Keith D. Paulsen and Huabei Jiang, "Enhanced frequency-domain optical image reconstruction in tissues through total-variation minimization," Appl. Opt. 35, 3447-3458 (1996)
Optical image reconstruction in heterogeneous turbid media is sensitive to noise, especially when the signal-to-noise ratio of a measurement system is low. A total-variation-minimization-based iterative algorithm is described in this paper that enhances the quality of reconstructed images with frequency-domain data over that obtained previously with a regularized least-squares approach. Simulation experiments in an 8.6-cm-diameter circular heterogeneous region with low- and high-contrast levels between the target and the background show that the quality of the reconstructed images can be improved considerably when total-variation minimization is included. These simulated results are further verified and confirmed by images reconstructed from experimental data by the use of the same geometry and optically tissue-equivalent phantoms. Measures of imaging performance, including the location, size, and shape of the reconstructed heterogeneity, along with absolute errors in the predicted optical-property values are used to quantify the enhancements afforded by this new approach to optical image reconstruction with diffuse light. The results show improvements of up to 5 mm in terms of geometric information and an order of magnitude or more decrease in the absolute errors in the reconstructed optical-property values for the test cases examined.
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Comparison of Image Errors for Simulated-Data Images with Differing Noise and Contrast Levels, both with and without the Total-Variation (TV) Minimizationa
Noise State and Contrast Level
Absolute Error
D Image (mm)
μa Image (10−2 mm−1)
Without TV
With TV
Without TV
With TV
Maximum
Average
Maximum
Average
Maximum
Average
Maximum
Average
No noise
2:1 Contrast
0.20
0.021
0.096
0.005
1.64
0.085
0.40
0.003
10:1 Contrast
0.21
0.045
0.092
0.009
1.83
0.12
0.51
0.005
10% Added noise
0.23
0.18
0.16
0.019
3.65
1.22
0.61
0.008
Image errors constitute the absolute difference between the true and the reconstructed values. Background values: μa = 0.006 mm−1, μs′ = 0.6 mm−1, i.e., D = 0.56 mm. Target values for the 2:1 contrast case: μa = 0.012 mm−1, μs′ = 1.2 mm−1, i.e., D = 0.28 mm. Target values for the 10:1 contrast case: μa = 0.06 mm−1, μs′ = 6.0 mm−1, i.e., D = 0.056 mm.
Table 2
Geometric Information for Reconstructed Simulated-Data Images with Differing Noise and Contrast Levels, both with and without the Total-Variation (TV) Minimizationa
The x, y coordinates (in millimeters) are those of the target center; EF and GH are the transect lengths (in millimeters) of the target region along the x and y directions, respectively (see Fig. 1). The case of 10% added noise without TV is not included because no identifiable target region was recovered [see Figs. 4(b) and 4(e)].
Note that EF ≠ GH for the exact case because the property profile is modeled as a linear interpolation across the jump discontinuity assumed in the optical properties [see Figs. 2(a) and 2(d)].
Table 3
Image Errorsa for Experimental-Data Images with Differing Contrast Levels between the Target and the Background, both with and without the Total-Variation (TV) Minimization
Contrast Level
Absolute Error
D Image (mm)
μa Image (10−2 mm−1)
Without TV
With TV
Without TV
With TV
Maximum
Average
Maximum
Average
Maximum
Average
Maximum
Average
2:1 Contrast
0.24
0.035
0.12
0.016
1.85
0.10
0.62
0.008
10:1 Contrast
0.32
0.19
0.18
0.019
3.96
1.22
0.68
0.011
Image errors constitute the absolute difference between the true and the reconstructed values. Background values: μa = 0.006 mm−1, μs′ = 0.6 mm−1, i.e., D = 0.56 mm. Target values for the 2:1 contrast case: μa = 0.012 mm−1, μs′ = 1.2 mm−1, i.e., D = 0.28 mm. Target values for the 10:1 contrast case: μa = 0.06 mm−1, μs′ = 6.0 mm−1, i.e., D = 0.056 mm. The off-center target is located at 3 o'clock, approximately 10 mm away from the center of the phantom.
Table 4
Geometric Information for Reconstructed Experimental-Data Images with Differing Contrast Levels, both with and without the Total-Variation (TV) Minimizationa
Contrast Level, and TV State
Target Characteristics
Location
Size
Shape
D-Image Coordinates
μa-Image Coordinates
D-Image Transect Length
μa-Image Transect Length
D-Image Transect Ratio
μa-Image Transect Ratio
x
y
x
y
EF
GH
EF
GH
EF/GH
EF/GH
Exact
10.4
0.0
10.4
0.0
25.0
25.0
25.0
25.0
1.0
1.0
2:1 Contrast without TV
15.4
0.7
7.4
0.6
23.5
30.6
29.8
31.6
0.77
0.94
2:1 Contrast with TV
10.2
−0.2
11.8
0.4
25.5
26.8
27.3
28.1
0.95
0.97
10:1 Contrast without TV
14.1
0.6
8.2
0.7
27.2
28.1
29.2
30.5
0.97
0.96
10:1 Contrast with TV
10.8
−0.4
11.2
0.5
24.8
26.7
25.9
26.0
0.93
0.99
The x, y coordinates (in millimeters) are those of the target center; EF and GH are the transect lengths (in millimeters) of the target region along the x and y directions, respectively (see Fig. 1).
Tables (4)
Table 1
Comparison of Image Errors for Simulated-Data Images with Differing Noise and Contrast Levels, both with and without the Total-Variation (TV) Minimizationa
Noise State and Contrast Level
Absolute Error
D Image (mm)
μa Image (10−2 mm−1)
Without TV
With TV
Without TV
With TV
Maximum
Average
Maximum
Average
Maximum
Average
Maximum
Average
No noise
2:1 Contrast
0.20
0.021
0.096
0.005
1.64
0.085
0.40
0.003
10:1 Contrast
0.21
0.045
0.092
0.009
1.83
0.12
0.51
0.005
10% Added noise
0.23
0.18
0.16
0.019
3.65
1.22
0.61
0.008
Image errors constitute the absolute difference between the true and the reconstructed values. Background values: μa = 0.006 mm−1, μs′ = 0.6 mm−1, i.e., D = 0.56 mm. Target values for the 2:1 contrast case: μa = 0.012 mm−1, μs′ = 1.2 mm−1, i.e., D = 0.28 mm. Target values for the 10:1 contrast case: μa = 0.06 mm−1, μs′ = 6.0 mm−1, i.e., D = 0.056 mm.
Table 2
Geometric Information for Reconstructed Simulated-Data Images with Differing Noise and Contrast Levels, both with and without the Total-Variation (TV) Minimizationa
The x, y coordinates (in millimeters) are those of the target center; EF and GH are the transect lengths (in millimeters) of the target region along the x and y directions, respectively (see Fig. 1). The case of 10% added noise without TV is not included because no identifiable target region was recovered [see Figs. 4(b) and 4(e)].
Note that EF ≠ GH for the exact case because the property profile is modeled as a linear interpolation across the jump discontinuity assumed in the optical properties [see Figs. 2(a) and 2(d)].
Table 3
Image Errorsa for Experimental-Data Images with Differing Contrast Levels between the Target and the Background, both with and without the Total-Variation (TV) Minimization
Contrast Level
Absolute Error
D Image (mm)
μa Image (10−2 mm−1)
Without TV
With TV
Without TV
With TV
Maximum
Average
Maximum
Average
Maximum
Average
Maximum
Average
2:1 Contrast
0.24
0.035
0.12
0.016
1.85
0.10
0.62
0.008
10:1 Contrast
0.32
0.19
0.18
0.019
3.96
1.22
0.68
0.011
Image errors constitute the absolute difference between the true and the reconstructed values. Background values: μa = 0.006 mm−1, μs′ = 0.6 mm−1, i.e., D = 0.56 mm. Target values for the 2:1 contrast case: μa = 0.012 mm−1, μs′ = 1.2 mm−1, i.e., D = 0.28 mm. Target values for the 10:1 contrast case: μa = 0.06 mm−1, μs′ = 6.0 mm−1, i.e., D = 0.056 mm. The off-center target is located at 3 o'clock, approximately 10 mm away from the center of the phantom.
Table 4
Geometric Information for Reconstructed Experimental-Data Images with Differing Contrast Levels, both with and without the Total-Variation (TV) Minimizationa
Contrast Level, and TV State
Target Characteristics
Location
Size
Shape
D-Image Coordinates
μa-Image Coordinates
D-Image Transect Length
μa-Image Transect Length
D-Image Transect Ratio
μa-Image Transect Ratio
x
y
x
y
EF
GH
EF
GH
EF/GH
EF/GH
Exact
10.4
0.0
10.4
0.0
25.0
25.0
25.0
25.0
1.0
1.0
2:1 Contrast without TV
15.4
0.7
7.4
0.6
23.5
30.6
29.8
31.6
0.77
0.94
2:1 Contrast with TV
10.2
−0.2
11.8
0.4
25.5
26.8
27.3
28.1
0.95
0.97
10:1 Contrast without TV
14.1
0.6
8.2
0.7
27.2
28.1
29.2
30.5
0.97
0.96
10:1 Contrast with TV
10.8
−0.4
11.2
0.5
24.8
26.7
25.9
26.0
0.93
0.99
The x, y coordinates (in millimeters) are those of the target center; EF and GH are the transect lengths (in millimeters) of the target region along the x and y directions, respectively (see Fig. 1).