We have done all the necessary preprocessing for the three-step algorithm, that is, low-rank approximation-based x-ray transmittance modeling and determining the regularization parameters and the number of iterations for the bias correction step. We apply the three-step algorithm with these preprocessed results for estimating the basis line-integrals (i.e., thicknesses of water and bone) of
FIGURE 20.9 Bias and SD of the estimated results (thicknesses of water and bone) using the three-step algorithm and the ML-based estimator. The object consists of water. 0-30 cm with 2 cm interval, without a bone.
simple slab-geometry consisting of water (0 cm to 30 cm w'ith 2 cm interval) and bone (0 cm or 3 cm); and, compare the results to those of the ML-based estimator. For the ML-based estimator, w'e used the derivative-free Nelder-Mead algorithm  using a software package (fminsearch: MATLAB® R2013a, The MathWorks Inc., Natick, MA) to solve (Eq. 20.6).
Figure 20.9 shows the bias (first column) and standard deviation (SD; second column) for the estimated thicknesses of water (first row) and bone (second row) using the three-step algorithm and the ML-based estimator when the bone is absent in the object. We also plot the ideal results (0 for the bias and the CRLB for the SD [25, 26]) to compare the estimate to the ground truth. As we can see in this figure, the results of the three-step algorithm are comparable to those of the ML-based estimator, almost unbiased and achieve the CRLB.
Figure 20.10 also shows the same results with those of Figure 20.9 but when the 3 cm of bone is present in the object. The tendency is similar to that of without bone (Figure 20.9): the results of the three-step algorithm are comparable to those of the ML-based estimator. Flow'ever, the bias of the ML-based estimator increases as the amount of water increases from 20 cm to 30 cm due to the lack of sufficient statistics from a large amount of attenuation . On the other hand, the three- step algorithm show's a consistent tendency regardless of the amount of attenuation due to that the implicit regularization in the first step stabilizes the estimation and also that the iterative bias correction method effectively reduces the bias [11, 12].