17,299 research outputs found
Recognizing Visibility Graphs of Polygons with Holes and Internal-External Visibility Graphs of Polygons
Visibility graph of a polygon corresponds to its internal diagonals and
boundary edges. For each vertex on the boundary of the polygon, we have a
vertex in this graph and if two vertices of the polygon see each other there is
an edge between their corresponding vertices in the graph. Two vertices of a
polygon see each other if and only if their connecting line segment completely
lies inside the polygon, and they are externally visible if and only if this
line segment completely lies outside the polygon. Recognizing visibility graphs
is the problem of deciding whether there is a simple polygon whose visibility
graph is isomorphic to a given input graph. This problem is well-known and
well-studied, but yet widely open in geometric graphs and computational
geometry.
Existential Theory of the Reals is the complexity class of problems that can
be reduced to the problem of deciding whether there exists a solution to a
quantifier-free formula F(X1,X2,...,Xn), involving equalities and inequalities
of real polynomials with real variables. The complete problems for this
complexity class are called Existential Theory of the Reals Complete.
In this paper we show that recognizing visibility graphs of polygons with
holes is Existential Theory of the Reals Complete. Moreover, we show that
recognizing visibility graphs of simple polygons when we have the internal and
external visibility graphs, is also Existential Theory of the Reals Complete.Comment: Sumbitted to COCOON2018 Conferenc
Creating Simplified 3D Models with High Quality Textures
This paper presents an extension to the KinectFusion algorithm which allows
creating simplified 3D models with high quality RGB textures. This is achieved
through (i) creating model textures using images from an HD RGB camera that is
calibrated with Kinect depth camera, (ii) using a modified scheme to update
model textures in an asymmetrical colour volume that contains a higher number
of voxels than that of the geometry volume, (iii) simplifying dense polygon
mesh model using quadric-based mesh decimation algorithm, and (iv) creating and
mapping 2D textures to every polygon in the output 3D model. The proposed
method is implemented in real-time by means of GPU parallel processing.
Visualization via ray casting of both geometry and colour volumes provides
users with a real-time feedback of the currently scanned 3D model. Experimental
results show that the proposed method is capable of keeping the model texture
quality even for a heavily decimated model and that, when reconstructing small
objects, photorealistic RGB textures can still be reconstructed.Comment: 2015 International Conference on Digital Image Computing: Techniques
and Applications (DICTA), Page 1 -
Reconstructing Generalized Staircase Polygons with Uniform Step Length
Visibility graph reconstruction, which asks us to construct a polygon that
has a given visibility graph, is a fundamental problem with unknown complexity
(although visibility graph recognition is known to be in PSPACE). We show that
two classes of uniform step length polygons can be reconstructed efficiently by
finding and removing rectangles formed between consecutive convex boundary
vertices called tabs. In particular, we give an -time reconstruction
algorithm for orthogonally convex polygons, where and are the number of
vertices and edges in the visibility graph, respectively. We further show that
reconstructing a monotone chain of staircases (a histogram) is fixed-parameter
tractable, when parameterized on the number of tabs, and polynomially solvable
in time under reasonable alignment restrictions.Comment: Appears in the Proceedings of the 25th International Symposium on
Graph Drawing and Network Visualization (GD 2017
Space-Time Trade-offs for Stack-Based Algorithms
In memory-constrained algorithms we have read-only access to the input, and
the number of additional variables is limited. In this paper we introduce the
compressed stack technique, a method that allows to transform algorithms whose
space bottleneck is a stack into memory-constrained algorithms. Given an
algorithm \alg\ that runs in O(n) time using variables, we can
modify it so that it runs in time using a workspace of O(s)
variables (for any ) or time using variables (for any ). We also show how the technique
can be applied to solve various geometric problems, namely computing the convex
hull of a simple polygon, a triangulation of a monotone polygon, the shortest
path between two points inside a monotone polygon, 1-dimensional pyramid
approximation of a 1-dimensional vector, and the visibility profile of a point
inside a simple polygon. Our approach exceeds or matches the best-known results
for these problems in constant-workspace models (when they exist), and gives
the first trade-off between the size of the workspace and running time. To the
best of our knowledge, this is the first general framework for obtaining
memory-constrained algorithms
- …