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Light Transport in Tissue


Evaluation of the inverse method

To evaluate the accuracy of the iteration method, diffuse reflection and total transmission were calculated using the adding-doubling technique for a variety of optical properties (Table 3.4 and 3.5). These calculated values were used in place of measured reflection and transmission, and consequently, the true optical properties characterizing each set of reflection and transmission are known. Two separate evaluations were performed: one for a known anisotropy gHG and one for known collimated transmission. Either the anisotropy or the collimated transmission was known accurately and any errors presented in Tables 6.1 and 6.2 do not include possible errors in determining these known values.

Figure 6.4: Detailed flowchart for the inversion algorithm when the collimated transmission is known. This flowchart illustrates the changes in variables necessary to vary all the parameters during each iteration.
\includegraphics [scale=0.973]{fig64.eps}

Figure 6.5: Detailed flowchart for the inversion algorithm when the anisotropy is known. This flowchart illustrates the changes in variables necessary to vary all the parameters properly during the iteration process.
\includegraphics [scale=0.973]{fig65.eps}


Table: Differences and percent errors between calculated and true optical thickness $(\tau =2^n)$ and albedo a. The Henyey-Greenstein anisotropy $g_{\protect\mathrm{HG}}$ is assumed known. The reflection and transmission used as truth were taken from Tables 3.4 and 3.5.
  g=0.0 g=0.5 g=0.875
n a=.6 a=.9 a=.99 a=.6 a=.9 a=.99 a=.6 a=.9 a=.99
                   
acalc-a
-2 0.04 -0.00 0.01 0.13 0.02 0.01 0.31 0.08 0.01
-1 -0.01 -0.02 0.01 0.05 -0.01 0.01 0.25 0.05 0.01
0 -0.03 -0.02 -0.00 0.01 -0.02 0.00 0.20 0.02 0.01
1 -0.02 -0.02 -0.00 0.01 -0.01 -0.00 0.17 0.00 0.00
2 -0.02 -0.01 -0.00 0.01 -0.01 -0.00 0.17 0.00 -0.00
3 -0.02 -0.01 -0.00 0.01 -0.01 -0.00 0.17 0.00 -0.00
4 -0.02 -0.01 -0.00 0.01 -0.01 -0.00 0.17 0.00 -0.00
                   
Percent Errors
-2 7 -0 1 21 3 1 52 9 1
-1 -2 -3 1 8 -1 1 42 6 1
0 -4 -3 -0 2 -2 0 33 2 1
1 -4 -2 -0 1 -2 -0 28 0 0
2 -3 -1 -0 1 -1 -0 28 0 -0
3 -3 -1 -0 1 -1 -0 29 0 -0
4 -3 -1 -0 1 -1 -0 29 0 -0
$\tau_{calc}-\tau$
-2 0.07 0.07 0.09 0.12 0.11 0.14 0.54 0.45 0.52
-1 0.07 0.07 0.07 0.10 0.09 0.11 0.56 0.38 0.43
0 0.06 0.05 0.05 0.08 0.07 0.09 0.63 0.27 0.29
1 0.03 0.00 -0.01 0.07 0.04 0.06 0.90 0.16 0.20
2 -0.08 -0.15 -0.12 0.00 -0.10 -0.07 1.71 0.18 0.25
3 -0.39 -0.52 -0.25 -0.22 -0.46 -0.25 3.28 0.25 0.29
4 -2.18 -1.42 -0.52 0.36 -1.21 -0.48 6.21 -0.13 -0.30
                   
Percent Errors
-2 29 29 36 46 44 57 217 181 208
-1 14 14 13 20 19 22 112 76 87
0 6 5 5 8 7 9 63 27 29
1 2 0 -0 3 2 3 45 8 10
2 -2 -4 -3 0 -2 -2 43 5 6
3 -5 -7 -3 -3 -6 -3 41 3 4
4 -14 -9 -3 2 -8 -3 39 -1 -2


Table: Differences and percent errors between calculated and true Henyey-Greenstein anisotropies $(g_{\protect\mathrm{HG}})$ and albedos (a). The collimated transmission is assumed known for various optical depths $(\tau =2^n)$. The reflection and transmission used as truth were taken from Tables 3.4 and 3.5.
  g=0.0 g=0.5 g=0.875
n a=.6 a=.9 a=.99 a=.6 a=.9 a=.99 a=.6 a=.9 a=0.99
                   
acalc-a
-2 0.12 0.10 0.01 0.01 -0.01 0.01 0.11 0.06 0.01
-1 0.01 0.01 0.01 -0.04 -0.03 0.01 0.08 0.02 0.01
0 -0.02 -0.02 0.01 -0.03 -0.03 0.00 0.06 0.00 0.01
1 -0.02 -0.02 -0.00 -0.01 -0.02 -0.00 0.06 -0.00 0.00
2 -0.00 -0.01 -0.00 0.01 -0.01 -0.00 0.06 -0.00 -0.00
3 0.00 -0.00 -0.00 0.02 -0.00 -0.00 0.07 -0.00 -0.00
4 0.00 -0.00 -0.00 0.04 -0.00 -0.00 0.07 0.00 -0.08
                   
Percent Errors
-2 20 11 1 2 -1 1 19 6 1
-1 1 1 1 -6 -4 1 13 2 1
0 -4 -2 1 -5 -3 0 10 0 1
1 -4 -2 -0 -2 -2 -0 10 -1 0
2 -0 -1 -0 1 -1 -0 11 -0 -0
3 0 -0 -0 4 -0 -0 11 -0 -0
4 0 -0 -0 7 -0 -0 12 0 -8
                   
gcalc-g
-2 0.00 0.00 0.00 -0.50 -0.36 -0.44 -0.46 -0.27 -0.30
-1 0.00 0.00 0.00 -0.23 -0.14 -0.14 -0.23 -0.11 -0.12
0 0.00 0.00 0.00 -0.09 -0.05 -0.06 -0.12 -0.04 -0.04
1 0.00 0.00 0.00 -0.03 -0.01 -0.02 -0.09 -0.01 -0.01
2 0.05 0.05 0.03 -0.00 0.01 0.01 -0.08 -0.01 -0.01
3 0.05 0.07 0.03 0.02 0.03 0.02 -0.07 -0.00 -0.00
4 0.05 0.09 0.03 0.05 0.04 0.02 -0.07 0.00 -0.84
                   
Percent Errors
-2 -- -- -- -100 -73 -87 -53 -31 -35
-1 -- -- -- -45 -27 -28 -26 -13 -14
0 -- -- -- -17 -10 -11 -14 -4 -4
1 -- -- -- -6 -3 -4 -10 -1 -2
2 -- -- -- -0 3 2 -9 -1 -1
3 -- -- -- 5 6 3 -8 -0 -1
4 -- -- -- 11 8 3 -8 0 -96

Tables 6.1 and 6.2 indicate how the approximate delta-Eddington model used in the iteration procedure affects the optical properties calculated. For both cases (fixed collimated transmission and fixed anisotropy) the errors are least when the albedos are large. Changes in the anisotropy have little affect upon the accuracy of the calculated optical properties. Increasing optical thickness tends to decrease percent error. The method should not be used with thin samples ($\tau<$1) because of the large errors in the calculated values. However for any albedo, any anisotropy, and optical thicknesses larger than one, the inverse method has intrinsic errors less than ten percent when the collimated transmission is known (Table 6.2).

S. A. Prahl."Light Transport in Tissue," PhD thesis, University of Texas at Austin, 1988.