This study investigates the impact of the invariance of feature vectors for partial-to-partial point set registration under translation and rotation of input point sets, particularly in the realm of techniques based on deep learning and Gaussian mixture models (GMMs). We reveal both theoretical and practical problems associated with such deep-learning-based registration methods using GMMs, with a particular focus on the limitations of DeepGMR, a pioneering study in this line, to the partial-to-partial point set registration. Our primary goal is to uncover the causes behind such methods and propose a comprehensible solution for that. To address this, we introduce an attention-based reference point shifting (ARPS) layer, which robustly identifies a common reference point of two partial point sets, thereby acquiring transformation-invariant features. The ARPS layer employs a well-studied attention module to find a common reference point rather than the overlap region. Owing to this, it significantly enhances the performance of DeepGMR and its recent variant, UGMMReg. Furthermore, these extension models outperform even prior deep learning methods using attention blocks and Transformer to extract the overlap region or common reference points. We believe these findings provide deeper insights into registration methods using deep learning and GMMs. Our source code and datasets are available at https://github.com/tatsy/DGRM-ARPS.git.

X-ray CT scanners, due to the transmissive nature of X-rays, have enabled the non-destructive evaluation of industrial products, even inside their bodies. In light of its effectiveness, this study intro-duces a new approach to accelerate the inspection of many mechanical parts with the same shape in a bin. The input to this problem is a volumetric image (i.e., CT volume) of many parts obtained by a single CT scan. We need to segment the parts in the volume to inspect each of them; however, random postures and dense contacts of the parts prohibit part segmentation using traditional template matching. To address this problem, we convert both the scanned volumetric images of the template and the binned parts to simpler graph structures and solve a subgraph matching problem to segment the parts. We perform a distance transform to convert the CT volume into a distance field. Then, we construct a graph based on Morse theory, in which graph nodes are located at the extremum points of the distance field. The experimental evaluation demonstrates that our fully automatic approach can detect target parts appropriately, even for a heap of 50 parts. Moreover, the overall com-putation can be performed in approximately 30 min for a large CT volume of approximately $2000\times 2000\times 1000$ voxels.