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In this article, we study rectifying curves in arbitrary dimensional Euclidean space. A curve is said to be a rectifying curve if, in all points of the curve, the orthogonal complement of its normal vector contains a fixed point. We…

For the problem of 3D object recognition, researchers using deep learning methods have developed several very different input representations, including "multi-view" snapshots taken from discrete viewpoints around an object, as well as…

In modern visual SLAM systems, it is a standard practice to retrieve potential candidate map points from overlapping keyframes for further feature matching or direct tracking. In this work, we argue that keyframes are not the optimal choice…

3D reconstruction is a core task in many applications such as robot navigation or sites inspections. Finding the best poses to capture part of the scene is one of the most challenging topic that goes under the name of Next Best View.…

The convex shape contained in a disk having prescribed area and maximal perimeter is completely characterized in terms of the area fraction. The solution is always a polygon having all but one sides equal. The lengths of the sides are…

We propose a probabilistic shape completion method extended to the continuous geometry of large-scale 3D scenes. Real-world scans of 3D scenes suffer from a considerable amount of missing data cluttered with unsegmented objects. The problem…

Recent advancements in 3D reconstruction, especially through neural rendering approaches like Neural Radiance Fields (NeRF) and Plenoxel, have led to high-quality 3D visualizations. However, these methods are optimized for digital…

Recent volumetric 3D reconstruction methods can produce very accurate results, with plausible geometry even for unobserved surfaces. However, they face an undesirable trade-off when it comes to multi-view fusion. They can fuse all available…

Real-time 3D mapping is a critical component in many important applications today including robotics, AR/VR, and 3D visualization. 3D mapping involves continuously fusing depth maps obtained from depth sensors in phones, robots, and…

Neural surface reconstruction aims to reconstruct accurate 3D surfaces based on multi-view images. Previous methods based on neural volume rendering mostly train a fully implicit model with MLPs, which typically require hours of training…

Incremental scene reconstruction is essential to the navigation in robotics. Most of the conventional methods typically make use of either TSDF (truncated signed distance functions) volume or neural networks to implicitly represent the…

We propose OpenVoxel, a training-free algorithm for grouping and captioning sparse voxels for the open-vocabulary 3D scene understanding tasks. Given the sparse voxel rasterization (SVR) model obtained from multi-view images of a 3D scene,…

To facilitate depth-based 3D action recognition, 3D dynamic voxel (3DV) is proposed as a novel 3D motion representation. With 3D space voxelization, the key idea of 3DV is to encode 3D motion information within depth video into a regular…

In this paper we consider a problem, called convex projection, of projecting a convex set onto a subspace. We will show that to a convex projection one can assign a particular multi-objective convex optimization problem, such that the…

Writing an uncomplicated, robust, and scalable three-dimensional convex hull algorithm is challenging and problematic. This includes, coplanar and collinear issues, numerical accuracy, performance, and complexity trade-offs. While there are…

We present a new method for visualizing implicit real algebraic curves inside a bounding box in the $2$-D or $3$-D ambient space based on numerical continuation and critical point methods. The underlying techniques work also for tracing…

Let X be a (possibly nodal) K-trivial threefold moving in a fixed ambient space P. Suppose X contains a continuous family of curves, all of whose members satisfy certain unobstructedness conditions in P. A formula is given for computing the…

Texture is an essential property of physical objects that affects aesthetics, usability, and functionality. However, designing and applying textures to 3D objects with existing tools remains difficult and time-consuming; it requires…

Real-time detection of objects in the 3D scene is one of the tasks an autonomous agent needs to perform for understanding its surroundings. While recent Deep Learning-based solutions achieve satisfactory performance, their high…

We solve the problem of characteristic numbers of elliptic curves in any dimensional projective space The answers are given in the form of effective recursions. Many numerical examples are provided. A C++ program implementing all the…