267,969 research outputs found
Regression Depth and Center Points
We show that, for any set of n points in d dimensions, there exists a
hyperplane with regression depth at least ceiling(n/(d+1)). as had been
conjectured by Rousseeuw and Hubert. Dually, for any arrangement of n
hyperplanes in d dimensions there exists a point that cannot escape to infinity
without crossing at least ceiling(n/(d+1)) hyperplanes. We also apply our
approach to related questions on the existence of partitions of the data into
subsets such that a common plane has nonzero regression depth in each subset,
and to the computational complexity of regression depth problems.Comment: 14 pages, 3 figure
Combinatorial Depth Measures for Hyperplane Arrangements
Regression depth, introduced by Rousseeuw and Hubert in 1999, is a notion that measures how good of a regression hyperplane a given query hyperplane is with respect to a set of data points. Under projective duality, this can be interpreted as a depth measure for query points with respect to an arrangement of data hyperplanes. The study of depth measures for query points with respect to a set of data points has a long history, and many such depth measures have natural counterparts in the setting of hyperplane arrangements. For example, regression depth is the counterpart of Tukey depth. Motivated by this, we study general families of depth measures for hyperplane arrangements and show that all of them must have a deep point. Along the way we prove a Tverberg-type theorem for hyperplane arrangements, giving a positive answer to a conjecture by Rousseeuw and Hubert from 1999. We also get three new proofs of the centerpoint theorem for regression depth, all of which are either stronger or more general than the original proof by Amenta, Bern, Eppstein, and Teng. Finally, we prove a version of the center transversal theorem for regression depth
Combinatorial Depth Measures for Hyperplane Arrangements
Regression depth, introduced by Rousseeuw and Hubert in 1999, is a notion
that measures how good of a regression hyperplane a given query hyperplane is
with respect to a set of data points. Under projective duality, this can be
interpreted as a depth measure for query points with respect to an arrangement
of data hyperplanes. The study of depth measures for query points with respect
to a set of data points has a long history, and many such depth measures have
natural counterparts in the setting of hyperplane arrangements. For example,
regression depth is the counterpart of Tukey depth. Motivated by this, we study
general families of depth measures for hyperplane arrangements and show that
all of them must have a deep point. Along the way we prove a Tverberg-type
theorem for hyperplane arrangements, giving a positive answer to a conjecture
by Rousseeuw and Hubert from 1999. We also get three new proofs of the
centerpoint theorem for regression depth, all of which are either stronger or
more general than the original proof by Amenta, Bern, Eppstein, and Teng.
Finally, we prove a version of the center transversal theorem for regression
depth.Comment: To be presented at the 39th International Symposium on Computational
Geometry (SoCG 2023
PoseCNN: A Convolutional Neural Network for 6D Object Pose Estimation in Cluttered Scenes
Estimating the 6D pose of known objects is important for robots to interact
with the real world. The problem is challenging due to the variety of objects
as well as the complexity of a scene caused by clutter and occlusions between
objects. In this work, we introduce PoseCNN, a new Convolutional Neural Network
for 6D object pose estimation. PoseCNN estimates the 3D translation of an
object by localizing its center in the image and predicting its distance from
the camera. The 3D rotation of the object is estimated by regressing to a
quaternion representation. We also introduce a novel loss function that enables
PoseCNN to handle symmetric objects. In addition, we contribute a large scale
video dataset for 6D object pose estimation named the YCB-Video dataset. Our
dataset provides accurate 6D poses of 21 objects from the YCB dataset observed
in 92 videos with 133,827 frames. We conduct extensive experiments on our
YCB-Video dataset and the OccludedLINEMOD dataset to show that PoseCNN is
highly robust to occlusions, can handle symmetric objects, and provide accurate
pose estimation using only color images as input. When using depth data to
further refine the poses, our approach achieves state-of-the-art results on the
challenging OccludedLINEMOD dataset. Our code and dataset are available at
https://rse-lab.cs.washington.edu/projects/posecnn/.Comment: Accepted to RSS 201
Frustum PointNets for 3D Object Detection from RGB-D Data
In this work, we study 3D object detection from RGB-D data in both indoor and
outdoor scenes. While previous methods focus on images or 3D voxels, often
obscuring natural 3D patterns and invariances of 3D data, we directly operate
on raw point clouds by popping up RGB-D scans. However, a key challenge of this
approach is how to efficiently localize objects in point clouds of large-scale
scenes (region proposal). Instead of solely relying on 3D proposals, our method
leverages both mature 2D object detectors and advanced 3D deep learning for
object localization, achieving efficiency as well as high recall for even small
objects. Benefited from learning directly in raw point clouds, our method is
also able to precisely estimate 3D bounding boxes even under strong occlusion
or with very sparse points. Evaluated on KITTI and SUN RGB-D 3D detection
benchmarks, our method outperforms the state of the art by remarkable margins
while having real-time capability.Comment: 15 pages, 12 figures, 14 table
Deterministic Sampling and Range Counting in Geometric Data Streams
We present memory-efficient deterministic algorithms for constructing
epsilon-nets and epsilon-approximations of streams of geometric data. Unlike
probabilistic approaches, these deterministic samples provide guaranteed bounds
on their approximation factors. We show how our deterministic samples can be
used to answer approximate online iceberg geometric queries on data streams. We
use these techniques to approximate several robust statistics of geometric data
streams, including Tukey depth, simplicial depth, regression depth, the
Thiel-Sen estimator, and the least median of squares. Our algorithms use only a
polylogarithmic amount of memory, provided the desired approximation factors
are inverse-polylogarithmic. We also include a lower bound for non-iceberg
geometric queries.Comment: 12 pages, 1 figur
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