Analysis of Farthest Point Sampling for Approximating Geodesics in a Graph

A standard way to approximate the distance between any two vertices $p$ and $q$ on a mesh is to compute, in the associated graph, a shortest path from $p$ to $q$ that goes through one of $k$ sources, which are well-chosen vertices. Precomputing the distance between each of the $k$ sources to all vertices of the graph yields an efficient computation of approximate distances between any two vertices. One standard method for choosing $k$ sources, which has been used extensively and successfully for isometry-invariant surface processing, is the so-called Farthest Point Sampling (FPS), which starts with a random vertex as the first source, and iteratively selects the farthest vertex from the already selected sources. In this paper, we analyze the stretch factor $\mathcal{F}_{FPS}$ of approximate geodesics computed using FPS, which is the maximum, over all pairs of distinct vertices, of their approximated distance over their geodesic distance in the graph. We show that $\mathcal{F}_{FPS}$ can be bounded in terms of the minimal value $\mathcal{F}^*$ of the stretch factor obtained using an optimal placement of $k$ sources as $\mathcal{F}_{FPS}\leq 2 r_e^2 \mathcal{F}^*+ 2 r_e^2 + 8 r_e + 1$, where $r_e$ is the ratio of the lengths of the longest and the shortest edges of the graph. This provides some evidence explaining why farthest point sampling has been used successfully for isometry-invariant shape processing. Furthermore, we show that it is NP-complete to find $k$ sources that minimize the stretch factor.Comment: 13 pages, 4 figure

Analysis of Farthest Point Sampling for Approximating Geodesics in a Graph

International audienceA standard way to approximate the distance between two vertices $p$ and $q$ in a graph is to compute a shortest path from $p$ to $q$ that goes through one of $k$ sources, which are well-chosen vertices. Precomputing the distance between each of the $k$ sources to all vertices yields an efficient computation of approximate distances between any two vertices. One standard method for choosing $k$ sources is the so-called Farthest Point Sampling (FPS), which starts with a random vertex as the first source, and iteratively selects the farthest vertex from the already selected sources.In this paper, we analyze the stretch factor $\mathcal{F}_{\text{FPS}}$ of approximate geodesics computed using FPS, which is the maximum, over all pairs of distinct vertices, of their approximated distance over their geodesic distance in the graph. We show that $\mathcal{F}_{\text{FPS}}$ can be bounded in terms of the minimal value $\mathcal{F}^\ast$ of the stretch factor obtained using an optimal placement of $k$ sources as $\mathcal{F}_{\text{FPS}}\leq 2 r_e^2 \mathcal{F}^\ast+ 2 r_e^2 + 8 r_e + 1$, where $r_e$ is the length ratio of longest edge over the shortest edge in the graph. We further show that the factor $r_e$ is not an artefact of the analysis by providing a class of graphs for which $\mathcal{F}_{\text{FPS}} \geq \frac{1}{2} r_e \mathcal{F}^\ast$

Análise da amostragem por ponto mais distante para a aproximação geodésica em um grafo: o estado da arte

The farthest point sampling algorithm has been vastly used in various applications involving image processing, surface mapping, among other purposes. A recent work, proposed by the team of prof. Sylvain Lazard, applies this algorithm to approximate geodesic distances in graphs associated to isometry-invariant surfaces, being that some points in the publishing showed to be a little nebulous. In this present work, will be offered a bibliographic complementation to Lazard et al’s project, intending to clarify some questions about the farthest point sampling method and other concepts used on the proposed application. Moreover, it’s intended to offer a satisfactory translation of the ideas in the referred work.O algoritmo de amostragem por ponto mais distante, ou farthest point sampling, tem sido amplamente utilizado em diversas aplicações que envolvam processamentos de imagem, mapeamentos de superfícies, entre outras finalidades. Um recente trabalho, proposto pela equipe do prof. Sylvain Lazard, aplica este algoritmo para aproximar distâncias geodésicas em grafos associados a superfícies isometricamente invariantes, sendo que alguns pontos na publicação se mostraram um pouco nebulosos. Neste presente trabalho, será oferecida uma complementação bibliográfica ao projeto de Lazard et al, no intuito de clarificar questões acerca do método de farthest point sampling e alguns outros conceitos utilizados na aplicação proposta. Além disso, tenciona-se oferecer uma tradução satisfatória das ideias do referido trabalho

Local k-NNs pattern in omni-direction graph convolution neural network for 3D point clouds

Effective representation of objects in irregular and unordered point clouds is one of the core challenges in 3D vision. Transforming point cloud into regular structures, such as 2D images and 3D voxels, are not ideal. It either obscures the inherent geometry information of 3D data or results in high computational complexity. Learning permutation invariance feature directly from raw 3D point clouds using deep neural network is a trend, such as PointNet and its variants, which are effective and computationally efficient. However, these methods are weak to reveal the spatial structure of 3D point clouds. Our method is delicately designed to capture both global and local spatial layout of point cloud by proposing a Local k-NNs Pattern in Omni-Direction Graph Convolution Neural Network architecture, called LKPO-GNN. Our method converts the unordered 3D point cloud into an ordered 1D sequence, to facilitate feeding the raw data into neural networks and simultaneously reducing the computational complexity. LKPO-GNN selects multi-directional k-NNs to form the local topological structure of a centroid, which describes local shapes in the point cloud. Afterwards, GNN is used to combine the local spatial structures and represent the unordered point clouds as a global graph. Experiments on ModelNet40, ShapeNetPart, ScanNet, and S3DIS datasets demonstrate that our proposed method outperforms most existing methods, which verifies the effectiveness and advantage of our work. Additionally, a deep analysis towards illustrating the rationality of our approach, in terms of the learned the topological structure feature, is provided. Source code is available at https://github.com/zwj12377/LKPO-GNN.git

Realidad aumentada en cirugía: una aproximación semántica mediante aprendizaje profundo

En este proyecto se ha desarrollado una herramienta que combina técnicas clásicas de visión por computador para tomar medidas precisas junto con técnicas de aprendizaje profundo. Esto permite crear un sistema capaz de entender la escena que está viendo, a la vez que la posiciona en el espacio de manera precisa, permitiendo incluso tomar medidas en verdadera magnitud. La fusión de estos dos tipos de tecnología permite abrir una nueva línea de trabajo, estableciendo las bases para la extracción semántica del interior de un paciente en un entorno quirúrgico.El principal reto abordado en el proyecto es la identificación de las regiones internas del paciente (el hígado en este caso) a partir de imágenes planas tomadas con cámaras monoculares estándar, como un endoscopio. El objetivo final es la estimación de la pose (posición con respecto a la cámara, compuesta de traslación y rotación) del hígado, que se utilizará para localizar partes internas no visibles, como vasos sanguíneos o tumores, sobre el órgano en realidad aumentada durante una intervención quirúrgica.Para el entrenamiento de la red neuronal se ha utilizado un modelo sintético del hígado, obtenido a partir de un simulador quirúrgico desarrollado en el Grupo AMB del Instituto de Investigación en Ingeniería de Aragón (I3A). La principal dificultad del proyecto radica en el entrenamiento de la red para la obtención de la pose de forma precisa. Para ello, se ha reentrenado un modelo de red neuronal con imágenes del hígado, tanto sobre fondos homogéneos como sobre fondos simulando una intervención laparoscópica, con el objetivo de realizar predicciones en condiciones lo más realistas posibles.Finalmente, la información obtenida mediante la red neuronal se ha incorporado a ORB-SLAM, para la obtención de resultados en tiempo real.La principal novedad introducida en este proyecto es el uso conjunto de redes neuronales con ORB-SLAM. Esto permite realizar estimaciones de pose y escalado automáticamente sin la necesidad de utilizar información adicional, de modo que la herramienta se puede utilizar directamente con cámaras de laparoscopia, sin tener que recurrir a sensores adicionales como acelerómetros o LIDAR.<br /