Translating Hausdorff is Hard: Fine-Grained Lower Bounds for Hausdorff Distance Under Translation

Computing the similarity of two point sets is a ubiquitous task in medical imaging, geometric shape comparison, trajectory analysis, and many more settings. Arguably the most basic distance measure for this task is the Hausdorff distance, which assigns to each point from one set the closest point in the other set and then evaluates the maximum distance of any assigned pair. A drawback is that this distance measure is not translational invariant, that is, comparing two objects just according to their shape while disregarding their position in space is impossible. Fortunately, there is a canonical translational invariant version, the Hausdorff distance under translation, which minimizes the Hausdorff distance over all translations of one of the point sets. For point sets of size $n$ and $m$, the Hausdorff distance under translation can be computed in time $\tilde O(nm)$ for the $L_1$ and $L_\infty$ norm [Chew, Kedem SWAT'92] and $\tilde O(nm (n+m))$ for the $L_2$ norm [Huttenlocher, Kedem, Sharir DCG'93]. As these bounds have not been improved for over 25 years, in this paper we approach the Hausdorff distance under translation from the perspective of fine-grained complexity theory. We show (i) a matching lower bound of $(nm)^{1-o(1)}$ for $L_1$ and $L_\infty$ (and all other $L_p$ norms) assuming the Orthogonal Vectors Hypothesis and (ii) a matching lower bound of $n^{2-o(1)}$ for $L_2$ in the imbalanced case of $m = O(1)$ assuming the 3SUM Hypothesis

Interactive Hausdorff distance computation for general polygonal models

Figure 1: Interactive Hausdorff Distance Computation. Our algorithm can compute Hausdorff distance between complicated models at interactive rates (the first three figures). Here, the green line denotes the Hausdorff distance. This algorithm can also be used to find penetration depth (PD) for physically-based animation (the last two figures). It takes only a few milli-seconds to run on average. We present a simple algorithm to compute the Hausdorff distance between complicated, polygonal models at interactive rates. The algorithm requires no assumptions about the underlying topology and geometry. To avoid the high computational and implementa-tion complexity of exact Hausdorff distance calculation, we approx-imate the Hausdorff distance within a user-specified error bound. The main ingredient of our approximation algorithm is a novel polygon subdivision scheme, called Voronoi subdivision, combined with culling between the models based on bounding volume hier-archy (BVH). This cross-culling method relies on tight yet simple computation of bounds on the Hausdorff distance, and it discards unnecessary polygon pairs from each of the input models alterna-tively based on the distance bounds. This algorithm can approxi-mate the Hausdorff distance between polygonal models consisting of tens of thousands triangles with a small error bound in real-time, and outperforms the existing algorithm by more than an order of magnitude. We apply our Hausdorff distance algorithm to the mea-surement of shape similarity, and the computation of penetration depth for physically-based animation. In particular, the penetration depth computation using Hausdorff distance runs at highly interac-tive rates for complicated dynamics scene

Apport de l'assistance par ordinateur lors de la pose d'endoprothèse aortique

The development of endovascular aortic procedures is growing. These mini-invasive techniques allow a reduction of surgical trauma, usually important in conventional open surgery. The technical limitations of endovascular repair are pushed to special aortic localizations which were in the past decade indication for open repair. Success and efficiency of such procedures are based on the development and the implementation of decision-making tools. This work aims to improve endovascular procedures thanks to a better utilization of pre and intraoperative imaging. This approach is in the line with the framework of computer-assisted surgery whose concepts are applied to vascular surgery. The optimization of endograft deployment is considered in three steps. The first part is dedicated to preoperative imaging analysis and shows the limits of the current sizing tools. The accuracy of a new measurement criterion is assessed (outer curvature length). The second part deals with intraoperative imaging and shows the contribution of augmented reality in endovascular aortic repair. In the last part, image guided surgery on soft tissues is addressed, especially the arterial deformations occurring during endovascular procedures which disprove rigid registration in fusion imaging. The use of finite element simulation to deal with this issue is presented. We report an original approach based on a predictive model of deformations using finite element simulation with geometrical and anatomo-mechanical patient specific parameters extracted from the preoperative CT-scan.Les techniques endovasculaires, particulièrement pour l’aorte, sont en plein essor en chirurgie vasculaire. Ces techniques mini-invasives permettent de diminuer l’agression chirurgicale habituellement importante lors de la chirurgie conventionnelle. Les limites techniques sont repoussées à certaines localisations de l’aorte qui étaient il y a encore peu de temps inaccessibles aux endoprothèses. Le succès et l’efficience de ces interventions reposent en partie sur l'élaboration et la mise en œuvre de nouveaux outils d'aide à la décision. Ce travail entend contribuer à l’amélioration des procédures interventionnelles aortiques grâce à une meilleure exploitation de l’imagerie pré et peropératoire. Cette démarche s’inscrit dans le cadre plus général des Gestes Médico-Chirurgicaux Assistés par Ordinateur, dont les concepts sont revisités pour les transposer au domaine de la chirurgie endovasculaire. Trois axes sont développés afin de sécuriser et optimiser la pose d'endoprothèse. Le premier est focalisé sur l’analyse préopératoire du scanner (sizing) et montre les limites des outils de mesure actuels et évalue la précision d’un nouveau critère de mesure des longueurs de l’aorte (courbure externe). Le deuxième axe se positionne sur le versant peropératoire et montre la contribution de la réalité augmentée dans la pose d’une endoprothèse aortique. Le troisième axe s’intéresse au problème plus général des interventions sur les tissus mous et particulièrement aux déformations artérielles qui surviennent au cours des procédures interventionnelles qui mettent en défaut le recalage rigide lors de la fusion d’images. Nous présentons une approche originale basée sur un modèle numérique de prédiction des déformations qui utilise la simulation par éléments finis en y intégrant des paramètres géométriques et anatomo-mécaniques spécifique-patient extraits du scanner préopératoire