### Point Matching Problem: Identifying Matching Corrosion Features Based on Consecutive ILIs
This is primarily due to the approach's strict correspondence criteria, which consider only features with a correspondence probability \(\geq 0.9\) as matches. Throughout this study, this model will be referred to as the \"annealing model\". To address these limitations, [PERSON] et al. [10] proposed an alternative model using the iterative closest point (ICP) algorithm, initially introduced by [PERSON] and [PERSON] [23]. This model incorporates Voronoi tessellations to identify mixed nearest neighbors between \(\widetilde{\mathrm{P}}\) (or \(\mathrm{P}\) in the absence of features near the unrolling reference point) and \(\mathrm{Q}\), as shown in Figure 5. Mixed nearest neighbors refer to the pair of features \(\widetilde{\mathrm{P}_{\mathrm{i}}}\) and \(\mathrm{q_{\mathrm{j}}}\) positioned within each other's Voronoi cells and separated by a distance less than the defect position uncertainty threshold (\(\delta\)), denoted as \(\mathrm{d}\!\left(\mathrm{q_{\mathrm{j}}}\widetilde{\mathrm{P}_{\mathrm{i}}} \right)<\delta\). Here, \(\delta=\sqrt{\delta_{\mathrm{x}}^{2}+\delta_{y}^{2}}\) or the vector (\(\delta_{\mathrm{x}}\), \(\delta_{\mathrm{y}}\)), where \(\delta_{\mathrm{x}}\) and \(\delta_{\mathrm{y}}\) denote the axial and circumferential (angular) location uncertainties, respectively. Figure 4: Pipeline unrolling and moving set double unrolling as proposed by [PERSON] and [PERSON] [13]– example problem. Figure 3: Pipeline segmentation as proposed by [PERSON] and [PERSON] [13]. This nearest neighbor selection strategy is then applied iteratively to determine the optimum affine transformation \(\mathrm{T}^{\prime}\). In each iteration, a new estimated affine transformation \(\mathrm{T}\) of the mixed nearest neighbors, as described by [PERSON] et al. [24] and formulated in Equation (2), is applied to \(\widetilde{\mathrm{P}}\), resulting in a transformed moving set that will be used in the next iteration to determine the mixed nearest neighbors. The iterative process converges when the mean square error (MSE) of the matchings between two consecutive iterations falls below a predefined convergence threshold (\(\tau\)). A value of \(\tau=0.001\) was found to provide a suitable balance between accuracy and convergence. According to [PERSON] et al. [24], if there are \(\ell\) pairs (\(\ell\geq 2\)) of potentially matching points, denoted as \(\mathrm{a_{i}}:=\left(\mathrm{x_{a_{i}}},\mathrm{y_{a_{i}}}\right)\leftrightarrow \mathrm{b_{i}}:=\left(\mathrm{x_{b_{i}}},\mathrm{y_{b_{i}}}\right)\) for \(\mathrm{i}=1\ldots\ell\), the affine transformation \(\mathrm{T}\) that minimizes the sum of squared residuals, \(\sum_{i=1}^{\ell}\left|\mathrm{T}(\mathrm{a_{i}})-\mathrm{b_{i}}\right|^{2}\), can be expressed as \(\mathrm{\overline{r}}=\left(\mathrm{t_{x}},\mathrm{t_{y}},\mathrm{Scos}\, \theta,\mathrm{Ssin}\,\theta\right)^{\mathrm{T}}\). The formula for \(\mathrm{\overline{r}}\) that provides the optimal least-squares match is derived as follows:  
\[\mathrm{\overline{r}}=\frac{1}{\det}\begin{bmatrix}\mathrm{I_{A}}&0&-\mu_{ \mathrm{X_{A}}}&\mu_{\mathrm{Y_{A}}}\\ 0&\mathrm{I_{A}}&-\mu_{\mathrm{Y_{A}}}&-\mu_{\mathrm{X_{A}}}\\ -\mu_{\mathrm{X_{A}}}&-\mu_{\mathrm{Y_{A}}}  
becomes a linear programming problem with linear constraints limited to binary correspondence and outlier solutions [25]. Here \(\rm c_{ij}\) is a binary correspondence between moving feature \(\rm\widetilde{P_{i}}\) and reference feature \(\rm q_{j}\), while \(\rm r_{i}\) and \(\rm s_{j}\) denote binary values, whether or not moving feature \(\rm\widetilde{P_{i}}\) and reference feature \(\rm q_{j}\) are classified as outliers, respectively. The outliers and mismatches controlling parameter \(\alpha\) can be linked to the proportion of outliers in both sets. Higher \(\alpha\) values influence the optimization to favor farther neighbors, while lower values enforce stricter proximity constraints, filtering out actual correspondences as outliers. \[\rm{argmin}_{C,r,s}\sum_{i=1}^{2n}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!