Chaima Soussi

Mathieu Aucejo
email: mathieu.aucejo@lecnam.net

Walid Larbi

Jean-François Deü

Numerical analyses of the sound transmission at low frequencies of a calibrated Insulating Glazing Unit

Keywords: Insulating Double Glazing, Transmission Loss, Experimental Modal Analysis, Numerical modeling

The acoustic performances of building elements such as windows are carried out in laboratory according to standards. In addition to the high cost of the experimental tests, the measurements at low frequencies face some difficulties such as the lack of reproducibility, the rooms' modal behaviour effect and the diffuseness of the acoustic field. To separately study each source of deviation from the ideal experimental situation, four numerical configurations, based on experimental conditions, are proposed to characterize the transmission loss (TL) of an Insulation Glazing Unit (IGU) below 500 Hz. The effect of the emitting and receiving rooms with a comparison to the ideal configuration which has a free field on both sides of the IGU are investigated. The numerical model used for the IGU is calibrated from an Experimental Modal Analysis. As expected, TL results show that there is a significant effect of the modal behavior and the acoustic properties of the rooms at low frequencies. Numerical results are compared to those obtained from experimental tests and a good agreement in the all frequency range of interest is observed. In addition, parametric studies are carried out to investigate the influence of the variation of some properties of the studied structure such as the structural damping and the panel's thickness.

Introduction

This research work concerns the vibro-acoustic response of a full scale domestic wooden windows at low frequencies. Since windows are mainly mounted in the exterior building facades, the present study is more particularly focused on the transmission of outdoor airborne noises which are generally emitted from the traffic corresponding to low-pitched noise [START_REF] Mathys | Low-frequency noise and acoustical standards[END_REF]. Due to the complexity of the structure, existing works only consider the simpler part of the system namely the Insulting Glass Unit (IGU).

The sound insulation of double partition units, like double panels, has been widely studied. The earlier analytic predictive models were developed by Beranek and Work [START_REF] Beranek | Sound Transmission through Multiple Structures Containing Flexible Blankets[END_REF] and London [START_REF] London | Transmission of reverberant sound through double walls[END_REF] who studied the sound insulation of infinite plates excited with plane waves. Since then, many other models have been developed to describe various physical phenomena such as the influence of the non-resonant vibration modes and resonances of a cavity [START_REF] Price | Sound transmission through double panels using statistical energy analysis[END_REF], the sound bridges between the two panels [START_REF] Sharp | A study of techniques to increase the sound insulation of building elements[END_REF] and the absorption in the acoustic medium [START_REF] Fahy | Sound and structural vibration: radiation, transmission, and response[END_REF]. However, Hongisto [START_REF] Hongisto | Sound insulation of double panels -Comparison of existing prediction models[END_REF] showed with a detailed study on seventeen models of double panels that there is a significant variation between them, which car reach 20 dB, and no single model was able to well predict the sound performance of such systems.

Practically, the sound insulation of a building element is based on its sound transmission loss (TL). The latter indicator is determined with experimental tests according to standards [START_REF]Acoustics -laboratory measurement of sound insulation of building elements: Part 1: Application rules for specific products; Part 2: Measurement of airborne sound insulation; Part 4: Measurement procedures and requirements; Part 5: Requirements for test facilities and equipment[END_REF]. The tests are carried out in an acoustic laboratory using two finite-sized rooms and one or many loudspeakers to create the sound field excitation with the assumption that the flanking transmission is neglected. The standards assume that the sound field in the two rooms is perfectly diffuse which means that the energy density is the same on all points of the volume [START_REF] Nélisse | Characterization of a diffuse field in a reverberant room[END_REF]. However, no diffuse field condition can be expected below the Schroeder frequency [START_REF] Schroeder | The "Schroeder frequency" revisited[END_REF], since at low frequency range, the few rooms modes lead to standing waves which dominate the acoustic field. Utley [START_REF] Utley | Single leaf transmission loss at low frequencies[END_REF] stated that the difference between the measured transmission loss and the mass low prediction depends on the characteristics of the rooms of the test laboratory rather than the intrinsic panel properties. Inter-comparisons of the transmission loss between ISO laboratories [START_REF] Kihlman | The effects of some laboratory designs and mounting conditions on reduction index measurements[END_REF] and ASTM laboratories [START_REF] Jones | Intercomparisons of laboratory determinations of airborne sound transmission loss[END_REF] evidenced that considerable differences up to 8 dB were found for frequencies below Schroeder frequency due to mounting conditions, frames and apertures.

As reported by many authors [START_REF] Fahy | Sound and structural vibration: radiation, transmission, and response[END_REF][START_REF] Pederson | Measurement of the low-frequency sound insulation of building components[END_REF][START_REF] Prato | Sound Insulation of Building Elements at Low Frequency: A Modal Approach[END_REF], at very low frequencies , theoretical and experimental approaches of the transmission loss through partition between rooms are highly dependent on the test conditions. In other words, it means that the predictive TL is valid only for the specific case under examination. To overcome this difficulty, numerical methods can be considered as a mean of introducing more representative measurement conditions to study the particularities of each case. In addition to the energetic approach such as Statistical Energy Analysis (SEA) or its improved version the Statistical modal Energy distribution Analysis (SmEdA), numerical methods such as the Boundary Element Method (BEM) and the Finite Element Method (FEM) are available to solve the problem of acoustic insulating. The choice of the adequate one depends on the computational cost and the frequency range of interest.

The SEA introduced in [START_REF] Unger | Statistical Energy Analysis of vibrating systems[END_REF][START_REF] Lyon | Statistical Energy Analysis of Dynamical Systems[END_REF] is based on the statistical energy exchanges of diverse subsystems. Due to the fact that underlying assumptions taken in this method which can be unrealistic in some cases, Maxit et Guyader [START_REF] Maxit | Statistical modal energy distribution analysis(smeda)[END_REF] extended it to SmEdA in order to take into account the modal energy distribution. Despite these improvements, the energetic approaches are reliable only at high and medium frequency range with the view to the significant uncertainties due to the few resonant modes in each of the subdomains [START_REF] Tadeu | Acoustic insulation of single panel walls provided by analytical expressions versus the mass law[END_REF]. To this end, an hybrid approach was developped by Shorter and Langley [START_REF] Shorter | Vibro-acoustic analysis of complex systems[END_REF] that combines the finite element method and statistical energy analysis in a single model. This method consists in modeling the deterministic components with the FEM while the statistical components are described in terms of their vibrational energy with SEA [START_REF] Shorter | Vibro-acoustic analysis of complex systems[END_REF]. However, at low frequency range, due to the high modal behavior of the subdomains in the building sector, the FEM and BEM provide a more accurate prediction of the sound transmission loss. Santos et Tadeu [START_REF] Santos | Acoustic insulation provided by a single wall separating two contiguous tunnels via BEM[END_REF] considered that the BEM approach could be a good tool to solve unbounded problems because it automatically satisfies the far field conditions. Sgard et al. [START_REF] Sgard | A numerical model for the low frequency diffuse field sound transmission loss of double-wall sound barriers with elastic porous linings[END_REF] studied the transmission loss through double partition using the FEM for different layers of porous-elastic material coupled to a variational BEM to account for fluid excitation. Only the FEM has been employed by Maluski and Gibbs [START_REF] Maluski | Application of a finite-element model to low-frequency sound insulation in dwellings[END_REF] to investigate the sound insulation of a wall mounted between two rooms and results pointed out a strong dependence between the TL and the modal behavior of both rooms and of the partition. Despite the numerous existed studies which focused on the sound insulation, the dilemma of the sensibility of this acoustic indicator to its environ-ment at low frequencies is to the best of our knowledge still unsolved. That is why ISO 12999-1 standard [START_REF]Acoustics -Determination and application of measurement uncertainties in building acoustics -Part 1: Sound insulation[END_REF] authorises deviations on TL.

Materials and methods

This section exposes the different steps of the proposed modeling strategy.

Sound transmission loss

Basically, the sound transmission loss characterizes the acoustic insulation of a structure and it is defined as:

T L = 10 log 10 Π inc Π rad , (1)
where Π inc and Π rad are the incident and the radiated sound power by the structure, respectively.

In experimental measurements, according to standard [START_REF]Acoustics -laboratory measurement of sound insulation of building elements: Part 1: Application rules for specific products; Part 2: Measurement of airborne sound insulation; Part 4: Measurement procedures and requirements; Part 5: Requirements for test facilities and equipment[END_REF], the acoustic tests are carried out in a laboratory composed of two reverberant rooms separated by a common wall containing an opening in which the test element is mounted. The incident and transmitted sound power are determined indirectly by measuring the average sound pressure levels L e and L r in emitting and receiving room, respectively, and the sound transmission loss is determined as:

T L = L e -L r + 10 log 10 S A , ( 2
)
where S is the area of the test specimen and A is the equivalent sound absorption area in the receiving room.

Numerical configurations of the problem

In the present paper, the Finite Element Method is used to better understand the rooms effect on the sound transmission loss through an Insulation Glazing Unit (IGU) at frequencies below 500 Hz. To this end, three different configurations based on the experimental conditions described in the standards are used to investigate the acoustic performances of IGU: in the first model, the two rooms are designed; in the two following configurations, either the receiving room or the source room is modeled. Results are compared to those obtained with the ideal free fields conditions in the emitting and receiving sides of the baffled IGU. In this last case, the two reverberant enclosures are removed and perfect fields in the two sides of the structure are considered. In total, the next four configurations are used;

• IGU mounted between two reverberant rooms (Room-IGU-Room);

• IGU excited by a diffuse sound field (DSF) and coupled to a receiving room (DSF-IGU-Room);

• IGU coupled to the source room and radiates in a free field (Room-IGU-Freefield);

• IGU mounted in an infinite baffle, excited by a diffuse field and radiates in a semi-infinite domain (DSF-IGU-Free-field).

Insulating Glazing Unit numerical model

The numerical model of the Insulating Glazing Unit used to study its vibro-acoustic response is calibrated from Experimental Modal Analysis. This step is essential to take into account all the required details of the structure and its dynamic behavior especially at low frequencies. To this end, the roving hammer approach is employed.

Results analysis 2.4.1. Theoretical/ Numerical comparison

First, the results of the sound reduction of a double glazing system calculated with the finite element method are compared with those of an infinite system determined with an analytical approach. Several approaches presented in the literature with the problem of sound transmission through double wall systems, among which, the impedance approach described by Heckl [START_REF] Heckl | The tenth sir richard fariry memorial lecture: sound transmission in buildings[END_REF] is chosen.

A characteristic phenomenon of the double-partition system is the massair-mass frequency. At this frequency, the two plates move out of phase and the effect of the cavity on the plates is mostly one of added stiffness [START_REF] Larbi | Vibroacoustic analysis of doublewall sandwich panels with viscoelastic core[END_REF]. This frequency can be calculated approximately by the next expression [START_REF] Heckl | The tenth sir richard fariry memorial lecture: sound transmission in buildings[END_REF]:

f mam ∼ = 1800 m 1 + m 2 d m 1 m 2 , ( 3
)
where d is the depth of the acoustic cavity between the panels in mm, and m 1 and m 2 are the surface masses of the panels in kg/m 2 .

Comparison of the four numerical configurations

To investigate the influence of the rooms, the first three configurations are compared to the forth one. Theoretically, Schroeder has shown that the acoustic response of an acoustic enclosure presents two zones: the modal region and the high modal density region [START_REF] Schroeder | The "Schroeder frequency" revisited[END_REF]. The limits between the two zones is known as the "Schroeder frequency f Sch " and depends on the acoustic absorption [START_REF] Schroeder | The "Schroeder frequency" revisited[END_REF]. For this, inter-comparison of the TL obtained by the four numerical configurations are discussed in two cases: (i) when perfect rigid walls are modeled for the rooms and (ii) when an acoustic absorption is taken into account.

In the first case, the rooms walls are considered perfectly reflecting. In the second case, to better reproduce the test conditions in laboratory, an absorption coefficient α is added to the walls of the rooms expect for the one in which the structure is mounted. It is supposed constant over the surfaces and the frequency range of interest. Its value is deducted from the equivalent sound absorption area A and the absorbing areas of the room S i . The values of these two parameters are issued from the data of the experimental tests carried out in CERIBOIS laboratory.

Numerical configurations of the problem

In this section, the four proposed numerical configurations used to analyze the vibro-acoustic response of the IGU are detailed.

Room-IGU-Room

In the first numerical configuration presented by Fig. 1, the actual laboratory conditions are considered, meaning that both reverberant rooms with rigid walls are modelled. Regarding the excitation side, a monopole source is placed in one of the upper corners of the emitting room in order to excite the maximum number of acoustic modes. The structure is mounted in the common wall in such a way that the left pane of IGU is coupled to the emitting room and the right one is coupled to the receiving one. The sound transmission through the structure between the two rooms is calculated by considering the sound field of the source and receiving chambers, the dynamic behavior of the structure and their coupling. A fullycoupled fluid-structure interaction model has been used which means that the vibration of the double-glazing is influenced by the acoustic pressure in the two rooms and these fields are coupled to the structural motion. In the finite element context, this vibro-acoustic problem is described in terms of structural displacement u and acoustic pressure p. The discretization of the variational formulation associated to this problem [START_REF] Larbi | A new finite element formulation for internal acoustic problems with dissipative walls[END_REF][START_REF] Larbi | Vibration of axisymmetric composite piezoelectric shells coupled with internal fluid[END_REF] leads to the following matrix system (see Ref. [START_REF] Larbi | Vibroacoustic analysis of doublewall sandwich panels with viscoelastic core[END_REF] for details):

K u -C 0 K p -ω 2 M u 0 C T M p U P = F Q , ( 4
)
where K u and M u are the structural stiffness and mass matrices, K p and M p are the associated acoustic matrices, C is the fluid-structure coupling matrix, F is the nodal force excitation applied on the structure and Q is the nodal acoustic excitation. The structural and acoustic domains are subdivided into finite elements whose sizes are controlled by the wavelength λ which depends on the frequency range of interest. For the fluid domain, we consider the acoustic wavelength defined as λ a = c 0 /f where c 0 is the sound speed. For the structure, the bending wavelength is

λ f = (2π/f )(D/M ) 1/4
where D is the bending stiffness and M is the surface mass density. It should be noted that, in order to reduce the computational costs, an incompatible mesh can be used at the structural-acoustic interface by enforcing continuous fields using Lagrange multipliers.

In this model, which is the most complete, the rooms dimension are chosen according to the characteristics of CERIBOIS laboratory. The emitting and receiving rooms dimensions are 5×4.5×3.25 m 3 and 5×4×3.25 m 3 , respectively as shown in Fig. 2.

DSF-IGU-Room

At low frequencies, to respect the condition of incident diffuse sound field, it is necessary to model a room whose dimensions are much larger than the wavelength, leading to an increase of the computational costs. To overcome this problem, the DSF can be modeled as superposition of an infinite plane waves random in phase, arriving uniformly from all the directions whose propagating vector is pointed toward the structure. In such case:

Π d inc = 2π 0 π/2 0 Π inc(θ,ϕ) sin(θ) dθ dϕ, (5)
where Π inc(θ,ϕ) is the incident power of one plane wave calculated as :

Π inc(θ,ϕ) = |A i | 2 S 2 ρ 0 c 0 cos(θ). (6)
where A i is the complex amplitude of the plane wave defined by the angle along the longitude θ and the angle according to the colatitude ϕ, S is the area of the plate and ρ 0 is the mass density of the fluid in the external acoustic domain.

For the receiving side, the clamped structure radiates in a rigid rectangular cavity. In the case of a plane structure, whose radiation surface is coupled to an acoustic enclosure, the radiated power Π rad of the structure in the acoustic domain is given by [START_REF] Pierce | Acoustics: An Introduction to Its Physical Principles and Applications[END_REF]:

Π rad = 1 2 Re S pv * n dS , (7)
where v n is the normal velocity of the structure, p is the wall pressure, * denotes the complex conjugate and Re is the real part of a complex number. The normal velocity v n is calculated from the finite element formulation (Eq. ( 4)) applied to the receiving room. The model is illustrated by Fig. 3.

Room-IGU-Free-field

Another interesting configuration consists to eliminate the effect of the receiving room and so to study only the effect of the emitting one. This is approximated by the situation when the receiving side is an anechoic room, and so, its modal behavior can not be observed. In such a configuration, presented by Fig. 4, the first pane of the double-glazing is coupled to the emitting room and the second one radiates in a half infinite domain.

For a plane vibrating structure mounted on a rigid baffle, the pressure p in the semi-infinite fluid domain can be obtained with the Rayleigh integral [START_REF] Pierce | Acoustics: An Introduction to Its Physical Principles and Applications[END_REF]. It is defined by:

p(M, ω) = iρ 0 ω 2π S v n (G, ω) e -ikr r dS, (8)
where ρ 0 is the fluid density, M is a point in the semi-infinite acoustic domain, G is a point on the radiating surface S, v n is the normal velocity at the point G defined as v n = v.n and n is the external normal to the structure. As described by Fahy [START_REF] Fahy | Sound and structural vibration: radiation, transmission, and response[END_REF], the baffled vibrating panel can be divided into discrete rectangular elements whose transverse motions are specified in terms of normal velocity at their central position. Assuming that the size S e of the elements is small compared to the acoustic wavelength. The total radiated sound power defined by Eq. (( 7)) can be formulated as the sum of the radiated powers of each element:

Π rad = S e 2 Re(v H n p), (9)
where H is the hermitian transpose, v n and p are the vector of complex normal velocity and acoustic pressure in all elements, respectively. In this formulation, the normal velocity vector of each of these elemental sources is related to the vector of sound pressure by the impedance relation:

p = Z v n , ( 10
)
where Z is a symmetric matrix of acoustic impedances. Substituting Eq. (( 10)) in the Eq. (( 9)), the total radiated power can be calculated from:

Π rad = v H n Rv n , (11)
in which R is the matrix of radiation resistances of the elements dividing the baffled panel and defined as:

R = S 2 e ω 2 ρ0 4πc0           1 sin(kr12) kr12 • • • sin(kr1R) kr1R sin(kr21) kr21 1 • • • sin(kr2R) kr2R . . . . . . . . . . . . sin(krR1) krR1 sin(krR2) krR2 • • • 1           , ( 12
)
where r ij is the distance between the centers of the i th and j th elements. This method can be applied to any plane structure mounted in an infinite baffle, regardless its boundary conditions.

DSF-IGU-Free-field

The fourth configuration showed by Fig. 5 presents the ideal case when the baffled structure is excited by a diffuse field on the emitting side and radiates in a free field in the receiving one. The sound transmission loss is determined from the equations presented in the previous sections. The incident sound power on the first panel is approximated by the summation of the excitation from each individual plane wave (Eq. ( 6)). In the other side, the radiated sound power from the second panel is given by the formulation in terms of the elemental radiators, Eq. (( 11)).

Calibration of the IGU from Experimental Modal Analysis

As mentioned in the introduction, due to the complexity of the wooden window, the four numerical configurations presented in the previous section will be only applied to the IGU. The latter structure, presented in Fig. 6, consists of two panels of glass separated by an acoustic cavity filled with air or a gaz. In this study, two double glazing systems are considered. The first one, designated by 6/18/4, is composed of two panels whose thicknesses are 6 mm and 4 mm, separated by 18 mm argon cavity. The second system, designated by 4/20/4, is composed of two panels with the same thickness (4 mm), separated by 20 mm argon cavity.

The panels are linked together along their perimeters by various edge seal systems. It consists of numerous components, including the spacer bar, the desiccant and the sealant. In fact, the stainless hollow steel spacer is half filled at least by the desiccant which is used to dry out the cavity of the IGU. Typically, the desiccants in the IGU are molecular sieves or a mix of silica gel with molecular sieves [START_REF] Meisel | Role of desiccants, complex subject made easy[END_REF]. The sealant which is a dual combination is commonly used. It is composed of a primary barrier placed between the spacer and the glass panels. For this purpose, a synthetic rubber, typically polyisobutylene (PIB), is usually used. The secondary barrier, which can be silicone, polyurethane or polysulphide [START_REF] Wolf | Factors governing the life expectancy of dual-sealed insulating glass units[END_REF], is applied around the perimetre of the IGU to join the spacer bar to the glass panels. So, the complexity of the IGU is located along its perimeter. Foret et all [START_REF] Foret | Effect of spacer designs on acoustic performance of windows[END_REF] tested different edge sealant systems of a nominally IGU and results stated that strong dispersion can be observed for the frequencies below 200 Hz and notably at the mass-air-mass resonance region. For this reason, the properties of the edge seal system of the studied IGU need to be carefully identified. To this end, the Experimental Modal Analysis (EMA) is used to determine the modal parameters of the IGU which are used later to calibrate an equivalent numerical model.

Experimental Modal Analysis set-up

The characterization of the modal parameters of the IGU such as the natural frequencies, the modal shapes and the damping ratios are obtained from an Experimental Modal Analysis (EMA) whose set-up is presented by Fig. 7. The application of this method requires the measurement of the Frequency Response Function (FRF) at different points of the structure after its excitation (Roving Hammer Test). It consists on using the impact hammer which is equipped with a force sensor and a nylon tip to excite the structure. For the response, three reference accelerometers are used to measure the acceleration. The suspended double-glazing (see Fig. 8), of size 0.581× 1.309 m 2 , is composed of two glass panels whose thickness are 6 mm and 4 mm, separated by 18 mm of argon. The IGU is meshed into 153 nodes in order to obtain a good visualization of the mode shapes. Excitation is done at all nodes with an average of three impacts and measurements are performed up to 400 Hz with a frequency resolution of 0.5 Hz.

For performing the EMA, a home-made Matlab toolbox has been developed and includes several approaches like LSCF [START_REF] Guillaume | A poly-reference implementation of the least-squares complex frequency-domain estimator[END_REF] and its polyreference version [START_REF] El-Kafafy | Polymax Plus estimator: better estimation of the modal parameters and their confidence bounds[END_REF]. This toolbox consists of several Matlab routines, allowing to establish the complete modal analysis from experimental data step by step. Since the Roving Hammer test is used for this EMA, we must be careful in the quality of tests. Typically, a fairly good and relatively flat Input Power Spectra is required [START_REF] Avitabile | Modal testing : A practitioner's guide[END_REF]. This recommendation is ensured during the tests as the excitation level presented in Fig. 9 is constant over the whole frequency range. The coherence and the FRF are also used as data quality tools. As we can see from the same figure, the coherence is close to 1 and the quality of the obtained FRF is good with the presence of peaks for all the frequency range. In addition, the principle of reciprocity must be verified since the structure is supposed to be linear. It means that the FRF measurements of two points must be the same regardless of which of them is the input or the output. For this, two cases are studied; the first one where the excitation and the measurement points, respectively, P1 and P2, are on the same panel glass. For the second case, the excitation point P1 and the measurement point P2 are in the opposite glasses. The results presented in Fig. 10 show that the system is linear since the curves are superposed following the permutation between the excitation and measurement points, therefore, a linear numerical model can be used for the IGU.

Numerical calibrated model

The IGU experimentally tested is modeled numerically with the finite element method. The objective is to find a numerical representation of the structure able to reproduce its dynamic modal behavior. The numerical model consists of two 2D shells representing the two glass panels separated by a three-dimensional cavity filled with argon (sound speed c 0 = 317 m/s and density ρ 0 = 1.6 kg/m 3 ). The mechanical properties of the glass are E g = 60 GPa; ρ g = 2450 kg/m 3 and ν g = 0.23. For the sealant system, an equivalent sub-domain is defined for the group composed of the spacer, the desiccant and the seal, which is called Equivalent Spacer for the rest of the paper. Its density is calculated from the masses and volumes of its different components. The Table 1 shows the properties of the aluminium for the spacer, Butyl and Silicone of the first and second sealing barrier, respectively. In addition, the indicative mass per linear meter of the desiccant is approximately 62 g/m. The equivalent density ρ es = 1523 kg/m 3 is thus obtained. Note that the EMA results will be used for the estimation of the Young Modulus of the Equivalent Spacer.

Components Aluminium Butyl Silicone

ρ (kg/m 3 ) 2700 1200 1200

V (m 3 ) 2,005×10 -4 1,323×10 -5 2,0412×10 -4
m (kg) 0,541 0,0159 0.245

For the meshing, quadrilateral shell elements for the glass panels and hexahedral elements for the Equivalent Spacer and the fluid are used in the finite element discretization (see Fig. 11). Since it is recommended to use 6 linear elements per acoustic wavelength, and 10 elements for the bending wavelength, the final model contains around 35600 degrees of freedom. At the interface between the solid and acoustic domains, structural-acoustic coupling conditions are then enforced. The structure is considered free in the space.

Argon

Equivalent Spacer Panels

Results of the model calibration

The calibration of the model of the Equivalent Spacer is done regarding the first eigenfrequency, but other natural frequencies are presented for the verification. The numerical results presented here are obtained for a Young's modulus of the Equivalent Spacer E es = 100 MPa and an equivalent Poisson ratio ν es = 0.49. The comparison of the first ten natural frequencies of the structure obtained from the EMA and the numerical analysis of the calibrated model is presented in Fig. 12. The difference of the first natural frequencies is almost zero. For a such complex structure, results are considered satisfying since the maximum difference of results is 6.8 %, observed in the 8 th mode.

The MAC (Modal Assurance Criterion) is also used out to evaluate the coherence of experimental and numerical modal vectors. The MAC calculated from the comparison of the five first modal shapes is presented in Fig. 13. As it can be seen, the diagonal terms are close to 1 and the offdiagonal terms are less than 0.25, which means that there is a consistent correspondence between the experimental and numerical results. The corresponding modes shapes are presented in Fig. 14. The first line presents the modal shapes obtained with the EMA and the second line presents the ones found by the FEM of the calibrated model of the IGU.

Regarding the comparison of the eigenfrequencies and the modal shapes, the calibrated numerical model of the IGU is satisying, and therefore, it is used in the next section for the prediction of the sound transmission loss with the four numerical configurations already discussed.

Results of the Transmission Loss

The analyses have been carried out to investigate the sound transmission loss through the IGU 6/18/4 calibrated in the previous section. To this end, the 6 mm glass panel is placed in the emitting side and 4 mm glass panel in the receiving side. A damping coefficient of 1 % is used for all the materials of the structure. The latter is fully clamped at its edges in the common wall of the rooms.

To compare the numerical cost of the four configurations, a comparison of the degrees of freedom (DOF) with considering the size of each model is given in Table 2. With respecting the meshing conditions for the structure and the fluid domains, the first case with the two acoustic rooms has 189 ×10 3 DOF which is around 5 times the DOF of the case with free field in the both sides of the double glazing.

Comparison of the TL of the four configurations

To study the intrinsic acoustic response of the IGU without any influence of environmental factors, the narrow band transmission loss of the structure predicted with the perfect free-fields configuration is presented in Fig. 15. The result is compared to the one obtained with the impedance approach [START_REF] Heckl | The tenth sir richard fariry memorial lecture: sound transmission in buildings[END_REF]. As it can be observed, the theoretical TL is higher than the numerical result except in the restricted frequency band around 175 Hz. This has been pointed out by Heckel [START_REF] Heckl | The tenth sir richard fariry memorial lecture: sound transmission in buildings[END_REF] who stated that the impedance theory can only be applied qualitatively in the case of the double glazing of windows. This was explained by the fact that, in addition to the small size of this type of structure, the argon cavity between the panels has a significant lateral resonances. The mass-air-mass frequency of the studied structure is around 174 Hz and the corresponding modal shape is illustrated in the Fig. 16. The other dips in the numerical TL (Fig. 15) correspond to the eigenfrequencies of the double glazing whose density increases as the frequency increases. The coupling of the structure to acoustic rooms change its response and so the transmission of the sound due to two related factors: the first one is the closeness of the eigenfrequencies of the rooms and those of the structure, the second is the spatial matching between the distributions of the displacements and the acoustic pressure over the structure [START_REF] Fahy | Sound and structural vibration: radiation, transmission, and response[END_REF]. In the first case, a perfect rigid walls are modeled for the rooms. Indeed, even if this case is an extreme situation, this study will highlight the significant effect of the acoustic absorption in the rooms. From the comparison between the four configurations in narrow band presented in Fig. 17a, it can be observed that in very low frequencies, there are visible and separated natural frequencies of the rooms. As a result, intense dips and fluctuations marked the TL in addition to those presenting the modal behavior of the structure. As the frequency increases, this behavior becomes less significant due to the increase of the modal density of the rooms, and so, the TL curves of the configurations with one or two rooms become smoother. From the 1/3-octave spectrum presented in Fig. 17b, it can seen that the results from all the configurations have the same trend for frequencies above 250 Hz, although, a significant difference is observed between them. For example, the difference between the case of the acoustic suit and the perfect free-fields reaches 14.5 dB in 500 Hz.

In the second case, an absorption coefficient α is added to take into account the acoustic dissipation on the rooms walls. The value of α is averaged from the laboratory's data and a value of about 0.084 is used for the simulation. It gives a Schroeder frequency around 300 Hz. The results of the transmission loss of the IGU calculated when it is mounted on an infinite baffle and other configurations are compared and presented in Fig. 18. For all configurations, comparing to the first case (perfect rigid walls), there is an attenuation and reduction of dips amplitudes and the modal and diffuse regions are clearly separated (Fig. 18a). The comparison between the TL predicted from the first and the fourth configuration shows that, even with treated walls and absorption coefficient respecting the recommendations of the standards [START_REF]Acoustics -laboratory measurement of sound insulation of building elements: Part 1: Application rules for specific products; Part 2: Measurement of airborne sound insulation; Part 4: Measurement procedures and requirements; Part 5: Requirements for test facilities and equipment[END_REF], a special care for the measurement of this indicator must be considered in different facilities at low frequencies. From the results of the DSF-IGU-room (2 nd config) and the perfect free-fields model (4 th configuration), it can be observed that the modal behavior of the receiving room alone do not significantly affect the transmission loss compared to the ideal case.

Finally, the comparison of the case with only the emitting room (3 rd configuration) and free-fields configuration shows that the modal behavior of the emitting room is still present at very low frequencies (presented by the dips on the TL curve). So that, the properties of this room dominate the TL calculated with the complete acoustic suite (1 st configuration). Otherwise, the stationary waves created at the natural frequencies of the source room amplify the mean square pressure in this room. This is can be explained by the sensitivity of the diffuseness of the acoustic field to many parameters such as the source position, the room geometry and the distribution of the absorption in the walls.

A synthesis of the TL calculated with the four configurations in 1/3octave bands is presented by Fig. 18b. For all the frequency range, TL curves have the same trend with a difference of 4 dB, for example, between the cases with one room (2 nd and 3 rd config) at 500 Hz. Although, in very low frequencies below 125 Hz, the TL is sensitive to the presence of the source room. To this end, a specific care in the numerical modeling of the room or in the carrying out of the acoustic test in laboratory is recommended in order to avoid its modal behavior and to be closer to the ideal free-fields case.

So in order to extrapolate this survey to other studies with taking into account only the intrinsic properties of the double glazing and not the properties of the rooms, in addition to the computational efficiency compared to the other configurations, the 4 th one will be used in the comparison with experimental data of TL.

Comparison with experimental measurements

In order to validate the 4 th configuration and the calibrated model of the sealing system of the double glazing, we compare the numerical results with experimental data from the literature. The frequency band depends on the available experimental data. We have chosen to predict the transmission loss of the system up to 500 Hz when the experimental data starts at 50 Hz, and up to 630 Hz when measurements are made for frequencies above 100 Hz.

The experimental data presented in this section are derived from measurements made in accordance with the standards [START_REF]Acoustics -laboratory measurement of sound insulation of building elements: Part 1: Application rules for specific products; Part 2: Measurement of airborne sound insulation; Part 4: Measurement procedures and requirements; Part 5: Requirements for test facilities and equipment[END_REF]. The typical installation of double glazing in the opening of the wall separating the two rooms is shown in Fig. 19. A layer of mineral wool is inserted into the partition of the wall and the wooden opening to reduce the solid sound transmission. After the installation of the double glazing, a compressible foam and a mastic seal are applied to the entire perimeter to eliminate sound leaks. The measurement results are presented in third-octave bands according to the recommendations of the standard [START_REF]Acoustics -laboratory measurement of sound insulation of building elements: Part 1: Application rules for specific products; Part 2: Measurement of airborne sound insulation; Part 4: Measurement procedures and requirements; Part 5: Requirements for test facilities and equipment[END_REF]. For the results issued from Foret et al. [START_REF] Foret | Effect of spacer designs on acoustic performance of windows[END_REF] work, the type of edge sealant system of the structure is detailed, however, it is unknown for the data issu from the work of Assaf [START_REF] Assaf | Analyse du comportement vibroacoustique des parois multicouches composites dans les constructions[END_REF]. In fact, the majority of the globally manufactured IGU are dual sealed especially in Europe with about 85-90 % of commercial glazing units [START_REF] Wolf | Silicone sealed insulating glass units[END_REF]. To this end, the calibrated properties of our model, which is a dual sealed one, are used for the present models.

The results from the work of Foret al. [START_REF] Foret | Effect of spacer designs on acoustic performance of windows[END_REF] are illustrated in by Fig. [START_REF] Shorter | Vibro-acoustic analysis of complex systems[END_REF]. In total, seven 4/16/4 IGU have been tested with normalized dimensions which are 1.48 m wide by 1.23 m high. The difference between them lies in the sealing system, i.e. the type of the spacer and the two sealing barriers (see Ref. [START_REF] Foret | Effect of spacer designs on acoustic performance of windows[END_REF] for further details). The Fig. 20a shows the TL of the different configurations tested from 100 Hz to 630 Hz. Results show that the acoustic performance of the system depends on the sealing system with a difference that can reach about 5 dB at 200 Hz, corresponding to the resonance region.

The first IGU that has been tested experimentally has the same sealant system of our model calibrated previously. For that, a comparison of the result of the numerical model and that of the first system is presented in Fig. 20b. A good agreement can be noticed between the two approaches over the whole frequency range with a maximum difference of 1.5 dB at 315 Hz. In order to asses the validity of our calibrated model, the numerical TL is compared to the average of the seven series of the experimental data (Fig. 20b). It can be seen that the numerical curve falls within the standard deviation from the mean of the experimental results except for the frequencies 125 Hz and 315 Hz. The degradation of the acoustic performance due to the effect of the resonance mass-air-mass can be clearly distinguished which is estimated at 228 Hz for the present system.

The second IGU studied is composed of two glass of 6 mm separated by 12 mm by a cavity fulled with argon and tested by Assaf [START_REF] Assaf | Analyse du comportement vibroacoustique des parois multicouches composites dans les constructions[END_REF]. The dimensions of the glass are 1.13 × 1.38 m 2 whose sealing system is not specified. Fig. 21 presents the comparison between numerical and experimental transmission loss. Even with the absence of information (type of the sealent system), the both curves follow the same trend over the frequency range of interest except for the 80 Hz frequency. For this glazing composition, the analytical mass-air-mass frequency gives a value of 215 Hz [START_REF] Fahy | Sound and structural vibration: radiation, transmission, and response[END_REF], which corresponds to the dip in the TL at 200 Hz for both results.

The comparison with experimental results of the two double glazing composition show a good agreement with the numerical transmission loss for the whole frequency range even for those below 100 Hz. As a consequence, the calibrated model of the Equivalent Spacer can be considered satisfying acoustically and the 4 th configuration leads to predict properly the acoustic performances of the structure independently to the rooms.

Parametric analyses

In this section, the ideal configuration that allows the investigation of the intrinsic characteristics of the IGU is used for parametric survey.

The effect of structural damping on transmission loss is studied here. This parameter, which introduces dissipation into the structure, is important in the analysis of vibrations. To this end, three structural damping coefficients are studied; 0 % (undamped system), 1 % and 2 %. Classically, a structural coefficient damping of 1 % is considered. So even if the value 2 % seems unrealistic, it allows to highlight the effect of damping on the evolution of the sound transmission loss. The TL results in the narrow and 1/3-octave band are presented in Fig. 22. From the comparison between both sets of results in narrow band, it can be seen that only around the natural frequencies differences can be observed and hardly influences the overall transmission loss. That explains why the influence of damping increases with the frequency due to the increase in the modal density. These results confirm the known fact that damping has an effect only on the resonances of the structure. The transmission loss below the first eigenfrequency is governed by the stiffness. For the studied structure, the lowest natural frequency is 56 Hz, that is why, it can be observed from the one 1/3-octave band result (Fig. 22b) that the curves are superposed for the center frequencies 40 Hz, 50 Hz and 63 Hz.

The second parameter considered is the glass thickness. For this purpose, the combination of a 6 mm glass plate and 18 mm argon cavity was preserved and the thickness of the second plate was varied from 4 mm to 10 mm with a 2 mm step. Theoretically, in the frequency range of interest, up to 600 Hz, for a such double-partition system, the two regions that can be observed in the TL results are the stiffness and resonance-controlled regions. The results in Fig. 23 show that the modes shift to lower frequencies with the decreasing of the total mass of the system. Below the first resonance frequency, as expected, the sound insulation is governed by the rigidity of the structure. For this, it can be seen that the heaviest structure 6/18/10 has the highest TL. Above this frequency there is no clear interpretation due to the influence of the mass-air-mass frequency of each system. For this frequency range, most of the insulation dips at the structural eigenfrequencies of the symmetric double-glazing 6/18/6 are greater than those with different glass thickness.

Conclusions

The sound Transmission Loss (TL) of an Insulating Glazing Unit (IGU) at low frequencies has been studied using the Finite Element Method. In order to study the effect of the rooms on the vibro-acoustic response of the considered structure, four different configurations have been developed that taking into account the acoustic laboratory conditions, either the receiving room or the emitting room and. Finally, the case where the two reverberant rooms are removed and a perfect field is considered in the both side.

The numerical model of the IGU has been calibrated with the Experimental Modal Analysis (EMA). This step was essential to built the numerical model of the structure due to geometrical and mechanical complexities of its edge sealant system components. The criteria used to validate the calibrated numerical model have been satisfied. In fact, the gap between the first numerical and experimental natural frequencies is around 0.3 % and the maximum difference of the first ten frequencies is 6.8 %. In addition, the corresponding mode shape show a good agreement with an almost diagonal MAC. The modal damping obtained from the EMA, not discussed here, will be used in future work using the modal approach.

The acoustic analyses pointed out that the results are sensitive to the presence of the rooms due to their modal behavior, especially the emitting one. When the rooms walls are rigid, the difference between the case of the acoustic suit (1 st configuration) and the perfect free-fields(4 th configuration) reaches about 14 dB in 500 Hz. The add of the acoustic absorption on the rooms walls improves significantly the results especially at low frequencies. In this case, at 500 Hz, the maximum TL difference of 4 dB is observed between the second ant the third configurations. The 4 th configuration that combines free-fields on both sides of the structure ensures a perfect diffuseness of the acoustic field even for the very low frequencies and avoids the modal behavior of the two rooms. Consequently, the problem of the lack of reproducibility will not be faced since this configuration evaluates only the intrinsic performances of the tested element.

In addition, due the computational efficiency of this configuration and in order to extrapolate the study to other IGU models, results issued from this FEM model have been compared to experimental results of different edge system models with two glass composition (4/16/4 and 6/12/6). Results showed that the IGU's TL depends on the sealing system and a gap of 5 dB can be reached between different models at the mass-air-mass resonance region. The comparison of the developed numerical model's results and the experimental ones, when the same edge system is employed, shows a very good agreement for the whole frequency range of interest. A maximum difference of 1.5 dB was observed at 315 Hz. The same configuration has been used to carry out parametric studies to investigate the influence of the variation of some properties such as the structural damping and the panel's thickness.

Further investigations are conceivable for this work. The immediate application concerns the analysis of the vibro-acoustic response of the full structure i.e. the wooden window with the two opening. Another line of research is the development of reduced order models to optimize the numerical calculation cost. Finally, extension of future window's model to other types of elements of construction could be considered, in particular wood window doors.

Figure 1 :

 1 Figure 1: Room-IGU-Room configuration.

Figure 2 :

 2 Figure 2: Emitting and receiving rooms connected through the IGU.

Figure 3 :

 3 Figure 3: DSF-IGU-Room configuration.

Figure 4 :

 4 Figure 4: Room-IGU-Free field configuration.

Figure 5 :

 5 Figure 5: DSF-IGU-free field configuration.

Figure 6 :

 6 Figure 6: IGU: (a) Nomenclature and (b) Edge sealant system.

Figure 7 :

 7 Figure 7: The set-up of Experimental Modal Analysis.

Figure 8 :

 8 Figure 8: Suspended double-glazing.

Figure 9 :

 9 Figure 9: Verification of the quality of the measurement.

Figure 10 :

 10 Figure 10: Verification of the system linearity. P1: excitation point-P2: measurement point

Figure 11 :

 11 Figure 11: Numerical meshing of the IGU.

Figure 12 :

 12 Figure 12: Comparaison of the first ten experimental and numerical eigenfrequencies of the IGU.

Figure 13 :Figure 14 :

 1314 Figure 13: MAC of experimental and numerical mode shapes of the IGU.

Figure 15 :

 15 Figure 15: Comparison of the TL between the impedance approach and the FEM (4 th configuration).

Figure 16 :

 16 Figure 16: Fluid pressure and panels normal displacement at the mass-air-mass frequency (174 Hz).

Figure 17 :

 17 Figure 17: Comparison of TL of the IGU calculated with the four configurations when the rooms walls are perfectly rigid.

Figure 18 :

 18 Figure 18: Comparison of TL of the IGU calculated with the four configurations when the rooms walls are not perfectly rigid.

Figure 19 :

 19 Figure 19: Mounting details of double glazing in the wall for the acoustic test.

  Effect of sealing systems (seven different systems numbered from 1 to 7 -see Ref.[START_REF] Foret | Effect of spacer designs on acoustic performance of windows[END_REF] for details) Average of all the experimental (Experimental-average), the experimental of system 1 (Experimental-sys 1) and Numerical(FEM) TL[START_REF] Foret | Effect of spacer designs on acoustic performance of windows[END_REF]

Figure 20 :Figure 21 :

 2021 Figure 20: Comparison of the numerical and experimental transmission loss of a normalized 4/16/4 IGU.

Figure 22 :

 22 Figure 22: Effect of structural damping on the TL of the IGU.

Figure 23 :

 23 Figure 23: Effect of the panel's thikness on the TL of the IGU.

Table 1 :

 1 Mechanical properties of the Equivalent Spacer components.

Table 2 :

 2 DOFs of the four configurations.

	Configuration	1	2	3	4
	DOFs (×10 3 )	189 110 115 36

Acknowledgements

The authors would like to express their thanks to CODIFAB (Comité professionnel de développement des industries françaises de l'ameublement et du bois) for its financial support.


Q Galvane

C Lino

M Christie

R Cozot

Directing the Photography: Combining Cinematic Rules, Indirect Light Controls and Lighting-by-Example

Keywords: CCS Concepts, Computing methodologies → Computational photography, •Human-centered computing → User centered design, •Applied computing → Media arts

Figure 1: The 3-point-lighting technique (on the left) is a classical cinematic rule to place lights around a subject. Playing with the lights position, size, and flux considerably impacts the aesthetics and atmosphere of a scene (in the middle). We propose a technique that automatically computes the parameters of a 3-point-lighting setup made of area lights, given the reference image in the middle, to generate the image on the right in a virtual environment.

Introduction

Lighting entities in 3D environments is a central concern for whoever wants to bring an aesthetic and emotional dimension to an image. In real photography and cinematography, there are many guiding principles to lighting (e.g. Rembrandt lighting, 3-point and 4-point lighting) [Wis12]. Surprisingly, most 3D modelling tools do not propose a strong support in this lighting stage other than di-rect interaction with each of the lights low level parameters. Hence, individually placing each light and controlling its parameters is a complex endeavor that requires advanced skills and expertise.

To ease this creative stage, the research community has been proposing interactive or automated light placement techniques. They commonly rely on an inverse-lighting approach: the user specifies the features of his desired lighting, and a solver detersubmitted to Pacific Graphics (2018) mines an appropriate set of lighting parameters (position, orientation, flux, color of lights). Approaches are considered as local when the user controls local image features such as the positions or sizes of shadows and highlights, or the shading color of lit areas [SDS * 93, [START_REF] Pellacini | Lighting with paint[END_REF][START_REF] Pellacini | envylight: An interface for editing natural illumination[END_REF]; or global when the user controls global image features, e.g. histogram of luminance or contrast [START_REF] Kawai | Radioptimization: Goal based rendering[END_REF][START_REF] Castro | Energy-saving light positioning using heuristic search[END_REF]FB14]. Lighting-by-example is another class of approaches that mixes local and global approaches; it relies on analyzing features from a reference image , and then compute light parameters to obtain a lighting style as close as possible to the reference image (e.g. encoded through an histogram of image luminance [HCB16] or an image decomposition into wavelets [HO07]).

The main issue of such inverse-lighting approaches is the computational cost of rendering and evaluating the image features each time a light parameter evolves. Optimization techniques require to extensively evaluate the cost function expressed as a distance in terms of these image features. This cost may be reduced through the use of simplified light sources (point, spot or directional lights) compatible with rasterization rendering, at the expense of lowering the realism of the computed images.

The challenge is hence to reduce the practical complexity of automated light placement, while at the same time considering more complex light models such as area lights commonplace in real lighting stages. To address this challenge, we first propose to reduce the general problem of light placement to a 3-point light setup arranged around a subject (see Figure 1). This setup is composed of a number of key-lights -they represent the main source of light -, fill-lightstheir role is to control shadows intensity of areas not lit by the keylights-, and back/rim-lights -their role is to highlight the silhouette of a subject. Second we propose to rely on complex light sources, namely area lights, for which real-time rendering techniques have been proposed [HDHN16, LDSM17], hence reducing the cost of evaluating image features. And third, we adopt a two stage solving process that consists in first computing light positions and orientations, and second optimizing lights size, opening angle and flux. Our process can (i) automatically compute how many fill-lights and back/rim-lights are needed, together with their best placement, by analyzing the scene layout and the 3D geometry of the subject; and (ii) automatically compute the size, opening angle and flux of each area light source, given a set of desired image features extracted from a reference image. In addition, we provide the user with an interactive control of the key-light position; each change triggers the re-computation of the fill-lights and back/rim-lights.

Our contributions are:

• an indirect manipulation technique that enables a quick repositioning and re-orienting of area lights using a 3-point cinematic lighting technique; • a two-stage computation method which automatically determines the number, position, orientation, size, opening angle and flux of fill-lights and back-lights, given one or more key-lights. It mixes a geometric analysis with a stochastic numerical optimization; • an objective function driven by a lighting-by-example approach.

It compares the distribution and gradient of image luminance.

Related work

Lighting is a key element in photography/cinematography [Ada56, DeM59, Mal92, Las00, Sip10] and computer-generated contents [Wis12]. Choices in lighting aesthetics conveys the mood or atmosphere in a picture or in a story. In real or virtual environments, professional visual content creators rely on complex lighting setups, the design of which is highly technical and tedious due to the large number of parameters to control and to coordinate for each light source (e.g. type of light, position, orientation, shape, size, aperture, flux, falloffs, color). The user must also determine how many lights are needed.

An extensive body of work has been devoted to tackling inverselighting problems, first introduced by Kawai et al. [START_REF] Kawai | Radioptimization: Goal based rendering[END_REF]. All inverse-lighting methods share the common goal of computing a lighting setup, that can produce an image that corresponds as closely as possible to a desired lighting. Their differences can be analyzed along their specification of a desired lighting [PP03]: (i) Global methods, (ii) Painting-based methods (which we also refer to as local methods), (iii) Lighting-by-example based methods and (iv) Perception-based methods.

Global methods often tackle light source placement or flux problems in architectural design applications [START_REF] Cassol | Multi-objective optimization as a new approach to illumination design of interior spaces[END_REF][START_REF] Castro | Energy-saving light positioning using heuristic search[END_REF]FB12,FB14]. In this context, the lighting aesthetics in the rendered image is not a priority. Instead, the desired lighting is often specified as a set of statistical lighting features [START_REF] Kawai | Radioptimization: Goal based rendering[END_REF] (e.g. mean radiance, maximum non-uniformity, or radiance variance). These methods proceed by minimizing an objective function built as a weighted linear combination of these statistical lighting features; the user role is then reduced to specifying all weights. Such global methods are not well-suited for the design of a precise image lighting aesthetics.

Painting-based methods allow a more precise control of lighting details. A user directly paints desired lighting effects onto 3D objects. In the method proposed by Schoeneman et al. [SDS * 93], the user inputs a single target radiosity vector by painting on the initial rendered image of the scene. Their fully automated optimization method then addresses the flux problem for a predefined number of lights (their positions and orientations are considered as fixed). Pelaccini [START_REF] Pellacini | Lighting with paint[END_REF] proposed an iterative painting-based approach where the user can interactively paint lighting effects (i.e. add radiance) onto objects, with brushes. He can then choose which light (either an added light or an existing light) and which parameters to consider in the optimization. This greatly reduces the computational complexity. Then according to each new painted stroke, the user can iteratively refine the desired lighting. Their system tackles both the flux and placement problems. However, their system greatly relies on the user artistic skills; in particular such a system is not able to automatically determine when to add or remove a light, or which sub-set of the parameters to optimize. Such interactive methods can then be time-consuming and tedious to use, as mentioned by Pellacini in [KP09]; according to him, such paintingbased methods can actually appear less intuitive to use than indirect control methods. Moreover, painting-based methods are not wellsuited for the lighting of animated contents. submitted to Pacific Graphics (2018) Lighting-by-example methods address the main limitations of global and local approaches. Their central idea is that the desired lighting is provided as a reference image (a real image or a render of another 3D scene) or directly as another, already lit, 3D scene. Zupko and El-Nasr [ZEN09] ask the user to choose an image of an illuminated sphere that describe their desired image aesthetics. The target image is thus independent of the scene but limited to an image of a single sphere in front of a uniform background. Ha et al. proposed to define the desired lighting aesthetics with any image [HLCO10]. They encode the input lighting aesthetics as a wavelet decomposition of the radiance in the image. Their system then tries to optimize the lights in a new scene so as, when rendering this scene, the rendered image has a similar wavelet decomposition. The success of their technique is however very dependent to the layout of objects on the screen. Léon et al. [LGCB14] instead extract statistical features (e.g. mean and variance luminance) from the input image. They then try to minimize the distance between the reference and output images according to these statistical features. Only the opening angle and flux of the light (a spotlight source) are accounted for in their optimization process. They use a classic 3-point-lighting placement setup, which they explain is enough to achieve most of lighting aesthetics. In a way similar, Hudon et al.

[HCB16] use a cost function computed as a cosine distance between the luminance histograms of the reference and rendered images. Their approach also differs in how they approach the choice of the number of lights. Their light source placement method is made of two steps: (i) they place a set of around ten lights onto a sphere surrounding the lit object, for which they optimize the flux, then (ii) they put off all lights for which the contribution is too small. Perception-based methods , such as [SL02, WVBT16] deal with inverse lighting problems to address non-aesthetic features, more related to the proper perception and understanding of the shape or fine details of an inspected 3D object. They are often targeting scientific visualization applications. Though we do not precisely target this type of applications, a method close to ours is Wambecke et al. [START_REF] Wambecke | Automatic lighting design from photographic rules[END_REF] which proposes a fast geometric heuristic inspired by 2-point-lighting techniques. They offer a simple heuristic, based on the analysis of an object's geometry, allowing to place both a key-light and a fill-light to maximize the perception of the object's shape.

Discussion

To our knowledge, we provide the first inverse-lighting technique to place area lights using a real-time rasterization engine. Our approach nicely mixes the advantages of indirect control proposed by painting-based methods and lighting-by-example methods, while optimizing in parallel area lights flux, size and position. First, the user can specify where to place the key-light in screen space (in a way similar to [START_REF] Pellacini | Lighting with paint[END_REF]). Second, and conversely to local methods, the user can provide any reference image as input to refine the desired aesthetics (in a way similar to [LGCB14]). Third, our method uses photographic guiding techniques (as in [START_REF] Wambecke | Automatic lighting design from photographic rules[END_REF]) to model lights placement.

Overview

Our overall objective is to assist creatives in the design of a lighting setup around a 3D subject. The inputs of our system are (i) a single Figure 2: The design of our 3-point light setup using area lights. The Back/Rim-light is designed in a way that it is placed behind the subject as done in real-life photography/cinematography. Note that our model is designed to handle multiple key-lights, back-lights or fill-lights.

3D model (e.g. a character or an object) with no background, (ii) a camera viewing the 3D subject and (iii) a reference image that represents the desired lighting style. Our system analyzes the reference image, the object's geometry and the current placement of one or multiple key lights, to automatically compute, as an output, the placement of a collection of lights (fill-lights and back-lights).

To solve this light placement problem, we first propose a compact representation of the 3-point-light setup (see Section 4). We then determine the number, positions and orientations of fill-lights in this setup, so that most of the object parts visible from the camera, but not lit by a key-light, are then lit by at least one fill-light. We analyze the object's geometry to select all normals of a mesh triangle that are both back-facing the key-light and front-facing the camera. For this set of normals, we then use a k-means clustering on these normals. We first determine the appropriate number of filllights and, for each computed cluster, we create one fill-light (see Section 4.2).

In a second stage, given the camera and object's positions, and the object's geometry, we place a specific back-light rig so that four back/rim-lights are placed (i) outside the camera's frustum, and (ii) behind the subject (see Section 4.3).

In a third stage, we use a numerical optimization process to automatically improve the object's lighting style. Given an example lighting style provided as a reference image, we optimize the size, opening angle area and flux of each light source (for both the keylights, the fill-lights and the back-lights, see 5.2). The cost function we minimize accounts for the distances between the histograms of luminance and the histogram of the gradient of luminance in the reference image and the render of the scene.

Light type

Manipulation steps Optimization steps Re-computed dynamically key-light d, θ, ϕ, a, s, λ, f , c f , s, λ w, h, q fill-light λ, c f , s, λ w, h, q rim-light c f w, h, θ, ϕ, q all rim-lights d, σ, λ d, σ, λ

Table 1: Our parametric model: all lights are rectangular area lights, parameterized through their position (distance d, azimuth θ and elevation ϕ), orientation q, width w and height h, flux f and color c. We also consider design parameters such as the light aspect a (as the width/height ratio), its size s (as a scaling factor enforcing this aspect ratio). Note that for rim-lights, instead of controlling the size and aspect ratio of each light individually, we use a span parameter σ; its role is to constrain the width of side rim-lights and the height of top/ bottom rim-lights to be equal to this span value.

Figure 3: 2D top view of our light setup. Key and fill-lights are defined relatively to the camera-subject axis, using spherical coordinates, with the back/rim-lights are tangent to the camera frustum.

Lights are implicitely oriented towards the subject.

A compact 3-point-light model

Our 3-point lighting setup is founded on several hypotheses. First, we use three types of light sources: key-lights, fill-lights and back/rim-lights. All three types are represented as rectangular area lights, of width w and height h. An area light has a flux f and a color c. Second, every light is oriented so that it always faces the subject (see Figure 3), which is abstracted as a single 3D point. Hence, in our model, a light position is represented through spherical coordinates -it is located at distance d, azimuth θ and elevation ϕ -, while its orientation q is determined in a way similar to a camera look-at rotation. Third, we consider an opening angle λ, that controls the solid angle of its emitted flux.

To provide users with a more intuitive control over the light setup, we introduce some design parameters: the light aspect ratio a, equal to the width/height ratio; the light size s, a scaling factor from which we can determine a width and height that enforces a fixed aspect ratio. For rim-light, we introduce a light span σ. As illustrated in 2, rim-lights are constrained in terms either of their height, or their width. The light span controls the value of the other unconstrained parameters; in our model, all rim lights are finally constrained to be at the same distance of the camera and have the same span.

Interactively placing key-lights

When manipulating a key light, the user automatically triggers the recomputation of all the other light positions. To first position a key light in the environment we rely on a spherical model centered on the subject. The radius of the sphere is controlled by the user to adjust the distance of the light. Manipulating the light then consists in moving it onto this sphere. Two simple metaphors are proposed: ArcBall Light Control . We take inspiration from the Arcball metaphor for controlling cameras, and we adapt it to offer an intuitive means to manipulate key-lights. As the user drags the mouse (i.e. left-right or up-down), the selected key-light is rotated around the subject accordingly. The controlled light naturally remains at the same distance to the subject, and is re-oriented to always face toward the sphere center.

Painting-based Light Control . We also take inspiration from previous painting-based lighting interfaces to offer a means to specify directly, in image space, the areas to be lit by a key-light. Our method is designed to dynamically updated the key-lights position on manipulation. As the user highlights an object area, we analyze its geometry to select the set of visible vertices that are inside the area to lit. The direction of the key light is simply computed as the mean of all normals at these selected vertices.

Automatically placing fill-lights

Fill lights aim at softening or removing shadows produced by key lights. In other words, they should lit areas visible by the camera but which remain unlit after the key lights placement stage. The process is composed of three steps. First, given the mesh geometry of the subject and the current placement of the key lights and the camera, we compute sub-meshes representing unlit areas. Second, we compute the number of fill lights needed to lit these sub-meshes. Last, we compute the placement of fill lights by using a clustering on the vertex normals of these sub-meshes.

Computing the unlit areas . We start by extracting all vertices, as well as their associated normals, from the subject's mesh. We then prune all vertices not visible from the camera (i.e. either backfacing or located outside the camera's frustum). We also prune all vertices that are already lit by the key light. Here, we do not check for self-occlusions of the mesh (note that this would be a costly problem by itself) but solely use the normals at remaining vertices and compare them against the direction of the key light; a vertex is considered as lit only if the dot product of its normal and the key light direction is negative.

Computing the number of fill lights Given the set of unlit vertices, a naive approach would be to place a fill light in the direction of the mean normal direction of all unlit vertices. However, in many cases the the subject's geometry is too complex, which prevents a single fill light from being able to lit all sub-meshes. To solve this problem of better covering unlit areas, we perform a standard kmeans clustering on this set of unlit normals. We here account for the variation of unlit normals, and compute the number of clusters (i.e. the number of fill lights, k) through the following formula

k = kmax + 1 N N ∑ i=1 n i -n
where n i is the normal at the i th unlit vertex, n is the mean normal vector at unlit vertices and kmax is a user-specified threshold on the number of clusters. We experimentally found that a maximum of three fill lights is enough in most cases. Figure 4 shows an example of clustering obtained with our technique.

Placing the fill lights Given a set of directions, the fill lights are placed at the same distance as the key light. Their sizes and flux are computed proportionally to the size and flux of the key light.

Automatically placing back/rim-lights

The back light's role is to highlight a subject's contour. Traditionally, there are two ways to place a back light:

• either behind the subject along the camera's axis so as to ensure it is remains hidden by the subject; • or behind the subject, on a side, so as to be fully outside of the camera's frustum.

Unfortunately, the magic of back-lights placed behind the target rims only really occurs on complex surfaces like hair or wool that capture the light and remain difficult to render in real time. Conversely, the second method is less sensitive to the quality of the models and the textures. Following this main guideline, we propose to model our rim light as a light rig, made ,of four area lights. These areas lights are placed behind the subject, so that one edge of each area light is touching the corresponding plane of the camera's frustum; they are also oriented so as to light-to the subject, as illustrated in figure 3. This light rig allows a fine control on the rim effect, by adjusting the distance from the subject, the width, aperture and flux of each individual rim light. While adjusting these design parameters, our system can automatically compute or re-compute the placement of the rim lights.

Automatic lighting style matching

Once a first 3-point-lighting setup has been computed, the user can now refine the lighting style, by manually adjusting the set of low-level light parameters such as their fluxes, sizes or opening angles. This however remains a tedious task, which could also rapidly break the 3-point-lighting guidelines. To remove the burden of manually tuning all parameters we here propose an optimization process which, based on our 3-point-lighting model, offers the user with an efficient means to reproduce a lighting style.

Encoding and comparing lighting styles

To encode the lighting style of an image, we rely on a comparison method similar to [HCB16]. They first define a desired lighting aesthetics by using a reference image. Then then perform an optimization on lights parameters by relying on a cosine distance between the histograms of luminance in the reference and rendered images. Their method gives good results in many scenarios but fails to capture some important geometric information. Figure 5 illustrates this limitation. While the two images have very similar histograms of luminance, their lighting aesthetics are clearly different.

To overcome this limitation, we extend their method by also using the gradient in the image to enrich the representation of a lighting aesthetics. We not only calculate an histogram of luminance but we also calculate two other histograms, respectively capturing the horizontal and vertical gradient of luminance. These additional histograms provide valuable geometric information on the object's shading in the image; indirectly, they also provide clues on the relative placement of lights. Figure 5(d) demonstrates the difference in gradient histograms.

Optimizing the lighting style

The main advantage of our 3-point-light model is that it allows manipulating or searching a lower-dimensional parameter space. This is valuable for interactively controlling a full 3-point-lighting setup by simply manipulating the key-light (as presented in Section 4). In this new optimization step, the parameter space to search is reduced to the following subset of parameters:

• The flux, opening angle and size for the key-light;

• The flux, opening angle and size for each fill-light separately;

• The distance, span and opening angle common to all rim lights;

• The flux of each rim-light separately;

The dimension of our search space is then N = 3k + 10, where k is the number of placed fill-lights. By considering that kmax = 3, the search space has at most 19 parameters.

We cast our style-matching problem into a minimization process in this sub-space. We define the objective function f (i, r) to minimize as a linear combination of the cosine distances between histograms H extracted from the reference (input) image r and the current rendered image i.

f (i, r) = w 0 .Ψ(H i L , H r L )+w 1 .Ψ(H i ∇v , H r ∇v )+w 2 .Ψ(H i ∇h , H r ∇h )
where H L , H ∇v and H ∇h are respectively the histogram of luminance, the histogram of vertical gradient and the histogram of horizontal gradient. Ψ is the cosine distance function, which takes two histograms as input, and outputs their dot product (an histogram being viewed as a real-valued vector). In the current implementation, weights w 0 , w 1 and w 2 have been set respectively to the obvious values of 1, 0.5 and 0.5. This equally balances the importance of the luminance and the gradient of luminance in the optimization process.

To account for lights colors, we also extend this optimization scheme by using a separate histogram on each channel, instead of a single histograms of luminance. In the above equation, we then replace

w 0 .Ψ(H i L , H r L ) by w 0,R .Ψ(H i R , H r R ) + w 0,G .Ψ(H i G , H r G ) + w 0,B .Ψ(H i B , H r B )
where H R , H G and H B are the histogram of the red, green and blue channels, respectively.

To solve this minimization problem, we rely on a Particle Swarm Optimization (PSO) solver. This allows efficient computations while avoiding many local minima. For comparison purposes, we also tested with other search algorithms such as the gradient descent or the Nelder Mead algorithms, but found that they often failed to avoid local minima. To improve the performances of our PSO solver, we strategically add a set of warm positions in addition to the set of randomly sampled initial particles positions. We generate this set of warm positions by successively turning on only one light while other lights are turned off. The results of this optimization stage are presented in section 9.

Implementation

We developed our lighting assistant within the Unity3D game engine. To speed up mesh and image processing, we heavily rely on general-purpose GPU programming; in the current implementation, we use the Unity's compute shaders features. To compute histograms, we only render the subject. This allows to be agnostic of the background and avoid interference. This hypothesis however implies that the input images must either be detoured or taken on a black background. All our experiments (detailed in section 9) were conducted with Unity 2018 on a Core i7 @ 3.1GHz and a NVidia Quadro M2200.

Results

We tested our lighting design system on a variety of scenes, with different subjects and different mesh resolutions. In this section, we evaluate the performances of our interactive controller. We also evaluate the quality and accuracy of our lighting style optimizer, and validate the convergence of the particle swarm optimizer in tested scenes. Last, we evaluate the quality of our output images. All our results are presented below. We also provide a companion video showing the tool in action † .

Qualitative evaluation To evaluate the quality of both our interactive tool and our optimizer, we tested our system on a large set of example meshes and initial lighting configurations. Figure 6 presents some of these results; additional examples are also shown in the following sections. Using different reference images from different viewpoints and different initial lighting setups as inputcreated by using our 3-point-lighting interactive tools -our results not only show that our 3-point-lighting model can efficiently assist the user in appropriately creating and placing complementary light sources, but also that our optimization process is successful in reproducing a desired lighting aesthetics.

Convergence and efficiency of the method As shown in Figure 7, our system can also properly improve an existing lighting setup to match the luminance histogram of the rendered image with the histogram of the reference image. Figure 7e shows the evolution of our objective function for each particle of our PSO. One can clearly observe that, by using our objective function, the optimization algorithm quickly converges toward a good solution after a few iterations only. † http://cinematography.inria.fr/publications/automatic-lighting/ Figure 8 presents another interesting result. Given the same reference image but different initial lights placements -with keylights placed differently and therefore fill-lights and back-lights also placed in another configuration -our optimization process is still able to converge to a final lights placement which is very close. Expert feedback Our system was demoed and tested by both a professional director of photography and a film director. They both appreciated the use of the three-point lighting technique and the effort put in the formalization of this principle. They were especially impressed with the results and thought of many different applications for our tool. The film director would take advantage of our system mostly for previsualization purposes, to test different lighting styles based on reference images -a common practice in the early stages of a movie production. The director of photography found the tool very promising and suggested to add additional fea-tures to be able to use light sources based on existing equipment (i.e. led panels, spots, reflectors, etc.), with their actual specifications. While they both agreed that the lighting and shading produced by the game engine was not completely realistic, they believe the results are sufficient and could prove very useful in their respective fields.

Computation times As shown in Figure 9, our two interactive light control tools (light painting and automatic fill-lights placement) run easily in real-time in most cases. When processing highly detailed meshes, the framerate may drop down during some manipulations, but remains interactive (around 10 frames per second in the worst cases). When processing the 86k vertices mesh, the average framerate was around 30 frames per second. When optimizing the lighting aesthetic, the computation time depends on the parameters of the PSO solver and especially the number of particles. In our case, we set the size of the swarm to 20 particles. As shown in Figure 7, with this configuration it only takes a few iterations to converge. On average, it takes between 5 to 10 iterations to reach convergence, where each iteration takes around 0.4 seconds to compute -on average, 21 ms per particle per iteration.

Limitations

As an implementation of the 3-point lighting method, the technique presented in this paper suffers from its limitations. When placing the lights and optimizing over their parameter space, our tool only focuses on the subject. While ignoring the background is a relatively common practice, extending our solution to the 4-point lighting method constitute a very interesting challenge to be addressed in future work. Regarding the optimization process, the use of the histogram of the gradient in the objective function proved to be be efficient in many cases. However, this method is sometimes not sufficient when dealing with very smooth shading. Other heuristics could be investigated to better account for the orientation of the light and its impact on the shading.

In this paper, we made the assumption that the object in the reference image and the 3D model share materials of similar nature. Using different materials will affect the gradient histograms of the images and consequently the results. With specular objects especially, slight variations in the materials property makes the optimization process less accurate as well as the visual comparison more difficult (see Figure 10).

Conclusion

This paper has presented an inverse-lighting technique allowing to easily manipulate area lights to lit 3D subjects in virtual environments, and to reproduce a given lighting style provided as a reference image. We have provided a compact representation for the placement of complementary lights, that encodes a practical and well-established lighting technique in photography (the 3-pointlighting). We have proposed a hybrid optimization solution that can efficiently compare the lighting features in a reference and a render of the current scene to properly place lights and optimize their parameters so as to reproduce the input lighting style. Our solution nicely couples interactive control techniques, 3D geometry and image processing techniques, together with recent real-time rendering of polygonal area lights. Our approach represents a first step toward the design of more complex lighting setups. In particular, we think it will open the perspective of computing lighting setups for more complex scenes, typically with fore-ground/back-ground objects, and for animated scenes.

Figure 4 :

 4 Figure 4: Creating the fill lights by (a) analyzing the geometry, (b) extracting unlit areas visible from the cameras and their normals (c) clustering the remaining normals (red and green normals are in two different clusters) and (d) placing the fill-lights

Figure 5 :

 5 Figure 5: Comparison of two images rendered under two different lighting conditions (a) and (b). The two images present a similar luminance histograms (c) but different gradient histograms (d).

  Figure 6: Example of images produced with our tool. The system takes as input (a) a reference image, (b) let the user interactively place the key light while optimizing the placement of fill and rim lights and (c) optimize the lighting parameters to match the style of the reference image.

Figure 7 :

 7 Figure 7: From a reference image (a) and an initial lighting configuration, our system is able to reproduce the desired lighting aesthetic (c) by matching luminance histograms (d). The optimization process quickly converge after a few iterations (e).

Figure 8 :

 8 Figure 8: Using the same reference image (a) but starting from different lighting conditions (b) our system is able to converge to similar solutions (c).

Figure 9 :

 9 Figure 9: Evolution of the computation time (in seconds), taken each frame, needed to place the key-light and the fill-lights. The graph displays the results for different subjects with different mesh resolutions.

Figure 10 :

 10 Figure 10: Lighting style optimized for a reference image (a) using different types of materials for the rendered object (b), (c) and (d).

submitted to Pacific Graphics(2018)