predictive modeling of AAA growth
6. Analysis of the choice of similarity measure using synthetic datameasure using synthetic data
6.2. Comparison of parameter estimates
synthetically applied measurement noiseζ. The consistent choice ofσN, which describes the measurement noise transformed into the space of currents, is not straightforward due to the complicated relation of the noisy push-forward relation of the associated cur-rentϕA(cϑ,P)+ζ(ΓO)(ω). Furthermore, normally distributed noise in the space of discrete currents is defined by a covariance structure on the normals defining the surface, see chapter 3.2. It is therefore unlikely that white Gaussian noise on the measurementsDΓO
transfers to the correct noise fitting into the assumptions defining the current likelihood (6.8). Hence, a comparison of the two different parameter estimation problems in terms of an absolute quantification of the deviation of the estimates from the ground truth is not possible.
However, the purpose of the comparison was to quantify the susceptibility of the applied measures of similarity with respect to the systematic error contained in the mea-surements. This can still be achieved since there is a systematic error free measurement DΓO and ΓA(cO ϑ,P) for each of the measurementsVR,ΓO andΓRO respectively. By simply replacing the measurements in (6.6) and (6.8) with their systematic error free counter-parts, the resulting estimation problems, or the corresponding estimates, constitute a reference solution for each of the two problems (6.6) and (6.8). The susceptibility of each of the problems can now be measured in terms of the deviation of the solution from the respective reference solution. Due to the relative character of this measure of susceptibility, the comparison is independent of the choice of the noise parameters σD and σN. In the following, these are set toσD = 0.18 mm andσN = 1.0 mmresulting in similar values for the log-likelihood functions (6.6) and (6.8) such that the effect of the prior assumptions is similar for each of the problems.
To complete the setup of the parameter identification problem, the TV prior (3.93) (αtv = 0.1,= 1.0e−2) is assumed over the parameters. Details on the graph structure defining the adjacency matrix are given in appendix F. In combination with the two different likelihood functions given by (6.6) and (6.8), this constitutes the two different parameter estimation problems to be compared.
6.2. Comparison of parameter estimates
The following analysis is carried out by means of the MAP estimates which are computed using the LBFGS algorithm 1 and 2 (convergence tolerance tol = 1.0e−6, storage size ms = 200), see chapter 4.4.1. The solutions are parametrized using the element-wise basis (2.105).
6.2.1. Point-wise measurements
The MAP estimate corresponding to (6.6) is denoted by cϑ,D. The associated reference solution, which is obtained by replacing VR,ΓO with DΓO, is denoted by ˜cϑ,D. Both solutions are depicted in figure 6.3.
It can be seen that the reference solution ˜cϑ,D clearly shows the 3 different patches from the ground truth, see also figure 6.4 (left). Due to the inverse-crime nature of this reference solution, the deviation from the ground truth is entirely defined by the synthetic noise vector ζ. Since the measurements VR,ΓO and DΓO differ solely by the
Figure 6.3.: Reference solution˜cϑ,D(left) and solutioncϑ,D(right). The clear difference between the two is entirely caused by the propagation of the systematic error contained in the measurementsVR,ΓO.
systematic error component, the deviation shown by cϑ,D is caused exclusively by this systematic error. As can also be seen in figure 6.4 (right), this leads to a drastic deviation from the ground truth. To quantify the influence of the systematic error component, a relative error ∆D is computed as
∆D = kcϑ,D−˜cϑ,Dk
k˜cϑ,Dk ≈0.41. (6.9)
0.05 1.0 2.0
ground truthcϑ,P[1/t]¯ 0.0
0.5 1.0 1.5 2.0 2.5 3.0
inversesolution˜cϑ,D[1/¯t]
0.05 1.0 2.0
ground truthcϑ,P[1/¯t]
0.0 0.5 1.0 1.5 2.0 2.5 3.0
inversesolutioncϑ,D[1/¯t]
Figure 6.4.: Reference solutionc˜ϑ,D(left) and solutioncϑ,D(right) in comparison to the ground truthcϑ,P. The estimated solutions are grouped according to the ground truth. The group-wise mean is indicated by the red lines. For visualization, random horizontal jitter is applied within each group.
6.2.2. Surface measurements
The MAP estimate corresponding to (6.8) is denoted by cϑ,W. The associated reference solution, which is obtained by replacing ΓRO with ΓA(cO ϑ,P), is denoted by c˜ϑ,W. Both solutions are depicted in figure 6.5.
6.3. Discussion
Figure 6.5.: Reference solution˜cϑ,W (left) and solutioncϑ,W (right). Although visually almost indistinguishable, the difference between the two is entirely caused by the propa-gation of the systematic error contained in the measurementsΓRO.
Both solutions, which are visually almost indistinguishable, show the patch-wise char-acter of the ground truth, see also figure 6.6. As for the displacement-based solutions, the difference, although small, is exclusively caused by the systematic error in which ΓRO differs fromΓA(cO ϑ,P). The propagation of this difference is measured in terms of the relative error ∆W as
∆W = kcϑ,W −c˜ϑ,Wk
k˜cϑ,Wk ≈0.02. (6.10)
0.05 1.0 2.0
ground truthcϑ,P[1/t]¯ 0.0
0.5 1.0 1.5 2.0 2.5
inversesolution˜cϑ,W[1/¯t]
0.05 1.0 2.0
ground truthcϑ,P[1/¯t]
0.0 0.5 1.0 1.5 2.0 2.5
inversesolutioncϑ,W[1/¯t]
Figure 6.6.: Reference solutionc˜ϑ,W (left) and solutioncϑ,W (right) in comparison to the ground truthcϑ,P. The estimated solutions are grouped according to the ground truth. The group-wise mean is indicated by the red lines. For visualization, random horizontal jitter is applied within each group. Both solutions clearly represent the patch-wise character of the ground-truth.
6.3. Discussion
The results presented in chapter 6.2.1 and chapter 6.2.2 indicate that the same system-atic error contained in both measurements VR,ΓO and ΓRO is propagated differently by
the different measures of similarity in (6.6) and (6.8). By measuring this propagation with respect to a similarity-specific reference solution, obtained from measure-ments free of systematic error, an objective comparison between the susceptibility of the respective measures of similarity is possible. Given the relative errors ∆D and ∆W, a quantitative measure of the error propagation is presented. It is revealed from (6.9) and (6.10) that the error propagation through the displacement-based similarity measure is an order of magnitude larger than the error propagation through the surface current based similarity measure.
On the one hand, this is due to the hard point-wise correspondences in terms of the component-wise associations {VR,ΓO}i ↔ {DΓO}i. This correspondence was already identified as a source of ill-posedness for the classical optimization problem (3.5). The effect shown here is just a manifestation of this ill-posed character, which is reflected in a high sensitivity of the MAP solution with respect to systematic errors. On the other hand, the surface current normk·kW∗ is specifically tailored to be insensitive with respect to in-surface errors by relaxing these point-wise correspondences. Since image registration is well-known to be more accurate in the direction of steep image gradients [see e.g. 162] - eventually segmented as surfaces - the use of the surface current based likelihood (6.8) is advocated in scenarios involving registration.
With respect to patient-specific data and the intention of a probabilistic analysis, the derivation of the likelihood-function (6.8) from the assumption of normally distributed random currents is an issue to be addressed. Computational methods, like the FEM, make heavy demands on the smoothness of model boundaries: to allow for the mesh generation on the one hand and to fit into the modeling assumptions imposed by the computational methods on the other hand. As a consequence, the resulting smooth representations of the model boundaries don’t reflect the variability in the available image data. Rather, the whole process of mesh generation is prone to the incorporation of systematic errors. This issue will be discussed in more detail in chapter 7.