4,763 research outputs found
Intersection of paraboloids and application to Minkowski-type problems
In this article, we study the intersection (or union) of the convex hull of N
confocal paraboloids (or ellipsoids) of revolution. This study is motivated by
a Minkowski-type problem arising in geometric optics. We show that in each of
the four cases, the combinatorics is given by the intersection of a power
diagram with the unit sphere. We prove the complexity is O(N) for the
intersection of paraboloids and Omega(N^2) for the intersection and the union
of ellipsoids. We provide an algorithm to compute these intersections using the
exact geometric computation paradigm. This algorithm is optimal in the case of
the intersection of ellipsoids and is used to solve numerically the far-field
reflector problem
The Relative Power of Composite Loop Agreement Tasks
Loop agreement is a family of wait-free tasks that includes set agreement and
simplex agreement, and was used to prove the undecidability of wait-free
solvability of distributed tasks by read/write memory. Herlihy and Rajsbaum
defined the algebraic signature of a loop agreement task, which consists of a
group and a distinguished element. They used the algebraic signature to
characterize the relative power of loop agreement tasks. In particular, they
showed that one task implements another exactly when there is a homomorphism
between their respective signatures sending one distinguished element to the
other. In this paper, we extend the previous result by defining the composition
of multiple loop agreement tasks to create a new one with the same combined
power. We generalize the original algebraic characterization of relative power
to compositions of tasks. In this way, we can think of loop agreement tasks in
terms of their basic building blocks. We also investigate a category-theoretic
perspective of loop agreement by defining a category of loops, showing that the
algebraic signature is a functor, and proving that our definition of task
composition is the "correct" one, in a categorical sense.Comment: 18 page
Virtual Rephotography: Novel View Prediction Error for 3D Reconstruction
The ultimate goal of many image-based modeling systems is to render
photo-realistic novel views of a scene without visible artifacts. Existing
evaluation metrics and benchmarks focus mainly on the geometric accuracy of the
reconstructed model, which is, however, a poor predictor of visual accuracy.
Furthermore, using only geometric accuracy by itself does not allow evaluating
systems that either lack a geometric scene representation or utilize coarse
proxy geometry. Examples include light field or image-based rendering systems.
We propose a unified evaluation approach based on novel view prediction error
that is able to analyze the visual quality of any method that can render novel
views from input images. One of the key advantages of this approach is that it
does not require ground truth geometry. This dramatically simplifies the
creation of test datasets and benchmarks. It also allows us to evaluate the
quality of an unknown scene during the acquisition and reconstruction process,
which is useful for acquisition planning. We evaluate our approach on a range
of methods including standard geometry-plus-texture pipelines as well as
image-based rendering techniques, compare it to existing geometry-based
benchmarks, and demonstrate its utility for a range of use cases.Comment: 10 pages, 12 figures, paper was submitted to ACM Transactions on
Graphics for revie
Fast algorithms for computing defects and their derivatives in the Regge calculus
Any practical attempt to solve the Regge equations, these being a large
system of non-linear algebraic equations, will almost certainly employ a
Newton-Raphson like scheme. In such cases it is essential that efficient
algorithms be used when computing the defect angles and their derivatives with
respect to the leg-lengths. The purpose of this paper is to present details of
such an algorithm.Comment: 38 pages, 10 figure
Smooth one-dimensional topological field theories are vector bundles with connection
We prove that smooth 1-dimensional topological field theories over a manifold
are the same as vector bundles with connection. The main novelty is our
definition of the smooth 1-dimensional bordism category, which encodes cutting
laws rather than gluing laws. We make this idea precise through a smooth
generalization of Rezk's complete Segal spaces. With such a definition in hand,
we analyze the category of field theories using a combination of descent, a
smooth version of the 1-dimensional cobordism hypothesis, and standard
differential geometric arguments.Comment: 20 pages. Comments and questions are very welcom
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