text: 10 root node r. This in turn deﬁnes a rooted geometric triangulation. In order to describe the behavior of this root under the reﬁning operations, we further equip the root simplex with a ﬂag structure, which consists simply in a choice of nondisjoint lower-dimensional simplices σ0 ⊂ · · · ⊂ σd. r With these structures, we can ﬁnally consider the inﬁnite set of ﬂagged rooted geometric tri- angulations of a given manifold Σ. On this set, we now choose the reﬁnement operations to (cid:74) (cid:75) be the Alexander moves. These Alexander moves reﬁne a given triangulation by subdividing a k-dimensional simplex, where k ∈ 1, d , with the addition of a new (embedded) vertex. As ex- plained in [18], if the root of a triangulation is subdivided by a move, the ﬂag structure deﬁnes in a canonical manner a new root for the reﬁned triangulation. (cid:48) A geometric triangulation ∆ is said to be ﬁner than a geometric triangulation ∆ if it can be obtained form ∆ by a (ﬁnite) sequence of reﬁning Alexander moves, and if its vertices after the (cid:48) moves have the same coordinates as those of ∆. We will denote this relation by ∆ ≺ ∆ . In oder to unambiguously specify paths γ in a graph Γ and to obtain a characterization of the fundamental cycles of this graph (and to perform a gauge-ﬁxing), it is convenient to introduce (maximal spanning) trees. A maximal or spanning tree T in a graph Γ is a connected subgraph of Γ which does not contain any closed cycles and includes all the nodes of Γ. In the rest of the paper we use ‘tree’ as short form for maximal connected tree. The links composing a tree are called branches and denoted by b, while the links of the graph that are not in the tree are called leaves and denoted by (cid:96). Although the leaves correspond to the links in Γ\T , by abuse of language we will refer to the leaves of the tree. The set of leaves in a tree will be denoted by L. A rooted tree with root r is a tree where a preferred node is identiﬁed and called the root. There is a one-to-one correspondence between the leaves of a tree and the fundamental cycles of the underlying graph. For a rooted tree, one way to characterize these cycles is to associate them with paths γ labeled by the leaves (cid:96) in L. A path γ starts at the root, goes along the branches of the tree until the source node of (cid:96) the leaf (cid:96), then goes along (cid:96), and back to the root along branches. The cardinality of the set of fundamental cycles is independent of the choice of tree. Explicitly, if we denote by |n| the number of nodes in the graph Γ and by |l| the number of links, then any given tree has |n| − 1 branches and card(L) = |(cid:96)| = |l| − |n| + 1 leaves (and fundamental cycles).

text: root

text: node

text: r.

text: This

text: in

text: turn

text: deﬁnes

text: a

text: rooted

text: geometric

text: triangulation.

text: In

text: order

text: to

text: describe

text: the

text: behavior

text: of

text: this

text: root

text: under

text: the

text: reﬁning

text: operations,

text: we

text: further

text: equip

text: the

text: root

text: simplex

text: with

text: a

text: ﬂag

text: structure,

text: which

text: consists

text: simply

text: in

text: a

text: choice

text: of

text: nondisjoint

text: lower-dimensional

text: simplices

text: σ0

text: σd.

text: r

text: With

text: these

text: structures,

text: we

text: can

text: ﬁnally

text: consider

text: the

text: inﬁnite

text: set

text: of

text: ﬂagged

text: rooted

text: geometric

text: tri-

text: angulations

text: of

text: a

text: given

text: manifold

text: Σ.

text: On

text: this

text: set,

text: we

text: now

text: choose

text: the

text: reﬁnement

text: operations

text: to

text: be

text: the

text: Alexander

text: moves.

text: These

text: Alexander

text: moves

text: reﬁne

text: a

text: given

text: triangulation

text: by

text: subdividing

text: a

text: k-dimensional

text: simplex,

text: where

text: k

text: d

text: with

text: the

text: addition

text: of

text: a

text: new

text: (embedded)

text: vertex.

text: As

text: ex-

text: plained

text: in

text: if

text: the

text: root

text: of

text: a

text: triangulation

text: is

text: subdivided

text: by

text: a

text: move,

text: the

text: ﬂag

text: structure

text: deﬁnes

text: in

text: a

text: canonical

text: manner

text: a

text: new

text: root

text: for

text: the

text: reﬁned

text: triangulation.

text: A

text: geometric

text: triangulation

text: is

text: said

text: to

text: be

text: ﬁner

text: than

text: a

text: geometric

text: triangulation

text: if

text: it

text: can

text: be

text: obtained

text: form

text: by

text: a

text: (ﬁnite)

text: sequence

text: of

text: reﬁning

text: Alexander

text: moves,

text: and

text: if

text: its

text: vertices

text: after

text: the

text: moves

text: have

text: the

text: same

text: coordinates

text: as

text: those

text: of

text: We

text: will

text: denote

text: this

text: relation

text: by

text: In

text: oder

text: to

text: unambiguously

text: specify

text: paths

text: γ

text: in

text: a

text: graph

text: Γ

text: and

text: to

text: obtain

text: a

text: characterization

text: of

text: the

text: fundamental

text: cycles

text: of

text: this

text: graph

text: (and

text: to

text: perform

text: a

text: gauge-ﬁxing),

text: it

text: is

text: convenient

text: to

text: introduce

text: (maximal

text: spanning)

text: trees.

text: A

text: maximal

text: or

text: spanning

text: tree

text: T

text: in

text: a

text: graph

text: Γ

text: is

text: a

text: connected

text: subgraph

text: of

text: Γ

text: which

text: does

text: not

text: contain

text: any

text: closed

text: cycles

text: and

text: includes

text: all

text: the

text: nodes

text: of

text: Γ.

text: In

text: the

text: rest

text: of

text: the

text: paper

text: we

text: use

text: ‘tree’

text: as

text: short

text: form

text: for

text: maximal

text: connected

text: tree.

text: The

text: links

text: composing

text: a

text: tree

text: are

text: called

text: branches

text: and

text: denoted

text: by

text: b,

text: while

text: the

text: links

text: of

text: the

text: graph

text: that

text: are

text: not

text: in

text: the

text: tree

text: are

text: called

text: leaves

text: and

text: denoted

text: by

text: Although

text: the

text: leaves

text: correspond

text: to

text: the

text: links

text: in

text: Γ\T

text: by

text: abuse

text: of

text: language

text: we

text: will

text: refer

text: to

text: the

text: leaves

text: of

text: the

text: tree.

text: The

text: set

text: of

text: leaves

text: in

text: a

text: tree

text: will

text: be

text: denoted

text: by

text: L.

text: A

text: rooted

text: tree

text: with

text: root

text: r

text: is

text: a

text: tree

text: where

text: a

text: preferred

text: node

text: is

text: identiﬁed

text: and

text: called

text: the

text: root.

text: There

text: is

text: a

text: one-to-one

text: correspondence

text: between

text: the

text: leaves

text: of

text: a

text: tree

text: and

text: the

text: fundamental

text: cycles

text: of

text: the

text: underlying

text: graph.

text: For

text: a

text: rooted

text: tree,

text: one

text: way

text: to

text: characterize

text: these

text: cycles

text: is

text: to

text: associate

text: them

text: with

text: paths

text: γ

text: labeled

text: by

text: the

text: leaves

text: in

text: L.

text: A

text: path

text: γ

text: starts

text: at

text: the

text: root,

text: goes

text: along

text: the

text: branches

text: of

text: the

text: tree

text: until

text: the

text: source

text: node

text: of

text: the

text: leaf

text: then

text: goes

text: along

text: and

text: back

text: to

text: the

text: root

text: along

text: branches.

text: The

text: cardinality

text: of

text: the

text: set

text: of

text: fundamental

text: cycles

text: is

text: independent

text: of

text: the

text: choice

text: of

text: tree.

text: Explicitly,

text: if

text: we

text: denote

text: by

text: |n|

text: the

text: number

text: of

text: nodes

text: in

text: the

text: graph

text: Γ

text: and

text: by

text: |l|

text: the

text: number

text: of

text: links,

text: then

text: any

text: given

text: tree

text: has

text: |n|

text: branches

text: and

text: card(L)

text: |l|

text: |n|

text: leaves

text: (and

text: fundamental

text: cycles).


text: The two faces of mimetic Horndeski gravity: disformal transformations and Lagrange ∗ multiplier Frederico Arroja1b, Nicola Bartolo2,3c, Purnendu Karmakar2,3d and Sabino Matarrese2,3,4e ∗∗ †† (Dated: May 27, 2020) We show that very general scalar-tensor theories of gravity (including, e.g., Horndeski models) are generically invariant under disformal transformations. However there is a special subset, when the transformation is not invertible, that yields new equations of motion which are a generalization of the so-called “mimetic” dark matter theory recently introduced by Chamsedinne and Mukhanov. These conclusions hold true irrespective of whether the scalar ﬁeld in the action of the assumed scalar- tensor theory of gravity is the same or diﬀerent than the scalar ﬁeld involved in the transformation. The new equations of motion for our general mimetic theory can also be derived from an action containing an additional Lagrange multiplier ﬁeld. The general mimetic scalar-tensor theory has the same number of derivatives in the equations of motion as the original scalar-tensor theory. As an application we show that the simplest mimetic scalar-tensor model is able to mimic the cosmological background of a ﬂat FLRW model with a barotropic perfect ﬂuid with any constant equation of state. I. INTRODUCTION The so-called ΛCDM model of cosmology has been very successful at explaining all cosmological observations with a minimal set of six cosmological parameters. These six parameters have recently been measured to an unprecedented accuracy with the Planck satellite [1, 2]. Several deviations from this simple model have been constrained to be relatively small [1–6]. This model has been called the standard model of cosmology. However, despite its successes, the ΛCDM model contains at its core two unknown components. One being a very small cosmological constant and the other the so-called dark matter. This dark matter component is assumed to be cold, collisionless and does not interact with the other particles of the standard model of particle physics except gravitationally. It amounts to about one quarter of the total energy density budget of the current universe [2]. Furthermore, despite the many experimental searches for these dark matter particles, both on Earth and in space, none was found up to now. See for example [7] and [8] (and reference therein) for reviews. The presence of these unknown energy components has motivated many studies that try to explain the phenomena that they give origin to, by modifying the law of gravitation without introducing new energy sources, see e.g. [9, 10]. One recent interesting attempt to explain the dark matter phenomenon by introducing a modiﬁcation to General Relativity is the so-called “mimetic” dark matter proposal [11]. In this proposal, General Relativity is reformulated in terms of an auxiliary metric which is conformally related with the original “physical” metric, where the conformal factor is a certain function of the new metric and the ﬁrst derivative of a scalar ﬁeld. In these new variables, the theory becomes invariant under Weyl rescaling of the new metric and therefore provides traceless equations of motion. This switches on a new conformal degree of freedom of gravity which behaves as an irrotational pressureless perfect ﬂuid, i.e. it can mimic a cold dark matter component. In this model, the observed cold dark matter energy density would, in general, be the sum of two unknown amounts of energy density contributions, one coming from hypothetical dark matter particles and the other from the “mimetic” dark matter which is only a gravitational eﬀect. In a subsequent work [12] it was shown that by introducing a potential for the new scalar ﬁeld one can mimic the gravitational behavior of almost any form of matter (see also [13] for earlier work). Refs. [12, 14, 15] considered the eﬀects of higher-derivative interactions on the cosmology of the “mimetic” dark matter model. In [14], the authors showed that the energy momentum tensor of the mimetic theory with higher-derivatives is actually of the imperfect ﬂuid type and the theory can support vorticity. Ref. [15] argues that these higher-derivative interactions could possibly help solving the small-scale problems of the cold dark matter model. Ref. [16] proposed an alternative conformal extension of General Relativity, using a vector ﬁeld, that can also support rotational ﬂows for the mimetic dark matter. b arroja@phys.ntu.edu.tw c nicola.bartolo@pd.infn.it d purnendu.karmakar@pd.infn.it ∗e sabino.matarrese@pd.infn.it ∗∗ †† ##LTLine##

text: The

label: title

text: two

text: faces

text: of

text: mimetic

text: Horndeski

text: gravity:

text: disformal

text: transformations

text: and

text: Lagrange

label: paragraph

text: multiplier

text: Frederico

label: author

text: Nicola

text: Purnendu

text: and

text: Sabino

text: (Dated:

text: May

label: date

text: We

label: abstract

text: show

text: that

text: very

text: general

text: scalar-tensor

text: theories

text: of

text: gravity

text: (including,

text: e.g.,

text: Horndeski

text: models)

text: are

text: generically

text: invariant

text: under

text: disformal

text: transformations.

text: However

text: there

text: is

text: a

text: special

text: subset,

text: when

text: the

text: transformation

text: is

text: not

text: invertible,

text: that

text: yields

text: new

text: equations

text: of

text: motion

text: which

text: are

text: a

text: generalization

text: of

text: the

text: so-called

text: “mimetic”

text: dark

text: matter

text: theory

text: recently

text: introduced

text: by

text: Chamsedinne

text: and

text: Mukhanov.

text: These

text: conclusions

text: hold

text: true

text: irrespective

text: of

text: whether

text: the

text: scalar

text: ﬁeld

text: in

text: the

text: action

text: of

text: the

text: assumed

text: scalar-

text: tensor

text: theory

text: of

text: gravity

text: is

text: the

text: same

text: or

text: diﬀerent

text: than

text: the

text: scalar

text: ﬁeld

text: involved

text: in

text: the

text: transformation.

text: The

text: new

text: equations

text: of

text: motion

text: for

text: our

text: general

text: mimetic

text: theory

text: can

text: also

text: be

text: derived

text: from

text: an

text: action

text: containing

text: an

text: additional

text: Lagrange

text: multiplier

text: ﬁeld.

text: The

text: general

text: mimetic

text: scalar-tensor

text: theory

text: has

text: the

text: same

text: number

text: of

text: derivatives

text: in

text: the

text: equations

text: of

text: motion

text: as

text: the

text: original

text: scalar-tensor

text: theory.

text: As

text: an

text: application

text: we

text: show

text: that

text: the

text: simplest

text: mimetic

text: scalar-tensor

text: model

text: is

text: able

text: to

text: mimic

text: the

text: cosmological

text: background

text: of

text: a

text: ﬂat

text: FLRW

text: model

text: with

text: a

text: barotropic

text: perfect

text: ﬂuid

text: with

text: any

text: constant

text: equation

text: of

text: state.

text: I.

text: INTRODUCTION

text: The

text: so-called

text: ΛCDM

text: model

text: of

text: cosmology

text: has

text: been

text: very

text: successful

text: at

text: explaining

text: all

text: cosmological

text: observations

text: with

text: a

text: minimal

text: set

text: of

text: six

text: cosmological

text: parameters.

text: These

text: six

text: parameters

text: have

text: recently

text: been

text: measured

text: to

text: an

text: unprecedented

text: accuracy

text: with

text: the

text: Planck

text: satellite

text: Several

text: deviations

text: from

text: this

text: simple

text: model

text: have

text: been

text: constrained

text: to

text: be

text: relatively

text: small

text: This

text: model

text: has

text: been

text: called

text: the

text: standard

text: model

text: of

text: cosmology.

text: However,

text: despite

text: its

text: successes,

text: the

text: ΛCDM

text: model

text: contains

text: at

text: its

text: core

text: two

text: unknown

text: components.

text: One

text: being

text: a

text: very

text: small

text: cosmological

text: constant

text: and

text: the

text: other

text: the

text: so-called

text: dark

text: matter.

text: This

text: dark

text: matter

text: component

text: is

text: assumed

text: to

text: be

text: cold,

text: collisionless

text: and

text: does

text: not

text: interact

text: with

text: the

text: other

text: particles

text: of

text: the

text: standard

text: model

text: of

text: particle

text: physics

text: except

text: gravitationally.

text: It

text: amounts

text: to

text: about

text: one

text: quarter

text: of

text: the

text: total

text: energy

text: density

text: budget

text: of

text: the

text: current

text: universe

text: Furthermore,

text: despite

text: the

text: many

text: experimental

text: searches

text: for

text: these

text: dark

text: matter

text: particles,

text: both

text: on

text: Earth

text: and

text: in

text: space,

text: was

text: found

text: up

text: to

text: now.

text: See

text: for

text: example

text: and

text: (and

text: reference

text: therein)

text: for

text: reviews.

text: The

text: presence

text: of

text: these

text: unknown

text: energy

text: components

text: has

text: motivated

text: many

text: studies

text: that

text: try

text: to

text: explain

text: the

text: phenomena

text: that

text: they

text: give

text: origin

text: to,

text: by

text: modifying

text: the

text: law

text: of

text: gravitation

text: without

text: introducing

text: new

text: energy

text: sources,

text: see

text: e.g.

text: One

text: recent

text: interesting

text: attempt

text: to

text: explain

text: the

text: dark

text: matter

text: phenomenon

text: by

text: introducing

text: a

text: modiﬁcation

text: to

text: General

text: Relativity

text: is

text: the

text: so-called

text: “mimetic”

text: dark

text: matter

text: proposal

text: In

text: this

text: proposal,

text: General

text: Relativity

text: is

text: reformulated

text: in

text: terms

text: of

text: an

text: auxiliary

text: metric

text: which

text: is

text: conformally

text: related

text: with

text: the

text: original

text: “physical”

text: metric,

text: where

text: the

text: conformal

text: factor

text: is

text: a

text: certain

text: function

text: of

text: the

text: new

text: metric

text: and

text: the

text: ﬁrst

text: derivative

text: of

text: a

text: scalar

text: ﬁeld.

text: In

text: these

text: new

text: variables,

text: the

text: theory

text: becomes

text: invariant

text: under

text: Weyl

text: rescaling

text: of

text: the

text: new

text: metric

text: and

text: therefore

text: provides

text: traceless

text: equations

text: of

text: motion.

text: This

text: switches

text: on

text: a

text: new

text: conformal

text: degree

text: of

text: freedom

text: of

text: gravity

text: which

text: behaves

text: as

text: an

text: irrotational

text: pressureless

text: perfect

text: ﬂuid,

text: i.e.

text: it

text: can

text: mimic

text: a

text: cold

text: dark

text: matter

text: component.

text: In

text: this

text: model,

text: the

text: observed

text: cold

text: dark

text: matter

text: energy

text: density

text: would,

text: in

text: general,

text: be

text: the

text: sum

text: of

text: two

text: unknown

text: amounts

text: of

text: energy

text: density

text: contributions,

text: one

text: coming

text: from

text: hypothetical

text: dark

text: matter

text: particles

text: and

text: the

text: other

text: from

text: the

text: “mimetic”

text: dark

text: matter

text: which

text: is

text: only

text: a

text: gravitational

text: eﬀect.

text: In

text: a

text: subsequent

text: work

text: it

text: was

text: shown

text: that

text: by

text: introducing

text: a

text: potential

text: for

text: the

text: new

text: scalar

text: ﬁeld

text: one

text: can

text: mimic

text: the

text: gravitational

text: behavior

text: of

text: almost

text: any

text: form

text: of

text: matter

text: (see

text: also

text: for

text: earlier

text: work).

text: Refs.

text: considered

text: the

text: eﬀects

text: of

text: higher-derivative

text: interactions

text: on

text: the

text: cosmology

text: of

text: the

text: “mimetic”

text: dark

text: matter

text: model.

text: In

text: the

text: authors

text: showed

text: that

text: the

text: energy

text: momentum

text: tensor

text: of

text: the

text: mimetic

text: theory

text: with

text: higher-derivatives

text: is

text: actually

text: of

text: the

text: imperfect

text: ﬂuid

text: type

text: and

text: the

text: theory

text: can

text: support

text: vorticity.

text: Ref.

text: argues

text: that

text: these

text: higher-derivative

text: interactions

text: could

text: possibly

text: help

text: solving

text: the

text: small-scale

text: problems

text: of

text: the

text: cold

text: dark

text: matter

text: model.

text: Ref.

text: proposed

text: an

text: alternative

text: conformal

text: extension

text: of

text: General

text: Relativity,

text: using

text: a

text: vector

text: ﬁeld,

text: that

text: can

text: also

text: support

text: rotational

text: ﬂows

text: for

text: the

text: mimetic

text: dark

text: matter.

text: b

text: arroja@phys.ntu.edu.tw

text: c

text: nicola.bartolo@pd.infn.it

text: d

text: purnendu.karmakar@pd.infn.it

text: ∗e

text: sabino.matarrese@pd.infn.it

text: ##LTLine##


text: 16 [21] A. Golovnev, Phys.Lett. B728, 39 (2014), 1310.2790. [22] P. Creminelli, J. Gleyzes, J. Noren˜a, and F. Vernizzi, Phys.Rev.Lett. 113, 231301 (2014), 1407.8439. [23] M. Minamitsuji, Phys.Lett. B737, 139 (2014), 1409.1566. [24] S. Tsujikawa, JCAP 1504, 043 (2015), 1412.6210. [25] Y. Watanabe, A. Naruko, and M. Sasaki, (2015), 1504.00672. [26] H. Motohashi and J. White, (2015), 1504.00846. [27] G. Dom`enech, A. Naruko, and M. Sasaki, (2015), 1505.00174. [28] R. Myrzakulov, L. Sebastiani, and S. Vagnozzi, (2015), 1504.07984. [29] G. W. Horndeski, Int.J.Theor.Phys. 10, 363 (1974). [30] C. Deﬀayet, X. Gao, D. A. Steer, and G. Zahariade, Phys. Rev. D84, 064039 (2011), 1103.3260. [31] T. Kobayashi, M. Yamaguchi, and J. Yokoyama, Prog. Theor. Phys. 126, 511 (2011), 1105.5723. [32] J. Gleyzes, D. Langlois, F. Piazza, and F. Vernizzi, Phys.Rev.Lett. 114, 211101 (2015), 1404.6495. [33] J. Gleyzes, D. Langlois, F. Piazza, and F. Vernizzi, JCAP 1502, 018 (2015), 1408.1952. [34] X. Gao, Phys.Rev. D90, 081501 (2014), 1406.0822. [35] Z. Haghani, T. Harko, H. R. Sepangi, and S. Shahidi, JCAP 1505, 022 (2015), 1501.00819.

label: reference

text: A.

text: Golovnev,

text: Phys.Lett.

text: P.

text: Creminelli,

text: J.

text: Gleyzes,

text: J.

text: Noren˜a,

text: and

text: F.

text: Vernizzi,

text: Phys.Rev.Lett.

text: M.

text: Minamitsuji,

text: Phys.Lett.

text: S.

text: Tsujikawa,

text: JCAP

text: Y.

text: Watanabe,

text: A.

text: Naruko,

text: and

text: M.

text: Sasaki,

text: H.

text: Motohashi

text: and

text: J.

text: White,

text: G.

text: Dom`enech,

text: A.

text: Naruko,

text: and

text: M.

text: Sasaki,

text: R.

text: Myrzakulov,

text: L.

text: Sebastiani,

text: and

text: S.

text: Vagnozzi,

text: G.

text: W.

text: Horndeski,

text: Int.J.Theor.Phys.

text: C.

text: Deﬀayet,

text: X.

text: Gao,

text: D.

text: A.

text: Steer,

text: and

text: G.

text: Zahariade,

text: Phys.

text: Rev.

text: T.

text: Kobayashi,

text: M.

text: Yamaguchi,

text: and

text: J.

text: Yokoyama,

text: Prog.

text: Theor.

text: Phys.

text: J.

text: Gleyzes,

text: D.

text: Langlois,

text: F.

text: Piazza,

text: and

text: F.

text: Vernizzi,

text: Phys.Rev.Lett.

text: J.

text: Gleyzes,

text: D.

text: Langlois,

text: F.

text: Piazza,

text: and

text: F.

text: Vernizzi,

text: JCAP

text: X.

text: Gao,

text: Phys.Rev.

text: Z.

text: Haghani,

text: T.

text: Harko,

text: H.

text: R.

text: Sepangi,

text: and

text: S.

text: Shahidi,

text: JCAP


text: 8 (cid:54) Using this and the fact that ∂νΨ = 0 at least for one index ν we can simplify Eq. (40) to ∇ − √Ω E + T 1 db(Ψ) [(E + T)b(Ψ)∂µΨ] −Ψ = . (42) µ g 2 b(Ψ) dΨ This is exactly the same equation as (37). So we have managed to show that Eq. (37) results from taking the ∇ covariant derivative of Eq. (38) and use Tµν = 0 and Eqs. (36) and (39). This proof is independent of using the µ Lagrange multiplier method or not and shows that in order to solve the dynamics of the system it is suﬃcient to consider Eqs. (36), (38) and (39). These three equations, when written in terms of the metric g , do not contain any µν more higher-order derivatives that the equations of motion that result from the non-mimetic theory deﬁned by the L L Lagrangian . It is also worth noting that the action (30) does not contain any more higher-order derivatives than . However, the new theory (30) does contain a new ﬁeld, the Lagrange multiplier λ. The three independent equations of motion when written in terms of the new metric (cid:96) may contain higher-order derivatives. For concreteness, we L µν can think of as being the Horndeski Lagrangian [29]3 and we would be considering the “mimetic” Horndeski theory. IV. NON-TRIVIAL EXAMPLES OF COSMOLOGY IN THE “MIMETIC” HORNDESKI MODEL As an application of the results of the preceding sections, in this section, we will present three simple examples of non-trivial cosmological solutions that arise in very simple “mimetic” Horndeski models. The most general class of 4D local scalar-tensor theories that contain second-order equations of motion and that can be derived from an action is known as t(cid:90)he Horndeski the(cid:90)ory [29]. Its action is (cid:88) √ √ − L − 3 L S = d4x g = d4x g , (43) H H n n=0 where L = K (X, Ψ) , (cid:104) (cid:105) (44) 0 L − (cid:3) = G (X, Ψ) Ψ, (45) 1 3 (cid:104) (cid:105) L (cid:3) − ∇ ∇ = G (X, Ψ) ( Ψ)2 ( Ψ)2 + R G (X, Ψ) , (46) 2 4,X µ ν 4 L −1 (cid:3) − (cid:3) ∇ ∇ ∇ ∇ ∇ ∇ = G (X, Ψ) ( Ψ)3 3 Ψ ( Ψ)2 + 2 ( Ψ)3 + G µ νΨ G (X, Ψ) , (47) 3 6 5,X µ ν µ ν µν 5 − ∇ ∇ ∇ ∇ ∇ ∇ ∇ ∇ ∇ ∇ ∇ ∇ ∇ ∇ ∇ ∇ and X = 1/2 Ψ µΨ, ( Ψ)2 = Ψ µ νΨ and ( Ψ)3 = Ψ µ ρΨ ν Ψ. The functions µ µ ν µ ν µ ν µ ν ρ K(X, Ψ), G (X, Ψ), G (X, Ψ) and G (X, Ψ) are free function of their two variables and deﬁne a particular theory 3 4 5 in the Horndeski class. The subscript denotes derivative with respect to X. In the next three subsections the ,X L L actions of the models considered will be Eq. (30) with = but with diﬀerent choices for the free functions in (cid:112) H each subsection. Notice that for a general mimetic Horndeski model, the free function b(Ψ) in the second term of Eq. (30) can be | | reabsorbed by deﬁning a new ﬁeld Φ as dΦ = b dΨ. Because the Horndeski Lagrangian is form invariant under ﬁeld redeﬁnitions of this type, this transformation just amounts to consider a diﬀerent starting Horndeski Lagrangian L . H A. A very simple example In our ﬁrst simple example we will consider the mimetic theory of a canonical kinetic term scalar ﬁeld with no L L potential coupled to Einstein’s gravity theory. The action of this model is Eq. (30) with = , S = 0 and with H m the choice K(X, Ψ) = c X, G (X, Ψ) = 0, G (X, Ψ) = 1/2, G (X, Ψ) = 0, (48) 2 3 4 5 3 One can also consider healthy extensions of Horndeski’s theory, like for instance the so-called G3 theories [32, 33] or even their extensions [34]. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

font: default

text: Using

text: this

text: and

text: the

text: fact

text: that

text: ∂νΨ

text: at

text: least

text: for

text: one

text: index

text: ν

text: we

text: can

text: simplify

text: Eq.

text: to

label: equation

text: √Ω

text: E

text: T

text: db(Ψ)

text: [(E

text: T)b(Ψ)∂µΨ]

text: −Ψ

text: µ

text: g

text: b(Ψ)

text: dΨ

text: This

text: is

text: exactly

text: the

text: same

text: equation

text: as

text: So

text: we

text: have

text: managed

text: to

text: show

text: that

text: Eq.

text: results

text: from

text: taking

text: the

text: covariant

text: derivative

text: of

text: Eq.

text: and

text: use

text: Tµν

text: and

text: Eqs.

text: and

text: This

text: proof

text: is

text: independent

text: of

text: using

text: the

text: µ

text: Lagrange

text: multiplier

text: method

text: or

text: not

text: and

text: shows

text: that

text: in

text: order

text: to

text: solve

text: the

text: dynamics

text: of

text: the

text: system

text: it

text: is

text: suﬃcient

text: to

text: consider

text: Eqs.

text: and

text: These

text: three

text: equations,

text: when

text: written

text: in

text: terms

text: of

text: the

text: metric

text: g

text: do

text: not

text: contain

text: any

text: µν

text: more

text: higher-order

text: derivatives

text: that

text: the

text: equations

text: of

text: motion

text: that

text: result

text: from

text: the

text: non-mimetic

text: theory

text: deﬁned

text: by

text: the

text: L

text: L

text: Lagrangian

text: It

text: is

text: also

text: worth

text: noting

text: that

text: the

text: action

text: does

text: not

text: contain

text: any

text: more

text: higher-order

text: derivatives

text: than

text: However,

text: the

text: new

text: theory

text: does

text: contain

text: a

text: new

text: ﬁeld,

text: the

text: Lagrange

text: multiplier

text: λ.

text: The

text: three

text: independent

text: equations

text: of

text: motion

text: when

text: written

text: in

text: terms

text: of

text: the

text: new

text: metric

text: may

text: contain

text: higher-order

text: derivatives.

text: For

text: concreteness,

text: we

text: L

text: µν

text: can

text: think

text: of

text: as

text: being

text: the

text: Horndeski

text: Lagrangian

text: and

text: we

text: would

text: be

text: considering

text: the

text: “mimetic”

text: Horndeski

text: theory.

text: IV.

text: NON-TRIVIAL

text: EXAMPLES

text: OF

text: COSMOLOGY

text: IN

text: THE

text: “MIMETIC”

text: HORNDESKI

text: MODEL

text: As

text: an

text: application

text: of

text: the

text: results

text: of

text: the

text: preceding

text: sections,

text: in

text: this

text: section,

text: we

text: will

text: present

text: three

text: simple

text: examples

text: of

text: non-trivial

text: cosmological

text: solutions

text: that

text: arise

text: in

text: very

text: simple

text: “mimetic”

text: Horndeski

text: models.

text: The

text: most

text: general

text: class

text: of

text: local

text: scalar-tensor

text: theories

text: that

text: contain

text: second-order

text: equations

text: of

text: motion

text: and

text: that

text: can

text: be

text: derived

text: from

text: an

text: action

text: is

text: known

text: as

text: Horndeski

text: Its

text: action

text: is

text: L

text: L

text: S

text: g

text: g

text: H

text: H

text: n

text: where

text: L

text: K

text: (X,

text: Ψ)

text: L

text: G

text: (X,

text: Ψ)

text: Ψ,

text: L

text: G

text: (X,

text: Ψ)

text: R

text: G

text: (X,

text: Ψ)

text: µ

text: ν

text: L

text: −1

text: G

text: (X,

text: Ψ)

text: Ψ

text: G

text: µ

text: νΨ

text: G

text: (X,

text: Ψ)

text: µ

text: ν

text: µ

text: ν

text: µν

text: and

text: X

text: Ψ

text: µΨ,

text: Ψ

text: µ

text: νΨ

text: and

text: Ψ

text: µ

text: ρΨ

text: ν

text: Ψ.

text: The

text: functions

text: µ

text: µ

text: ν

text: µ

text: ν

text: µ

text: ν

text: µ

text: ν

text: ρ

text: K(X,

text: Ψ),

text: G

text: (X,

text: Ψ),

text: G

text: (X,

text: Ψ)

text: and

text: G

text: (X,

text: Ψ)

text: are

text: free

text: function

text: of

text: their

text: two

text: variables

text: and

text: deﬁne

text: a

text: particular

text: theory

text: in

text: the

text: Horndeski

text: class.

text: The

text: subscript

text: denotes

text: derivative

text: with

text: respect

text: to

text: X.

text: In

text: the

text: next

text: three

text: subsections

text: the

text: ,X

text: L

text: L

text: actions

text: of

text: the

text: models

text: considered

text: will

text: be

text: Eq.

text: with

text: but

text: with

text: diﬀerent

text: choices

text: for

text: the

text: free

text: functions

text: in

text: H

text: each

text: subsection.

text: Notice

text: that

text: for

text: a

text: general

text: mimetic

text: Horndeski

text: model,

text: the

text: free

text: function

text: b(Ψ)

text: in

text: the

text: second

text: term

text: of

text: Eq.

text: can

text: be

text: reabsorbed

text: by

text: deﬁning

text: a

text: new

text: ﬁeld

text: Φ

text: as

text: dΦ

text: b

text: dΨ.

text: Because

text: the

text: Horndeski

text: Lagrangian

text: is

text: form

text: invariant

text: under

text: ﬁeld

text: redeﬁnitions

text: of

text: this

text: type,

text: this

text: transformation

text: just

text: amounts

text: to

text: consider

text: a

text: diﬀerent

text: starting

text: Horndeski

text: Lagrangian

text: L

text: H

text: A.

text: A

text: very

text: simple

text: example

text: In

text: our

text: ﬁrst

text: simple

text: example

text: we

text: will

text: consider

text: the

text: mimetic

text: theory

text: of

text: a

text: canonical

text: kinetic

text: term

text: scalar

text: ﬁeld

text: with

text: no

text: L

text: L

text: potential

text: coupled

text: to

text: Einstein’s

text: gravity

text: theory.

text: The

text: action

text: of

text: this

text: model

text: is

text: Eq.

text: with

text: S

text: and

text: with

text: H

text: m

text: the

text: choice

text: K(X,

text: Ψ)

text: c

text: X,

text: G

text: (X,

text: Ψ)

text: G

text: (X,

text: Ψ)

text: G

text: (X,

text: Ψ)

text: One

text: can

text: also

text: consider

text: healthy

text: extensions

text: of

text: Horndeski’s

text: theory,

text: like

text: for

text: instance

text: the

text: so-called

text: theories

text: or

text: even

text: their

text: extensions

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 9 where c is a constant which may have either sign. In the non-mimetic theory, if c is negative it is well known 2 2 that the scalar ﬁeld has the wrong sign in the kinetic term and is a ghost. In the present mimetic model it would be interesting to study perturbations and determine what are the conditions for the absence of ghost and other instabilities. We leave this work for the near future. With the same notation of the previous section and for a ﬂat FLRW background, the objects that appear there are − 1 E = 3H2 + c Ψ˙ 2, (49) 00 2 2 1 E = E = E = a2(3H2 + 2H˙ + c Ψ˙ 2), (50) xx yy zz (cid:16) 2 2(cid:17) E = 12H2 + 6H˙ + c Ψ˙ 2, (51) 2 − − Ω = a3 3c HΨ˙ c Ψ¨ , (52) Ψ 2 2 where a is the scale factor, H = a˙/a and dot denotes derivative with respect to cosmic time. x, y, z denote the comoving spatial coordinates. The equations of motion for this simple mimetic model, i.e. Eq. (36), the time and spatial components of Eq. (38) and Eq. (37), are respectively (cid:104) (cid:16) (cid:17)(cid:105) b(Ψ)Ψ˙ 2 + 1 = 0, (53) Ψ˙ 2 − 3H2 = c 2b(Ψ) 12H2 + 6H˙ + c Ψ˙ 2 , (54) (cid:104) 2 2 2 (cid:105) 6H2 + 4H˙ + c Ψ˙ 2 = 0, (55) − 2 (cid:18) − −(cid:19) (cid:104) − (cid:105) b(Ψ) 6H(6H2Ψ˙ + 7Ψ˙ H˙ + 2HΨ¨) 6H˙ Ψ¨ 6Ψ˙ H¨ 3c Ψ˙ 2(HΨ˙ + Ψ¨) 2 (cid:48) 1 − − − c +c (3HΨ˙ + Ψ¨) + b (Ψ) + 2Ψ˙ 2 6H2 3H˙ 2 Ψ˙ 2 = 0, (56) 2 b(Ψ) 2 where prime denotes derivative with respect to the ﬁeld Ψ. Eqs. (54) and (56) are not independent from Eqs. (53) and (55) because they can be derived from them. It is easy to check that Eqs. (53) and (55)(cid:114)admit the following solution √ a(t) = t3(12+ω) , Ψ(t) = ± −cα2 log tt0 , b(Ψ) = −Ψ˙12 = cα2 t2 = cα2 t20e±2 −cα2 Ψ, (57) − where t is an integration constant, the parameter α is α = 8ω , where ω is a constant parameter. This 0 3(1+ω)2 expansion law is the same as the one given by a perfect ﬂuid universe with a constant equation of state ω. If c is 2 positive then the equation of state ω has to be positive too. This shows that this simple mimetic scalar ﬁeld model can mimic the background evolution of a perfect ﬂuid universe with a constant equation of state. For diﬀerent ω the value of α changes but the functional form of b(Ψ) does not change. It is obvious that this new solution is not a solution of the Einstein plus Klein-Gordon (with zero potential) ﬁeld theory. There ω is necessarily unity. By adjusting the function b(Ψ) accordingly (note that b(Ψ) < 0 for a time-like scalar velocity), this simple model can mimic the expansion history of almost any model. To be concrete, we can mimic the expansion history of a perfect − ﬂuid model with a ﬁxed sign of the pressure. In that case, 6H2 + 4H˙ = 2p, where p is the pressure of the perfect ﬂuid. So from the independent equation of motion (55) one can see that the pressure cannot change sign. The fact that one can have almost any expansion history desired is somewhat similar to the minimal extension of the original mimetic dark matter model proposed in [12]. See also [13] for an earlier work where models similar to our present one were considered. B. Mimetic cubic Galileon In this subsection, we will consider the mimetic cubic Galileon model as a further example of a simple mimetic Horndeski model. The mimetic cubic Galileon model with c = 0 (and also including the other Galileon interactions) 2 was previously studied in [35] but for the case of a constant b(Ψ). Here we allow the function b to depend on Ψ. The L L action of the model is Eq. (30) with = , S = 0 and with the free functions chosen as H m ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: where

text: c

text: is

text: a

text: constant

text: which

text: may

text: have

text: either

text: sign.

text: In

text: the

text: non-mimetic

text: theory,

text: if

text: c

text: is

text: negative

text: it

text: is

text: well

text: known

text: that

text: the

text: scalar

text: ﬁeld

text: has

text: the

text: wrong

text: sign

text: in

text: the

text: kinetic

text: term

text: and

text: is

text: a

text: ghost.

text: In

text: the

text: present

text: mimetic

text: model

text: it

text: would

text: be

text: interesting

text: to

text: study

text: perturbations

text: and

text: determine

text: what

text: are

text: the

text: conditions

text: for

text: the

text: absence

text: of

text: ghost

text: and

text: other

text: instabilities.

text: We

text: leave

text: this

text: work

text: for

text: the

text: near

text: future.

text: With

text: the

text: same

text: notation

text: of

text: the

text: previous

text: section

text: and

text: for

text: a

text: ﬂat

text: FLRW

text: background,

text: the

text: objects

text: that

text: appear

text: there

text: are

text: E

text: c

text: Ψ˙

text: E

text: E

text: E

text: c

text: Ψ˙

text: xx

text: yy

text: zz

text: E

text: c

text: Ψ˙

text: Ω

text: HΨ˙

text: c

text: Ψ¨

text: Ψ

text: where

text: a

text: is

text: the

text: scale

text: factor,

text: H

text: a˙/a

text: and

text: dot

text: denotes

text: derivative

text: with

text: respect

text: to

text: cosmic

text: time.

text: x,

text: y,

text: z

text: denote

text: the

text: comoving

text: spatial

text: coordinates.

text: The

text: equations

text: of

text: motion

text: for

text: this

text: simple

text: mimetic

text: model,

text: i.e.

text: Eq.

text: the

text: time

text: and

text: spatial

text: components

text: of

text: Eq.

text: and

text: Eq.

text: are

text: respectively

text: b(Ψ)Ψ˙

text: Ψ˙

text: c

text: c

text: Ψ˙

text: c

text: Ψ˙

text: b(Ψ)

text: 7Ψ˙

text: H˙

text: Ψ¨

text: 6Ψ˙

text: H¨

text: Ψ˙

text: Ψ¨)

text: c

text: +c

text: Ψ¨)

text: b

text: (Ψ)

text: 2Ψ˙

text: Ψ˙

text: b(Ψ)

text: where

text: prime

text: denotes

text: derivative

text: with

text: respect

text: to

text: the

text: ﬁeld

text: Ψ.

text: Eqs.

text: and

text: are

text: not

text: independent

text: from

text: Eqs.

text: and

text: because

text: they

text: can

text: be

text: derived

text: from

text: them.

text: It

text: is

text: easy

text: to

text: check

text: that

text: Eqs.

text: and

text: the

text: following

text: solution

text: a(t)

text: Ψ(t)

text: −cα2

text: log

text: b(Ψ)

text: −Ψ˙12

text: −cα2

text: Ψ,

text: where

text: t

text: is

text: an

text: integration

text: constant,

text: the

text: parameter

text: α

text: is

text: α

text: 8ω

text: where

text: ω

text: is

text: a

text: constant

text: parameter.

text: This

text: expansion

text: law

text: is

text: the

text: same

text: as

text: the

text: one

text: given

text: by

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text: 1 Joint Source-Channel Coding for Real-Time Video Transmission to Multi-homed Mobile Terminals Xiaoyan Gao School of Computer Science and Technology, North China Institute of Science and Technology Email: xiaoyan gao@ncist.edu.cn. Abstract— This study focuses on the mobile video delivery from Location 3 a video server to a multi-homed client with a network of het- Location 2 erogeneous wireless. Joint Source-Channel Coding is effectively used to transmit video over bandwidth-limited, noisy wireless Location 1 networks. But most existing JSCC methods only consider single Trajectory path video transmission of the server and the client network. The problem will become more complicated when consider multi- Cellular Base path video transmission, because involving low-bandwidth, high- Station drop-rate or high-latency wireless network will only reduce the video quality. To solve this critical problem, we propose a novel Path Adaption JSCC (PA-JSCC) method that contain below Cellular Base WLAN WiMAX characters: (1) path adaption, and (2) dynamic rate allocation. Station Controller We use Exata to evaluate the performance of PA-JSCC and Experiment show that PA-JSCC has a good results in terms of PSNR (Peak Signal-to-Noise Ratio). mobile video delivery; heterogeneous wireless networks; Video Server Internet multi-homing; joint source channel coding Fig. 1. Illustration of a mobile video streaming service in a heterogeneous I. INTRODUCTION wireless network. Nowadays handheld devices (e.g., smartphone and iPad) is explosive increasing, on the other hand, mobile video server and client to transfer mobile video. The problem streaming (e.g., Hollywood [1] and Metacafe [2]) has became becomes more complicated when consider multi-path video one of the most popular portion of network ﬂow. According transmission in heterogenous wireless network. Fig. 1 shows to the Cisco Visual Index [3] report, the percentage of mobile the problem, in location 1, the user encounter a video problem video streaming is 57% in 2012 and will increase to 69% because the cellular link not transfer the streaming well. In by the year 2017. As a result, focus on mobile video quality location 2, the user requests to the video server and in the guarantee study is meaningful. meanwhile access WLAN network, but because of the unstable With the progress of technology mobile devices users could WLAN link, the video quality reduce more. In location 3 access network with various options (e.g., WLAN, 3G/4G and which the place user could connect to WiMAX and then the WiMAX), but problem still existing because bandwidth lim- perceived quality became better than in location 1 and 2. This itation of single wireless causing bottleneck of mobile video can be summarized as the user-perceived video quality will quality promotion. Contemporary WLAN systems could not only be reduced when involving an unreliable wireless access guarantee video streaming quality because of small coverage network. area and limited bandwidth when there are number of mo- To improve the JSCC approach for mobile video transmis- bile users [4-5]. Meanwhile, cellular networks (e.g., UMTS) sion in heterogeneous wireless networks, we propose a path could provide more reliable service to mobile users, but their adaption JSCC (PA-JSCC) in this study. The term ’path adap- bandwidth is ﬁnite for high-quality video transmission [6]. tion’ means dynamically selecting the appropriate wireless The 4G LTE and WiMAX could provide higher transmission path and allocating the appropriate data rate to each of them. rate and larger coverage area, however, they are not widely First, the video source rate adaption scheme is designed to applied and bandwidth limitation also existing as the number meet the delay requirements of real-time video transmission. of mobile users increases. Above all, the limitation of single Second, the Forward Error Correction (FEC) redundancy esti- wireless network turn research attentions to multi-path video mation is employed to decrease the tolerable loss rate. Third, transmission. a effective ﬂow allocation algorithm is proposed to reduce the To solve the existing problem, researchers proposed joint end-to-end video distortion. source-channel coding (JSCC) and proved it was an effec- tive approach in noisy wireless networks. But existing JSCC Speciﬁcally, the contributions of this paper can be summa- approaches (e.g., [7-8]) only consider single path between rized in the following: ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Joint

text: Source-Channel

text: Coding

text: for

text: Real-Time

text: Video

text: Transmission

text: to

text: Multi-homed

text: Mobile

text: Terminals

text: Xiaoyan

text: Gao

text: School

text: of

text: Computer

text: Science

text: and

text: Technology,

text: North

text: China

text: Institute

text: of

text: Science

text: and

text: Technology

text: Email:

text: xiaoyan

text: gao@ncist.edu.cn.

text: Abstract—

text: This

text: study

text: focuses

text: on

text: the

text: mobile

text: video

text: delivery

text: from

text: Location

text: a

text: video

text: server

text: to

text: a

text: multi-homed

text: client

text: with

text: a

text: network

text: of

text: het-

text: Location

text: erogeneous

text: wireless.

text: Joint

text: Source-Channel

text: Coding

text: is

text: effectively

text: used

text: to

text: transmit

text: video

text: over

text: bandwidth-limited,

text: noisy

text: wireless

text: Location

text: networks.

text: But

text: most

text: existing

text: JSCC

text: methods

text: only

text: consider

text: single

text: Trajectory

text: path

text: video

text: transmission

text: of

text: the

text: server

text: and

text: the

text: client

text: network.

text: The

text: problem

text: will

text: become

text: more

text: complicated

text: when

text: consider

text: multi-

text: Cellular

text: Base

text: path

text: video

text: transmission,

text: because

text: involving

text: low-bandwidth,

text: high-

text: Station

text: drop-rate

text: or

text: high-latency

text: wireless

text: network

text: will

text: only

text: reduce

text: the

text: video

text: quality.

text: To

text: solve

text: this

text: critical

text: problem,

text: we

text: propose

text: a

text: novel

text: Path

text: Adaption

text: JSCC

text: (PA-JSCC)

text: method

text: that

text: contain

text: below

text: Cellular

text: Base

text: WLAN

text: WiMAX

text: characters:

text: path

text: adaption,

text: and

text: dynamic

text: rate

text: allocation.

text: Station

text: Controller

text: We

text: use

text: Exata

text: to

text: evaluate

text: the

text: performance

text: of

text: PA-JSCC

text: and

text: Experiment

text: show

text: that

text: PA-JSCC

text: has

text: a

text: good

text: results

text: in

text: terms

text: of

text: PSNR

text: (Peak

text: Signal-to-Noise

text: Ratio).

text: mobile

text: video

text: delivery;

text: heterogeneous

text: wireless

text: networks;

text: Video

text: Server

text: Internet

text: multi-homing;

text: joint

text: source

text: channel

text: coding

text: Fig.

text: Illustration

text: of

text: a

text: mobile

text: video

text: streaming

text: service

text: in

text: a

text: heterogeneous

text: I.

text: INTRODUCTION

text: wireless

text: network.

text: Nowadays

text: handheld

text: devices

text: (e.g.,

text: smartphone

text: and

text: iPad)

text: is

text: explosive

text: increasing,

text: on

text: the

text: other

text: hand,

text: mobile

text: video

text: server

text: and

text: client

text: to

text: transfer

text: mobile

text: video.

text: The

text: problem

text: streaming

text: (e.g.,

text: Hollywood

text: and

text: Metacafe

text: has

text: became

text: becomes

text: more

text: complicated

text: when

text: consider

text: multi-path

text: video

text: one

text: of

text: the

text: most

text: popular

text: portion

text: of

text: network

text: ﬂow.

text: According

text: transmission

text: in

text: heterogenous

text: wireless

text: network.

text: Fig.

text: shows

text: to

text: the

text: Cisco

text: Visual

text: Index

text: report,

text: the

text: percentage

text: of

text: mobile

text: the

text: problem,

text: in

text: location

text: the

text: user

text: encounter

text: a

text: video

text: problem

text: video

text: streaming

text: is

text: in

text: and

text: will

text: increase

text: to

text: because

text: the

text: cellular

text: link

text: not

text: transfer

text: the

text: streaming

text: well.

text: In

text: by

text: the

text: year

text: As

text: a

text: result,

text: focus

text: on

text: mobile

text: video

text: quality

text: location

text: the

text: user

text: requests

text: to

text: the

text: video

text: server

text: and

text: in

text: the

text: guarantee

text: study

text: is

text: meaningful.

text: meanwhile

text: access

text: WLAN

text: network,

text: but

text: because

text: of

text: the

text: unstable

text: With

text: the

text: progress

text: of

text: technology

text: mobile

text: devices

text: users

text: could

text: WLAN

text: link,

text: the

text: video

text: quality

text: reduce

text: more.

text: In

text: location

text: access

text: network

text: with

text: various

text: options

text: (e.g.,

text: WLAN,

text: and

text: which

text: the

text: place

text: user

text: could

text: connect

text: to

text: WiMAX

text: and

text: then

text: the

text: WiMAX),

text: but

text: problem

text: still

text: existing

text: because

text: bandwidth

text: lim-

text: perceived

text: quality

text: became

text: better

text: than

text: in

text: location

text: and

text: This

text: itation

text: of

text: single

text: wireless

text: causing

text: bottleneck

text: of

text: mobile

text: video

text: can

text: be

text: summarized

text: as

text: the

text: user-perceived

text: video

text: quality

text: will

text: quality

text: promotion.

text: Contemporary

text: WLAN

text: systems

text: could

text: not

text: only

text: be

text: reduced

text: when

text: involving

text: an

text: unreliable

text: wireless

text: access

text: guarantee

text: video

text: streaming

text: quality

text: because

text: of

text: small

text: coverage

text: network.

text: area

text: and

text: limited

text: bandwidth

text: when

text: there

text: are

text: number

text: of

text: mo-

text: To

text: improve

text: the

text: JSCC

text: approach

text: for

text: mobile

text: video

text: transmis-

text: bile

text: users

text: Meanwhile,

text: cellular

text: networks

text: (e.g.,

text: UMTS)

text: sion

text: in

text: heterogeneous

text: wireless

text: networks,

text: we

text: propose

text: a

text: path

text: could

text: provide

text: more

text: reliable

text: service

text: to

text: mobile

text: users,

text: but

text: their

text: adaption

text: JSCC

text: (PA-JSCC)

text: in

text: this

text: study.

text: The

text: term

text: ’path

text: adap-

text: bandwidth

text: is

text: ﬁnite

text: for

text: high-quality

text: video

text: transmission

text: tion’

text: means

text: dynamically

text: selecting

text: the

text: appropriate

text: wireless

text: The

text: LTE

text: and

text: WiMAX

text: could

text: provide

text: higher

text: transmission

text: path

text: and

text: allocating

text: the

text: appropriate

text: data

text: rate

text: to

text: each

text: of

text: them.

text: rate

text: and

text: larger

text: coverage

text: area,

text: however,

text: they

text: are

text: not

text: widely

text: First,

text: the

text: video

text: source

text: rate

text: adaption

text: scheme

text: is

text: designed

text: to

text: applied

text: and

text: bandwidth

text: limitation

text: also

text: existing

text: as

text: the

text: number

text: meet

text: the

text: delay

text: requirements

text: of

text: real-time

text: video

text: transmission.

text: of

text: mobile

text: users

text: increases.

text: Above

text: all,

text: the

text: limitation

text: of

text: single

text: Second,

text: the

text: Forward

text: Error

text: Correction

text: (FEC)

text: redundancy

text: esti-

text: wireless

text: network

text: turn

text: research

text: attentions

text: to

text: multi-path

text: video

text: mation

text: is

text: employed

text: to

text: decrease

text: the

text: tolerable

text: loss

text: rate.

text: Third,

text: transmission.

text: a

text: effective

text: ﬂow

text: allocation

text: algorithm

text: is

text: proposed

text: to

text: reduce

text: the

text: To

text: solve

text: the

text: existing

text: problem,

text: researchers

text: proposed

text: joint

text: end-to-end

text: video

text: distortion.

text: source-channel

text: coding

text: (JSCC)

text: and

text: proved

text: it

text: was

text: an

text: effec-

text: tive

text: approach

text: in

text: noisy

text: wireless

text: networks.

text: But

text: existing

text: JSCC

text: Speciﬁcally,

text: the

text: contributions

text: of

text: this

text: paper

text: can

text: be

text: summa-

text: approaches

text: (e.g.,

text: only

text: consider

text: single

text: path

text: between

text: rized

text: in

text: the

text: following:

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text:  (cid:124) (cid:123)(cid:122) α (cid:125) (cid:124) ·(cid:123)(cid:122) ∗(cid:125)  (cid:80) Dtotal = D0 + V − V0 + βDcπhlB , (3) V · n/k · (cid:88)Riω=r1 ωi < µr, for 1 ≤ r < R, Dsrc subject to: R · ≤ V n/k µ , in which α, V , D and β are constants for a speciﬁc video r 0 0 codec and video sequence. Since this model takes into account r=1 (4) the effects of intra coding and spatial loop ﬁltering, it provides accurate estimates for end-to-end distortion [20]. This is a nonlinear optimization problem with linear con- C. Forward Error Correction straints. With regard to the computational cost and conver- gence, it is impractical to derive the exact solution for the In this paper, we employ systematic Reed-Solomon (RS) minimal video distortion. In the next section, we will show block erasure code to eliminate channel losses. Generically, a how to resolve this optimization problem in the design of the FEC block of n data packets includes k source packets and proposed PA-JSCC. − n k redundant packets. Most of time the receiver can fully IV. DESIGN OF FLOW RATE ALLOCATION BASED JOINT reconstruct the original k data packets if at least k packets SOURCE CHANNEL CODING of the FEC block are correctly received. Particularly, the FEC In this section, we describe the overall design of the code rate n/k needs to be dynamically chosen based on the proposed PA-JSCC and outline the functionality of its major loss requirement and channel status. components. The architecture of the system framework is In this paper, Reed-Solomon code is adopted to construct presented in Fig. 4. The system design includes components a virtual connection by aggregating multiple heterogeneous implemented in both the server and client side, respectively. In wireless networks. Actually, the uncertainty of the resulting order to solve the optimization problem, at the server side we virtual connection may increase although its total bandwidth is add two components: (1) path selector, (2) ﬂow rate allocation. higher as the number of integrated wireless networks increases. Particularly, the server side include a data distributor compo- Fountain code is very useful to overcome the uncertainty of nents, it responsibilities is split encoded video streaming into a virtual path since its code rate is not ﬁxed in a priori multiple available wireless networks and dispatching the FEC way and can be adjusted ﬂexibly according to the wireless data packets onto different channels. channel states. In general, a large source block size can At the client side, the video frames will be stored in the reduce sensitivity to the wireless channel uncertainty at the playback buffer after the FEC decoding process. The inter- cost of increased delay. In addition, fountain code has some frame resequence step aims at reordering the video frames in advantages for video transmission over wireless networks due case they arrive at the client out-of-order. As each video frame to its low encoding/decoding processing demand and high is associated with a decoding deadline, the overdue frames will coding efﬁciency. be discarded and concealed by copying from the last received Actually, the frame-level [21], GoP-level [22] or sub-GoP ones. And we add an information feedback components to level [23] FEC coding is often applied for video data protec- send channel status information back to server side. Next, we tion. In this work, the GoP-level FEC (see Fig. 3) is employed will describe the key components in the system design and for the data protection. their major functions. D. Problem Formulation Sender Receiver Multi-path data We are now ready to formulate the problem of ﬂow rate FEC Data transmission Packet Packet allocation combining the JSCC for video delivery in heteroge- Encoder Distributor Receiver Reordering neous wireless networks. Note that, it is not practical for the Sub-flow level video encoder to trace the frequent variation in source rate. network Flow Rate Path information Information Data Flow Therefore, we adapt the source rate in units of GoP, based on Allocator Selector Feedback Reassemble the channel status, FEC code rate and delay requirements. To Data flow Information flow Proposed component allow fast adaptation of the source rate to abrupt changes in the video content, this parameter is updated for each group of Fig. 2. System architecture for performance evaluation. pictures (GOP) in the encoded video sequence, typically once every 0.25 second (with J = 8 frames, F = 30 fps). The A. Path Selector objective is to minimize the summation of the total distortion According to information theory [24], the video source dis- Dtotal subject to loss, delay and bandwidth constraints. tortion can be minimized by increasing the effective encoding For each GoP, determine the value of Φ, Ω, V, n rate. On the other hand, the increasing encoding rate will to minimize (cid:124) (cid:123)(cid:122) (cid:125) lead to higher transmission rate which impose heavier load on (cid:124) (cid:123)(cid:122) (cid:125) α · ∗ channels. If the imposed load exceeds the network capacity, it Dtotal = Dtotal = D0 + V − V + β πB , will in turn result in longer delay and packet loss due to the 0 Dchl network congestion. There is a inherent conﬂict between the Dsrc source and channel distortion. The functions of path selector ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: α

text: Dtotal

text: V

text: βDcπhlB

text: V

text: n/k

text: ωi

text: µr,

text: for

text: r

text: R,

text: Dsrc

text: subject

text: to:

text: R

text: V

text: n/k

text: µ

text: in

text: which

text: α,

text: V

text: D

text: and

text: β

text: are

text: constants

text: for

text: a

text: speciﬁc

text: video

text: r

text: codec

text: and

text: video

text: sequence.

text: Since

text: this

text: model

text: takes

text: into

text: account

text: the

text: effects

text: of

text: intra

text: coding

text: and

text: spatial

text: loop

text: ﬁltering,

text: it

text: provides

text: accurate

text: estimates

text: for

text: end-to-end

text: distortion

text: This

text: is

text: a

text: nonlinear

text: optimization

text: problem

text: with

text: linear

text: con-

text: C.

text: Forward

text: Error

text: Correction

text: straints.

text: With

text: regard

text: to

text: the

text: computational

text: cost

text: and

text: conver-

text: gence,

text: it

text: is

text: impractical

text: to

text: derive

text: the

text: exact

text: solution

text: for

text: the

text: In

text: this

text: paper,

text: we

text: employ

text: systematic

text: Reed-Solomon

text: (RS)

text: minimal

text: video

text: distortion.

text: In

text: the

text: next

text: section,

text: we

text: will

text: show

text: block

text: erasure

text: code

text: to

text: eliminate

text: channel

text: losses.

text: Generically,

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text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

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text: VARIATIONAL INFERENCE FOR BACKGROUND SUBTRACTION IN INFRARED IMAGERY K. Makantasis1, A. Doulamis2, N. Doulamis2 1Technical University of Crete, University campus, Kounoupidiana, 73100, Chania, Greece 2National Technical University of Athens, Zographou campus, 15780, Athens, Greece kmakantasis@isc.tuc.gr; adoulam@cs.ntua.gr; ndoulam@cs.ntua.gr ABSTRACT We propose a Gaussian mixture model for background subtraction in infrared imagery. Following a Bayesian approach, our method automatically estimates the number of Gaussian components as well as their parameters, while simultaneously it avoids over/under ﬁtting. The equations for estimating model parameters are analytically derived and thus our method does not require any sampling algorithm that is computationally and memory inefﬁcient. The pixel density estimate is followed by an efﬁcient and highly accurate updating mechanism, which permits our system to be automatically adapted to dynamically changing operation conditions. Experimental results and comparisons with other methods show that our method outperforms, in terms of precision and recall, while at the same time it keeps computational cost suitable for real-time applications. 1. INTRODUCTION Pixels values of infrared frames correspond to the relative differences in the amount of thermal energy emitted or reﬂected from objects in the scene. Due to this fact, infrared cameras are equally applicable for both day and night scenarios, while at the same time, compared to visual-optical cameras, are less affected by changing illumination or background texture. Furthermore, infrared imagery eliminates any privacy issues as people being depicted in the scene can not be identiﬁed [13]. These features make infrared cameras prime candidate for persistent video surveillance systems. Although, infrared imagery can alleviate several problems associated with visual-optical videos, it has its own unique chal- lenges such as a) low signal-to-noise ratio (noisy data) and b) almost continuous pixel values that model objects’ temperature. Both issues complicate pixel responses modeling. An example of raw thermal responses is presented in Figure 1, where pixel values are ﬂoating point numbers ranging from 293 to 299 Kelvin degrees. Due to this peculiarity most of conventional com- puter vision techniques, that successfully used for visual-optical data, can not be applied straightforward on thermal imagery. For many high-level vision based applications, either they use visual-optical videos [4,21,24,25] or infrared data [16,17,26], the task of background subtraction constitutes a key component, as this is one of the most common methods for locating moving objects, facilitating search space reduction and visual attention modeling. Background subtraction techniques applied on visual-optical videos model the color properties of depicted objects [3, 15] and can be classiﬁed into three main categories [10]: basic background modeling [19, 28], statistical background modeling [11, 27] and background estimation [20, 23]. The most used methods are the statistical ones due to their robustness to critical situations. In order to statistically represent the background, a probability distribution is used to model the history of pixel values intensity over time. Towards this direction, the work of Stauffer and Grimson [22], is one of the best known approaches. ##LTLine##

text: VARIATIONAL

text: INFERENCE

text: FOR

text: BACKGROUND

text: SUBTRACTION

text: IN

text: INFRARED

text: IMAGERY

text: K.

text: A.

text: N.

text: University

text: of

text: Crete,

text: University

text: campus,

text: Kounoupidiana,

text: Chania,

text: Greece

text: Technical

text: University

text: of

text: Athens,

text: Zographou

text: campus,

text: Athens,

text: Greece

text: kmakantasis@isc.tuc.gr;

font: DCMXJG+NimbusMonL-Regu

text: adoulam@cs.ntua.gr;

text: ndoulam@cs.ntua.gr

text: ABSTRACT

text: We

text: propose

text: a

text: Gaussian

text: mixture

text: model

text: for

text: background

text: subtraction

text: in

text: infrared

text: imagery.

text: Following

text: a

text: Bayesian

text: approach,

text: our

text: method

text: automatically

text: estimates

text: the

text: number

text: of

text: Gaussian

text: components

text: as

text: well

text: as

text: their

text: parameters,

text: while

text: simultaneously

text: it

text: avoids

text: over/under

text: ﬁtting.

text: The

text: equations

text: for

text: estimating

text: model

text: parameters

text: are

text: analytically

text: derived

text: and

text: thus

text: our

text: method

text: does

text: not

text: require

text: any

text: sampling

text: algorithm

text: that

text: is

text: computationally

text: and

text: memory

text: inefﬁcient.

text: The

text: pixel

text: density

text: estimate

text: is

text: followed

text: by

text: an

text: efﬁcient

text: and

text: highly

text: accurate

text: updating

text: mechanism,

text: which

text: permits

text: our

text: system

text: to

text: be

text: automatically

text: adapted

text: to

text: dynamically

text: changing

text: operation

text: conditions.

text: Experimental

text: results

text: and

text: comparisons

text: with

text: other

text: methods

text: show

text: that

text: our

text: method

text: outperforms,

text: in

text: terms

text: of

text: precision

text: and

text: recall,

text: while

text: at

text: the

text: same

text: time

text: it

text: keeps

text: computational

text: cost

text: suitable

text: for

text: real-time

text: applications.

text: INTRODUCTION

text: Pixels

text: values

text: of

text: infrared

text: frames

text: correspond

text: to

text: the

text: relative

text: differences

text: in

text: the

text: amount

text: of

text: thermal

text: energy

text: emitted

text: or

text: reﬂected

text: from

text: objects

text: in

text: the

text: scene.

text: Due

text: to

text: this

text: fact,

text: infrared

text: cameras

text: are

text: equally

text: applicable

text: for

text: both

text: day

text: and

text: night

text: scenarios,

text: while

text: at

text: the

text: same

text: time,

text: compared

text: to

text: visual-optical

text: cameras,

text: are

text: less

text: affected

text: by

text: changing

text: illumination

text: or

text: background

text: texture.

text: Furthermore,

text: infrared

text: imagery

text: eliminates

text: any

text: privacy

text: issues

text: as

text: people

text: being

text: depicted

text: in

text: the

text: scene

text: can

text: not

text: be

text: identiﬁed

text: These

text: features

text: make

text: infrared

text: cameras

text: prime

text: candidate

text: for

text: persistent

text: video

text: surveillance

text: systems.

text: Although,

text: infrared

text: imagery

text: can

text: alleviate

text: several

text: problems

text: associated

text: with

text: visual-optical

text: videos,

text: it

text: has

text: its

text: own

text: unique

text: chal-

text: lenges

text: such

text: as

text: a)

text: low

text: signal-to-noise

text: ratio

text: (noisy

text: data)

text: and

text: b)

text: almost

text: continuous

text: pixel

text: values

text: that

text: model

text: objects’

text: temperature.

text: Both

text: issues

text: complicate

text: pixel

text: responses

text: modeling.

text: An

text: example

text: of

text: raw

text: thermal

text: responses

text: is

text: presented

text: in

text: Figure

text: where

text: pixel

text: values

text: are

text: ﬂoating

text: point

text: numbers

text: ranging

text: from

text: to

text: Kelvin

text: degrees.

text: Due

text: to

text: this

text: peculiarity

text: most

text: of

text: conventional

text: com-

text: puter

text: vision

text: techniques,

text: that

text: successfully

text: used

text: for

text: visual-optical

text: data,

text: can

text: not

text: be

text: applied

text: straightforward

text: on

text: thermal

text: imagery.

text: For

text: many

text: high-level

text: vision

text: based

text: applications,

text: either

text: they

text: use

text: visual-optical

text: videos

text: or

text: infrared

text: data

text: the

text: task

text: of

text: background

text: subtraction

text: constitutes

text: a

text: key

text: component,

text: as

text: this

text: is

text: one

text: of

text: the

text: most

text: common

text: methods

text: for

text: locating

text: moving

text: objects,

text: facilitating

text: search

text: space

text: reduction

text: and

text: visual

text: attention

text: modeling.

text: Background

text: subtraction

text: techniques

text: applied

text: on

text: visual-optical

text: videos

text: model

text: the

text: color

text: properties

text: of

text: depicted

text: objects

text: and

text: can

text: be

text: classiﬁed

text: into

text: three

text: main

text: categories

text: basic

text: background

text: modeling

text: statistical

text: background

text: modeling

text: and

text: background

text: estimation

text: The

text: most

text: used

text: methods

text: are

text: the

text: statistical

text: ones

text: due

text: to

text: their

text: robustness

text: to

text: critical

text: situations.

text: In

text: order

text: to

text: statistically

text: represent

text: the

text: background,

text: a

text: probability

text: distribution

text: is

text: used

text: to

text: model

text: the

text: history

text: of

text: pixel

text: values

text: intensity

text: over

text: time.

text: Towards

text: this

text: direction,

text: the

text: work

text: of

text: Stauffer

text: and

text: Grimson

text: is

text: one

text: of

text: the

text: best

text: known

text: approaches.

text: ##LTLine##


text: (cid:18) (cid:19) ∗ ∗ − ∗ | ln q (τ ) = ln q (µ , τ ) ln q (µ τ ) = (35a) k k k k k (cid:18)N − − = a + k (cid:0) 1 ln τ (cid:1) 0 2 k − 1 − τ β(cid:88)µ (cid:0)m 2+ (cid:1) 2 k 0 k 0 N (cid:19) − 2− + 2b(cid:0)+ r (cid:1) x µ 0 nk n k n=1 − − C ⇒ β µ m 2 + (35b) k k k ∗ | q (τ ) = Gam(τ a , b ) (35c) k k k k The parameters a and b are given by k k (cid:18) (cid:19) N (cid:0) (cid:1) a = a + k (36a) k 0 2 1 β N − b = b + N σ + 0 k x¯ m 2 (36b) k 0 2 k k β + N k 0 (cid:80) 0 k − where σ = 1 N (x x¯ )2. k Nk n=1 n k B. REFERENCES [1] C. Bishop. Pattern Recognition and Machine Learning (Information Science and Statistics). Springer, Oct. 2007. [2] S. Boyd and L. Vandenberghe. Convex Optimization. Cambridge University Press, Mar. 2004. [3] S. Brutzer, B. Hoferlin, and G. Heidemann. Evaluation of background subtraction techniques for video surveillance. In 2011 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pages 1937–1944, June 2011. [4] S.-C. S. Cheung and C. Kamath. Robust background subtraction with foreground validation for urban trafﬁc video. EURASIP Journal on Advances in Signal Processing, 2005(14):726261, Aug. 2005. [5] C. Dai, Y. Zheng, and X. Li. Pedestrian detection and tracking in infrared imagery using shape and appearance. Computer Vision and Image Understanding, 106(2–3):288–299, May 2007. [6] S. Dasgupta and D. Hsu. On-line estimation with the multivariate gaussian distribution. In Proceedings of the 20th Annual Conference on Learning Theory, COLT’07, pages 278–292, Berlin, Heidelberg, 2007. Springer-Verlag. [7] J. Davis and V. Sharma. Fusion-based background-subtraction using contour saliency. In IEEE Computer Society Conference on Computer Vision and Pattern Recognition - Workshops, 2005. CVPR Workshops, pages 11–11, June 2005. [8] J. W. Davis and V. Sharma. Robust background-subtraction for person detection in thermal imagery. In Proceedings of the 2004 Conference on Computer Vision and Pattern Recognition Workshop (CVPRW’04) Volume 8 - Volume 08, CVPRW ’04, pages 128–, Washington, DC, USA, 2004. IEEE Computer Society. [9] J. W. Davis and V. Sharma. Background-subtraction in thermal imagery using contour saliency. International Journal of Computer Vision, 71(2):161–181, Feb. 2007. [10] F. El Baf, T. Bouwmans, and B. Vachon. Fuzzy statistical modeling of dynamic backgrounds for moving object detection in infrared videos. In IEEE Computer Society Conference on Computer Vision and Pattern Recognition Workshops, 2009. CVPR Workshops 2009, pages 60–65, June 2009. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

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text: q

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text: N

text: (x

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