1,036,009 research outputs found
Duality and separation theorems in idempotent semimodules
We consider subsemimodules and convex subsets of semimodules over semirings
with an idempotent addition. We introduce a nonlinear projection on
subsemimodules: the projection of a point is the maximal approximation from
below of the point in the subsemimodule. We use this projection to separate a
point from a convex set. We also show that the projection minimizes the
analogue of Hilbert's projective metric. We develop more generally a theory of
dual pairs for idempotent semimodules. We obtain as a corollary duality results
between the row and column spaces of matrices with entries in idempotent
semirings. We illustrate the results by showing polyhedra and half-spaces over
the max-plus semiring.Comment: 24 pages, 5 Postscript figures, revised (v2
The Perfect Perspective: A Mathematical Analysis of Perspective Using Tools Available to Middle School Students
This paper examines the basic properties of perspective drawings, the history of perspective drawings, and the basic mathematics of perspective. Using a side view and a top view of a three-dimensional projection, similar triangles can be used to find distances from the axes and vanishing point in a projection. By breaking the three-dimensional projection into two, two-dimensional planes, one can recreate projections based on actual figures, or create placements of figures in real space based on a projection. Using this method, one \u27can change a projection based on the changing position of the vanishing point. This simple approach to perspective makes it accessible to students of different ability levels, as well as creating a strong connection between art and mathematics
SalsaNet: Fast Road and Vehicle Segmentation in LiDAR Point Clouds for Autonomous Driving
In this paper, we introduce a deep encoder-decoder network, named SalsaNet,
for efficient semantic segmentation of 3D LiDAR point clouds. SalsaNet segments
the road, i.e. drivable free-space, and vehicles in the scene by employing the
Bird-Eye-View (BEV) image projection of the point cloud. To overcome the lack
of annotated point cloud data, in particular for the road segments, we
introduce an auto-labeling process which transfers automatically generated
labels from the camera to LiDAR. We also explore the role of imagelike
projection of LiDAR data in semantic segmentation by comparing BEV with
spherical-front-view projection and show that SalsaNet is projection-agnostic.
We perform quantitative and qualitative evaluations on the KITTI dataset, which
demonstrate that the proposed SalsaNet outperforms other state-of-the-art
semantic segmentation networks in terms of accuracy and computation time. Our
code and data are publicly available at
https://gitlab.com/aksoyeren/salsanet.git
3D Point Cloud Denoising via Deep Neural Network based Local Surface Estimation
We present a neural-network-based architecture for 3D point cloud denoising
called neural projection denoising (NPD). In our previous work, we proposed a
two-stage denoising algorithm, which first estimates reference planes and
follows by projecting noisy points to estimated reference planes. Since the
estimated reference planes are inevitably noisy, multi-projection is applied to
stabilize the denoising performance. NPD algorithm uses a neural network to
estimate reference planes for points in noisy point clouds. With more accurate
estimations of reference planes, we are able to achieve better denoising
performances with only one-time projection. To the best of our knowledge, NPD
is the first work to denoise 3D point clouds with deep learning techniques. To
conduct the experiments, we sample 40000 point clouds from the 3D data in
ShapeNet to train a network and sample 350 point clouds from the 3D data in
ModelNet10 to test. Experimental results show that our algorithm can estimate
normal vectors of points in noisy point clouds. Comparing to five competitive
methods, the proposed algorithm achieves better denoising performance and
produces much smaller variances
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