Realizability of Free Spaces of Curves

The free space diagram is a popular tool to compute the well-known Fr\'echet distance. As the Fr\'echet distance is used in many different fields, many variants have been established to cover the specific needs of these applications. Often, the question arises whether a certain pattern in the free space diagram is "realizable", i.e., whether there exists a pair of polygonal chains whose free space diagram corresponds to it. The answer to this question may help in deciding the computational complexity of these distance measures, as well as allowing to design more efficient algorithms for restricted input classes that avoid certain free space patterns. Therefore, we study the inverse problem: Given a potential free space diagram, do there exist curves that generate this diagram? Our problem of interest is closely tied to the classic Distance Geometry problem. We settle the complexity of Distance Geometry in $\mathbb{R}^{> 2}$, showing $\exists\mathbb{R}$-hardness. We use this to show that for curves in $\mathbb{R}^{\ge 2}$, the realizability problem is $\exists\mathbb{R}$-complete, both for continuous and for discrete Fr\'echet distance. We prove that the continuous case in $\mathbb{R}^1$ is only weakly NP-hard, and we provide a pseudo-polynomial time algorithm and show that it is fixed-parameter tractable. Interestingly, for the discrete case in $\mathbb{R}^1$, we show that the problem becomes solvable in polynomial time.Comment: 26 pages, 12 figures, 1 table, International Symposium on Algorithms And Computations (ISAAC 2023

Spline projection-based volume-to-image registration

This thesis focuses on the rigid-body registration of a three-dimensional model of an object to a set of its two-dimensional projections. The main contribution is the development of two registration algorithms that use a continuous model of the volume based on splines, either in the space domain or in the frequency domain. This allows for a well-defined gradient of the dissimilarity measure, which is a necessary condition for efficient and accurate registration. The first part of the thesis contains a review of the literature on volume-to- image registration. Then, we discuss data interpolation in the space domain and in the frequency domain. The basic concepts of our registration strategy are given in the second part of the thesis. We present a novel one-step approach for fast ray casting to simulate space-based volume projections. We also discuss the use of the central-slice theorem to simulate frequency-based volume projections. Then, we consider the question of the registration robustness. To improve the robustness of the space-based approach, we apply a multiresolution optimization strategy where spline-based data pyramids are processed in coarse-to-fine fashion, which improves speed as well. To improve the robustness of the frequency-based registration, we apply a coarse-to-fine strategy that involves weights in the frequency domain. In the third part, we apply our space-based algorithm to computer-assisted orthopedic surgery while adapting it to the perspective projection model. We show that the registration accuracy achieved using the orthopedic data is consistent with the current standards. Then, we apply our frequency-based registration to three-dimensional electron-microscopy application. We show that our algorithm can be used to obtain a refined solution with respect to currently available algorithms. The novelty of our approach is in dealing with a continuous space of geometric parameters, contrary to the standard methods which deal with quantized parameters. We conclude that our continuous parameter space leads to better registration accuracy. Last, we compare the performance of the frequency-based algorithm with that of the space-based algorithm in the context of electron microscopy. With these data, we observe that frequency-based registration algorithm outperforms the space-based one, which we attribute to the suitability of interpolation in the frequency domain when dealing with strictly space-limited data

Comparing Features of Three-Dimensional Object Models Using Registration Based on Surface Curvature Signatures

This dissertation presents a technique for comparing local shape properties for similar three-dimensional objects represented by meshes. Our novel shape representation, the curvature map, describes shape as a function of surface curvature in the region around a point. A multi-pass approach is applied to the curvature map to detect features at different scales. The feature detection step does not require user input or parameter tuning. We use features ordered by strength, the similarity of pairs of features, and pruning based on geometric consistency to efficiently determine key corresponding locations on the objects. For genus zero objects, the corresponding locations are used to generate a consistent spherical parameterization that defines the point-to-point correspondence used for the final shape comparison