Niall J English
email: niall.english@ucd.ie

Damian A Mooney
email: damian.mooney@ucd.ie

Stephen O'brien

Ionic Liquids in external electric and electromagnetic fields: a molecular dynamics study

Keywords: Ionic Liquids, Electric Fields, Electromagnetic Fields, Conductivity, Diffusion

The non-thermal effects of external electric and electromagnetic fields in the microwave to far-infrared frequency range and at (r.m.s.) electric field intensities of 10 -3 to 0.25 V/Å r.m.s. on neat salts of 1-3-dimethyl-imidazolium hexafluorophosphate

have been investigated by means of non-equilibrium molecular dynamics simulation.

Significant alterations in molecular mobility were found vis-à-vis zero-field conditions. Using Green-Kubo and transient time correlation function (TTCF) analysis, the electrical conductivity of these ionic liquids has been estimated. Results indicate that ionic liquids respond most significantly to frequencies much lower than those of smaller polar solvents such as water, although the mechanism of field response is almost exclusively translational.

Introduction

Ionic liquids (here referred to as ionic liquids, or ILs) are organic salts in the liquid state, which offer considerable possibilities in the chemical process and allied industries due to their unique physical and chemical properties. They are excellent solvents for a wide variety of organic and inorganic substances, often bringing together otherwise immiscible reagents and, in many cases, maintaining or enhancing reaction rates, extents of reaction and selectivities. In addition, ILs are essentially non-volatile and provide an opportunity for replacing traditional organic solvents, thereby potentially reducing the emissions and energy requirements of unit operations used for separations. Those ILs which are liquids at room temperature ('room temperature' ILs), are of particular interest since they provide for more cost-effective handling and processing. Therefore, ILs are regarded as very promising 'green' solvents and, as such, the impetus for their introduction into existing processes is significant. 1,2 From the earliest studies showing the utility of microwave fields in synthesis, 3,4 there has been significant growth in their application to a wide variety of reactions and processes, accelerated by advances in microwave technology. 5 For the case of ILs, their ionic nature, and, to a lesser extent, dipolar polarization, make them ideal candidates for use with microwave technology. One of the earliest investigations on microwaves in IL synthesis was in the solvent-free production of a series of 1-alkyl-3methyl-imidazolium (AMIM) halides, by the reaction of 1-methylimidazole with alkyl halides/terminal dihalides using a household microwave oven. 6 In this work, improvements in reaction rates were observed. To date, the utility of microwave fields has been exploited in three areas of synthetic chemistry, namely in the synthesis of ILs themselves, as already mentioned, as an aid for heating dipolar substances and when ILs are used as solvents and reactants and as an aid in the processing of reactions carried out in the presence of ILs. 7 Given the increasing importance of ILs, the use of electric and electromagnetic (e/m) fields to exploit their physical and catalytic properties further constitutes a valuable avenue of research. ILs possess important properties making them amenable to electric and e/m field processing, in particular their ionic nature, which offers many intriguing questions and potential applications. Enhancing the mass and heat transfer characteristics of ILs, and of other solvents in ILs, by e/m field exposure would result in smaller reactor volumes and improved, less expensive distillation and liquid-liquid extraction systems. Similarly, the use of microwave irradiation to improve the synthesis efficiency of ILs themselves, and for organic reactions therein, could potentially offer a means of process intensification in the chemical industry. 1 This would serve to maximize the use of resources by their efficient use, necessary for the long-term future of chemical and allied sectors. This work builds on our earlier non-equilibrium molecular dynamics (MD) studies of the behavior of binary mixtures of water and ILs in external e/m fields. 8,9 In those cases, the presence of water molecules led to the consideration of field-induced dipolar alignment as an important consideration in characterizing the response to the fields. In this work, we have chosen to investigate similar, imidazolium-based ionic liquids, specifically hexafluorophosphate ([PF 6 ] -) salts of 1-3-dimethyl-imidazolium and 1-butyl-3-methyl-imidazolium -[dmim] + and [bmim] + , respectively. This study seeks to study in more detail the molecular mechanisms of pure IL species' behavior in electric and e/m field conditions, which is dominated by induced translational motion for neat IL systems, given the absence of strongly dipolar species such as water. This work examines non-thermal field effects on IL properties, especially those Using Green-Kubo and transient time correlation function (TTCF) analysis, the electrical conductivity of these ionic liquids has been estimated, at lower applied electric field intensities than possible in previous molecular simulation studies (10 -3 ).

A goal of this study is to provide insights into the optimization of the use of electric and e/m fields in processes involving ILs.

Methodology

There exist a large number of force-fields developed especially for ILs, where the trend has been toward a generic force-field model adaptable to a large number of possible IL variants. [10][11][12] In this work, we have employed the well documented and tested force-field of Lopes et al. for imidazolium salts, 10 which itself is based on the AMBER/OPLS 13,14 force-fields, with the general form given by (1) More recent work 15 has attempted to refine the force-field of Lopes et al., 10 specifically the non-bonded contributions, in order to replicate more closely hydrogen bonding between cations and anions, as well as transport, including self-diffusion, coefficients. The most notable differences to the force-field of Lopes et al. is the inclusion of a reduced charge total on each ion (=/-0.8e). Despite these differences, Waals interactions were treated by the twin range method, 16 with cut-off radii of 12 and 15Å, respectively. The smooth particle mesh Ewald method 17 was used to handle long-range electrostatics. The screening parameters and number of wave vectors were chosen so as to ensure that the relative error in the Ewald energy was less than 1×10 -6 ;

E tot = 4ε ij σ ij r ij       12 - σ ij r ij       6         + q i q j r ij           + j >i N ∑ i =1 N -1 ∑ k r r -r o ( ) bonds ∑ 2 + k θ θ -θ o ( ) 2 angles ∑ + k 1 + cos nφ -δ ( )     dihedrals ∑ + k ϕ ϕ -ϕ o ( )
in practice this lead to the product of the cut-off distance and screening parameter to be approximately 3.2.

Initial structures were generated first by constructing a diamond lattice arrangement of each IL, followed by relaxation under NPT conditions at 450K and 1 bar with Melchionna's modified form of the Hoover barostat using isotropic cell fluctuations, 18 for a period of up to 2 ns depending on the IL, using time-constants of 1 ps and 2 ps for the thermal and pressure reservoirs, respectively. For all production runs (post NPT relaxation), we employed the configurational thermostat of Braga and Travis 19,20 which has been applied recently to the calculation of conductivity within the TTCF formalism. 21 We have found that this configurational-based thermostat gives rise (with regard to the system under investigation here) to a conductivity that is 15% higher than that determined using a kinetic-based Nosé-Hoover thermostat. This arises because, in these non-equilibrium simulations, kinetic-based thermostats are unable to distinguish between thermal motion and that flow which is induced by the applied field (flow of anions and cations).

As the core purpose of this paper is to assess the influence of electric and e/m fields (with varying field frequency and r.m.s. field intensity) on IL behavior, primarily in terms of properties governed by translational response, it was decided to conduct all Uniform electric and e/m fields were applied with the electric and magnetic components E and B in the z-and y-directions, i.e., or E(t) = E 0 k (static case) and , or B(t) = 0 k for a static electric field 23,2420,21 +q ( ) ( )

i i i i i i m t q = + × r f E v B t  (2)
where denotes the charge and the force on site i due to the potential. Classical mechanics was considered valid to model the electric and e/m field absorption, since the experimental spectrum of ionic liquids is continuous. Fields of frequency = 2.45 to 200 GHz and of r.m.s. intensity from = 0.001 to 0.25 V/Å r.m.s. were studied, along with static electric fields of 0.001 V/Å. It has been found from equilibrium MD simulation that electric field intensities in condensed water phases are in the range of around 1.5 to 2.5 V/Å, 2421 and the lowest field intensities investigated in this work is two orders of magnitude lower than those of previous MD studies of IL-water mixtures in e/m fields 8,9 and close to the intensities available in experimental and industrial magnetrons. 25,2622,23 In addition, the electric field force, qE, acting on the ions' centers-of-mass is ~0.1 % of the typical force magnitudes ). Meanwhile, qE (from eqn. 2) scales linearly with intensity, so that a 0.05 to 0.1 V/Å r.m.s. field corresponds to an electric force of between 5 and 10% of typical cohesive forces, and for the purpose of this work the criterion used for identification of the linear response régime was observed to be the point at which the applied field intensity resulted in an induced flow with a characteristic Reyonold's number of <0.1. This is in line with previous simulation studies of non-thermal effects of static electric fields on molten NaCl, 2724 . 2.45 GHz corresponds to the frequency of commercially available microwave reactors. 25,2622,23 Field frequencies below this pose excessive computational demands since multiple field cycles are require to obtain statistically meaningful results. For example, a field of frequency of 2.45 GHz has a period of ~408 ps, requiring 3-5 ns of simulation to generate satisfactory statistics for results (cf. Results).

E(t) = E max cos ωt ( )k B(t) = B max cos ωt ( )j i q i f ν rms E Field
The external fields were applied in conjunction with a Nosé-Hoover NVT coupling with a relaxation time of 1 ps, 16 and are referred to as non-equilibrium NVT (NNVT) simulations. 23,2820,25 These were performed in order to isolate athermal effects as much as possible from thermal effects. The electric (and e/m) fields induce anisotropic translational motion of the anions and cations, i.e., in opposite directions (at any instant in time, cf. eqn. 1), but there is no net mass flow, so the temperature fixed by the Nosé-Hoover thermostat assumes that thermal motion is defined with respect to the overall center-of-mass velocity. A series of simulations were carried out at 450 K for up to 5 ns in fields of frequency 2.45, 10, 20 50, 100, 200 GHz with r.m.s. intensities from 0.001 to 0.25 V/Å r.m.s. , as well as static fields with intensity of 0.001 V/Å, using the relaxed systems as starting configurations. Longer simulation times were required for lower field frequencies to ensure that an adequate number of to investigate e/m field effects, as gauged by the development of Fickian diffusion within this time period. A reasonable number of cycles of the field can also be applied in this time (with a maximum period of ~408 ps for the 2.45 GHz field). Such a broad range of frequencies were chosen because the expected frequency response are different to that of water, as confirmed in our previous studies of IL-water mixtures:

ILs respond to lower frequencies, owing to translational response and their greater mass vis-à-vis water molecules. 2421 Equilibrium, zero-field simulations were also carried out for each system for 5 ns, as a basis for comparison to assessing the effects of the various field conditions.

For each simulation, the cation and anion self-diffusion coefficients were evaluated using the Einstein relation, along with the total system dipole (z-component), and percentage dipole alignment, which will be defined below. Also, the electrical conductivites, arising from translational motion in response to the fields, were estimated in the static fields using the TTCF approach, 29,3026,27 and verified with respect to zero-field Green-Kubo analysis. 27,3124,28

Analysis

In order to rationalize the results of these simulations, it is useful to consider in the first instance the effect of electric and e/m fields on charged, dipolar bodies from a mechanistic perspective, although dielectric relaxation spectroscopy reveals that the response is essentially due to Ohmic loss attributable to translational motion in response to the applied field. 3229 To confirm this in this study, it is however of use to define the dipolar alignment. The overall dipole moment P s over a set N s ions may be of mass) which leads to its purely translation (Ohmic) response to the applied fields.

In addition, there is no centrally defined origin, being purely local for each ion, coinciding to each ion's centre of mass.

The proportional maximal alignment of a species, which we shall denote as α, may be

determined from 0, max / z s s s z N p α = P (4)
i.e., as the time-average of the species' z-dipole extrema along + and -directions as a proportion of the maximum theoretical z-component alignment (for a sufficiently large number of sampled complete field cycles) vis-à-vis a hypothetical state of complete field alignment of each monomer's dipole vector for the entire species, taken as a time-average for our purposes.

For the translational response, the acceleration due to the field (i.e., acting along the + z-direction) is determined as a function of the force acting on ions' centers-of-mass and a retarding force due to friction. Provided that the Reynolds number, given by va ρ µ , is less than around 0.1, in which limiting (translational) velocity is t v ± k , the characteristic radius is a and the (mass) density of the medium ( ) ( ) ( )

max cos 6 E t qE t a t m ω π υ   = -   a k (5)
In the case of a static field, the cosine-dependence would not be present. In the case of this study, an (r.m.s.) electric field intensity of 0.001 V/Å r.m.s. led to laminar flow conditions, consistent with the linear-response régime (see 'Methodology'). At fields above 0.001 V/Å r.m.s the linear regime was not applicable. Here, is the instantaneous velocity of the ion center-of-mass, which reaches 'terminal' values in either direction, . As mentioned previously, there is an anisotropic distribution of induced velocities at each instant of field application, with anions' and cations' induced velocities in opposite directions, without any overall mass flow in the system. In the case of a static electric field, this will lead to essentially constant induced flow of anions and cations in opposite directions, whereas in the e/m case, the field oscillates back and forth along the z-axis, leading to induced velocities changing direction in response to the field after a frictional lag time. Integrating eqn.

5 to obtain , assuming initial ions' centers-of-mass velocities of zero (or one which is certainly small in comparison to ), one obtains 3431

( ) ( ) ( ) ( ) ( ) 2 2 max 1 cos sin B t qE m B m t t m ω ω ω ω   = + +     v k (6)
where B denotes . For a static electric field, this (constant) velocity t v k is given by ( )

0 qE B k .
Integrating further to obtain the z-axis displacement of the center-of-mass vis-à-vis its initial position gives This induced flow response is manifested in the definition of the electrical conductivity, σ. This was computed in the absence and presence of static fields to study the translational field response further. In both cases, the charge density J was defined over all ions as

( ) ( ) ( ) ( ) ( ) 2 2 max 1 1 sin cos B t qE m B m t t m ω ω ω ω   = + -     z k (7) ( ) v t k t v ± k ( ) v t t v 6 a πµ
( ) ( ) cm, 1 n i i i t q t = = ∑ J v (8)
Under zero-field conditions, the electrical conductivity may be computed from a

Green-Kubo integral of the charge density's autocorrelation function 27,3124,28 ( ) ( )

0 B 1 0 3 t Vk T σ ∞ = ⋅ ∫ J J (9)
Using the TTCF approach in static electric fields, the electrical conductivity may also be estimated using 2724 ( ) ( )

0 B 1 0 z z J t J Vk T σ ∞ = ⋅ ∫ ( 10
)
where the correlation function has now been estimated from the z-component of the charge density directly from the non-equilibrium simulation. This approach has the distinct advantage that it may be used in direct non-equilibrium simulations in electric fields of arbitrarily high or low intensity. 2724 The Green-Kubo and TTCF correlation functions were sampled every 50 fs, following observation from previous studies that timescales for appreciable variation of charge density ACF's are in this range, 3128 and cubic spline interpolation used to compute intermediate points in constructing correlation functions. 3633 The validity of this method was checked by running a shorter simulation (0.25 ns) where the data was stored at half the intervals, so another 5 ps ACF was constructed but with double the number of points at half the time-spacing between discrete points. Upon application of spline interpolation, the results for the intermediate ACF points were the essentially identical. Correlation function lengths of 10, 20 and 50 ps were defined, based on 5 ns simulations, and all were found to give similar results for the electrical conductivity.

Equation Chapter 1 Section 1Results and Discussion

In order to ascertain the limited extent of dipolar orientation in terms of field [bmim] + , one may observe that dipole alignment is more pronounced at lower field frequencies. This is more significant for [bmim] + , which diminishes significantly at field frequencies above 10 GHz. This frequency range for optimal dipolar alignment is significantly lower than that of water in binary IL-water mixtures at similar conditions to this study, as concluded in previous work. 8,9 In this work, the higher moments of inertia of the [dmim] + and [bmim] + species vis-à-vis water necessitates a longer field period (i.e. lower frequency) to achieve the high levels of dipolar alignment observed in Fig. 2 at low frequency. 8,9 In order to probe the translational response to the fields directly, the mobility of the ions' centers-of-mass was assessed from self-diffusion coefficients (D) determined using the Einstein relation (valid at long times) (11) where is termed the ensemble-average mean-squared displacement (MSD), with as the position of the centre-of-mass of the molecule at time t. 16 The zero-field and NNVT simulations were performed for up to 5 ns each, so as to ensure that the Fickian régime had been reached, a pre-requisite for obtaining meaningful self-diffusion coefficients. As pointed out in previous work on liquid water, 3734 the MSD as computed under NNVT conditions are a reasonable approximation to the self-diffusivity. Nevertheless, to verify this for the present study, selected NNMD simulations (i.e. NVE with the field applied) were performed in fields and the 'local' MSD time-derivatives (divided by six) for the period of the simulation where the temperature was within 5% of 450 K were found to be within 20% or better of the NNPT results presented here.. As can be observed from Figure 4, diffusion is Fickian beyond ~200 ps (indicative of enhanced mobility under such field strengths) and beyond this region the diffusion coefficients were determined. The oscillation in the plot is most pronounced at 2.45 GHz (and 10 GHz), but appears to be damped at long times. In determining diffusion coefficients, care was taken to ensure that calculations incorporated a whole number of damped oscillations. Similarly, Fig. 5 illustrates the effect of r.m.s. field intensity on molecular mobility for both ionic liquids studied.

2tD = 1 3 r i (t) -r i (0) 2 r i (t) -r i (0) 2 r i (t)
As will be seen later, only field strengths above 0.05 V/Å r.m.s. witnessed an appreciable increase in diffusion within the time-scales of these simulations.

Self-diffusion coefficients, calculated from eqn. 11, are shown in Fig. 6 and reported in Tables 1 and2. Here, these are plotted against the applied e/m field frequency for 0.1 and 0.25 V/Å r.m.s. (Fig. 5a), while those for [bmim][PF 6 ] are shown in (Fig. 6b). Also shown in these graphs are the self-diffusion coefficients for cations and anions under zero-field conditions. This highlights a number of results. Firstly, the imposition of e/m-fields results in self-diffusion coefficients for both cation and anion that are ~15 to ~50 times greater than the zero-field case (0.1 and 0.25 V/Å r.m.s. , respectively), a dramatic translational response vis-à-vis induced dipolar rotational motion. This is consistent with eqns. 7 and 11, where increased mobility results from reduces by 14% (similar for for [PF 6 ] -) over the same frequency range. This is consistent with the results shown in Figure 2, where the greatest dipole alignment was observed for lower e/m field frequencies, i.e., those frequencies for which significant translational and, to a lesser extent, rotational effects occur. Finally, it should be noted that the difference in mobility between anions and cations under similar e/m field conditions is much lower than is the case for zero-field conditions, most likely a result of the disruption of the underlying ionic liquid hydrogen bonding network structure allowing much more free ion transport, i.e., a lower de facto viscosity (cf. eqns. 5-7).

The effect of r.m.s. field intensity on diffusion is shown in temperatures at which studies were conducted (namely 450K) ensure that there was little difference in computed conductivities between runs (<5%). Here, the conductivity results for 20 and 50 ps CF durations were equivalent, with optimal sampling statistics for the 20 ps CF's (i.e., 250:1). Averaged over five independent simulation s of 5ns each, the (Green-Kubo) electrical conductivities were found to be around 1. Sm -1 . However, if one tentatively extrapolates this to 450K, a conductivity of 20 Sm - 1 is suggested. Taken together, it is reasonable to assume that the conductivity at 450K is closer to a value of 20 Sm -1 (even though it is obviously beyond the limits of the applicability). In this case, as we have seen for diffusion coefficients, the best agreement with experiment is achieved using the force-field of Bhargava et al. 15 The TTCF results may also be seen to be essentially equivalent to the Green-Kubo results. This confirms the utility for use of the TTCF method for gauging properties in external fields of arbitrary intensity, 2724 which is especially true for the relatively weak field used in this study (in comparison to typical force magnitudes acting in zero-field situations due to the force-field). Table 1: Calculated self-diffusion co-efficient for [dmim] + and [PF 6 ] -under zero and e/m field conditions. In order to make your re-review easier, I have reiterated each of the comments made by the reviewer and followed these by describing the changes applied to the paper. These are enumerated as follows:

Conclusions

1. "I was surprised to see that the authors chose to define the thermal motion with respect to the overall center-of-mass velocity. While the authors are correct in stating that there is no net mass flow, the electric field will still induce the flow of cations and anions and this cannot be counted as thermal motion. The thermostat used by the authors artificially decreases the current and hence the conductivity since the current itself is identified as thermal motion by the thermostat. I would suggest to the authors to run a few tests using a thermostat free of this kind of assumption such as e.g. the configurational thermostat of Braga and Travis (JCP 123, 134101 -2005) . We have added this to the Methodology section with a short discussion of why this would be a superior thermostat method. We have also added the reference and renumbered all of them accordingly. It was found that the results for the electrical conductivity are up to 15% larger than previously, but little appreciable change was observed for the dipolar orientation or diffusion results. Therefore, Fig. 8 has been replaced accordingly (vide infra).

2. "The definition of the overall dipole moment Ps over a set of Ns ions (Equation 3) is very confusing. Equation 3 can only be applied to ions with a non-zero dipole moment (it does not apply to the ion [PF6]considered in this work for instance) and the choice for the overall origin is not specified. Only the first non-zero moment of an electrostatic distribution is independent from the choice of an origin. For a set of ions, the overall dipole moment will depend on the choice of the origin. This needs to be addressed."

We thank the referee for this insightful comment. We chose this definition because the lack of local dipole moment for [PF6] -ions (according to our definition, w.r.t. the ions' individual centres of mass) leads to a purely translational (i.e., Ohmic) response in motion for each of these ions to the applied electric fields -thus our definition, which casts the [PF6] -ions as non-dipolar, is entirely appropriate. The definition of the origin is a purely local one for each ion, and coincides with each ion's center of mass, to determine each ion's dipole vector. There is no central definition of an origin for the entire system. Remarks to this effect have been added to the paper.

"The authors state that the z-component of Ps is indicative of alignment. However, in the definition of

the proportional alignment of a species (Equation 4), the authors use the norm of Ps and not its zcomponent. This also needs to be addressed."

Thank you for pointing that out. This is purely confusion in terms of symbols, and we apologise for this. We have specified the z-component, and have stated that the norm brackets refer here to the absolute value of the z-component.

4. "There are many typos such as 'Reyonld's number' -> 'Reynolds' number' (page 7) or 'simulation sof' -> 'simulations of' (page 15)."

Typos have been corrected and document re-read to eliminate others (where they might exist).

I trust that the above is to your satisfaction. Please do get back to me if there is anything else you require in advancing this paper.

With great respect

Damian A. Mooney

Senior Lecturer (US Associate Professor)

  response, such as electrical conductivity and the variation of the de facto self-diffusivity with frequency, as well as the comparative field response of anions and cations, with simulations conducted over wide range of frequencies (2.45 GHz to 200 GHz) and e/m field intensities (10 -3 to 0.25 V/Å r.m.s. ).

  statistical sampling. These simulations were also of sufficient duration

  the individual dipoles (defined with respect to their centers-ofmass) (3) and the z-component of P s vis-à-vis the overall magnitude taken as indicative of alignment. This definition was chosen because of the lack of dipole moment for [PF 6 ] -(as a result of individual dipole moments defined according to the ions' centre

  then laminar flow conditions prevail, and one may invoke Stoke's' law for the frictional force, 3330 i.e.

  response vis-à-vis translational motion, the z-component dipole alignment, α, from eqn.4. Fig.1illustrates the relative extent and periodicity of dipole z-component alignment for both [dmim] + and [bmim] + , respectively, under the same e/m field intensity, but with varying field frequency. Frequencies of 10, 50 and 200 GHz correspond to periods of 100, 20 and 5 ps, respectively, with this periodicity evident from this figure. Dipolar lag-times are of the order of 3-5 ps for very limited alignment. The dipole moments of [dmim] + and [bmim] + are centered narrowly about 0.87 D (±0.012 D) and 6.17 D (±0.043 D);8, 9 the dipole moment of [bmim] + from the force-field of Bhargava et al.15 was calculated to be only marginally lower than that of Lopes et al.10 (5.84 ± 0.038 D). The difference in dipole moments between[dmim] + and [bmim] + is apparent from Fig.1in terms of the absolute magnitude of the z-component alignment. In order to quantify the extent of alignment, the maximal α, given by equation (4) above, for [dmim]+ and [bmim] + at two field intensities (0.1 and 0.25 V/Å r.m.s. ) is shown in Fig. 2. For both [dmim] + and

Figure 4

 4 Figure 4 shows plots of the MSD of the centre-of-mass of anions and cations in (a) [dmim][PF 6 ] and (b) [bmim][PF6] ionic liquids at 2.45, 10 and 100 GHz for a field intensity of 0.1 V/Å r.m.s . It must be remembered that these de facto molecular mobilities are in the absence of any temperature rise and any effects are athermal in nature. As can be observed from Figure 4, diffusion is Fickian beyond ~200 ps

Fig 7 ,

 7 at a single field frequency of 2.45GHz. This figure shows that the (dramatic) onset of e/m field effects (at the frequency at which these ionic liquids are most susceptible to their action) takes place between 0.05 and 0.1 V/ Å r.m.s. .; a range of r.m.s. intensities that corresponds to the field electric force of 5-10% of the cohesive inter-molecular forces present in these systems.In order to probe direct translational response further, time-evolution of the conductivity integrals via the Green-Kubo and TTCF approaches (cf. eqns. 9 and 10) is shown in Fig.8. . In line with the work of Delhommelle et al.2724 , we have carried out five non-equilibrium TTCF runs for each systems studied, taking each starting point from a long equilibrium trajectory, each 0.5 ns apart. The relatively high

  Non-equilibrium MD simulation of [dmim][PF 6 ] and [bmim][PF 6 ] ionic liquid systems have been conducted in the NNVT case, as a means of assessing and understanding the non-thermal effects of electric and e/m fields (in the microwave to frequency) primarily on translational response, in the guise of molecular mobility and electrical conductivity. Results indicate that these ionic liquids responded most significantly to field frequencies of 2.45 GHz to 10 GHz for all measured properties including self-diffusion coefficients, and, less importantly, dipole alignment. This is in agreement with dielectric relaxation spectroscopy,3229 and confirms Ohmic loss as the dielectric relaxation mechanism. The use of the TTCF method to allow the more statistically robust estimation of the electrical conductivity at lower field intensities with a poor signal-to-noise ratio rather than by direct computation of ( ) 0 t VE J leads to electrical conductivity results essentially in agreement with a Green-Kubo treatment. This work suggests that (non-equilibrium) classical molecular simulation offers a viable and potentially valuable way of assessing optimal e/m field design strategies for ionic liquid exposure in terms of solvent recovery, but these techniques could be applied to other exciting new areas, such as e/m field promotion of reaction rates.

Figure 1 :

 1 Figure 1: z-component alignment total dipole (in Debyes) of (a) [dmim] + and (b) [bmim] + under different field frequencies, namely 10, 50 and 200 GHz corresponding to periods of 100, 20 and 5 ps, respectively for r.m.s. field intensity of 0.25 V/Å r.m.s. . For (a), 10 GHz and 200 GHz plots were offset by -100 and +100 D, respectively, for clarity, while for (b) 10 GHz and 200 GHz plots were offset by -1200 and +1200 D.

Figure 2 :

 2 Figure 2: Maximal z-component alignment of (a) [dmim] + in [dmim][PF 6 ] and (b) [bmim] + in [bmim][PF 6 ] as a proportion of maximum possible alignment (see eqn. 6) for neat ionic liquid systems. The calculated dipole moments for isolated molecules using the force-fields in this study were 0.87 D and 6.71 D for [dmim] + and [bmim] + , respectively.

Figure 3 :

 3 Figure 3: Mean squared displacement under zero-field conditions for all model ionic liquid systems studied in this work (anion and cation), including [dmim] + , [bmim] + and corresponding [PF 6 ] -species. (a) [dmim] + and [PF 6 ] -ions in [dmim][PF 6 ] and (b) [bmim] + and [PF 6 ] -ions in [bmim][PF 6 ]. Graphs for the [bmim][PF 6 ] are shown for both the force-field of Lopes et al.10 and of Bhagarva et al.15 .

Figure 4 :

 4 Figure 4: Mean squared displacements of (a) [dmim] + and [PF 6 ] -ions in [dmim][PF 6 ] and (b) [bmim] + and [PF 6 ] -ions in [bmim][PF 6 ] at 0.1 V/ Å r.m.s. at three different field frequencies (2.45, 10 and 100 GHz).

Figure 5 : 23 Figure 6 :

 5236 Figure 5: Mean squared displacements of (a) [dmim] + and [PF 6 ] -ions in [dmim][PF 6 ] and (b) [bmim] + and [PF 6 ] -ions in [bmim][PF 6 ] at 2.45GHz at three different r.m.s. field intensities (0.05, 0.10 and 0.25 V/ Å r.m.s. ).

Figure 7 :

 7 Figure 7: Self-diffusion coefficient of (a) [dmim] + and [PF 6 ] -ions in [dmim][PF 6 ] and [bmim] + and [PF 6 ] -ions in [bmim][PF 6 ] as a function of r.m.s. field intensities (from 0.001 to 0.25 V/ Å r.m.s. ) at a field frequency of 2.45 GHz.

Figure 8 :

 8 Figure 8: Development of the electrical conductivity, i.e., time-evolution of eqns. 9 and 10, for (a) zero-field conditions (Green-Kubo) and (b) static-field (0.001 V/Å) conditions (TTCF). Results are shown for a typical 5ns simulation.

  the force-field of Lopes et al10 for which we expect no qualitative differences in behavior under the external fields applied in this work.

			5
	we have utilized All simulations were conducted for 216 ion pairs, which for [PF 6 ] -salts of [dmim] +
	and [bmim] + systems corresponded to 4968 and 6912 ions respectively. van der
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	6
	simulations at a single temperature. This temperature was chosen to be 450 K to
	ensure that reasonable sampling could be achieved within feasible simulation times of
	up to 5 ns. In addition, the choice of the cations and anion used in this work was based
	on the availability of well-tuned force-field models. Our goal is not an exhaustive
	study of electric and e/m field response over a wide range of temperatures and F different types of IL cation and anion pairs, but rather a study of response of particular o r ILs to electric fields and to a spectrum of e/m field frequencies and intensities which
	may provide an insight into the behavior of a broader range of ionic liquid types. For P example, a worthwhile study could consider the new ionic liquid cations and/or e r e anions, including 'FAP' (tris(perfluoroalkyl)trifluorophosphate). 221919
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	7
	acting on them due to interactions with all other ions (determined from comparison of
	qE with center-of-mass forces in zero-field simulations) for the lowest field intensity
	applied (0.001 V/Å r.m.s.
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  In this study, the motivation is to study non-thermal field effects on systems in the linear-response régime close to experimentally and industrially available intensities: for 0.001 V/Å r.m.s. , a direct

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	estimate of σ from	J	( ) t VE	0	is highly unreliable due to a weak signal-to-noise
	ratio, which has also been found in previous attempts to estimate electrical
	conductivities directly at similar ranges of field intensity. 3532	Field Code Changed

  7 Sm -1 for [bmim][PF 6 ] and 1.9 Sm -1 for [dmim][PF 6 ], for the Lopes et al. force-field, 10 while a conductivity of ~5.7 Sm -1 , is observed for [bmim][PF 6 ] using the force-field of Bhargava et al. 15 While experimental values for conductivities at 450K are not available, it is worthwhile to present available data for [bmim][PF 6 ] which is presented by Suarez et al. 3835 Here, a correlation for conductivity is present for temperatures up to 353K. At 353K, the conductivity of [bmim][PF 6 ] equals to 2.6

Table 2 :

 2 Calculated self-diffusion co-efficient for [bmim] + and [PF 6 ] -under zero and e/m field conditions. Results (under zero-field conditions) are also reported from simulations using the force-field of Bhargava et al.15 Results of Bhargava et al. force-field15 shown in brackets.

				20
	Table 1
	Conditions	[dmim] + × 10 10 m 2 s -1	[PF 6 ] -× 10 10 m 2 s -1
	Zero-Field	0.81		0.51
		0.1 V/A
	2.45 GHz	14.50		11.02
	10 GHz 20 GHz 50 GHz 100 GHz 200 GHz	18.12 16.11 F 14.11 9.39 o 4.81 r 0.25 V/Å r.m.s.	12.05 12.04 7.86 6.46 4.81
	2.45 GHz	56.90		53.21
	10 GHz 20 GHz 50 GHz 100 GHz	61.16 61.86 48.03 49.62	43.27 45.25 P 49.41 e 49.85
	200 GHz	48.70		r e 38.60
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  , which has been recently applied to the calculation of conductivity within the transient-time correlation function formalism by Desgranges and Delhommelle (PRE 77, 027701 -2008)"We implemented the configurational (canonical-ensemble) thermostat of Braga and Travis for holonomic constraints, [K.P. Travis KP and C. Braga, J. Chem. Phys., 128, 014111 (2008)]

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Acknowledgements

N.E. acknowledges very fruitful discussions of the TTCF method with Dr. Billy Todd. The authors also thank the Irish Centre for High End Computation for use of their facilities.

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E Bosc

L Morin

M Pressigout

A content based metho

Keywords: 3DTV, compression, multi-view video plus depth 3D

Multi-view Video plus Depth (MVD) data refer to a set of conventional color video sequences and an associated set of depth video sequences, all acquired at slightly different viewpoints. This huge amount of data necessitates a reliable compression method. However, there is no standardized compression method for MVD sequences. H.264/MVC compression method, which was standardized for Multi many adaptations to MVD. However, it has been shown that MVC is

We propose a novel option as for compression of MVD data. consistency. The originality of the proposed method relies on the use of the decoded color data a associated depth compression. This is meant to motivated by previous studies of artifacts occurring in synthesized views: most annoying distortions are located strong depth discontinuities and these distortions are due to misalignment of depth and color edges in decoded images. Thus the method is meant to preserve edges and to ensure consistent localization of color edges and depth edges.

To ensure compatibility, colored sequences are encoded with H.264. Depth maps compression image codec, namely LAR (Locally adapted Resolution). It consists in a quad quad-tree representation contributes in the p The adopted strategy is meant to be more perceptually driven than state compared to H.264 encoding of depth images. O method, and visual quality of synthesized views is improved with the proposed approach.

INTRODUCTION

, such as 3D Television (3DTV) or Free Viewpoint Television (FTV), require the use of several conventional video sequences to ensure depth sensation, or to offer novel views of a scene. For these purposes, the use of color and geometry information of the scene is the key. MVD data refer to a specific representation of an observed scene and meet this need. They consist in a set of conventional color video sequences and an associated set of depth video sequences, all acquired at slightly different viewpoints. issue refers to the need for an efficient MVD compression method, considering the huge amount of data to be Up to now, there is no standardized compression method for MVD sequences. Most of the proposed compression methods rely on the extension of state-of-the-art 2D codecs. The most popular is H264/AVC View-Video representation, MVV), namely H.264/MVC for Multi , has been the subject of many adaptations for MVD compression [START_REF] Merkle | Efficient prediction structures for multiview video coding[END_REF] . However, the exploitations of the spatial of data turn out to be insufficient in particular cases. For instance, observed that in case of large disparity between the different views of multi-view sequences, the predictions structures n an improved coding efficiency.

A second issue refers to the synthesis of novel views from decoded data. New intermediate viewpoints can be generated through Depth-Image-Based-Rendering [START_REF] Fehn | Depth-image-based rendering (DIBR), compression and transmission for a new approach on 3D-TV[END_REF] (DIBR) methods. Previous studies already pointed

joint color/depth

Coesmes, 35708 Rennes view Video plus Depth (MVD) data refer to a set of conventional color video sequences and an associated set of depth video sequences, all acquired at slightly different viewpoints. This huge amount of data necessitates a reliable However, there is no standardized compression method for MVD sequences. H.264/MVC Video representation (MVV), has been the subject of not well adapted to encode multi-view depth data.

ts main purpose is to preserve joint color/depth The originality of the proposed method relies on the use of the decoded color data as a prior for the consistency in both types of data after decoding. Our strategy is motivated by previous studies of artifacts occurring in synthesized views: most annoying distortions are located around strong depth discontinuities and these distortions are due to misalignment of depth and color edges in decoded images. Thus the method is meant to preserve edges and to ensure consistent localization of color edges and depth edges.

bility, colored sequences are encoded with H.264. Depth maps compression is based on a 2D still tree representation of the images. The reservation of edges in both color and depth data. art methods. The proposed approach is .264 and with the proposed iewpoint Television (FTV), require the use of several iews of a scene. For these purposes, the use of data refer to a specific representation of an observed scene associated set of depth video MVD compression method, considering the huge amount of data to be ression method for MVD sequences. Most of the proposed art 2D codecs. The most popular is H264/AVC 3 whose 3D Video representation, MVV), namely H.264/MVC for Multi-view Video the exploitations of the spatial sufficient in particular cases. For instance, Merkle et al. [START_REF] Merkle | Efficient prediction structures for multiview video coding[END_REF] view sequences, the predictions structures ew intermediate viewpoints can be generated Previous studies already pointed out the impact of depth encoding on the synthesized frames. Compression-related artifacts that may be imperceptible in depth maps cause important distortion during the synthesis process [START_REF] Merkle | The effects of multiview depth video compression on multiview rendering[END_REF] .

Many methods have been proposed recently in order to address the aforementioned issues. Various encoding strategies are possible to achieve depth map compression. Several studies have proposed bit-rate-control methods [START_REF] Morvan | Joint depth/texture bit-allocation for multi-view video compression[END_REF][START_REF] Daribo | Motion Vector Sharing and Bit-rate Allocation for 3D Videoplus-Depth Coding[END_REF] , relying on the objective quality of the resulting synthesized views, or on a distortion model [START_REF] Liu | Joint video/depth rate allocation for 3D video coding based on view synthesis distortion model[END_REF] . A popular and efficient strategy is the post-processing of depth maps after decoding [START_REF] De Silva | A Depth Map Post-Processing Framework for 3D-TV Systems based on Compression Artifact Analysis[END_REF] . Depth-adapted encoding methods [START_REF] Morvan | Platelet-based coding of depth maps for the transmission of multiview images[END_REF][START_REF] Graziosi | Compressing depth maps using multiscale recurrent pattern image coding[END_REF][START_REF] Sarkis | Depth map compression via compressed sensing[END_REF] have also been proposed. Section 0 gives a review of these methods. Our work is in line with the depth-adapted encoding strategy since the method proposed in this paper relies on a content-based representation of the depth map.

The main purpose of this novel framework is to preserve the consistency between color and depth data. Our strategy is motivated by previous studies [START_REF] Merkle | The effects of multiview depth video compression on multiview rendering[END_REF] of artifacts occurring in synthesized views: most annoying distortions are located around strong depth discontinuities [START_REF] Bosc | Towards a new quality metric for 3D synthesized view assessment[END_REF] and these distortions are due to misalignment of depth and color edges in decoded images. Thus the method is meant to preserve edges and to ensure consistent localization of color edges and depth edges. It is based on a 2D still image codec, namely LAR [START_REF] Pasteau | Locally Adaptive Resolution (LAR) codec[END_REF] (Locally adapted Resolution). The LAR codec is based on a quad-tree representation of the images. In this quad-tree, the smaller the blocks, the higher the probability of the presence of a depth discontinuity. Analogously, big blocks correspond to smooth areas. The quad-tree representation contributes in the preservation of depth transitions when target bit-rate decreases. Another original contribution of the proposed method relies on the use of the decoded color data as an anchor for the enhancement of the associated decoded depth, together with information provided by the quad-tree structure. This is meant to ensure consistency in both types of data after decoding.

This paper is organized as follows: Section 2 introduces the compression issues for MVD data. Section 3 presents the proposed method. Section 4 defines the experimental protocol and gives the results. Finally Section 5 concludes the paper.

COMPRESSION OF MVD SEQUENCES AND QUALITY OF SYNTHESIZED VIEWS

This section presents the main issues related to MVD compression and a review of the proposed methods addressing these problems in the literature.

Most of the proposed compression methods for MVD data rely on the extension of state-of-the-art 2D codecs. Sets of color and depth sequences can be separately encoded through existing 2D methods. This is an evident encoding strategy because depth maps, being monochromatic signals, are considered as conventional sequences. However, depth maps are not natural images. They provide structural information of the scene: large and smooth regions often belong to the same depth plane. The closer the depth plane is from the acquiring camera, the lighter the region. This leads to smooth areas with sharp edges. The edges correspond to depth transitions.

Previous studies [START_REF] Merkle | Multi-view video plus depth representation and coding[END_REF][START_REF] Bosc | Bit-rate allocation for multi-view video plus depth[END_REF][START_REF] Vetro | Towards a 3D video format for auto-stereoscopic displays[END_REF] have shown that coding artifacts on depth data can dramatically influence the quality of the synthesized view. Particularly, the sharp edges of the depth maps are prone to synthesis errors even when depth maps are uncompressed. As pointed out in a recent study [START_REF] Bosc | Towards a new quality metric for 3D synthesized view assessment[END_REF] , the synthesis process, with DIBR methods, induces specific artifacts located around the edges of objects. These errors are notably due to depth map inaccuracy, numerical rounding, hole filling method in DIBR, or both. Consequently, errors occurring in these specific critical areas of the depth maps are enhanced by coarse compression. The impacts of depth compression on visual quality of synthesized views can be explained by the fact that 2D codecs are optimized for human visual perception of color images. Thus, artifacts, that may be imperceptible when visualizing the depth map, produce distortions because during the synthesis, the warping process relies on wrong depth values. The impacts of depth compression was observed in different studies [START_REF] Merkle | The effects of multiview depth video compression on multiview rendering[END_REF][START_REF] Bosc | Bit-rate allocation for multi-view video plus depth[END_REF] .

Consequently, efforts have been directed in order to propose depth compression methods more adapted to the special features of depth maps. Morvan et al. [START_REF] Morvan | Platelet-based coding of depth maps for the transmission of multiview images[END_REF] proposed to represent the depth map thanks to platelets (piecewise linear functions). The depth map is first divided through quad-tree decomposition and each block is approximated by a platelet. The platelet-based compression outperformed JPEG2000 in the study. An additional interesting comparison would be that against H.264/AVC. Moreover, in this study, the gain is evaluated with respect to the depth distortion (in PSNR). This protocol of validation is questionable because since the artifacts in the two compared methods are different, their impact on the synthesis may also be different. Yet, the quality of the synthesized views generated from the decoded depth maps is not presented. Graziosi et al. [START_REF] Graziosi | Compressing depth maps using multiscale recurrent pattern image coding[END_REF] also proposed a block partioning method associated to a least-square prediction for depth map compression. In this method, the validation is also achieved by comparing the depth map distortion from different compression scheme (JPEG2000 and H.264 intra). The method includes the use of a dictionary, containing concatenations of scaled versions of previously encoded image blocks. Sarkis et al. [START_REF] Sarkis | Depth map compression via compressed sensing[END_REF] proposed a depth compression method based on a subsampling in the frequency domain followed by a reconstruction using a non-linear conjugate gradient minimization scheme. The method also meant to preserve the particular features of the depth map. The method outperformed JPEG and JPEG2000.

As Morvan et al. and Graziosi et al., our method relies on a block partioning. Contrary to the aforementioned methods, we choose to evaluate the performances of our method against H.264, whose artifacts in the depth map induce less annoying distortions than JPEG2000 (Gibbs effect in the depth map induce disastrous distortions in the synthesized view). The quantization used in our method is also different since we choose to modify the block partioning of the depth map according to the target bit-rate. The next section will present our proposed method that is meant to preserve object edges to maximize the synthesized view quality.

PROPOSED METHOD

Since color and depth sequences can be encoded through two different schemes, the proposed method enables the use of any compression method for the color sequences. In our case, we use H.264 for color sequence encoding since it has proved its efficiency for conventional color media compression. Moreover, encoding color with a standard codec enables backward compatibility with classical 2D video. This section presents the depth map encoding strategy.

Depth map encoding method

To address the first constraint regarding the preservation of the depth map edges, a content-based representation and encoding is required. We choose to base our method on the LAR method because the quad-tree representation of this method matches the characteristics of the depth maps. The LAR method is based on the assumption that an image can be considered as the combination of two components: the global information and the details, which are respectively the flat image and the local texture. The flat image and the local texture both rely on the same quad-tree representation. Each pixel in the flat image is assigned the mean value of the pixels of the block it belongs to. Each pixel in the local texture is then assigned the compensated error. Figure 1 depicts this principle.

Since depth maps do not contain high frequency areas, the local texture (that is to say the details) is not essential and represents an avoidable additional cost of compression. Thus, only the flat image is considered and encoded in the method we propose.

Quad-tree decomposition

The quad-tree decomposition is dependent on the local gradient of the depth image. Given a threshold ܻ for the local gradient, the image is split into blocks: the higher the local activity, the more splits. This leads to small blocks around object edges and bigger ones in continuous areas.

+ =

Flat image (low resolution) Texture image (details) Original Image We denote ܲ ሾே ೌೣ …ே ሿ the quad-tree partition with ܰ ௫ and ܰ the maximal and minimal allowed block sizes, expressed as power of 2. Let ܫ be an image and ,ݔሺܫ ݕሻ a pixel of ܫ with coordinatesሺ,ݔ ݕሻ. The block ܾ ே ሺ݅, ݆ሻ in ܫ is noted ܫሺܾ ே ሺ݅, ݆ሻሻ, expressed as:

ܾ ே ሺ݅, ݆ሻ ൌ ൛ሺ,ݔ ݕሻ א ܰ ௫ ൈ ܰ ௬ | ܰ ൈ ݅ ݔ ൏ ܰ ൈ ሺ݅ 1ሻ, ܰ ൈ ݆ ݕ ൏ ܰ ൈ ሺ݆ 1ሻ, ܰ א ሾܰ … ܰ ௫ ሿൟ (1)
As explained before, the quad-tree partition ܲ ሾே ೌೣ …ே ሿ relies on the analysis of the local gradient. Then, the decomposition can be expressed as:

 ܫ൫ܾ ே ሺ݅, ݆ሻ൯, ܰ ൌ ە ۖ ۔ ۖ ۓ ܰ ᇱ א ሾܰ ௫ … ܰ ሾ݂݅ max ሺܫ ቀܾ ே ᇲ ሺ݅, ݆ሻቁ െ min ሺܫሺܾ ே ᇲ ሺ݅, ݆ሻሻ ܻ, ܽ݊݀ ݂݅ ሺ݇, ݉ሻ א ሼ0,1ሽ ଶ |max ሺܫ ൭ܾ ே ᇲ ଶ ሺ݅ ݇, ݆ ݉ሻ൱ െ min ሺܫሺܾ ே ᇲ ଶ ሺ݅ ݇, ݆ ݉ሻሻ ܻ ܰ ݁ݏ݅ݓݎ݄݁ݐ (2)
The value of the threshold ܻ strongly influences the final representation of the image. Figure 2 gives an example of quad-tree decomposition for the first frame of Book Arrival sequence.

Compression scheme

The compression scheme in the LAR method is based on a pyramidal decomposition [START_REF] Babel | Interleaved S+ P pyramidal decomposition with refined prediction model[END_REF] . The pyramid, built from ,ܫ consists of a set of images, noted as ሼܮ ሽ ୀ ୀ ೌೣ , as a multi-resolution representation of the image, where ݈ ௫ is the top of the pyramid and ݈ ൌ 0 is the lowest level, i.e. the full resolution image. At each level, the image is expressed by:

ቐ ݈ ൌ 0, ܮ ሺ,ݔ ݕሻ ൌ ,ݔሺܫ ݕሻ ݈ 0, ܮ ሺ,ݔ ݕሻ ൌ ܮ ିଵ ሺ2,ݔ 2ݕሻ ܮ ିଵ ሺ2ݔ ݕ2,1 1ሻ 2 (3)
The LAR method allows the prediction of each level of the pyramid, from top to bottom. For each level, the associated image of errors, also relying on the quad-tree decomposition, can be transmitted to compensate the prediction errors. At the decoder side, from the top to the bottom, the image is reconstructed.

Compression cost is mainly due to the encoding of small blocks. This is why in our proposed method small blocks are not transmitted (those are blocks whose size is such as ܰ ൌ ܰ ). This is achievable thanks to the pyramidal decomposition. The encoding of small blocks is related to the image of errors corresponding to the lowest level, i.e. ܮ . The lowest level is not encoded in the method we propose, and the image will be refined at the decoder side thanks to the analysis of the values of the nearest neighbor blocks whose size is such as ܰ ܰ : they will be predicted, depending on the values of their closest larger blocks. This allows bit-rate savings. The pseudo code of this prediction is given in Figure 3.

Depth map Quad-tree with Y=8

Rate control in depth map

Pasteau et al. [START_REF] Pasteau | Locally Adaptive Resolution (LAR) codec[END_REF] suggested applying a quantization step depending on the block sizes, in the case of conventional images.

Our experiments revealed that in the case of depth map compression, this was not an adequate strategy because the smaller the blocks, the coarser was the quantization (this allowed bit rate savings because small block are costly). Yet, small blocks correspond to strong depth discontinuities and errors occurring in these areas may have disastrous effect at the synthesis step. Figure 4 shows the impact of the quantization as suggested in Pasteau et al. [START_REF] Pasteau | Locally Adaptive Resolution (LAR) codec[END_REF] (first column) at 0.06 bpp. Depth transitions are highly degraded and will result in errors in the synthesized frame (third column, crumbling artifacts around the head and around the legs of the chair). The synthesized frames obtained in Figure 4 are generated from original color data and decoded depth maps in order to visually assess only the impact of depth quantization (i.e. not the combined effect of both color and depth compression).

Prediction of lowest level of the pyramidal decomposition

ܮ ෨ is the estimated representation of the image at the decoder side, for level ݈. ܲ ሾே ೌೣ …ே ሿ is the quad-tree partition.

For ݈=݈ ௫ … ݈ ଵ Estimate ܮ ෨ as in the LAR method End for For each block of P such as

ܰ ൌ ܰ ௫ … 2 כ ܰ Given ܲ ሾே ೌೣ …ே ሿ , then ܮ ෨ ሺܾ ே ሺ݅, ݆ሻሻ ൌ ܮ ෨ ଵ ሺቒ ଶ ቓ , ቒ ଶ ቓሻ

End For

For each block of ܲ such as ܰ ൌ ܰ ܮ ෨ ൫ܾ ே ሺ݅, ݆ሻ൯ ൌ Mean value of the closest block ܾ ே of ܲ such as ܰ ܰ End For Thus, we propose a quantization achieved through the evolution of the quad-tree representation of the image. Small blocks are costly and a way to reduce the bit-rate is to reduce the number of small blocks. This implies that the quad-tree representation can change according to the target bit-rate. The number of small blocks is directly related to the value of the threshold ܻ. Thus, an increasing threshold ܻ decreases the bit-rate, so that the representation of the image contains larger blocks. This corresponds to a spatial quantization that affects the values of the depth. It results in assigning the same depth to object that were not formerly in the same depth plane. The dynamic range of the depth is reduced but the global structure is preserved. Figure 4 shows that the proposed method (second column) renders sharp depth transitions. The synthesized frame in Figure 4, fourth column, shows improvements compared to the previous strategy, third column. Figure 5 gives the quad-tree representations and the resulting depth maps using two different thresholds for the quad-tree decomposition. It shows that the semantic information of the image is preserved.

Decoded depth when applying our proposed strategy.

Decoded depth when applying quantization suggested in Pasteau et al.

Frame synthesized from decoded depth when applying our proposed strategy.

Frame synthesized from decoded depth when applying quantization suggested in Pasteau et al. In this study, we empirically determined a model allowing the choice of ܻ depending on the target bit-rate. Based on the analysis of the synthesized view quality scores obtained for various values of ܻ, and according to the corresponding bitrate ܴ of the encoded frames, we opted to an exponential model such as:

ܻ ൌ ܽ݁ ோ
(4) ܴ is the target bit-rate, ܽ and ܾ are two constants. Previous experiments showed that ܽ ൌ 30 and ܾ ൌ െ12 gave good results for the tested sequence. Note that these values will differ from one MVD sequence to another, since the representation of the quad-tree depends on the structure of the sequence, thus ܻ model depends on the depth structure of the sequence.

Depth reconstruction at decoder side

To address the second constraint regarding the consistency between color and depth edges, reconstruction step is included at the depth map decoder side, right after the first estimation of the smallest blocks, as explained in Section 3.1.2. The additional step described in this section can be considered as a second pass of the depth reconstruction. It consists of a multi-lateral filtering aided by the quad-tree representation whose principles are partially based on the description of Lai et al. method [START_REF] Lai | Depth map processing with iterative joint multilteral filtering[END_REF] . In our proposed method, the decoded associated color image is used to enhance only the blocks smaller or equal to 4x4 in the depth map. Small blocks are likely to be located around depth discontinuities. Thus, it is believed that improving the accuracy in these regions, according to the decoded associated color, will ensure consistency between color and depth edges. Let ܥ ሚ be the decoded associated color image, and ܮ ෨ the lowest level image of the depth pyramidal decomposition. Let Ω be the set of pixels such as:

Ω ൌ ܮ ෨ ሺ,ݔ ݕሻหܮ ෨ ሺ,ݔ ݕሻ א ܮ ෨ ൫ܾ ே ሺ݅, ݆ሻ൯, ܰ א ሾܰ … 4ሿ (5)
The reconstruction, noted ܮ ෨ ሺ,ݔ ݕሻ, of any pixel belonging to Ω is expressed as: Figure 6 gives the overview of the proposed method. In this figure, at the encoding step, black blocks correspond to non transmitted blocks.

 ܮ ෨ ሺ,ݔ ݕሻ א Ω, ܮ ෨ ሺ,ݔ ݕሻ ൌ ܮ ෨ ሺሻ ൌ ଵ ∑ ܮ ෨ ሺሻ א ݁ ି ԡషԡ మ ݁ ି ฮಽ ෩ బ ሺሻషಽ ෩ బ ሺሻฮ మ ೞ ݁ ି ԡಽೠೌሺሻషಽೠೌሺሻԡ మ , ( 6
)
ܭ ൌ ∑ ݁ ି ԡషԡ మ ݁ ି ฮಽ ෩ బ ሺሻషಽ ෩ బ ሺሻฮ మ ೞ ݁ ି ԡಽೠೌሺሻషಽೠೌሺሻԡ మ א , (7)

EXPERIMENTS 4.1 Protocol

The proposed method is compared to state-of-the-art codec H.264 in intra mode. As preliminary studies, the experiments concern only still images. First frames of views 6 and 10 from Book Arrival were encoded through both encoding methods. Afterwards, decoded color and depth maps were used to compute the intermediate view 8, through the reference software, VSRS 3.5 [START_REF] Tanimoto | Reference Softwares for Depth Estimation and View Synthesis[END_REF] . Since view 8 is among the originally acquired views, it is considered as a ground truth for the quality assessments. In this paper, the quad-tree decomposition parameters are ܰ ൌ 1 and ܰ ௫ ൌ 12. In Equation ( 4), ܽ ൌ 30 and ܾ ൌ െ12. Finally, in Equation 7, ߪ ௗ ൌ 4, ߪ ௦ ൌ 10, ߪ ൌ 3. The color images are encoded with a QP varying from 0 to 50. Figure 7 gives an overview of the experimental protocol.

Results

For the performance comparisons, a pixel-based metric (PSNR) and a more perception-oriented metric VIF (Visual Information Fidelity 23 ) are considered. Figure 8 depicts the rate-distortion curve obtained by computing the PSNR scores, and the VIF scores of the synthesized views, with respect to the original acquired view. At high bit-rates (higher than 2bpp), the proposed method obtains better PSNR scores. However, under 2bpp, H.264 performs better.

The curve based on VIF scores shows that H.264 and the proposed method give similar results at high bit-rates (higher than 2bpp). However, contrary to the curve based on PSNR, the curve based on the perception-oriented VIF shows that the proposed method performs better at low bit-rates.

A visual appreciation is also useful to evaluate the methods. Figure 9 gives snapshots of the obtained synthesized views for 0.1bpp and 0.9bpp. Ghosting effect is perceptible with both methods behind the head of the man. However, the quadtree based method preserves better the vertical edges: the vertical dark lines of the posters are better rendered with the data encoded with the proposed method. At low bit-rate (0.1bpp), Figure 9 gives snapshots of the synthesized views. Although, PSNR score shows lower performances for the proposed method at low bit-rate, the observation of Figure 9 shows improvements around the edges of the synthesized objects. The ghosting effect around the head of the man is less strong with the proposed method. The crumbling artifacts occurring around the leg of the chair at 0.1bpp with H.264 are no longer perceptible with the proposed method. representation of the images. The quad-tree representation contributes in the preservation of edges in depth data. The originality of the proposed method relies on the proposed quantization method and the use of the decoded color data as an anchor for the associated depth enhancement at the decoder side. The proposed method showed visual performances similar to H.264 at high bit-rates and some improvements at lower bit-rates because it preserves better the object edges. Future work should focus on the use of a more perception-oriented criterion for the quad-tree decomposition. A method to choose automatically the ܻ model, for any sequence, should also be investigated. Finally, the method should be extended to exploit temporal redundancies in the whole sequence.

Figure 1 :

 1 Figure 1: Assumption of LAR method.

Figure 2 :

 2 Figure 2: Quad-tree decomposition (Book Arrival).

Figure 3 :

 3 Figure 3: Pseudo code for Prediction Algorithm of lowest level of the pyramidal decomposition.

Figure 4 :

 4 Figure 4: Comparison of two decoded depth maps at 0.06bpp, using the LAR method or the proposed method of rate control.

bFigure 5 :

 5 Figure 5: Quantization of the depth map.

Figure 6 :

 6 Figure 6: Overview of the proposed method.

Figure 7 :Figure 8 :

 78 Figure 7: Overview of the experimental protocol

Figure 9 :

 9 Figure 9: Snapshots of synthesized views from data encoded with H.264 and from data encoded with the proposed method.

AKNOWLEDGMENTS

This work is partly supported by the French ANR-PERSEE project n° ANR-09-BLAN-0170, and ANR-CAIMAN project n° ANR-08-VERS-002.