arXiv subjects

Tom Kelly

Publications and source records attributed to Tom Kelly.

43 records · Page 3Linked to original sources

arXiv2019

FrankenGAN: Guided Detail Synthesis for Building Mass-Models Using Style-Synchonized GANs

Coarse building mass models are now routinely generated at scales ranging from individual buildings through to whole cities. For example, they can be abstracted from raw measurements, generated procedurally, or created manually. However, these models typically lack any meaningful semantic or texture details, making them unsuitable for direct display. We introduce the problem of automatically and realistically decorating such models by adding semantically consistent geometric details and textures. Building on the recent success of generative adversarial networks (GANs), we propose FrankenGAN, a cascade of GANs to create plausible details across multiple scales over large neighborhoods. The various GANs are synchronized to produce consistent style distributions over buildings and neighborhoods. We provide the user with direct control over the variability of the output. We allow her to interactively specify style via images and manipulate style-adapted sliders to control style variability. We demonstrate our system on several large-scale examples. The generated outputs are qualitatively evaluated via a set of user studies and are found to be realistic, semantically-plausible, and style-consistent.

cs.GR↗

arXiv2018

Simplifying Urban Data Fusion with BigSUR

Our ability to understand data has always lagged behind our ability to collect it. This is particularly true in urban environments, where mass data capture is particularly valuable, but the objects captured are more varied, denser, and complex. To understand the structure and content of the environment, we must process the unstructured data to a structured form. BigSUR is an urban reconstruction algorithm which fuses GIS data, photogrammetric meshes, and street level photography, to create clean representative, semantically labelled, geometry. However, we have identified three problems with the system i) the street level photography is often difficult to acquire; ii) novel façade styles often frustrate the detection of windows and doors; iii) the computational requirements of the system are large, processing a large city block can take up to 15 hours. In this paper we describe the process of simplifying and validating the BigSUR semantic reconstruction system. In particular, the requirement for street level images is removed, and greedy post-process profile assignment is introduced to accelerate the system. We accomplish this by modifying the binary integer programming (BIP) optimization, and re-evaluating the effects of various parameters. The new variant of the system is evaluated over a variety of urban areas. We objectively measure mean squared error (MSE) terms over the unstructured geometry, showing that BigSUR is able to accurately recover omissions from the input meshes. Further, we evaluate the ability of the system to label the walls and roofs of input meshes, concluding that our new BigSUR variant achieves highly accurate semantic labelling with shorter computational time and less input data.

cs.GR↗

arXiv2018

Bounding $χ$ by a fraction of $Δ$ for graphs without large cliques

The greedy coloring algorithm shows that a graph of maximum degree at most $Δ$ has chromatic number at most $Δ+ 1$, and this is tight for cliques. Much attention has been devoted to improving this "greedy bound" for graphs without large cliques. Brooks famously proved that this bound can be improved by one if $Δ\geq 3$ and the graph contains no clique of size $Δ+ 1$. Reed's Conjecture states that the "greedy bound" can be improved by $k$ if the graph contains no clique of size $Δ+ 1 - 2k$. Johansson proved that the "greedy bound" can be improved by a factor of $Ω(\ln(Δ)^{-1})$ or $Ω\left(\frac{\ln(\ln(Δ))}{\ln(Δ)}\right)$ for graphs with no triangles or no cliques of any fixed size, respectively. Notably missing is a $\textit{linear}$ improvement on the "greedy bound" for graphs without large cliques. In this paper, we prove that for sufficiently large $Δ$, if $G$ is a graph with maximum degree at most $Δ$ and no clique of size $ω$, then $$χ(G) \leq 72Δ\sqrt{\frac{\ln(ω)}{\ln(Δ)}}.$$ This implies that for sufficiently large $Δ$, if $ω^{(72c)^2} \leq Δ$ then $χ(G) \leq Δ/c$. This bound actually holds for the list-chromatic and even the correspondence-chromatic number (also known as the DP-chromatic number). In fact, we prove what we call a "local version" of it, a result implying the existence of a coloring when the number of available colors for each vertex depends on local parameters, like the degree and the clique number of its neighborhood. Our result simultaneously implies the linear improvement over the "greedy bound" and the two aforementioned results of Johansson.

math.CO↗

arXiv2018

Size of the Largest Induced Forest in Subcubic Graphs of Girth at least Four and Five

In this paper, we address the maximum number of vertices of induced forests in subcubic graphs with girth at least four or five. We provide a unified approach to prove that every 2-connected subcubic graph on $n$ vertices and $m$ edges with girth at least four or five, respectively, has an induced forest on at least $n-\frac{2}{9}m$ or $n-\frac{1}{5}m$ vertices, respectively, except for finitely many exceptional graphs. Our results improve a result of Liu and Zhao and are tight in the sense that the bounds are attained by infinitely many 2-connected graphs. Equivalently, we prove that such graphs admit feedback vertex sets with size at most $\frac{2}{9}m$ or $\frac{1}{5}m$, respectively. Those exceptional graphs will be explicitly constructed, and our result can be easily modified to drop the 2-connectivity requirement.

math.CO↗

arXiv2018

Induced 2-degenerate Subgraphs of Triangle-free Planar Graphs

A graph is $k$-degenerate if every subgraph has minimum degree at most $k$. We provide lower bounds on the size of a maximum induced 2-degenerate subgraph in a triangle-free planar graph. We denote the size of a maximum induced 2-degenerate subgraph of a graph $G$ by $α_2(G)$. We prove that if $G$ is a connected triangle-free planar graph with $n$ vertices and $m$ edges, then $α_2(G) \geq \frac{6n - m - 1}{5}$. By Euler's Formula, this implies $α_2(G) \geq \frac{4}{5}n$. We also prove that if $G$ is a triangle-free planar graph on $n$ vertices with at most $n_3$ vertices of degree at most three, then $α_2(G) \geq \frac{7}{8}n - 18 n_3$.

math.CO↗

arXiv2016

Minimum Size of Feedback Vertex Sets of Planar Graphs of Girth at least Five

A feedback vertex set of a graph is a subset of vertices intersecting all cycles. We provide tight upper bounds on the size of a minimum feedback vertex set in planar graphs of girth at least five. We prove that if $G$ is a connected planar graph of girth at least five on $n$ vertices and $m$ edges, then $G$ has a feedback vertex set of size at most $\frac{2m-n+2}{7}$. By Euler's formula, this implies that $G$ has a feedback vertex set of size at most $\frac{m}{5}$ and $\frac{n-2}{3}$. These results not only improve a result of Dross, Montassier and Pinlou and confirm the girth-5 case of one of their conjectures, but also make the best known progress towards a conjecture of Kowalik, Lužar and Škrekovski and solves the subcubic case of their conjecture. An important step of our proof is providing an upper bound on the size of minimum feedback vertex sets of subcubic graphs with girth at least five with no induced subdivision of members of a finite family of non-planar graphs.

math.CO↗

arXiv2016

Exponentially Many 4-List-Colorings of Triangle-Free Graphs on Surfaces

Thomassen proved that every planar graph $G$ on $n$ vertices has at least $2^{n/9}$ distinct $L$-colorings if $L$ is a 5-list-assignment for $G$ and at least $2^{n/10000}$ distinct $L$-colorings if $L$ is a 3-list-assignment for $G$ and $G$ has girth at least five. Postle and Thomas proved that if $G$ is a graph on $n$ vertices embedded on a surface $Σ$ of genus $g$, then there exist constants $ε,c_g > 0$ such that if $G$ has an $L$-coloring, then $G$ has at least $c_g2^{εn}$ distinct $L$-colorings if $L$ is a 5-list-assignment for $G$ or if $L$ is a 3-list-assignment for $G$ and $G$ has girth at least five. More generally, they proved that there exist constants $ε,α>0$ such that if $G$ is a graph on $n$ vertices embedded in a surface $Σ$ of fixed genus $g$, $H$ is a proper subgraph of $G$, and $ϕ$ is an $L$-coloring of $H$ that extends to an $L$-coloring of $G$, then $ϕ$ extends to at least $2^{ε(n - α(g + |V(H)|))}$ distinct $L$-colorings of $G$ if $L$ is a 5-list-assignment or if $L$ is a 3-list-assignment and $G$ has girth at least five. We prove the same result if $G$ is triangle-free and $L$ is a 4-list-assignment of $G$, where $ε=\frac{1}{8}$, and $α= 130$.

math.CO↗