and background regions have been used. Note that (5:2) and (10:5) in the 8th–11th columns express target comparisons of contrast pairs 5:1 and 2:1 and contrast pairs 10:1 and 5:1, respectively.
Table 4
Image Errors (Absolute Difference between the True and Reconstructed Values) for Images from Simulated Data with a Centered Target and Different Modulation Frequencya
Diffusion Coefficient (mm)
Absorption Coefficient (mm−1)
Modulation Frequency
Maximum
Average
Maximum
Average
50 MHz
0.1717
0.02521
0.03416
0.002561
200 MHz
0.1479
0.02501
0.03572
0.002646
300 MHz
0.1714
0.02819
0.02914
0.002319
The contrast level used for this table is 5:1 between the target and the background medium. The true values for the background are μa = 0.006 mm−1 and
(i.e., D = 0.56 mm); for the target region they are μa = 0.03 mm−1 and
(i.e., D = 0.112 mm).
Table 5
Image Errors (Absolute Difference between the True and Reconstructed Values) for Images from Experimental Data with Centered and Off-Centered Target Locationsa
Diffusion Coefficient (mm)
Absorption Coefficient (mm−1)
Target Location
Maximum
Average
Maximum
Average
Centered
0.1939
0.03456
0.02121
0.003671
Off-Centered
0.2278
0.03202
0.02365
0.003645
The true values for the background are μa = 0.006 mm−1 and
(i.e., D = 0.56 mm); for the target region they are μa = 0.012 mm−1 and
(i.e., D = 0.28 mm). The off-centered target is located at 3 o’clock (approximately 10 mm away from the center of the phantom).
Table 6
Geometric Information for Reconstructed Images from Experimental Data with Centered and Off-Centered Target Locationsa
Target Location
Target Size
Target Shape
D Image
μa Image
D Image
μa Image
D Image
μa Image
X
Y
X
Y
EF
GH
EF
GH
EF/GH
EF/GH
CenteredE
0.0
0.0
0.0
0.0
25.0
25.0
25.0
25.0
1.00
1.00
CenteredR
0.0
0.0
0.0
0.0
34.2
35.0
17.8
19.0
0.98
0.94
Off-centeredE
10.4
0.0
10.4
0.0
25.0
25.0
25.0
25.0
1.00
1.00
Off-centeredR
10.2
0.0
8.0
0.25
22.6
27.2
21.2
21.6
0.83
0.98
X and Y refer to the x and y coordinates (in millimeters) of the target center, respectively. EF and GH are the transect length (in millimeters) of the target region along the x and y directions, respectively [see Fig. 1(a)]. This table is for the case with 2:1 contrast level between the target and the background medium. CenteredE and CenteredR indicate the centered target for exact and reconstructed data, respectively, and Off-centeredE and Off-centeredR indicate the off-centered target for exact and reconstructed data, respectively.
Table 7
Reconstructed Contrast Levels between the Target and the Background and Optical Property Ratios of the Target between the Different Contrast Levels for Images Obtained from Experimental Data, Where the Target Is Located Off-Centera
μa
μa
μa
D
D
D
μa(5:2)
μa(10:5)
D(5:2)
D(10:5)
Exact
2:1
5:1
10:1
1:2
1:5
1:10
2.5:1
2:1
1:2.5
1:2
Reconstructed
2.5:1
3.8:1
4.2:1
1:1.1
1:1.2
1:1.2
1.6:1
1.2:1
1:1.02
1:1.02
Average values for optical properties in the target and background regions have been used. Note that (5:2) and (10:5) in the 8th–11th columns express target comparisons of contrast pairs 5:1 and 2:1 and contrast pairs 10:1 and 5:1, respectively.
Table 8
Image Errors (Absolute Difference between the True and Reconstructed Values) for Images from Experimental Data with Centered Target and Different Modulation Frequencya
Diffusion Coefficient (mm)
Absorption Coefficient (mm−1)
Modulation Frequency
Maximum
Average
Maximum
Average
50 MHz
0.3406
0.05257
0.04273
0.005152
200 MHz
0.3441
0.05543
0.03748
0.004831
300 MHz
0.3807
0.05236
0.04546
0.004703
The contrast level used for this table is 5:1 between the target and the background medium. The true values for the background are μa =0.006 mm−1 and
(i.e., D = 0.56 mm); for the target region they are μa = 0.03 mm−1 and
(i.e., D = 0.112 mm).