text:
### Talks in 2020

#### Strukturtheorie-Seminar

Title: Three recreational tales
Speaker: Dr. Florian Lehner (Thomas) (TU Graz)
Date: Thursday, 29 October 2020, 11:30
Room: Lecture room P2 (Physics building) + webex
Abstract:

Mathematical puzzles often lead to interesting and challenging research questions. In this talk I will outline three instances of problems rooted in recreational mathematics each of which has inspired a substantial amount of research: pursuit-evasion games, asymmetric graph colouring, and graph reconstruction. For each of these problems I will highlight open research questions, survey some recent results, and (depending on time) provide rough proof sketches.

The talk is aimed at a non-specialist audience, no previous knowledge will be required.

Thursday, 29 Oct, 2020 11:15 CET

Webex meeting number: 137 992 7734

Password: HEeNZ3Vhp42

https://tugraz.webex.com/tugraz/j.php?MTID=m888653d1ba2a8be86ed61f2fa8528e03

The talk starts at 11:30.

The lecture room can host up to 51 persons.

#### Seminar für Kombinatorik und Optimierung

Title: Decompositions of Hypergraphs
Speaker: Felix Joos (Universität Heidelberg)
Date: Friday 23rd October 14:15
Room: Online meeting (Webex)
Abstract:

I'll present various results on approximate decompositions of
hypergraphs. In all of the results, the key technique are random
processes that yield the desired results if combined suitably.

Meeting link:
$\text{ https://tugraz.webex.com/tugraz/j.php?MTID=m1cd0904285a119237aa9a7ce985ad803}$

Meeting number:
137 149 1265

Password:
JYc3B3dunG2

#### Strukturtheorie-Seminar

Title: WEIGHTED GRAPHS OVER ORDERED FIELDS
Speaker: Dr. Anna Muranova (TU Graz)
Date: Thursday, 22 October 2020, 13:00 on time
Room: Seminar room AE06 + webex
Abstract:

A weighted graph is a graph where each edge is given a weight. In the classical theory of weighted graphs, this weight is a positive real number. We consider more general weights, namely, positive elements of any ordered field. We prove the existence and uniqueness of the solution of a Dirichlet problem on finite graphs and investigate some properties of infinite graphs. Classical weighted
graphs are related to electrical networks with resistors. In a similar way, one can relate weighted graphs over the field of rational functions with electrical networks with coils, capacitors, and resistors. The ordered field of rational functions is the most known non-Archimedean field. Its Cauchy completion is the Levi-Civita field R. Therefore, we consider some known infinite electrical networks (e.g. Feynman ladder) over R.

No pre-knowledge on ordered fields is assumed in this talk.

Please note the atypical time. The talk is also accessible via webex mode.

Meeting number: 137 044 3093
Password: M2jDZr53emd

https://tugraz.webex.com/tugraz/j.php?MTID=mddc8c646cc16cfa1c71847fc3f5341b8

Webex meeting hosted by Christian Lindorfer, beginning at 12:45 (middle European time); the talk starts at 13:00

#### Vortrag

Title: Hopf Algebras and Diagonal Harmonics
Speaker: Cesar Ceballos (Inst. f. Geometrie, TU Graz)
Date: 16.10.2010, 11:00
Room: HS G, Kopernikusgasee 24, bzw. Videokonferenz
Abstract:

The theory of Hopf algebras is a fundamental area in
mathematics which originated in the 1940’s and 1950’s motivated by work of
Hopf on algebraic topology and of Diedonné on algebraic groups. Diagonal
harmonics, on the other hand, is a more recent and apparently unrelated
area initiated by Garsia and Haiman in the early 1990’s, which has
remarkable connections to Macdonald polynomials, algebraic geometry,
representation theory, knot theory, and mathematical physics.

In this talk, I will give an insight to these fascinating areas, mainly
through a series of examples and without many technicalities. The main
purpose is to present some unexpected connections arising in the study of a
Hopf algebra structure on pipe dreams, certain discrete objects that
provide a combinatorial understanding of Schubert polynomials.

The talk is addressed to a general mathematical audience and no previous
knowledge of Hopf algebras or diagonal harmonics will be assumed.

#### Strukturtheorie-Seminar

Title: Operator algebras for random walks
Speaker: Dr. Adam Dor-On (University of Copenhagen)
Date: Thursday, 15 October 2020, 10:30
Room: webex meeting
Abstract:

We present some newly emerging connections between the theory of random walks and operator algebras. More specifically, to each random walk P we can associate several operator algebras that capture various kinds of behaviors of P. This yields new insight on random walks and provides operator algebraic interpretations for the strong ratio limit property of random walks. No knowledge on operator algebras will be assumed in the talk.

webex meeting

Thursday, 15 Oct, 2020 10:15 Rome, Stockholm, Vienna

Meeting number: 137 600 7164

Password: f3AbF427pUs

https://tugraz.webex.com/tugraz/j.php?MTID=m2ab4cfe9ac172dc045d6dd6512a479da

[tugraz.webex.com]

(Start time is 10:15, the talk will start at 10:30)

#### Zahlentheoretisches Kolloquium

Title: On the maximum of inclomplete Kloosterman sums
Speaker: Dr. Dante Bonolis (IST Wien)
Date: 08.10.2020, 14:30 Uhr
Room: HS E, Kopernikusgasse 24, 1. OG
Abstract:

\begin{document}

Let $t:\mathbb{F}_{p}\rightarrow\mathbb{C}$ be a complex valued function on $\mathbb{F}_{p}$. A classical problem in analytic number theory is bounding the maximum
$M(t):=\max_{0\leq H<p}\Big|\frac{1}{\sqrt{p}}\sum_{0\leq n < H}t(n)\Big|$
of the absolute value of the incomplete sums $\frac{1}{\sqrt{p}}\sum_{0\leq n < H}t(n)$. In this very general context one of the most important results is the P\'olya-Vinogradov bound
$M(t)\leq \left\|\hat{t}\right\|_{\infty}\log 3p,$
where $\hat{t}:\mathbb{F}_{p}\rightarrow\mathbb{C}$ is the normalized Fourier transform of $t$. In this talk, we provide a lower bound for certain incomplete Kloosterman sums, namely we prove that there exists a subset of $a\in\mathbb{F}_{p}^{\times}$ such that
$M( e((ax+\overline{x})/p))\geq \left(\frac{2}{\pi}+o(1)\right)\log\log p,$
as $p\rightarrow \infty$. We prove this by studying the growth of the moments of $\{M(e((ax+\overline{x})/p))\}_{a\in\mathbb{F}_{p}^{\times}}$. This is a joint work with Pascal Autissier and Youness Lamzouri.

\end{document}

#### Seminar für Kombinatorik und Optimierung

Title: Cutvertices in random planar maps
Speaker: Benedikt Stufler (TU Wien)
Date: Friday 16th October 14:15
Room: Online meeting (Webex)
Abstract:

We study the number of cutvertices in a random planar map as the number of edges tends to infinity. Interestingly, the combinatorics behind this seemingly simple problem are quite involved.

(Joint work with Marc Noy and Michael Drmota)

Meeting link:
$\text{ https://tugraz.webex.com/tugraz/j.php?MTID=m1cd0904285a119237aa9a7ce985ad803}$

Meeting number:
137 149 1265

Password:
JYc3B3dunG2

#### Seminar für Kombinatorik und Optimierung

Title: Non-concentration of the chromatic number and the Zig-zag conjecture
Speaker: Annika Heckel (Department Mathematik, LMU München)
Date: Friday 2nd October 14:15
Room: Online meeting (Webex)
Abstract:

The chromatic number of a graph is the minimum number of colours we need to colour all vertices so that adjacent vertices receive different colours.

What can we say about the chromatic number of a random graph $G(n,p)$? One main direction of past research has been the likely value of this random variable, i.e. proving that certain upper and lower bounds hold with high probability (whp). The other main direction of research has been the following question: how sharp is the concentration of the chromatic number? In other words, what is the length of the shortest interval (or rather sequence of intervals) which contains the chromatic number whp?

The starting point is a classic result of Shamir and Spencer who showed that the chromatic number of $G(n,p)$ is whp contained in some sequence of intervals of length at most about $n^{1/2}$. For sparse random graphs, this can be improved dramatically: Alon and Krivelevich proved that the chromatic number of $G(n,p)$ is two-point concentrated whenever $p < n^{-1/2 - \epsilon}$.

In view of strong concentration results, Bollobás and Erdős asked the opposite question: can we find any examples where the chromatic number is not very narrowly concentrated? Specifically, can we show that the chromatic number of $G(n, 1/2)$ is not whp concentrated on 100 integers?

In this talk, I will present a recent result showing that, at least for some values $n$, the chromatic number of $G(n, 1/2)$ is not concentrated on fewer than $n^{1/2- o(1)}$ consecutive values, almost matching Shamir and Spencer's upper bound. I will also discuss and give evidence for a recent conjecture on the correct concentration interval length, which seems to depend on n.

\newpage

Meeting link:
$\text{ https://tugraz.webex.com/tugraz/j.php?MTID=m1cd0904285a119237aa9a7ce985ad803}$

Meeting number:
137 149 1265

Password:
JYc3B3dunG2

#### Mathematisches Kolloquium

Title: Monotone chains in multiplicative sets
Speaker: Dr. Oleksiy Klurman (Max-Planck-Institut für Mathematik, Bonn)
Date: 12.08.2020, 14:30 Uhr
Room: Seminarraum Analysis-Zahlentheorie, Kopernikusgasse 24, 2.OG
Abstract:

It is believed that given a typical multiplicative function $f:\mb{N} to\mb{R}$ and admissible integers $a_1... a_k$ each possible arrangement $f(n+a_1)<f(n+a_2)<....<f(n+a_k)$ occurs for infinitely many n. This problem is widely open in general.

In this talk, we describe a new approach to deal with questions of this type and present various applications.

#### Zahlentheoretisches Kolloquium

Title: Dirichlet Series with Periodic Coefficients and their Value-Distribution near the Critical Line
Speaker: Athanasios Sourmelidis (TU Graz)
Date: 17.7.2020, 13:30
Room: HS BE01, Steyrergasse 30
Abstract:

The class of Dirichlet series associated with a periodic arithmetical function $f$ includes the Riemann zeta-function as well as Dirichlet $L$-functions to residue class characters.

We study the value-distribution of these Dirichlet series in a neighbourhood of the critical line (which is the abscissa of symmetry of the related Riemann-type functional equation).

In particular, for a fixed complex number $a\neq 0$, we consider for even or odd periodic $f$ the values taken at the $a$-points of the $\Delta$-factor of the functional equation, prove the existence of the mean-value of these points, show the uniform distribution of their ordinates, and obtain a related discrete universality theorem. This is joint work with Jörn Steuding and Ade Irma Suriajaya.

Remark: Athanasios Sourmelidis is a new member of the Institute of Analysis and Number Theory, where he will work as a PostDoc researcher.

#### Seminar für Kombinatorik und Optimierung

Title: Keeping a graph connected via nonpreemptive edge scheduling (Maximizing the minimum load time in a graphic matroid)
Speaker: Lasse Wulf (Institut für Diskrete Mathematik, TU Graz)
Date: 2.7.2020, 16:15 (talk), 15:50-16:15 informal chat
Room: online meeting (Webex)
Abstract:

Consider the following process over time: Given a graph $G = (V,E)$ with positive integral edge weights $w(e)$, we choose for each edge e exactly one time point $t(e) \in [0, \infty)$. This causes $e$ to be active during the
interval $[t(e), t(e) + w(e)]$. We now ask how we can choose the times $t(e)$ such that the subgraph of active edges is spanning for a maximal amount of time. The problem is related to the classical problems of spanning
tree packing and Menger's problem. It can also be seen as a generalization of maximizing the minimum load time
in nonpreemptive scheduling. In this talk, we show that the problem is NP-complete, even if $G=K_{2,n}$ or if
$w(e) \in \{1, ... , 6\}$. Furthermore, if P $\ne$ NP, the problem can not be approximated in polynomial time by a factor
better than 7/6. On the other hand, if both the treewidth of the input graph and the edge weights are bounded
by a constant, we give a linear time algorithm.

{

Webex meeting info

Meeting number (access code): 137 382 3722

Meeting password: 8diN3Pd9q34

https://tugraz.webex.com/tugraz/j.php?MTID=m01485e57708c11c02cfe8673af6dcd11}
}

#### Strukturtheorie-Seminar

Title: Planar Cayley graphs and Kleinian groups
Speaker: Prof. Agelos Georgakopoulos (University of Warwick)
Date: Wednesday, July 1st, 2020, 16:15 (virtual coffee break beginning 15:45)
Room: Webex meeting
Abstract:

Kleinian groups are a classical topic. I will give an overview
and explain their relationship to groups having planar Cayley graphs.
Moreover, I will show that if a finitely generated group G acts faithfully
and properly discontinuously by homeomorphisms on a planar surface, then G
admits such an action that is in addition co-compact.

Link to preprint: https://arxiv.org/pdf/1905.06669}

Webex meeting number: 137 170 6111

Password: iUMCF5Pb6M5

Meeting-Link:

https://tugraz.webex.com/tugraz-de/j.php?MTID=m31a117d32de0fca9218c0731e7d60b47}

#### Seminar für Kombinatorik und Optimierung

Title: Some combinatorial aspects of pop-stack sorting
Speaker: Andrei Asinowski (Institut für Mathematik, Universität Klagenfurt)
Date: 25.6.2020, 15:00 (talk), informal chat starts at 14:50
Room: online meeting (Webex)
Abstract:

Pop-stack sorting is a natural procedure for sorting permutations, where at
each iteration all maximal descending strings are reversed (for example
T(526314) = 251364). It can be seen as a variation of the classical stack
sorting. I will present some structural, enumerative, and algorithmic results
and conjectures related to the pop-stack sorting, including links to finite
automata, lattice paths and random permutations.

Joint work with Cyril Banderier (University of Paris North) and Benjamin Hackl (University of Klagenfurt)

{
Webex meeting info

Meeting number (access code): 137 215 5408

Meeting password: dT6yWAuqu64

https://tugraz.webex.com/webappng/sites/tugraz/meeting/info/487199bb6fd74508b490443deb606df7}

#### UPDATE: Strukturtheorie-Seminar

Title: Cayley-Abels graphs of totally disconnected locally compact groups
Speaker: Prof. Rögnvaldur Möller + Dr. Waltraud Lederle (University of Iceland + Université Catholique de Louvain)
Date: Wednesday 10 June 2020, 16:15 (virtual coffee break beginning 15:45)
Room: Webex meeting
Abstract:

The Cayley-Abels graph of a compactly generated totally disconnected locally compact group is an analogue of the ordinary Cayley graph for a finitely generated group.   Given a compactly generated totally disconnected locally compact group G, what is the lowest possible valency of a Cayley-Abels graph?  How does the valency relate to other properties of the group?  Is there something special about graphs that have this lowest possible valency?

Joint work with Arnbjörg Soffia Arnadottir (Waterloo, Canada)

Webex meeting number: 321 855 198

Password: ZNpTHv9PT27

tugraz.webex.com/tugraz/j.php?MTID=m77665a58f92ac664ce28fc666b6d1f5d

Join by video system:
Dial 321855198@tugraz.webex.com

#### Probevortrag Habilitationsverfahren

Title: Der Fundamentalsatz der Algebra
Speaker: Oliver Cooley (Institut für Diskrete Mathematik, TU Graz)
Date: 27.05.2020, 16:00
Room: WebEx Meeting
Abstract:

Der Link fuer den Vortrag wird rechtzeitig bekanntgegeben.

#### Strukturtheorie-Seminar

Title: Poisson type limit theorems for a noncommutative independence associated with positive symmetric cones
Speaker: Lahcen Oussi (Intitut für Diskrete Mathematik, TU Graz)
Date: Mittwoch, 13.5.2020, 16:15
Room: https://tugraz.webex.com/meet/lahcen.oussi
Abstract:

We present an analogue of the classical Law of Small Numbers, formulated for a noncommutative independence (the bm-independence), where the random variables are indexed by elements of positive symmetric cones in Euclidean spaces, including $\mathbb{R}^{d}_{+}$, the Lorentz cone in Minkowski spacetime and positive definite real symmetric matrices. The geometry of the cones plays an important role in the study as the volume characteristic sequences of each cone, related to the growth of volumes of intervals in the cone, appears in our final formulas. Also the combinatorics of ordered partitions is crucial for our study as one of the main tools for performed computations.

#### Mathematisches Kolloquium

Title: Hypergraph containers with applications in discrete geometry
Speaker: Dr. Oliver Roche-Newton (RICAM, Linz, Austria)
Date: 17.04.2020, 14:00 Uhr
Room: Seminarraum Analysis-Zahlentheorie, Kopernikusgasse 24, 2.OG
Abstract:

In recent years, the newly developed theory of hypergraph containers has resulted in several remarkable results in graph theory and extremal combinatorics. Essentially, this theory gives detailed information about the independent sets of hypergraphs, provided that the edges are distributed reasonably well. I will discuss recent joint work with Audie Warren, in which these tools were applied to problems in discrete geometry. In particular, an upper bound for the number of subsets of the finite plane with no collinear triples is given.

#### Mathematisches Kolloquium

Title: Minimality of the rock-salt structure and Universal Optimality for multi-component lattice systems
Speaker: Dr. Laurent Betermin (Universität Wien)
Date: 20.03.2020, 14:00 Uhr
Room: Seminarraum Analysis-Zahlentheorie, Kopernikusgasse 24, 2.OG
Abstract:

The mathematical justification of crystallization phenomena is usually a challenging problem and very few results exist about the optimality of ionic crystals. In this talk, I will present new analytical and numerical results obtained with Markus Faulhuber (University of Vienna) and Hans Knüpfer (University of Heidelberg) about the optimality of the rock-salt structure among lattices and charges distributions. These results are based on optimality results for special lattice functions arising in Number Theory: the Epstein zeta functions and the lattice theta functions. Many open problems will be presented, including the Universal Optimality for alternating species that we have already obtained in dimension 2 for the triangular lattice with Markus Faulhuber.

#### Colloquium of the Institute of Discrete Mathematics

Title: Infinite bridges for tree-valued Markov chains
Speaker: Prof. Anton WAKOLBINGER (Universität Frankfurt)
Date: Wednesday, 11 March 2020, 16:15
Room: Lecture room BE01, Steyrergasse 30, ground floor
Abstract:

For a few examples of tree-valued Markov chains (Rémy’s tree growth chain, the radix sort chain and the PATRICIA chain), we discuss representations of their Doob-Martin boundary that are obtained from their extremal “bridges to infinity”
(and thus from a conditioning on their remote future).

The talk is based on joint work with Steve Evans (Berkeley) and Rudolf Grübel (Hannover).

EGW, Doob-Martin boundary of Rémy's tree growth chain, Ann. Probab. 45b (2017), 225-277.

EW1, Radix sort trees in the large, Electron. Commun. Probab. 22 (2017).

EW2, PATRICIA bridges. In: Genealogies of Interacting Particle Systems, eds.
M. Birkner, R. Sun and J. Swart, pp. 233-267, World Scientific 2020.

#### Selection procedure for assistant position

Title: Short talks by several candidates
Speaker: K. Heuer / K. Kolesko / A. Muranova / C. Alves / F. Tonti ()
Date: 29 + 30. + 31.1.2020, afternoon
Room:
Abstract:

Assistant position (with PhD) - selection procedure

20 minutes' scientific talks (titles below) followed by

20 Minuten Lehrvortrag (Deutsch), Thema
Die Eulersche Zahl e''

Wednesday, 29.1.2020, seminar room A306 (Steyrergasse 30, 3rd floor):

14:00 Karl HEUER (Berlin): Forcing Hamiltonicity in locally finite graphs via forbidden induced subgraphs

15:30 Konrad KOLESKO (Innsbruck): Limit theorems in branching processes

Thursday, 30.1., seminar room AE06 (Steyrergasse 30, ground floor):

13:30 Anna MURANOVA (Bielefeld): On the notion of effective impedance for networks

15:00 Caio ALVES (Leipzig): Decoupling inequalities in loop percolation

Friday, 31.1., seminar room AE02 (Steyrergasse 30, ground floor):

13:30 Fabio TONTI (Vienna): A new proof of Thoma's theorem

#### Zahlentheoretisches Kolloquium

Title: How to quantify the randomness of a real-valued sequence?
Speaker: Niclas Technau (Tel Aviv University)
Date: 28.1.2020, 13:30
Room: SR Analysis-Zahlentheorie, Kopernikusgasse 24, 2nd floor
Abstract:

This talk concerns local spacing statistics
which are used to quantify the randomness
of a given real-valued sequence.
We survey past and recent developments,
focusing on pair correlation statistics,
and explain the connection to (arithmetic) quantum chaos.
Furthermore, we report on joint work with Zeev Rudnick
about lacunary sequences.

#### Vorstellungsvortrag im Rahmen des Habilitationsverfahrens

Title: Linear differential equations and difference algebraic groups
Speaker: Dr. Michael Wibmer (TU Graz)
Date: 24.01.2020, 14:00 Uhr
Room: Seminarraum Analysis-Zahlentheorie, Kopernikusgasse 24, 2.OG
Abstract:

Classical differential Galois theory associates a linear algebraic group to a linear differential equation. This group measures the algebraic relations among the solutions. In joint work with L. Di Vizio and Ch. Hardouin I developed a Galois theory that measures the difference algebraic relations among the solutions of a linear differential equation. In this Galois theory the Galois groups are linear difference algebraic groups, i.e., subgroups of the general linear group defined by algebraic difference equations in the matrix entries. This Galois theory is helpful for understanding the behavior of the solutions under a transformation of the independent variable and also applies to linear differential equations depending on a parameter.

Because of their role in the study of linear differential equations it is desirable to have a comprehensive structure theory for linear difference algebraic groups. In this talk we will discuss some progress in this direction.

#### Vortrag

Title: Statistical methods and risk modeling in insurance companies
Speaker: Sanela Omerovic (FMA, Finanzmarktaufsicht Österreich)
Date: Freitag, 24. Jänner 2020, 15:15 Uhr
Room: SR für Statistik (NT03098), Kopernikusgasse 24/III
Abstract:

Abstract:

The implementation of Solvency II has introduced a set of risk-based principles for the solvency capital requirement (SCR) of insurance companies. Undertakings are therefore required to calculate their SCR by means of a holistic balance sheet approach. These requirements entail the use of sophisticated mathematical models. This presentation addresses different statistical methods and basic legal aspects for risk modeling in insurance companies.

#### Mathematisches Kolloquium

Title: Suprema in spectral spaces and density
Speaker: Carmelo A. Finocchiaro (University of Catania)
Date: 17.01.2020, 15 Uhr c.t.
Room: Seminarraum Analysis-Zahlentheorie, Kopernikusgasse 24, 2.OG
Abstract:

Let $X$ be a spectral space. As it is well known, the spectral topology of $X$ can be refined to another topology, the so called constructible topology}, which remains spectral and becomes Hausdorff. Since any spectral space is Kolmogoroff, $X$ admits a canonical partial order $\leq$, usually called the specialization order}, defined by setting, for any $x,y\in X$, $x\leq y$ if $y\in \overline{\{x\}}$. We will provide conditions for a subset $Y$ of the partially ordered set $(X,\leq)$ in order that the supremum of $Y$ (in $X$) exists and belongs to the closure of $Y$ in the constructible topology. Algebraic applications of such topological results will concern density properties of some spaces of rings and ideals. Moreover, we will provide topological characterizations of distinguished classes of domains, in terms of certain properties of their ideals. Joint work with D. Spirito.

#### Mathematisches Kolloquium

Title: BOUNDED EXPONENTIAL SUMS
Speaker: Reynold Fregoli (Royal Holloway University of London)
Date: 17.01.2020, 14:00 Uhr
Room: Seminarraum Analysis-Zahlentheorie, Kopernikusgasse 24, 2.OG
Abstract:

Let A $\subset$ $\mathbb{N}$, $\alpha$ $\in$ (0, 1), and for x $\in$ $\mathbb{R}$ let e(x) := e$^2$^\pi$$^i$$^x$. We set S$_A$($\alpha$,N) :=$\sum$n$\in$A$ n$\le$N e(n$\alpha$)

Recently, Lambert A’Campo proposed the following question: is there an infinite non-cofinite set A $\subset$ $\mathbb{N}$ such that for all $\alpha$ $\in$ (0, 1) the sum S$_A$($\alpha$,N) has bounded modulus as N $\rightarrow$ +$\infty$? In this talk I will give an idea of why such sets do not exist. To show this, I use a theorem by Duffin and Schaeffer on complex power series. The above result can also be extended to prove that if the sum S$_A$($\alpha$,N) is bounded in modulus on an arbitrarily small interval and on the set of rational points, then the set A has to be either finite or cofinite. On the other hand, it can be shown that there are infinite non-cofinite sets A such that $\mid$S$_A$($\alpha$,N)$\mid$ is bounded for all $\alpha$ $\in$ $E$ $\subset$ (0, 1), where $E$ has full Hausdorff dimension and $\mathbb{Q}$ $\cap$ (0, 1) $\subset$ $E$.

#### Geometrie-Seminar

Title: Using multi-cover persistent homology as a fingerprint for periodic crystals
Speaker: Theresa Heiss (ISTA)
Date: Thursday 16.1.2020, 10:30
Room: Seminarraum 2, Kopernikusgasse 24/IV
Abstract:

As the atoms in periodic crystals are arranged periodically,
such a crystal can be modeled by a periodic point set, i.e. by the union
of several translates of a lattice. Two periodic point sets are considered
equivalent if there is a rigid motion from one to the other. A periodic
point set can be represented by a finite cutout s.t. copying this cutout
infinitely often in all directions yields the periodic point set. The fact
that these cutouts are not unique creates problems when working with them.
Therefore, material scientists would like to work with a complete,
continuous invariant instead. We conjecture that a variant of persistent
homology, namely the sequence of $k$-fold cover persistence diagrams for all
positive integers $k$, is such a complete, continuous invariant for
equivalence classes of periodic point sets

#### Mathematisches Kolloquium

Title: On the Skolem Problem for parametric families of linear recurrence sequences and some G.C.D. problems
Speaker: Alina Ostafe (The University of New South Wales)
Date: 16.01.2020, 14:00 Uhr
Room: Seminarraum Analysis-Zahlentheorie, Kopernikusgasse 24, 2.OG
Abstract:

In this talk we discuss a parametric version of the Skolem Problem about decidability of the existence of a zero in a linear recurrence sequence.
We then connect this problem to studying the greatest common divisor of two linear recurrence sequences of polynomials.

#### Seminar für Kombinatorik und Optimierung

Title: Cycles in random planar graphs
Speaker: Michael Missethan (TU Graz)
Date: Friday 10th January 14:15
Room: Seminarraum AE06, Steyrergasse 30, EG
Abstract:

In this talk we discuss recent results on largest and shortest cycles in random planar graphs. More precisely, we consider the following questions in random planar graphs. What is the order of the longest cycle? Is the longest cycle in the largest component? Is there even a threshold phenomenon such that all cycles longer than a certain value belong to the largest component and all shorter cycles lie outside? We use structural properties of a random planar graph (e.g. order of the complex part, core and kernel) to find answers to these questions. But also a version of the P\'{o}lya urn model plays a crucial role in our approach to obtain bounds on the cycle lengths.

This talk is based on joint work with Mihyun Kang.


text:
# Order of an element and $\mathbb{Z_n}$

Consider the group $$\mathbb{Z_8}^*$$. This group contains only the following elements, 1,3,5,7. It was suggested to me that this group is not cyclic because there does not exist an element in the set with order 4.

I do not see how I can prove that for an element g in $$\mathbb{Z_n}^*$$ the integer n in $$g^n$$ gives me the number of elements in the set that is generated by g.

My thoughts on this are as follows: I believe that it may be the case that the order of an element in a group g tells me the number of elements in the generator of g, because G is closed and the order is the smallest positive integer n that gives the identity.

• Are you asking only if the order of a cyclic group equals the order of its generator, or something further? That surely has been asked in the past so it will end up being a duplicate question (always search first before asking). – Bill Dubuque Feb 2 '19 at 18:42
• Well, you know $|Z_8^*|=4$ because you can count the elements. 1-> one, 3-> two, 5-> three, 7-> four. There are four of them. So if $Z_8^*$ is cyclic then there is a $g$ so that $\{g, g^2, g^3, g^4, g^5,.........\} = \{1,3,5,7\}$. Since $\{1,3,5,7\}$ has four different elements then $\{g,g^2,g^3,g^4,g^5....\}$ must have four differen elements. So $\{g,g^2,g^3,g^4,g^5....\} = \{g,g^2,g^3,1\}$ and $g^4 = 1$ and $g^2 \ne g$ and $g^2 \ne 1$. and $g^3 \ne 1$. and $g^3 \ne g^2$. And it's just a matter of testing all the elements. There is no such element. – fleablood Feb 2 '19 at 19:22

By Lagrange's Theorem we know that for any group $$G$$ with $$\left| G \right| = n$$ we have $$g^n = 1$$ for any $$g \in G$$. If the group $$G$$ is cyclic then there must be an element $$g$$ such that $$g$$ has order $$n$$ (by definition of cyclicity). So if you check that no element has order $$4$$ for $$\mathbb{Z}_8^*$$ then the group is not cyclic. Does this help?

I think what you are asking is if $$g\in G$$ and $$g^k = 1$$ and that $$k$$ is the smallest positive integer, $$k$$ so that $$g^k =e$$ (i.e. the definition $$|g| = k$$) then $$|| = k$$.

Pf: $$ = \{g, g^2, g^3,......., g^l,.....\}$$

If $$m > k$$ then $$g^m = g^{k + (m-l)} = g^k*g^{m-1} = e*g^{m-1} = g^{m-l}$$ so $$g^m$$ is not distinct and $$$$ will have at most $$k$$ elements.

And $$ = \{g, g^2, g^3, ....., g^{k-1}, g^k=e\}$$.

Now suppose those elements are not distinct, that is, suppose $$g^i = g^j$$ but $$0 < i < j < k$$.

The $$g^i = g^j$$

$$g^i*(g^{-1})^i = g^j*(g^{-1})^i$$

$$e = g^{j-i}$$.

But $$0 < j-i < k$$ and that contradicts that $$k$$ was the smallest such positive integer where $$g^k = e$$.

So $$ = \{g, g^2, .... ,g^{k-1}, 1\}$$ and $$|| = k$$.

...

That was probably already proven in your class and is buried somewhere in your notes. You definitely used that when you proved Lagrange Theorem.

Just do it.

If $$\mathbb Z_8^*$$ is cyclic there is a $$g\in \mathbb Z_8^*$$ so that $$\{g,g^2,g^3,g^4\} = \{1,3,5,7\}$$

$$\{1,1^2, 1^3, 1^4\} = \{1,1,1,1\} = \{1\} \ne \{1,3,5,7\}$$

$$\{3, 3^2, 3^3, 3^4\} = \{3,1,3,1\} = \{1,3\} \ne \{1,3,5,7\}$$

$$\{5, 5^3, 5^3, 5^4\} = \{5,1,5,1\} = \{1,5\} \ne \{1,3,5,7\}$$

$$\{7,7^2, 7^3, 7^4\} = \{7,1,7,1\} = \{1,7\}\ne \{1,3,5,7\}$$

So there is no such $$g$$ so it isn't cyclic.

I do not see how I can prove that for an element g in Zn∗ the integer n in gn gives me the number of elements in the set that is generated by g.

Assuming $$n$$ is the same as the index i $$\mathbb Z_n$$ then $$g^n =1$$. This is a direct consequence of Lagrange theorem: If $$G$$ is a finite group of order $$n$$, then $$|g|=$$ divides $$|G| = n$$. And so $$g^n = (g^k)^{\frac nk} = 1^{\frac nk} = 1$$.

My thoughts on this are as follows: I believe that it may be the case that the order of an element in a group g tells me the number of elements in the generator of g, because G is closed and the order is the smallest positive integer n that gives the identity.

Well that's obvious.

If $$|g| = k$$ then BY DEFINITION $$g^k = 1$$ and $$g^{i; 0 so $$g, g^2,g^3, g^4, ......, g^k =1$$ will be the element in $$$$ and if you count them there are $$k$$ of them.

Okay, we have to prove then $$g^i \ne g^j$$ if $$0 \le i < j < k$$ but that's clear: If $$g^i = g^j$$ then $$g^i*(g^{-1})^i = g^j*(g^{-1})^i$$ so $$1 = g^{j-i}$$ but $$0 < j-i < k$$. But that contradicts $$|g| = k$$.

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Svoboda, Toby, Peter Irvine, Daniel Callies, and Masahiro Sugiyama. “The potential for climate engineering with stratospheric sulfate aerosol injections to reduce climate injustice.” Journal of Global Ethics (2019). Publisher's VersionAbstract
Climate engineering with stratospheric sulfate aerosol injections (SSAI) has the potential to reduce risks of injustice related to anthropogenic emissions of greenhouse gases. Relying on evidence from modeling studies, this paper makes the case that SSAI could have the potential to reduce many of the key physical risks of climate change identified by the Intergovernmental Panel on Climate Change. Such risks carry potential injustice because they are often imposed on low-emitters who do not benefit from climate change. Because SSAI has the potential to reduce those risks, it thereby has the potential to reduce the injustice associated with anthropogenic emissions. While acknowledging important caveats, including uncertainty in modeling studies and the potential for SSAI to carry its own risks of injustice, the paper argues that there is a strong case for continued research into SSAI, especially if attention is paid to how it might be used to reduce emissions-driven injustice.
Irvine, Peter, Kerry Emanuel, Jie He, Larry Horowitz, Gabriel Vecchi, and David Keith. “Halving warming with idealized solar geoengineering moderates key climate hazards.” Nature Climate Change (2019). Publisher's VersionAbstract

Solar geoengineering (SG) has the potential to restore average surface temperatures by increasing planetary albedo, but this could reduce precipitation. Thus, although SG might reduce globally aggregated risks, it may increase climate risks for some regions. Here, using the high-resolution forecastoriented low ocean resolution (HiFLOR) model—which resolves tropical cyclones and has an improved representation of present-day precipitation extremes—alongside 12 models from the Geoengineering Model Intercomparison Project (GeoMIP), we analyse the fraction of locations that see their local climate change exacerbated or moderated by SG. Rather than restoring temperatures, we assume that SG is applied to halve the warming produced by doubling CO2 (half-SG). In HiFLOR, half-SG offsets most of the CO2-induced increase of simulated tropical cyclone intensity. Moreover, none of temperature, water availability, extreme temperature or extreme precipitation are exacerbated under half-SG when averaged over any Intergovernmental Panel on Climate Change (IPCC) Special Report on Extremes (SREX) region. Indeed, for both extreme precipitation and water availability, less than 0.4% of the ice-free land surface sees exacerbation. Thus, while concerns about the inequality of solar geoengineering impacts are appropriate, the quantitative extent of inequality may be overstated.

Horton, Joshua B.Parametric Insurance as an Alternative to Liability for Compensating Climate Harms.” Carbon & Climate Law Review 12, no. 4 (2018): 285-296. Publisher's VersionAbstract
Interstate compensation for climate change based on legal liability faces serious obstacles. Structural incongruities related to causation, time, scope, and scale impede application of tort law to climate change, while political opposition from developed countries prevents intergovernmental consideration of liability as a means of compensating for climate damages. Insurance, however, in particular parametric insurance triggered by objective environmental indices, is emerging as a promising alternative to liability. This is manifest in the UNFCCC and the Paris Agreement, which ruled out recourse to legal liability, and in the formation and expansion of regional sovereign climate risk insurance schemes in the Caribbean, Africa, and the Pacific. Theory and early practice suggest that parametric insurance exhibits five key advantages compared to legal liability in the climate change context: (1) it does not require that causation be demonstrated; (2) it has evolved to provide catastrophic coverage; (3) it is oriented toward the future rather than the past; (4) it is contractual, rather than adversarial, in nature; and (5) it provides a high degree of predictability. Compensation based on parametric insurance represents a novel climate policy option with significant potential to advance climate politics.
Smith, Wake, and Gernot Wagner. “Stratospheric aerosol injection tactics and costs in the first 15 years of deployment.” Environmental Research Letters 13 (2018). Publisher's VersionAbstract
We review the capabilities and costs of various lofting methods intended to deliver sulfates into the lower stratosphere. We lay out a future solar geoengineering deployment scenario of halving the increase in anthropogenic radiative forcing beginning 15 years hence, by deploying material to altitudes as high as ~20 km. After surveying an exhaustive list of potential deployment techniques, we settle upon an aircraft-based delivery system. Unlike the one prior comprehensive study on the topic (McClellan et al 2012 Environ. Res. Lett. 7 034019), we conclude that no existing aircraft design—even with extensive modifications—can reasonably fulfill this mission. However, we also conclude that developing a new, purpose-built high-altitude tanker with substantial payload capabilities would neither be technologically difficult nor prohibitively expensive. We calculate early-year costs of ~$1500 ton−1 of material deployed, resulting in average costs of ~$2.25 billion yr−1 over the first 15 years of deployment. We further calculate the number of flights at ~4000 in year one, linearly increasing by ~4000 yr−1. We conclude by arguing that, while cheap, such an aircraft-based program would unlikely be a secret, given the need for thousands of flights annually by airliner-sized aircraft operating from an international array of bases.


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# ac generator winding wind turbines kw rpm

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AC4YD In-Person W5YI Test Session

Date: Apr 10, 2021 (Sat) 10:00am EDT 2 hours AC4YD $14.00 200 Maryland Ave, Winchester, KY 40391, USA view map W5YI-VEC Pre-registration is required, a Photo-ID is required and any license copies or CSCE's, if they apply. Cash is required. The test fee is$14.00 and if you have larger bills, they can be broken down to pay your fee. Covid-19 CDC guidelines will be followed. Your temperature will be checked, social-distancing will be observed, and a mask shall be worn at all times in the EOC (Emergency Operation Center). 0 / 15 slots claimed


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Notes On Development of Cricket in England - CBSE Class 9 History
Nearly 500 years ago, cricket developed in rural England out of several stick and ball games. Till the 18th century, cricket bats were curved like hockey sticks. The game was originally played on unfenced land in English villages with no defined boundaries. The laws of cricket were first written in 1744. These laws specified the dimensions of the stumps and the weight of the ball, but nothing about the shape and size of the bat and the distance of the boundaries from the wicket. The Marylebone Cricket Club, which was formed in 1787-88 revised the laws, which gradually made cricket a more skillful game. Bowlers now started bowling over-arm instead of the conventional under-arm style. While many important rules were introduced and changed in the 19th century, cricket matured during the industrial revolution. Timeless test matches were replaced by five-day test matches. Cricketing equipment is still made by hand, using natural pre-industrial material. The bat, stumps and bails are still made of wood, while the ball is made of leather twine and cork. Cricket developed as a Victorian gentlemanly game that embodied all Englishness. Professionals or players, played cricket for a living, while the gentlemen or amateurs played it for pleasure. Cricket was also introduced in public schools as part of physical training for boys. It was used to teach them discipline and the importance of hierarchy, and to groom them to run the British Empire in future. Women from the upper class played a slow-paced game called croquet. It was only by the end of the 19th century that women began to participate in cricket but they were discouraged from participating in competitions.

#### Summary

Nearly 500 years ago, cricket developed in rural England out of several stick and ball games. Till the 18th century, cricket bats were curved like hockey sticks. The game was originally played on unfenced land in English villages with no defined boundaries. The laws of cricket were first written in 1744. These laws specified the dimensions of the stumps and the weight of the ball, but nothing about the shape and size of the bat and the distance of the boundaries from the wicket. The Marylebone Cricket Club, which was formed in 1787-88 revised the laws, which gradually made cricket a more skillful game. Bowlers now started bowling over-arm instead of the conventional under-arm style. While many important rules were introduced and changed in the 19th century, cricket matured during the industrial revolution. Timeless test matches were replaced by five-day test matches. Cricketing equipment is still made by hand, using natural pre-industrial material. The bat, stumps and bails are still made of wood, while the ball is made of leather twine and cork. Cricket developed as a Victorian gentlemanly game that embodied all Englishness. Professionals or players, played cricket for a living, while the gentlemen or amateurs played it for pleasure. Cricket was also introduced in public schools as part of physical training for boys. It was used to teach them discipline and the importance of hierarchy, and to groom them to run the British Empire in future. Women from the upper class played a slow-paced game called croquet. It was only by the end of the 19th century that women began to participate in cricket but they were discouraged from participating in competitions.

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# Is there a way to use lazy constraints with Baron?

I am solving a non-linear mixed-integer programme with BARON. The objective function looks like $$\big( \sum_i x_i \big) \cdot \big(\prod_i e^{-y_i}\big)$$ (binary $$x$$ and real-valued $$y$$) and it has some quadratic constraints.

My problem is that I have an exponential number of linear constraints. I would like to add them in a way analogous to the Lazy Constraint callbacks of CPLEX and Gurobi. Does Baron support anything like this? Are there other solvers which allow lazy constraints and also have out-of-the-box support for weird non-linearities such as the above ones?

• No for BARON .. Oct 28, 2020 at 18:56
• Hello Alberto. Why do you have an exponential number of constraints? Would be nice if you give some words about the original problem behind the model you described. Just to understand if another formulation can be of interest. And also because we would like to benchmark LocalSolver on this interesting, challenging problem. Oct 30, 2020 at 20:53
• @LocalSolver a typical example would be subtour elimination constraints in a routing problem. Alternative formulations (such as MTZ) have notoriously poor continuous relaxations. Oct 30, 2020 at 21:16
• Sorry to be so curious, but what is the definition of the y variables in the model? Oct 31, 2020 at 8:49


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A non-local boundary value problem method for the Cauchy problem for elliptic equations.(English)Zbl 1170.35555

Let $$H$$ be a Hilbert space with norm $$\|\cdot\|$$, $$A:D(A)\subset H\to H$$ a positive definite, self-adjoint operator with compact inverse on $$H$$, and $$T$$ and $$\varepsilon$$ given positive numbers. The ill-posed Cauchy problem for elliptic equations
$u_{tt}= Au, \quad 0< t< T, \qquad \|u(0)-\varphi\|\leq\varepsilon, \qquad u_t(0)=0,$
is regularized by the well-posed non-local boundary value problem
$u_{tt}= Au, \quad 0< t< aT, \qquad \|u(0)+\alpha u(aT)=\varphi, \qquad u_t(0)=0,$
with $$a\geq1$$ being given and $$\alpha>0$$ the regularization parameter. A priori and a posteriori parameter choice rules are suggested which yield order-optimal regularization methods. Numerical results based on the boundary element method are presented and discussed.

MSC:

 35R25 Ill-posed problems for PDEs 65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
Full Text:


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# Recursion Warmup - An Odd Sum

What is the sum of the first 13 positive odd numbers?

×

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# Derivative of number e

#### wishmaster

##### Active member
I have to derivate this function:

$$\displaystyle x^3-5x^2+4x-e^x$$ where $x$ on e is actualy $x^0$,but i dont know how to write it here.....
i know how to derivate other terms,just this e is suspicious,because its a complement of the function if i am right.....

Another derivate i have to calculate is:

$$\displaystyle ln(x+\sqrt{1+x^{-25}}$$

Thank you all for the help!

#### MarkFL

Staff member
Just so we're clear, you are given:

$$\displaystyle f(x)=x^3-5x^2+4x-e^{x^0}$$

Right?

Note: In $\LaTeX$, to create an exponent having more than 1 character, enclose that exponent in curly braces, in the above I used the code e^{x^0}.

#### wishmaster

##### Active member
Just so we're clear, you are given:

$$\displaystyle f(x)=x^3-5x^2+4x-e^{x^0}$$

Right?

Note: In $\LaTeX$, to create an exponent having more than 1 character, enclose that exponent in curly braces, in the above I used the code e^{x^0}.

Yes,you are right! So i have learned something new

#### MarkFL

Staff member
Can you simplify $e^{x^0}$?

Staff member

#### MarkFL

Staff member
So then $x$ is zero,and $e$ is also $0$ ?
No, $e$ is a transcendental constant (like $\pi$), but what I was getting at, can we say:

$$\displaystyle e^{x^0}=e$$

when $x=0$ ?

In other words, can we state $0^0=1$ ?

#### wishmaster

##### Active member
No, $e$ is a transcendental constant (like $\pi$), but what I was getting at, can we say:

$$\displaystyle e^{x^0}=e$$

when $x=0$ ?

In other words, can we state $0^0=1$ ?
So then remains just $e$ when i derivate?

#### MarkFL

Staff member
So then remains just $e$ when i derivate?
No, apply the exponential and chain rules:

$$\displaystyle \frac{d}{dx}\left(e^{f(x)} \right)=e^{f(x)}\frac{d}{dx}\left(f(x) \right)$$

What do you find?

#### wishmaster

##### Active member
No, apply the exponential and chain rules:

$$\displaystyle \frac{d}{dx}\left(e^{f(x)} \right)=e^{f(x)}\frac{d}{dx}\left(f(x) \right)$$

What do you find?
$$\displaystyle e * 0$$ ?

#### MarkFL

Staff member
$$\displaystyle e * 0$$ ?
Correct, and any constant times zero is zero.

You would get the same result if you assume $e^{x^0}=e$, since the derivative of a constant is zero, but I wanted you to be careful, as I felt the intent of the author of the problem was to not make that assumption.

#### wishmaster

##### Active member
Correct, and any constant times zero is zero.

You would get the same result if you assume $e^{x^0}=e$, since the derivative of a constant is zero, but I wanted you to be careful, as I felt the intent of the author of the problem was to not make that assumption.
Just to mention, $e$ represents natural number.

My solution for the first one is $$\displaystyle 3x^2-10x+4$$

Can you help me with second one?


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# How are very low temperatures(-180k) measured?

1. Feb 24, 2010

### aliz_khanz

okie so cyrogenic processor is one thing but there has to be some phenomenon which is used to measure cyrogenic temperature???

2. Feb 24, 2010

### f95toli

I am not sure I understand your question.
All you need in order to measure low temperatures is a thermometer.
A thermometer designed to be used at or near room temperature (300K) will obviously not work very well at cryogenic temperatures, but that is to a large extent because it is not calibrated (and the materials used are not ideal for very low temperatures).
Some temperature sensors that are used at low temperatures (silicon diodes), work quite OK even well above room temperature.

Now, measuring VERY low temperatures (below say 0.3K) is somewhat trickier, but -180 degrees C can be measured even with something as simple as a 50p schottky diode as long as you calibrate it first, all you need to do is to bias it as some current and then measure how the voltage changes with temperature.

edit: Take a look at the following webpage
http://www.lakeshore.com/temp/sen/smindex.html [Broken]

Lakeshore is one of the biggest suppliers of temperature sensors and e.g. their DT470 Si diode is widely used (I am in fact installing one in a probe tomorrow)

Last edited by a moderator: May 4, 2017
3. Feb 24, 2010

### luke1970

Using a piece of equiptment delicate enough to observe the electron movement is one way, and probably the most effective.

4. Feb 24, 2010

### f95toli

Eh? I have no idea what you are referring to here.
There are many different types of thermometers that works well at low temperatures. But single electron transport isn't a very practical starting point for thermometry at any temperature.
It is possible to do perform temperature measurements using what is known as Coloumb blockade thermometry (which is sort of based on single electron transport); but it is a complicated technique and is never -as far as I know- used routinely.

Also, -180 degrees C is NOT a low temperature; it is not even liquid nitrogen temperature. Even a thermocouple will work reasonably well at 77K.

5. Feb 24, 2010

### mgb_phys

The main difficulty in cryogenic thermometers is limiting the heat leak in from the thermometer.
A thermocouple has fairly large, fairly high conductivity wires, if you had a very small cold part the heat leakage down the wires could be significant.

6. Feb 24, 2010

### truth is life

This is what my thermal physics book has to say about the subject:

Beyond that discussion of practical laboratory thermometry, it delves a bit into the theoretical underpinnings of thermometry earlier, during which it hints at how ultra-low temperature measurements are made--via electronic or nuclear spins (which it says are used to below 1 millikelvin).

7. Feb 25, 2010

### aliz_khanz

but can't i find a proper method , on when to claibrate , which special thermometers or diodes are used and how they function ? Thanks for your help !

Last edited by a moderator: May 4, 2017
8. Feb 25, 2010

### f95toli

There are several books about cryogenic thermometry. However, if you want a practical guide I would recommend Pobell's "Matter and Methods at Low Temperatures".

That said, it is not really that complicated. Just buy a sensor suitable for the temperature range you are interested in from e.g. Lakeshore (see the link above). A sensor that follows a standard resistance vs. temperature curve (which is usually accurate enough) costs something like $200. It is convenient to read out the sensor using a dedicated temperature controller, but you can also use e.g. an ordinary resistance bridge or something similar. For temperatures this high even a normal bench-top multimeter might work (self-heating isn't really an issue at temperatures as high as 100K). Last edited: Feb 25, 2010 9. Feb 25, 2010 ### aliz_khanz Thank you so so so much !!!! I really appreciate your help. I am trying to find this book on internet but it seems like i have to pay$25 bucks for this thing and i only want one chapter from it. Thanks again.

10. Feb 25, 2010

### Gokul43201

Staff Emeritus
If you don't need 50mK accuracy and repeatability, you could probably find a Pt RTD for about half that price.


text:
Journal of Multimedia Information System
Korea Multimedia Society
Section

# View Synthesis and Coding of Multi-view Data in Arbitrary Camera Arrangements Using Multiple Layered Depth Images

Seung-Uk Yoon1, Yo-Sung Ho2,*
1Computer Graphics Research Section, Creative Content Research Division, SW·Content Research Laboratory, Electronics and Telecommunications Research Institute (ETRI), Daejeon, Republic of Korea, suyoon@etri.re.kr
2School of Information and Communications, Gwangju Institute of Science and Technology (GIST), Gwanjgu, Republic of Korea, hoyo@gist.ac.kr
*Corresponding Author: Yo-Sung Ho, Gwangju Institute of Science and Technology (GIST), 123 Chemdan-gwagiro, Buk-gu, Gwangju500-712, Republic of Korea, +82-62-715-2211, hoyo@gist.ac.kr.

© Copyright 2014 Korea Multimedia Society. This is an Open-Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/4.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

Received: Aug 31, 2014 ; Revised: Sep 03, 2014 ; Accepted: Sep 15, 2014

Published Online: Sep 30, 2014

## Abstract

In this paper, we propose a new view synthesis technique for coding of multi-view color and depth data in arbitrary camera arrangements. We treat each camera position as a 3-D point in world coordinates and build clusters of those vertices. Color and depth data within a cluster are gathered into one camera position using a hierarchical representation based on the concept of layered depth image (LDI). Since one camera can cover only a limited viewing range, we set multiple reference cameras so that multiple LDIs are generated to cover the whole viewing range. Therefore, we can enhance the visual quality of the reconstructed views from multiple LDIs comparing with that from a single LDI. From experimental results, the proposed scheme shows better coding performance under arbitrary camera configurations in terms of PSNR and subjective visual quality.

Keywords: Free-viewpoint TV; layered depth image; multi-view color and depth data coding; view synthesis

## I. INTRODUCTION

As demands for high-quality visual services are increasing, three-dimensional (3-D) information is widely used in various multimedia applications, such as free-viewpoint video (FVV), free-viewpoint TV (FTV), 3DTV, games, virtual reality (VR), etc. Since these applications try to provide consumers with depth impression and an immersion, the free-viewpoint navigation is becoming one of the essential functionalities [1]. View interpolation, view synthesis, and intermediate view reconstruction (IVR) have been important research areas to meet various requirements and needs for those applications. Moreover, they get more attention since multi-view display technologies are being developed rapidly nowadays.

As one of the intermediate steps to realize free-viewpoint applications, researches on multi-view systems are active recently. The multi-view camera system captures the same scene at different viewpoints. If we acquire multi-view images from multiple cameras, we can generate scenes at virtual view positions. It means that consumers can change their viewpoints and can feel visible depth with their view interactions. However, a viewing range depends on camera arrangements, the distance between cameras, and the total number of cameras. Most multi-view camera systems have confined configurations such as equally spaced one-dimensional (1-D) parallel, 1-D convergent, 1-D arc, two-dimensional (2-D) array, 2-D cross, etc [2]. Usually, parallel types are used to generate stereoscopic views since stereo matching can be employed easily. Moreover, it is easy to compress data acquired from parallel cameras because views are linearly correlated. While an arc or a convergent formation can provide a wider viewing range, occlusions cause problems.

Most view synthesis methods highly depend on regular camera arrangements even though they target multi-view displays or FTV. Those target environments, however, require more flexible viewpoint changes at arbitrary viewing positions. Conventional IVR algorithms assuming linear correlations between multi-views have limitations. It is necessary to synthesize virtual views in arbitrary camera arrangements in order to achieve natural free-viewpoint navigation.

The other problem is the number of cameras to meet the goal. It is well known that the quality of intermediate views improves as cameras get closer. It is not practical to set up cameras at all possible viewpoints, since the size of data increases as the number of cameras does. We need to reduce the number of cameras considering the visual quality of synthesized views. In order to resolve this problem, we use a hierarchical representation based on layered depth image (LDI) [3] using multi-view color and depth images [4]. Since 3-D information or approximate geometry can reduce the number of cameras, we can generate virtual views of tolerable visual quality with less number of cameras. Unlike the previous approach making a single LDI from all multi-view color and depth data [4], we select multiple reference locations using a clustering technique and generate multiple LDIs at those positions. Since the FTV system requires depth maps for every viewpoint [5], we assume that color and depth data are available for all cameras with camera parameters.

In this paper, we focus on the design of an intermediate view generation scheme for coding of multi-view color and depth data in arbitrary camera arrangements using multiple LDIs. The main contributions are as follows.

First, our view synthesis scheme can handle sparse number of cameras using a hierarchical representation based on LDI, constructed from multiple color and depth images. Unlike the proposed approach, most view synthesis algorithms depend on linear correspondences among dense cameras.

Second, our approach can generate synthesized views with the better visual quality than others, especially under irregular camera arrangements. It is less dependent on linear correlations between adjacent views. We select reference camera locations using a clustering technique. Then, we construct multiple LDIs at reference positions and generate a new view from the closest LDI to the selected viewpoint. Our scheme is also applicable to the regular camera arrangements since clustering techniques can be applied to any camera configurations.

Third, we can enhance the visual quality of reconstructed views comparing with the previous approach using a single LDI. Since the distance between cameras is far and many pixels are overlapped at the same pixel location, it is difficult to select proper pixels to fill holes. By reducing the number of active layers considering pixel distributions, we reduce artifacts and improve the coding performance.

Finally, the proposed approach is appropriate for immerging applications requiring free-view navigation such as FTV. We can synthesize intermediate views with a tolerable visual quality at arbitrary viewpoints while employing small number of cameras. Therefore, we can set up a FTV system cost-efficiently.

The rest of this paper is organized as follows. We briefly overview the FTV system, view synthesis algorithms, and multi-view video coding activities in MPEG in Section II. We describe our view synthesis and coding schemes in Section III. The performance of the proposed technique is shown in Section IV. The paper is concluded with some discussions in Section V.

## II. REVIEW OF RELATED WORKS

In this section, we describe the FTV system, view synthesis techniques based on image-based rendering methods, and review multi-view video coding activity in MPEG.

1. Free-viewpoint TV (FTV) System

Television consumers perceive a distant world in real-time and have been more than a display device. It has drawn the most important visual information and communications technologies up to date. However, consumers only can get a single view of a 3-D world with conventional TVs. The view is determined not by consumers but by a camera placed in the 3-D world. This is different from what we experience in our real world.

FTV is an innovative visual media that enables us to view a distant 3-D world by freely changing our viewpoints as if we are there. It has much potential since such as functionality has not been achieved by conventional TV technologies [6].

ISO/IEC JTC1/SC29/WG11 Moving Picture Experts Group (MPEG) has recognized the importance of FTV technologies. Since FTV was proposed to MPEG in 2002, there have been consistent efforts to realize it until now.MPEG issued a call for proposals (CfP) on 3-D video coding for high-efficiency video coding (HEVC) [7] in 2011. In 2012, MPEG and Video Coding Experts Group (VCEG) of ITU-T have started a joint collaboration on 3-D video coding extension (JCT-3V) [8].

FTV realizes free navigation of consumers due to its new function of view generation. FTV can generate views at infinite number of viewpoints from finite number of camera images. FTV is implemented as the real-time complete chain from multi-view capturing to free-viewpoint display. The consumer can freely control the viewpoint for a real dynamic 3-D scene. In order to achieve free navigation functionality, depth information is required in addition to video signals.

Figure 1 shows an example of FTV system that transmits multi-view video with depth information (MVD) [5]. The content may be produced in a number of ways with input sources from a stereo camera, a depth camera, a multi-camera setup, or 2-D/3-D conversion. At the receiver, depth image based rendering can be performed to generate multiple views adapting to various types of displays.

Fig. 1. An example of the FTV system.
2. Image-based Rendering (IBR) for View Synthesis

Image-based rendering (IBR) techniques have been proposed to generate new viewpoints with photorealism, perceptually indistinguishable rendering, using two or more images without heavy rendering of 3-D models or scenes. They use 2-D images as primitives to render an arbitrary view of the 3-D scene. In particular, their rendering time does not depend on the scene complexity, but rely on image resolutions. Various IBR techniques can be classified into three categories based on how much geometry information is used [9]: rendering with no geometry; rendering with implicit geometry; and rendering with explicit geometry. Since it has shown that the approximate geometry can improve the output visual quality and reduce a number of input images in [10], [11], there has been trade-off between the total number of image samples and the amount of geometry data used in IBR [9], [12].

Chen and Williams [13] reconstructed intermediate views by calculating pixel-by-pixel correspondences between input images. Their approach works on closely located images with high spatial correlation. Only synthetic scenes were examined in their experiments. Seitz and Dyer [14] proposed view morphing, which can generate 2-D transitions between two images. However, they choose corresponding line features between two images manually. Droese et al. [15] proposed a view interpolation method in disparity domain using a block matching algorithm. Since their approach use fixed block size, errors are caused in boundary regions. Recently, Lee et al. [16] modified this approach using variable block size and pixel-level disparity estimation. However, their method is only applicable to parallel camera arrangements.

3. Multi-view Video Coding Activity in MPEG

As one of the intermediate step for FTV, an AHG on 3-D audio and visual (3DAV) has established in December 2001 [17]. Four main exploration experiments (EE) on 3DAV were performed from 2002 to 2004: EE1 on omni-directional video, EE2 on FTV, EE3 on coding of stereoscopic video using multiple auxiliary components (MAC) in MPEG-4, and EE4 on depth/disparity coding for 3DTV and intermediate view interpolation [18]. After MPEG called interested parties to bring evidences on MVC technologies in October 2004 [19], some evidences were recognized in January 2005 [20] and a CfP on MVC has been issued in July 2005 [2]. Through a series of efforts on multi-view video coding for several years, a CfP for HEVC-based 3-D video coding [7] issued in 2011.

Most multi-view video coding algorithms in MPEG focus only on reduction of data size of multiple color videos. They are based on the H.264/AVC video coding standard, a predictive coding scheme for a single video. Although numerous algorithms on 3D-HEVC efficiently reduce the size of multi-view texture and depth videos [21], they rarely use 3-D geometry information contained in multi-view video. In addition, they do not consider problems of efficient representation for multi-view data and coding of multiple depth sequences. In order to solve these problems, Yoon and Ho [4] proposed a framework and coding schemes of multiple color and depth video using a hierarchical data representation. In this approach, an LDI is generated for all multi-view data, so it has visual artifacts due to overlapped pixels at the same pixel location. In this paper, we try to reduce those artifacts and enhance the coding efficiency based on the proposed view synthesis and coding technique using multiple LDIs.

## III. VIEW SYNTHESIS AND CODING OF MULTI-VIEW DATA IN ARBITRARY CAMERA ARRANGEMENTS

In this section, we explain the proposed intermediate view reconstruction algorithm based on a hierarchical representation. Since our approach is based on LDI, we first describe the concept, the generation procedure, and problems before presenting our algorithms in detail.

Among a variety of IBR techniques, LDI is one of the efficient rendering methods for 3-D objects with complex geometries. It represents the current scene using an array of pixels viewed from a single camera position. However, each LDI pixel consists of color, depth between the camera and the pixel, and other data that support rendering of LDI. Three key characteristics of LDI are: (1) it contains multiple layers at each pixel location, (2) the distribution of pixels in the back layer is sparse, and (3) each pixel has multiple attribute values [4]. Because of these special features, LDI enables us to render arbitrary views of the scene at new camera positions. Moreover, the rendering operation can be performed quickly with the list-priority algorithm proposed by McMillan [22].

When the rays are emanating from a reference viewpoint (an LDI camera), it is possible to store intersecting points between rays and an object. Each intersecting point contains color and depth information.

Figure 2 represents the conceptual diagram of LDI [4]. As shown in Fig. 2, the first intersecting points construct the first layer of LDI, the second ones build up the second layer, and so on. Consequently, each layered depth pixel (LDP) has different number of depth pixels (DPs), which contain color, depth, and auxiliary information for reconstruction of multiple views. For example, LDP 3 in Fig. 2 has four layers, which contain intersecting points between Ray C and the object [4].

Fig. 2. The concept of layered depth image.

Since rays cannot go through a physical object, this concept is applicable to computer graphics (CG) models only. Therefore, we generate LDI from multiple color and depth images using a 3-D warping technique [4]. Pixels of multi-view images are moved to the LDI view, a reference location, and then layers are created by comparing depth values at that camera location. In our previous approach, we have constructed a single LDI by warping pixels from all other views. For example, we perform 3-D warping for eight images to the reference location when we have eight-view color and depth images. However, this can cause many holes when the distance between cameras is far as shown in Fig. 3.

Fig. 3. 3-D warping to the reference camera 4.

Fig. 3 shows the result of 3-D warping from other views to the reference camera for the “Ballet” dataset [23], [24]. The scene was captured from eight cameras with the 1-D arc arrangement. The distance between cameras is 20cm. In this example, we set the reference camera as the center one, camera number 4. There are lots of holes in the warped results because the distance from the left-most camera to the right-most camera is over 100cm. In the parallel camera arrangement, the distance between two extremes is even longer than the arc configuration.

In order to solve this problem, we separate cameras into several clusters and generate an LDI for each cluster in this paper. The first step is to assign 3-D coordinates to cameras so that we can treat them as 3-D vertices. Then, we divide those points into M (<Nmax, Nmax is the total number of cameras) clusters by considering minimum and maximum number of cameras. After constructing LDIs, we can generate virtual views selected by the consumer from the closest LDI to the views. Let us describe the procedures step by step.

1. Camera Positions in the 3-D World Space

Extrinsic camera parameters present translation and rotation of each camera based on a reference camera. The reference camera is set when the scene is captured from a multi-view camera system. For example, if a camera in the center is selected as reference, relative positions of others are calculated by the camera calibration.

In order to convert relative positions to absolute locations in the 3-D space, we set the 3-D world coordinates. The position of the reference camera Cr is represented by

(1)

where O is the origin of the world space and D is the offset or displacement of Cr from the origin in XYZ coordinates. Then, the locations of other cameras Ci, i is the camera number, are calculated again in this world coordinates.

Figure 4 shows an example on how we set the world coordinates and the position of the reference camera. The basic assumption is that we already know the camera parameters for all cameras. Since the extrinsic camera parameters indicate relative translation of each camera from the pre-defined reference, we change the relative values to the absolute 3-D points in the 3-D world space. Now, we select new multiple reference cameras among those 3-D points, that have (x, y, z) values, using a clustering technique.

Fig. 4. Position of the reference camera in the world space.
2. Multi-seed Clustering of Camera Positions

At this step, we partition 3-D points into M clusters. M is a natural number and the range of M is defined by

(2)

where M is greater than Nmin and less than Nmax. Nmax is the total number of cameras.

In order to determine Nmin, we need to examine the characteristics of LDI. One of the main advantages of using LDI is that LDI can resolve the occlusion problem by storing multiple depth pixels (DPs) at each pixel location [3]. Each DP contains color, depth, and auxiliary information. When we generate virtual views, the DPs in the back layers are used to fill holes caused by disocclusion. Although some areas cannot be covered by the DPs only, it is reasonable to collect pixels from other viewpoints and to use them for virtual view synthesis.

However, the problem is to select proper DP to fill the holes. If we have many DPs in the back layers, we should determine which one is appropriate for the current empty pixel. Table 1 lists the pixel distribution at each layer of the constructed LDI from the first eight views of the “Ballet” dataset.

Table 1. Pixel distribution at each layer (depth threshold: 3.0).
Layer Pixel Occupation [%] Layer Pixel Occupation [%]
1 100.0 5 15.8
2 99.0 6 5.0
3 68.2 7 1.1
4 38.4 8 0.1

Corresponding to Table 1, Figure 5 represents the layer images of the constructed LDI. From Table 1 and Figure 4, we can observe that the pixel distribution becomes lower than 50% after the third layer. Similar experiments performed before also show that the occupancy of pixels at each layer drops below 50% after the third or fourth layer [4], [25].

Fig. 5. Layer images of the constructed LDI.

Those mean that the effective number of DPs that could be used to fill holes is up to three or four depending on the distance between cameras. In the case of the 20cm spacing, we need at least three views to construct LDI. Based on these experiments, we have set the Nmin as two for the “Ballet” and “Breakdancers” dataset [23], [24] and for the “Dooyoung” dataset [26], [27]. We describe the characteristics of the test data in our experimental section in detail.

Then, we partition the 3-D points into M clusters using the K-means clustering technique [28] based on the Eq. (2). If the partitioned cluster has two points, we add one more point nearest to the seed point of the cluster. Finally, the seed points after clustering are set as the reference locations to generate LDIs.

3. Calculation of Camera Parameters

After we generate LDIs at multiple reference locations, we have to calculate camera parameters at the viewpoint of the consumer. First, intrinsic parameters of the selected virtual camera are interpolated as average values of those of cameras included in each cluster.

The translational component can be easily computed from the new viewpoint. Since we define all viewpoints and camera locations in the world coordinates, we can convert the 3-D location of the viewpoint to the relative camera position. The reference is already set by the partitioning and clustering in the early stage.

For the rotational component, we compute the single axis rotation angle between the reference camera and the newly selected virtual camera [14]. When the normal vector of the image plane of the reference camera is denoted as NR and that of the virtual camera is NV, the rotation axis NN and the angle θ between these two vectors is obtained by

(3)
4. Viewing Range of the Single LDI

The viewing range that can be covered by a single LDI is confined by that of input images. If we have an LDI constructed from the infinite number of input images at all viewpoints, there are no limits of viewing range. However, it is impractical to use a huge number of cameras and acquire depth information for all those views. Therefore, we need to assess proper viewing range of one LDI.

There are mainly two factors that can affect viewing range of the single LDI. One is the distance between cameras. As we mentioned before, the minimum number of input images to generate LDI can be changed based on it. The other is the performance of the hole filling technique used in LDI rendering. If we can fill dis-occluded areas within tolerable visual quality, the number of input images can be reduced further. Recently, Cheng et al. [29] proposed a hole filling scheme to reduce the effect of contaminated pixels from multiple DPs. If background colors are far different from foreground, visual artifacts occur at pixels along the boundary of objects because of contributions from both the background and foreground colors. Thus, they do not use the average pixel value of all DPs but that of clustered DPs.

However, we can remove those effects by generating multiple LDIs containing smaller layers. Experiments on pixel distributions show the rationale. Since we can find closest LDI to any viewpoint, whole areas can be covered by the proposed multiple LDI-based view generation approach.

## IV. EXPERIMENTAL RESULTS

In order to examine the performance of the proposed scheme, we use the following multi-view datasets: “Ballet”, “Breakdancers”, and “Dooyoung”.

Figure 6 shows the test data used in our experiments. Figure 6(a) and Figure 6(b) are captured from eight cameras with 1-D arc arrangements. Figure 6(c) is an example scene captured from 16 cameras. The cameras configurations used to capture these test data are depicted in Figure 7 [23], [27].

Fig. 6. Multi-view test datasets: color and depth.
Fig. 7. Camera configurations for the test datasets.

Figure 7(a) is used for “Ballet” and “Breakdancers”. Figure 7(b) and Figure 7(c) is for “Dooyoung”. The “Dooyoung” test dataset was captured from irregular camera arrangements with large spacing between cameras.

1. Generation of Multiple LDIs

Figure 8 presents the clustering results of the regular camera arrangement, e.g., for eight cameras. Since we have set the minimum number of cameras for each cluster is greater than two in Eq. (2), cluster three has overlapped camera. In Figure 8, the left-most camera is camera 0 and the right-most one is camera 7. Cluster 2 and 3 share the camera 5. Figure 8 shows one of the clustering results, so other cameras can be shared also.

Fig. 8. Clustered cameras for the regular camera arrangement.

Figure 9 represents the clustering results of the irregular camera arrangement, e. g., for 16 cameras. Because of the viewpoint, camera 12 is not seen in Fig. 8. Camera 12 is clustered with camera 14, 15, and 16 together. Reference cameras of these clusters are camera 13, camera 4, camera 7, camera 9, and camera 15.

Fig. 9. Clustered cameras for the irregular camera arrangement.

When the selected viewpoint is between two clusters, e. g., between cluster 1 and cluster 2 in Figure 7, we have calculated Euclidean distance between positions by

$D=\text{\hspace{0.17em}}‖{C}_{adj}\left({x}_{adj},{y}_{adj},{z}_{adj}\right)-{C}_{sel}\left({x}_{sel},{y}_{sel},{z}_{sel}\right)‖\text{\hspace{0.17em}}={\left({\left({x}_{adj}–{x}_{sel}\right)}^{2}+{\left({y}_{adj}–{y}_{sel}\right)}^{2}+{\left({z}_{adj}–{z}_{sel}\right)}^{2}\right)}^{1/2}$
(4)

where D means the distance between camera positions, Cadj is the 3-D position of an adjacent camera and Csel is that of the new viewpoint. Then, we generate the virtual view from the LDI constructed in the closer cluster.

If the selected view is shared by two clusters, e. g., camera 5 in Figure 8, we compute the distance from the chosen view to the reference cameras of overlapped clusters. Finally, the view is reconstructed from the LDI generated from the cluster, which has closer reference camera.

2. View Synthesis for Regular Camera Arrangements

Figure 10 shows the generated intermediate views between camera 0 and camera 2. The pictures in Figure 10(a) have been reconstructed from a single LDI, whose reference view is camera 4. Since the reference is far from these view positions, the results show visible artifacts in dancer’s hands especially.

Fig. 10. Synthesized views for the regular camera arrangement.

However, it is not easy to find noticeable artifacts in the results generated from multiple LDIs as shown in Fig. 10(b). In this case, the reference is camera 1.The center image is almost the same as the original view of camera 1.

In Figure 11, we have compared peak signal to noise ratio (PSNR) for each view for two approaches: view generation from a single LDI and that from multiple LDIs. The thick line is the proposed scheme. As we can observe in Figure 11, the proposed approach shows better PSNR performance since it has multiple references. The PSNR reduces as the viewpoint is far from the reference camera location in the previous method using a single LDI.

Fig. 11. PSNR curves for the synthesized views.
3. View Synthesis for Irregular Camera Arrangements

In Figure 12, we have constructed the LDI from camera 6, 7, and 8. In most linear intermediate view generation algorithms, it is difficult to calculate linear relationships between those three views because they are not linearly located. Although the long distance between cameras cause visible artifacts, it is possible to estimate views at arbitrary positions within the triangle connecting three camera points. The reference camera is set by camera 7. From the LDI at the location of camera 7, we have reconstructed views at camera 6 and 8 as shown in Fig. 12.

Fig. 12. Synthesized views for the irregular camera arrangement.

Figure 12(a) and Figure 12(c) are the original views at camera 6 and camera 8. Figure 12(b) and Figure 12(d) show the synthesized views from the constructed LDI at camera 7. Although the results are blurred a little, subjective qualities are acceptable.

In addition, we have compared the objective quality for the existing views in terms of PSNR. Since we have the original view 6 and view 8, we can compute PSNR between the original views and the synthesized ones from the proposed approach. PSNR of Figure 12(b) to Figure 12(a) is 31.23 dB and that of Figure 12(d) to Figure 12(c) is 30.52 dB. Since the warped object is not aligned in the same location of the original view, the PSNR is rather low. However, subjective visual quality is tolerable as depicted in Figure 12.

4. Coding Results for Regular Camera Arrangements

In order to show that the proposed scheme has benefits in terms of coding, we have shown comparison results in terms of PSNR Y vs. bitrate for the “Breakdancers” sequence in Figure 13. The proposed method is compared with the previous approach using a single LDI and other MVC schemes [4].

Fig. 13. PSNR curves for the regular camera arrangement.

The important fact in Figure 13 is that all other methods encode color components only, but the proposed one compresses color, depth, NOL, and residual data. In other words, the PSNR curve in Fig. 16 for the proposed method is the result of the total bitrates for depth, NOL, and residual as well as color data. However, the other curves present the PSNR Y vs. bitrate for the color component only. The coding results are displayed for three rate points: low, middle, and high bitrates [4]. In Figure 13, the anchor means that all views are encoded separately using H.264/AVC with specified parameters [2]. All PSNR curves represent the average values over all views.

As shown in Figure 13, we can observe that the proposed approach based on multiple LDIs shows better coding performance than that of using a single LDI. For the LDI data, the same encoding scheme as the case of the single LDI has been used in this experiments. We have taken the average of three LDIs for the “Breakdancers” sequence in this experiment.

5. Coding Results for Irregular Camera Arrangements

In order to show that the proposed scheme has benefits in terms of coding in irregular camera arrangements, we have shown comparison results in terms of PSNR Y vs. bitrate for the “Dooyoung” sequence in Figure 14. We have compared our approach with the previous approach using a single LDI. The selected views are view 6, 7, 8, 9, and 10 since we have synthesized those views in Figure 12. The total number of frames is 100. The curves represent the average PSNR for 100 frames at each bitrates. The other coding conditions are the same as the case of Fig. 13.

Fig. 14. PSNR curves for the irregular camera arrangement.

In Fig. 14, joint multi-view video model (JMVM), the reference software released by MPEG [30], has been used as the anchor. As shown in Fig. 14, the proposed scheme shows better performance in terms of PSNR. The improvement is about 1.8 dB on average at low bitrate for this test sequence.

## V. CONCLUSION

In this paper, we have proposed a new view synthesis technique for coding of multi-view color and depth data in arbitrary camera arrangements. We treat each camera position as a 3-D point in world coordinates and make clusters of those vertices. Color and depth data within a cluster are gathered into one location using a hierarchical representation based on the concept of layered depth image (LDI). Since the viewing range covered by a camera is limited, we set numerous reference positions where multiple LDIs are generated to cover the whole viewing range. Therefore, we have enhanced the visual quality of the reconstructed views from multiple LDIs compared with that from a single LDI. From experimental results, the proposed scheme has shown better coding performance under arbitrary camera configuration in terms of both objective and subjective visual qualities. The improvement was about 1.8 dB on average at low bitrate for the test sequence captured from the irregular camera arrangement.

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