### Point Matching Problem: Identifying Matching Corrosion Features Based on Consecutive ILIs  
In feature matching, the task closely resembles point pattern matching, benefiting from the availability of relatively accurate defect positions reported by ILI tools. Hence, the problem can be described as a matching problem involving two sets of points: a moving set \(\mathrm{P}=\left\{\mathrm{p}_{\mathrm{i}}\in\mathrm{W},\mathrm{i}=1,2,\ldots,n\right\}\) and a reference set \(\mathrm{Q}=\left\{\mathrm{q}_{\mathrm{j}}\in\mathrm{W},\mathrm{j}=1,2,\ldots, m\right\}\), each representing corrosion features from older and newer ILIs, respectively. Each \(\mathrm{p}_{\mathrm{i}}:=\left(\mathrm{x}_{\mathrm{p}_{\mathrm{i}}},\mathrm{ y}_{\mathrm{p}_{\mathrm{j}}}\right)\) and \(\mathrm{q}_{\mathrm{j}}:=\left(\mathrm{x}_{\mathrm{q}},\mathrm{y}_{\mathrm{q}}\right)\) indicates the location of a feature on the pipe, while \(\mathrm{W}=\left(\mathrm{A}\times\pi\mathrm{D}\right)\in\mathbb{R}^{2}\) denotes the plane defined by the abscissa A and perimeter \(\pi\mathrm{D}\), where \(\mathrm{D}\) is the diameter of the pipeline, internal or external. The problem is then formulated to find an affine transformation \(\mathrm{T}\) for each \(\mathrm{p}_{\mathrm{i}}\), as shown in Equation (1). Here, \(\mathrm{S}\) is the scaling factor, \(\mathrm{\theta}\) is the rotation angle, and \(\mathrm{t}_{\mathrm{x}}\) and \(\mathrm{t}_{\mathrm{y}}\) are the translations along the axial and circumferential directions, respectively [18]. This transformation maps the moving set \(\mathrm{P}\) to a transformed moving set \(\mathrm{P}^{\prime}\), thereby facilitating accurate correspondence with set \(\mathrm{Q}\), as illustrated in Figure 2.  
Feature matching between ILIs is usually performed manually, making it both time-consuming and error prone. Automating this process is therefore essential to improve accuracy and scalability. For instance, [PERSON] and [PERSON] [13] proposed an automated framework for matching corrosion defects across multiple in-line inspections. This framework simplifies the feature matching problem by transforming it into a series of manageable matching problems consisting of five steps:  
1. Reducing the multiple ILI matching problem to a series of two consecutive ILI matching problems. 2. Splitting the pipeline into segments between consecutive girth welds with an overlap of 0.3 m with adjacent joints at both ends, as shown in Figure 3.
3. Mapping feature locations to a two-dimensional plane by unrolling the pipeline axially and double unrolling the moving set \(\mathrm{P}\) to account for features near the unrolling reference point (ideally the 12 o'clock position), as shown in Figure 4. Double unrolling of the moving set is performed by extending the pipeline halves at the opposite ends, duplicating only the moving set in the process. This approach results in a new extended moving set \(\widetilde{\mathrm{P}}=\left\{\widetilde{\mathrm{P}}_{\mathrm{i}}\in\mathrm{W},\mathrm{i}=1,2,\ldots,2n\right\}\) that stacks both the moving and double unrolled moving sets, respectively. Figure 2: Illustration of affine transformation on a two-dimensional plane, demonstrating how translation, scaling, and rotation facilitate correspondence between moving and reference sets. 4. Matching corrosion features using annealing and the soft-assign mixed point matching approach [19; 20; 21], derived from the thin plate spline robust point matching (TPS-RPM) method introduced by [PERSON] and [PERSON] [22]. This approach aims to find an optimum affine transformation that is close to the identity transformation matrix, leveraging ILI tools' accuracy and controlling transformation from overdrifting. 5. Transforming the results back to the original feature matching problem. The proposed point matching approach (step 4) performs best when an identifiable displacement pattern (affine transformation) exists between features of the two sets. However, its accuracy declines when displacements are highly irregular [10].