P Barla

R Pacanowski

P Vangorp

A Composite BRDF Model for Hazy Gloss

Keywords:

We introduce a bidirectional reflectance distribution function (BRDF) model for the rendering of materials that exhibit hazy reflections, whereby the specular reflections appear to be flanked by a surrounding halo. The focus of this work is on artistic control and ease of implementation for real-time and off-line rendering. We propose relying on a composite material based on a pair of arbitrary BRDF models; however, instead of controlling their physical parameters, we expose perceptual parameters inspired by visual experiments [VBF17]. Our main contribution then consists in a mapping from perceptual to physical parameters that ensures the resulting composite BRDF is valid in terms of reciprocity, positivity and energy conservation. The immediate benefit of our approach is to provide direct artistic control over both the intensity and extent of the haze effect, which is not only necessary for editing purposes, but also essential to vary haziness spatially over an object surface. Our solution is also simple to implement as it requires no new importance sampling strategy and relies on existing BRDF models. Such a simplicity is key to approximating the method for the editing of hazy gloss in real-time and for compositing.

Introduction and related work

Real-world materials often exhibit reflections that may be described as "hazy", where more or less sharp specular reflections appear to be flanked by a halo. At the physical microscopic level, haziness may be due to various causes, such as partial polishing, diffraction effects, or multiple layers. As detailed in the following, two different approaches have been taken in the literature to design Bidirectional Reflectance Distribution Functions (BRDFs) that produce such hazy reflections: the aforementioned physical causes may be directly modeled, providing a predictive approach to material appearance; or a mathematical model with sufficient degrees of freedom (d-o-f) to reproduce haze effects may be provided. In this paper, we focus on the latter since artistic manipulation of hazy gloss is our central objective.

Physical models One way to produce hazy reflections is to make use of multi-layered BRDF models (e.g., [START_REF] Weidlich | Arbitrarily layered micro-facet surfaces[END_REF][START_REF] Marschner | A comprehensive framework for rendering layered materials[END_REF]). They require the specification of many parameters such as the number of layers, their refractive indices, interface roughnesses and medium absorption and scattering properties. Hazy gloss may even be produced with a single interface thanks to diffraction effects. Existing models [CTL89, Sta99, LKYU12, HP17] are controlled by statistical properties of surface irregularities such as height roughness or auto-correlation distance, which have complex effects on the final appearance since they affect the BRDF in a wavelength-dependent manner. From a purely artistic point of view, these physicallyaccurate approaches provide too many d-o-f, while their parameters only indirectly control the final hazy appearance. In addition, they remain costly to apply in real-time, or even in production where rendering times directly affect budget costs.

Microfacet models The microfacet theory [START_REF] Cook R | A reflectance model for computer graphics[END_REF] has been widely adopted by the Computer Graphics community for its combination of physical plausibility and artistic ease-of-use. It relies on normal distribution functions, where different distributions lead to different specular highlight appearances. Until recently, the Beckmann distribution [START_REF] Beckmann | The scattering of electromagnetic waves from rough surfaces[END_REF] was widely used; but the GGX distribution [TR75, WMLT07] gained favor over it due to its heavier tails. The main visual difference is that the GGX distribution produces slightly hazier reflections compared to Beckmann's; however, it does not provide any control over haze. Later work has introduced distributions with one additional d-o-f that controls distribution tails [LKYU12, BSH12, Bur12, RBMS17]. Unfortunately, none of them provides a satisfactory control over hazy gloss. The Shifted Gamma Distribution [BSH12], the peak of the distribution is strongly affected when attempting to modify its tails. The GTR [START_REF] Burley | Physically based shading at Disney[END_REF] and Student-t [RBMS17] distributions only permit achieving slightly hazier results than the GGX distribution. The ABC model [START_REF] Löw | Brdf models for accurate and efficient rendering of glossy surfaces[END_REF] produces a halo of very large extent that cannot be controlled by the artist.

Another simpler approach is to combine two or more distributions, which has oftentimes been deemed necessary for fitting BRDFs (e.g., [LFTG,[START_REF] Durand | Experimental analysis of BRDF models[END_REF]). Combination may also be performed manually for gaining more control over the shape of the specular term, as done by artists at Pixar or ImageWorks for instance [HMC * ]. However, the manipulation of haze is only indirect in this case and requires trial and error. In particular, it is extremely difficult to create materials where haze varies spatially over a surface without affecting other material properties. When dealing with anisotropic materials, control becomes even more tedious since the number of parameters is increased.

Perception of haze Recent work in visual perception [VBF17] has

shown that humans are visually sensitive to haziness in some specific configurations. The authors employ a specular BRDF based on two Ward BRDF components [START_REF] Ward | Measuring and modeling anisotropic reflection[END_REF], one of narrow extent (i.e., small roughness), the other of wide extent (i.e., large roughness). Their perceptual experiments show that haziness is a dimension of gloss that is distinct from contrast or distinctness of image [HH87, PFG00], and is more complex than the haze measurement retained by the ASTM [START_REF]ASTM: Standard Test Methods for Measurement of Gloss of High-Gloss Surfaces by Goniophotometry[END_REF]. Most importantly, none of the BRDF component parameters is able to account for perceived haziness: hence the perception of hazy gloss is not directly dependent on physical parameters. It is shown instead that a post-hoc decomposition of the BRDF into a specular core and a surrounding halo yields high correlation with subjective haziness ratings. Recent work (e.g., [SGM * 16]) provides perceptual parameters to navigate the space of material appearance, yet none provides a direct control over haziness.

Our approach Throughout the paper, we focus on specular BRDFs and leave the diffuse term untouched. Our approach takes inspiration from the core/halo decomposition of Vangorp et al. [VBF17] to introduce a BRDF model that grants direct control over hazy effects in a physically-correct manner. We introduce material parameters that manipulate not only the haze magnitude but also its extent, which in practice exposes an additional d-o-f compared to distributions with controllable tails [LKYU12,BSH12, Bur12, RBMS17]. In particular, our method affects the distribution tails while leaving the specular core mostly unchanged (even with anisotropic materials), and permits producing a much more noticeable hazy gloss appearance compared to existing alternatives. This is achieved by using a sum of two arbitrary BRDFs having the same model (e.g., microfacet-based with GGX or Beckmann distribution), whose physically-based parameters are obtained by a mapping from our perceptually-based core and halo parameters, while guaranteeing reciprocity, positivity and energy conservation [NRH * 77].

This simple approach has many benefits. It does not require any change in the rendering system, since only material parameters are modified; in particular, there is no need to introduce any new importance sampling strategy. It is also independent of the choice of distribution model, which makes it a viable solution for most rendering engines (such as those working with microfacet models); the method may even be implemented in the interface of a material editor. In spite of its simplicity, our method provides an efficient solution to a problem that would be hard to solve with a more complex single-component specular BRDF model.

Hazy gloss model

We first introduce a physical BRDF model made of a sum of two specular components (Section 2.1), which we then reorganize into a sum of specular core and halo components following recent research in visual perception (Section 2.2). We next describe a parametrization that guarantees that material components stay within the space allowed by physical constraints (Section 2.3), before providing the full haze mapping from perceptual to physical parameters in algorithmic form (Section 2.4). Table 1 lists the symbols and notations used throughout this section.

Physical BRDF model

We consider a composite BRDF fr having a pair of components sharing the same Fresnel term:

fr(ω i , ωo) = (1 -β) fn(ω i , ωo) + β fw(ω i , ωo) Fr(θ d ), (1)
where ω i and ωo are incoming and outgoing directions, Fr is the Fresnel term, θ d = acos(h • ω) is the difference angle with h = ωi+ωo ωi+ωo the halfway vector [START_REF] Rusinkiewicz | A new change of variables for efficient BRDF representation[END_REF], fn and fw are functions characterizing narrow and wide reflections respectively, and β ∈ [0, 1) linearly blends between the two functions. A common way to define fn and fw is through microfacet theory, in which case they model distributions of small and large roughness respectively † . The mixture ratio β then corresponds to the relative area occupied by the microfacets belonging to the distribution of large roughness. Whether the composite BRDF fr is physically-based obviously depends on whether fnFr and fwFr are themselves physicallybased, but also on their potential interactions through shadowing and masking (see Section 3). 2, even though the model of Equation 1 yields a more complex specular BRDF, it does not provide an independent control over distribution tails. Indeed, when the intensity of the wide component is modified through β (Figure 2b), not only the tails are modified, but the whole shape of the BRDF changes; in particular, the intensity in the mirror direction is affected. The same side-effect occurs when the roughness of the wide component is changed (Figure 2c). Having a direct and separate control over the peak and tails of the specular BRDF is not only desired to provide an artist-friendly handle on hazy gloss; it is also necessary if one wants to vary haziness across an object surface in a controllable manner.

Perceptual decomposition

Vangorp et al.

[VBF17] have suggested that the perception of hazy gloss relies on a decomposition of the specular BRDF into a specular core and a surrounding halo. In particular, they show that the energy of their halo component is strongly correlated to subjective haziness ratings obtained through a perceptual experiment. However, they only propose a post-hoc analysis and do not discuss how to design a physically-based BRDF model providing a perceptually-relevant control over haze.

We take inspiration from their work to re-express the model of Equation 1 as the sum of a specular core and a surrounding halo:

fr(ω i , ωo) = Fc(θ d ) fn(ω i , ωo) + K h (θ d )g h (ω i , ωo). (2)
More precisely, the specular core has the same shape fn as the narrow component of Equation 1, but is affected by a different Fresnel term Fc. The halo component is affected by an intensity K h and we propose to control its shape by g h = fw -P fn, where P ∈ [0, 1] is computed so that the contribution of the halo in the specular direction is minimized. As illustrated in the right column of Figure 2, the purpose of this decomposition is to grant independent control over the peak and tails of the distribution.

Equation 2 is obtained from Equation 1 by adding βFrP fn to the narrow component, and subtracting it from the wide component:

fr = (1 -β) fnFr + βP fnFr + β fwFr -βP fnFr = (1 -β) + βP Fr fn + βFr( fw -P fn).
By comparison with Equation 2 we obtain:

Fc(θ d ) = 1 -(1 -P(θ d ))β Fr(θ d ), (3)
K h (θ d ) = βFr(θ d ). (4)
We emphasize that unlike fn or fw, g h is not a valid BRDF function as it may become negative; neither is K h a valid Fresnel term. This is not problematic since Equation 2 is never directly evaluated P. Barla & R. Pacanowski & P. Vangorp / A Composite BRDF Model for Hazy Gloss but instead used for artistic control, after which its parameters are mapped back to physical parameters controlling Equation 1, which is evaluated. The only case where Equations 1 and 2 have the same form is when β = 0: the halo term then vanishes (K h = 0) and the specular core becomes identical to the narrow component (Fc = Fr).

An important consequence of the decomposition of Equation 2 is that for dielectric materials, the intensity of the specular core is uniquely determined by the θ d = 0 configuration. Indeed, the Fresnel term Fc is then uniquely determined by rc = Fc(0). ‡ This means that Equation 3 only needs to be evaluated at θ d = 0, hence knowing p = P(0) is sufficient to yield a full bijection between Equations 1 and 2. Our goal is to make g h vanish in the specular direction, as shown in cyan in Figure 2d-f. In the θ d = 0 configuration, this translates to g h (n, n) = 0, which is achieved by setting p = fw(n,n) fn(n,n) . We may thus regard p as the ratio of peak values between the wide and narrow components in the specular direction at normal incidence. The case of conductor materials may be treated similarly, as detailed in Section 2.4.

The exact analytic formula for the peak removal function P actually depends on the choice of underlying BRDF model. As detailed in the supplementary document for the case of microfacet theory, when θ d tends toward π 2 , P tends toward 1. As a result, the separation between the specular core and halo is not perfect at grazing angles, which is necessary to obtain a valid BRDF. Fortunately, this has little visible impact on results since reflections are severely compressed close to object contours. We emphasize that the analysis of P is not needed to implement the model, as we only require p = P(0) to obtain a full bijection between Equations 1 and 2.

Haze parametrization

The specular core component of Equation 2 may be directly controlled by an artist, for instance by adjusting its reflectivity rc and its roughness in fn. Our next goal is to find artistic parameters for controlling the halo component: its intensity K h and the extent of its shape g h .

Haze intensity

We want to provide a simple control over haze intensity K h , while at the same time guaranteeing physical plausibility. One may suggest to let the artist directly control β in Equation 4. However, this would be equivalent to modifying the mixture weight in Equation 1, which does not provide an independent control over haziness as shown in Figure 2b. We must thus instead provide a control that modifies both β and Fr in Equation 4, while ensuring that the reciprocity, energy conservation and positivity constraints of the BRDF model are always ensured. The first of these constraints, reciprocity, is guaranteed provided that fn and fw are themselves reciprocal. The positivity constraint (denoted P) must apply to each of the narrow and wide components in Equation 1, yielding (1β)Fr ≥ 0 and βFr ≥ 0. The energy conservation constraint (denoted E) is obtained by requiring that Fr ≤ 1. § ‡ The corresponding refractive index is given by ηc =

1+ √ rc 1- √ rc . § This is equivalent to enforcing Fr ∈ [0, 1] and β ∈ [0, 1].
In order to write the P and E constraints in terms of K h , we reexpress the global Fresnel term Fr in terms of our decomposition:

Fr(θ d ) = (1 -β)Fr(θ d ) + βFr(θ d ), = Fc(θ d ) -βFr(θ d )P(θ d ) + βFr(θ d ), = Fc(θ d ) + (1 -P(θ d ))K h (θ d ), (5)
where we have used Equation 3 in order to obtain the second line and Equation 4 for the third line.

We next observe that if the reflectivity r = Fr(0) remains in the [0, 1] range, then Fresnel Equations ouput values Fr(θ d ) ∈ [0, 1] as well for all

θ d ∈ [0, π 2 ]
. This means that we may safely restrict ourselves to the case where θ d = 0 once again, and express the positivity and energy conservation constraints in terms of k h = K h (0):

P: 0 ≤ k h ≤ rc p and E: k h ≤ 1 -rc 1 -p . ( 6
)
The new positivity constraint is obtained by applying Equation 3 to (1β)r ≥ 0, yielding rc ≥ βrp; then using Equation 4 to replace βr by k h . The new energy conservation constraint directly follows from Equation 5.

We visualize the constraints on k h as a function of rc using a pair of red lines in Figure 3-left. The two constraints intersect when rc = p yielding k h = 1, which should be considered a limiting case since it amounts to having a single wide specular component. In order to control k h , we propose to make use of a simple linear interpolation of k h from 0 to the physical bounds (black lines in Figure 3-left), which fully spans the space of valid BRDFs:

k h = β h rc p if rc ≤ p β h 1-rc 1-p otherwise, (7)
where β h ∈ [0, 1) is a haziness parameter that controls the proportion of available haze effect, given rc and p. We show the effect of β h on hazy gloss in Figure 3 for dielectric and conductor materials. Note that in the case of dielectrics, we use smaller values for β h ; indeed, higher values would yield a global reflectivity r more typical of a conductor. As a rule of thumb, we usually set β h ∈ [0, 0.1] for dielectrics, even though it may be increased for artistic purposes. We detail in the supplemental document the relationship between β h and the halo energy introduced by Vangorp et al.

[VBF17] to explain subjective haziness ratings.

In some situations, the C 1 discontinuity of Equation 7 occurring at rc = p may be an issue. For instance, for smooth surface variations of the specular reflectivity rc, the haze effect might exhibit visual discontinuities. We address this issue by replacing the piecewise linear form of Equation 7 with a quadratic rational Bézier curve (gray curves in Figure 3) that interpolates three control points: p 0 = (0, 0), p 1 = (p, β h ) and p 2 = (1, 0) with weights {1, w, 1}. It is guaranteed to remain in the (p 0 , p 1 , p 2 ) triangle. The additional parameter w ∈ R + controls the smoothness of the interpolation. Such a smooth interpolation comes at a price: only a subset of the space of parameters is then reachable. Equation 7 is obtained in the limit of w → ∞. The formula for k h using rational Bézier interpolation is provided in the Appendix. In order to avoid the C 1 discontinuity at rc = p, one may compute k h using a rational Bézier curve controlled by the points {p 0 , p 1 , p 2 } and weights {1, w, 1}. Four such curves with weights w ∈ {1, 2, 4, 8} are shown in gray. Right: each row of rendered spheres shows materials with increasing haziness β h for a given core reflectivity rc typical of a dielectric on top, and of a conductor at bottom. We use αn = 0.01 and λ h = 7 in both cases, and an additional greenish Lambertian term for the dielectric material.

Haze extent

The angular extent of the halo component depends on g h , and thus on the roughnesses of fn and fw. Different BRDF models use different notations and formula for roughness; we will follow the most common notation and denote roughness by α. For models that make use of different conventions (e.g., Ashikhmin-Shirley [START_REF] Ashikhmin | An anisotropic Phong BRDF model[END_REF]), their parameter should be remapped to α.

Let us first consider the isotropic case. By construction the roughness of the wide component αw must be greater than the roughness of the narrow component αn. We choose to define the former using αw = αn(1 + λ h ), where λ h ∈ R + controls the extent of the halo component in units of αn. This way, the extent of the halo component is relative to the extent of the specular core component. The configuration where λ h = 0 should be considered a limiting case, since the wide component then merges with the narrow one into a single-component BRDF. The first three columns of Figure 4 illustrate the effect of different values of λ h on the extent of the halo component.

For the general case of an anisotropic BRDF model, we denote by α x n and α y n the roughnesses of the narrow component in the tangential and binormal directions respectively, while α x w and α y w are the roughnesses of the wide component. We could have used equations similar to the isotropic case independently for each of the roughnesses along the tangential and binormal directions, but this would have had the effect of expressing haze extents in different units along different directions when α x n = α y n . We opt instead for the following definition:

α x w α y w = α x n α y n + λ x h λ y h α x n α y n , (8)
where λ x h ∈ R + and λ y h ∈ R + control the spread of the halo com-ponent in the same units along tangential and binormal directions respectively. When all components are isotropic, Equation 8 naturally reverts to the previously defined isotropic configuration. Our model also accommodates configurations where only one of the two components is isotropic, as shown in the last two columns of Figure 4. The supplemental document shows additional combinations of anisotropic roughnesses and haze extents.

Haze mapping

To recap, our approach essentially consists in a mapping from perceptual to physical parameters. It may actually be applied in the material editor of a rendering engine, either interactively when the artist is editing perceptual parameters, or as a batch process to convert textures storing variations of these parameters. The artist thus controls parameters of the specular core (rc, α x n and α For conductors, the refractive index is given by a complex number η + iκ. The additional d-o-f represented by the extinction coefficient κ thus makes the mapping under-constrained. Ideally, we would like a solution that reverts to the case of dielectrics when In practice, the artist provides a reflectivity rc and an edge-tint g for the specular core. Our haze mapping converts rc to r as before, but leaves g unmodified. We finally apply Gulbrandsen's mapping to retrieve η and κ and use them in the Fresnel equations for conductors. Since κ vanishes when g vanishes, this approach generalizes the case of dielectrics as desired.

The method works for both achromatic and chromatic materials. In the latter case though, we must decide on the color of the halo component. We suggest that it should have the same chroma as the specular core component, which we achieve with the following approach. We first rewrite reflectivities as r and rc to denote color vectors. We then define their chroma vector as c = r r = rc rc , with r (resp. rc) the color coefficient of maximum intensity in r (resp. rc). We then use r = rc = rc + (1p) kh c in lieu of Equation 5, where kh is obtained by replacing every occurrence of rc by rc in Equation 7. As shown in Figure 5, this gives the halo component the same color as the specular core, while leaving control over the edge tint color g. This approach is compatible with a physicallybased BRDF, which requires β to be a scalar: β = kh c r = kh r . As a summary, the general haze mapping is provided in the following Algorithm. In the next section, we explain how to instantiate it using various BRDF models. 2 provides a list of common BRDF models that may be used to instanciate both fn and fw, along with the corresponding formula for the peak ratio p (see the supplementary document for derivations). The Blinn-Phong [Bli77] and Ashikhmin-Shirley [AS00] models use shininess s {n,w} instead of roughness α {n,w} , which yield somewhat complex for-Figure 6: Our approach is modular in that it may be applied to a wide diversity of existing BRDF models. Here we show three such models, without and with haziness on each half. We fix rc = 0.05, αn = 0.02 (or sn = 5000 for Ashikhmin-Shirley) and λ h = 5. mula for p. In contrast, when using models involving either Beckmann [START_REF] Cook R | A reflectance model for computer graphics[END_REF][START_REF] Ward | Measuring and modeling anisotropic reflection[END_REF] or GGX [START_REF] Walter | Microfacet models for refraction through rough surfaces[END_REF] distributions, the peak ratio has a particularly concise formula. Note that our approach is compatible with models that have a coupled diffuse term designed to ensure energy conservation, such as Ashikhmin-Shirley [START_REF] Ashikhmin | An anisotropic Phong BRDF model[END_REF].

Figure 6 shows the result of our approach when using the models of Ashikhmin and Shirley [AS00], Ward [START_REF] Ward | Measuring and modeling anisotropic reflection[END_REF] and Walter et al. [START_REF] Walter | Microfacet models for refraction through rough surfaces[END_REF]. Observe in particular how hazy gloss is consistently affected by modifying β h in the same manner irrespectively of the choice of underlying model. The Ward model is also used in Figures 3,4, 5, 9 and 12; and the Walter et al. model in Figures 1, 11 and 13. We have not included BRDF models with additional degrees of freedom [Bur12, LKYU12, BSH12, RBMS17] since our goal is to provide an alternative solution.

Our approach is also compatible with legacy BRDF models that do not make use of a Fresnel term (e.g., [Bli77,[START_REF] Ward | Measuring and modeling anisotropic reflection[END_REF]). Equation 1 may then be written as fr = kn fn +kw fw, where kn and kw are scalar coefficients for the narrow and wide components respectively, with kn + kw ≤ 1 to preserve energy. This amounts to replacing the Fresnel term Fr by the reflectivity r, and setting kn = (1β)r and kw = βr. The rest of our approach remains essentially unchanged: we only need to output r = r rc rc instead of the refractive indices η and κ in the haze mapping. reflectance of a microsurface resulting from the mixture of two microfacet distributions. One may assume the mixture to be constructed of relatively large patches (on a micro-scale) of the component microsurfaces. The masking-shadowing effects across different patches may then be considered as negligible. In contrast, if the two distributions are intertwined, then it becomes necessary to consider a compound masking-shadowing term.

(sw+2)(sw+4)(2 -sn /2 +sn) (sn+2)(sn+4)(2 -sw/2 +sw) Ashikhmin-Shirley [AS00] √ (s x w +1)(s y w+1) √ (s x n +1)(s y n +1) Cook-Torrance [CT82] α 2 n α 2 w Ward [War92]
For most microfacet-based models, the masking-shadowing term does not depend on roughness parameters; the compound term is then identical for narrow and wide components. An exception is Smith's model [Smi67], which offers the most physically-realistic option [START_REF] Heitz | Understanding the masking-shadowing function in microfacet-based BRDFs[END_REF]. In its separable form, it is given by G

2 (ω i , ωo) = G 1 (ω i , h)G 1 (ωo, h), with G 1 (ω, h) = χ + (ω•h)
1+Λ(ω) where χ + (x) equals 1 if x is positive and 0 otherwise. In our case, Λ(ω) = (1β)Λn(ω) + βΛw(ω), where Λn and Λw correspond to integrals over slopes of the narrow and wide distributions respectively.

The use of a large-patch or intertwined mixture of distributions is a matter of choice, and our approach is independent of this choice. Indeed, we rely on the peak ratio p in our haze mapping, which only considers the configuration (θ h , θ d ) = (0, 0), in which case the masking-shadowing term is equal to 1 (all microfacets are visible at normal incidence). We discuss the visual impact of using a compound masking-shadowing term in the supplementary document.

Results

We first compare our compound BRDF to existing models in Section 4.1. More complex rendering results -including spatial material variations, global illumination and transmission -are presented in Section 4.2. We also consider approximations that permit adapting our approach to the constraints of real-time or compositing applications in Section 4.3.

Comparisons

There is no easy way to compare different BRDF models since they rely on different material parameters; we thus resort to a quali- tative evaluation. To support this comparison, we provide GLSL shaders to be used in BRDF Explorer [Dis11] for each of the considered models, and encourage the reader to edit their parameters and compare their expressivity and ease of use. We have used these shaders in Figure 7 and attempted to find sets of parameters for the STD [RBMS17], ABC [START_REF] Löw | Brdf models for accurate and efficient rendering of glossy surfaces[END_REF] and SGD [BSH12] models that would yield similar visual appearances before editing (left halves).

We start with the STD model [RBMS17] in the left of Figure 7. It is similar to the GTR distribution [START_REF] Burley | Physically based shading at Disney[END_REF], but provides an analytical masking-shadowing term. Moreover, it generalizes both the GGX and Beckmann distributions through a single parameter γ that controls distribution tails. In particular, the left half of the figure is made identical to the left half of Figure 6(GGX) by setting γ = 2. As shown in the right half of the figure, setting γ to its minimal feasible value produces a slightly hazier but also significantly darker appearance. This has been acknowledged by the authors.

The ABC model is shown in the middle of Figure 7. The formulation of Löw et al. [START_REF] Löw | Brdf models for accurate and efficient rendering of glossy surfaces[END_REF] is inspired by diffraction-based models [START_REF] Church | The prediction of BRDFs from surface profile measurements[END_REF] but rewritten as a microfacet-based BRDF. Even though their masking-shadowing term is not physically-based, the model retains interest as it permits control of distribution tails through the C parameter. As shown in the right half of the figure, the hazy appearance is much stronger when increasing C, but at the same time it exhibits a very large extent. Since no other material parameter is provided to adjust haze extent, the model remains limited for artistic editing.

The SGD model [BSH12] is shown at right in Figure 7. It has originally been introduced for the fitting of BRDF data; here we use a simplified version of the model for the purpose of parameter editing (see the supplemental document for details). Unfortunately, as opposed to the two previously discussed models, a desired hazy appearance requires the modification of two parameters instead of one. This not only implies significant trial and error, but also precludes the use of the model for spatially-varying haziness. The model also shares the limitation of the ABC model in that it does not provide control over haze extent, even though its extent is less pronounced.

In contrast, our composite model provides a direct control over both haziness and haze extent through dedicated parameters (i.e., β h and λ h ) while retaining physical validity; this may be seen by comparing Figures 6 and7. One might wonder whether the mixture parameter β of the physical model of Equation 1 could be used to control haziness as well. As shown in Figure 8, while a linear variation of β h produces a consistent change in haziness throughout variations, the same linear variation of β makes the material look hazy at low values, but makes it look mostly very rough at higher values. We again invite the reader to compare the two BRDF models using the provided shaders for BRDFExplorer (we also include executables for ease of use, as well as video captures).

Rendering

As with any other BRDF model, rendering is achieved through the reflected radiance equation [Kaj86]:

Lr(x, ωo) = Ω fr(x, ω i , ωo) L i (x, ω i ) ω i • n dω i , (9)
with Lr and L i the reflected and incoming radiances, x a surface point and Ω the hemisphere centered around the normal n. In our case, Equation 9 takes the form Lr = (1β)Ln + βLw where Ln and Lw are obtained by replacing fr by fnFr and fwFr respectively. As a result, there is no need for any new importance sampling strategy: we sample one randomly chosen component with probability 1β for Ln and β for Lw.

A key benefit of our perceptual decomposition is that it grants spatial variations of both haziness β h and haze extent λ h independently of the specular core. This is shown in Figure 9, where we compare variations of the physical and perceptual parameters: only in the latter case does hazy gloss vary in a consistent and controllable way. Figure 10 shows our model applied with more complex spatial variations. We also show in the supplemental document how our model behaves with variations of the core reflectivity rc. Figure 11 shows a more complex scene that showcases various hazy materials: dielectrics and conductors, isotropic and anisotropic. The material parameters are listed in Table 3. Our model is not limited to opaque materials as shown in the left of Figure 13. Rendering with Bidirectional Transmittance Distribution Functions (BTDF) actually requires no modification to our approach -another advantage of relying on existing material models through a simple haze mapping. We simply use the same physical parameters (mixture ratio β, refractive index η + iκ and roughnesses α {x,y} {n,w} ) in a two-component BTDF of a form similar to Equation 1.

Approximations

In video games, the need for balancing high performance, low memory consumption and realistic appearance has led to the use of local lighting environments pre-filtered for a range of material parameters [KVHS00]. In order to avoid precomputing and storing high-dimensional lookup tables, recent work (e.g., [Kar13]) has proposed to re-organize Schlick's approximation [Sch94] of the Fresnel term. Omitting function variables for clarity, the Fresnel term is then re-expressed as: Figure 12: For real-time rendering, we compute four pre-filtered environment maps L 0 n , L 1 n , L 0 w and L 1 w , visualized on the four spheres on the left (we use αn = 0.012, λ h = 6.4). As shown on the right, this permits the manipulation of haziness in real-time (we fix rc = 0.5), either for the whole object or locally using a texture map (finger prints applied to the side of the face). The supplemental video shows a live capture.

Fr = (1 -∆)r + ∆, with ∆ = (1 -cos(θ d )) 5 . (10)
Inserting Equation 10 inside Equation 1 then 9 yields:

Lr = (1 -β)(L 0 n r + L 1 n ) + β(L 0 w r + L 1 w ), (11)
where L 0 {n,w} and L 1 {n,w} are reflected radiance functions obtained by replacing fr in Equation 9 with f {n,w} (1 -∆) and f {n,w} ∆ respectively.

The main benefit of Equation 11 is that Lr becomes a linear function controlled by physical parameters β and r. Artist-controlled perceptual parameters may thus be converted to physical parameters using our haze mapping, and Equation 11 evaluated on the fly. This is shown in Figure 12 and the supplemental video, where we demonstrate interactive spatial variations of haziness.

Another advantage of this approach is that it might be used to control hazy gloss at the compositing stage. The idea is to output render buffers for L 0 n , L 1 n , L 0 w and L 1 w , as well as auxiliary buffers holding material parameters β, r and p. The compositing artist then edits β h and rc obtained from auxiliary buffers, which are then interactively converted back to modified β and r buffers and plugged into Equation 11 to yield a new composite. Preliminary compositing results are shown in the supplemental document.

Discussion and future work

We have introduced a simple yet effective approach to the artistic control of hazy gloss for physically-based shading. The haze effect is controlled independently of the BRDF peak, in particular granting control over spatial variations of haze. method relies on a mapping from perceptual to physical parameters that only requires a few lines of code. Since it works with existing BRDF models, it may be easily integrated in existing rendering pipelines without much effort. Its simplicity is also key to its integration in real-time rendering applications or compositing pipelines.

In terms of performance, our haze mapping adds a negligible overhead compared to the evaluation of Equation 1. Using a composite model is obviously more costly compared to a singlecomponent specular BRDF. However, this is already used in production [HMC * ], which shows that the improved control over the shape of reflections is worth the increased computation time.

One effect of our haze mapping is to affect the global reflectivity r; hence it also indirectly modifies the refractive index η + iκ. We have seen in Figure 3 that in the case of dielectrics, the haziness β h should thus be restricted to limit r to small values, and hence η to physically-plausible indices. However, we prefer to leave this decision up to the artist, and only guarantee energy conservation (β h < 1). Another consequence is that transmitted radiance is affected with increased haziness, as shown in Figure 13-left. In the case of conductors, β h need not be restricted; however, for very bright metals, the high core reflectivity rc does not leave much room for the halo component. There is no work-around this limitation since it is due to the energy conservation constraint. Our approach also inherits a limitation of Gulbrandsen's remapping [Gul14] for colored conductors. As the reflectivity is increased, the edge tint has less and less effect on material appearance: colors near occluding contours tend to become a blend between the edge tint and the reflectivity color. Since an increase in haziness results in an increase in global reflectivity, we obtain the same kind of color alteration near object boundaries as shown in Figure 13-right.

We have assumed throughout that the narrow and wide components fn and fw of Equation 1 are defined using the same type of BRDF function. One may want to use two different types of functions, but this would complicate the peak ratio formula p without adding much more expressivity to the model. We have also limited our composite BRDF to a pair of components to be in accordance with existing perceptual studies on hazy gloss [VBF17]. Adding a third component would bring two additional degrees of freedom, but it is not clear how they might affect perception: would the new component be perceived independently or as part of the halo? As shown in the supplemental document, our preliminary tests favor the latter interpretation; however, further perceptual experimentations are required to evaluate whether this is always the case.

One might wonder how our perceptual model performs on the fitting of measured BRDF data. Since our perceptual and physical models are connected by a bijection, the space of BRDFs they span is by definition the same; hence fitting (like rendering) can be performed using Equation 1. This has already been addressed in previous work [START_REF] Durand | Experimental analysis of BRDF models[END_REF], where it is reported that using a pair of Cook-Torrance components reduces the fitting error by more than 25% for 26 out of 100 measured BRDFs of the MERL database.

Nevertheless, the only valid physical interpretation of our model is that of a mixture of distributions such as partially polished singlelayer materials. Our composite BRDF is thus not adapted to the modeling of bi-layered materials, as we assume a common Fresnel term for both narrow and wide components. Yet it would be interesting to provide control over hazy gloss in multi-layered materials through an inverse design approach. This represents a challenge as it is not clear how different layer configurations might affect perceived haziness. In addition, other perceptual dimensions are likely to emerge with more complex materials: in particular, some layers may be disentangled visually while others might not -a property we may call "layeredness". Finding explicit correspondences between physical and perceptual properties of multi-layered materials is an exciting direction of research for future work.

Figure 1 :

 1 Figure 1: A material probe rendered with our BRDF model in global illumination. The insets show the same object with haziness removed. Our method works with both dielectrics and conductors.

  The use of multiple components was first suggested in the seminal work of Cook and Torrance [CT82] on microfacet theory. It is nowadays routinely used in production [HMC * ], where such a combination of functions is primarily aimed at better shaping the † Each function is of the form f = D G 4| cos θ i || cos θo| where D and G are distribution and masking-shadowing terms respectively.

  c 2018 The Author(s) Computer Graphics Forum c 2018 The Eurographics Association and John Wiley & Sons Ltd.

  K h ∈ [0, 1] halo intensity (with k h = K h (θ d = 0)) P ∈ (0, 1] peak removal function (with p = P(θ d = 0)) Variables ω {i,o} ∈ S 2 {incoming, outgoing} direction {n, h} ∈ S 2 {normal, halfway} direction θ {h,d} ∈ [0, π/2] {halfway, difference} angle Physical parameters η ∈ (1, ∞) real part of refractive index κ ∈ R + imaginary part of refractive index r ∈ [0, 1] global reflectivity (r = Fr(θ d = 0)) β ∈ [0, 1] mixture ratio of physical components α {x,y} {n,w} ∈ R + {narrow, wide} roughness along {tangent, binormal} directions Perceptual parameters rc ∈ [0, 1] specular core reflectivity (rc = Fc(θ d = 0)) g ∈ [0, 1] edge tint [Gul14] β h ∈ [0, 1) haziness (proportion of available haze) λ {x,y} h ∈ R + haze extent along {tangent, binormal} directions specular component. However, as shown in the left column of Figure

Figure 2 :

 2 Figure 2: Comparison between the physical BRDF of Equation 1 and the perceptual decomposition of Equation 2. The narrow and wide components of the former are shown in blue and green respectively, while the specular core and surrounding halo of the latter are shown in magenta and cyan. Insets show renderings of a sphere with a directional light source. The two decompositions initially yield the same composite BRDF (in red), as shown in (a,d). The BRDF peak is altered when the parameters of the physical decomposition are modified to yield higher (b) or wider (c) tails. When the parameters of the perceptual decomposition are changed, the peak is not affected while the tails are modified as desired in (e,f).

Figure 3 :

 3 Figure3: Left: the haze intensity k h at θ d = 0 is given as a function of the core reflectivity rc. The increasing (resp. decreasing) red line corresponds to the positivity P (resp. energy conservation E) BRDF constraint. Valid values for k h (black lines) are obtained by linearly interpolating between 0 and the BRDF constraints, via a user-controlled parameter β h ∈ [0, 1). In order to avoid the C 1 discontinuity at rc = p, one may compute k h using a rational Bézier curve controlled by the points {p 0 , p 1 , p 2 } and weights {1, w, 1}. Four such curves with weights w ∈ {1, 2, 4, 8} are shown in gray. Right: each row of rendered spheres shows materials with increasing haziness β h for a given core reflectivity rc typical of a dielectric on top, and of a conductor at bottom. We use αn = 0.01 and λ h = 7 in both cases, and an additional greenish Lambertian term for the dielectric material.

yn(recall that α x n and α y n

 y ) and of the halo component (β h , λ x h and λ y h ). Next k h is computed using either Equation 7 or the smoothly-varying version using Equation 15 (see Appendix). The perceptual parameters are finally converted into the physical parameters of Equation 1, namely r, α x w , α y w , and β = k h r remain unchanged). Equation 1 is then evaluated during rendering.For dielectrics, the global reflectivity r might either be directly used to control Schlick's approximation [Sch94] to the Fresnel term, or converted to a refractive index η = 1+ √ r 1-√ r to be used in the exact Fresnel equations.

Figure 4 :

 4 Figure 4: Each row shows materials with different narrow roughnesses αn and varying haze extents λ h (with rc = 0.03 and β h = 0.1 fixed). The first column shows no haze; the next two columns show isotropic haze effects of different extents and the last two columns show anisotropic haze effects. Please zoom in to see haze effects.

Figure 5 :

 5 Figure 5: In the case of colored conductors (here rp = 0.5, αn = 0.01 and λ h = 5.75), we assign the same color to the core and halo components. Compared to dielectric materials, this leaves a d-o-f that is controlled by g, an edge tint parameter [Gul14] that may be different from the reflectivity color c as seen in the rightmost image where reflected radiance takes on a purple tint near contours.

  ← rc ∞ ⊲ Max color coeff if smoothVar then u ← paramCoord(rc, p) ⊲ Equations 12-14 kh ← hazeIntensity(u, β h , w) ⊲ Equation 15 else kh ← hazeIntensity(rc, p, β h ) ⊲ Equation 7 end if r ← rc + (1p) kh ⊲ Equation 5 η, κ ← colorMapping( r rc rc, g)

w

  Compound masking-shadowing In the context of microfacet theory[START_REF] Cook R | A reflectance model for computer graphics[END_REF], Equation 1 should be interpreted as describing the

Figure 7 :

 7 Figure 7: Recent BRDF models offer control over distribution tails thanks to an additional parameter; in each image, the left and right halves show renderings before and after parameter editing, as indicated on top. The Student-t distribution (STD) achieves slightly hazier but also significantly darker results. The ABC model produces strong haziness but with no control over its large extent. Our simplified version of the Shifted Gamma distribution (SGD) requires tedious manipulations of two parameters.

Figure 8 :

 8 Figure 8: We compare variations of the mixture weight β in the physical model (top row) with variations of haziness β h in the perceptual model (bottom row). Both parameters are uniformly sampled in the [0, 0.95] range; other parameters are chosen to get identical images in the left column (αn = 0.012, r = rc = 0.5, αw = 0.088 or equivalently λ h = 6.4). The visual differences are mostly apparent in the three right-most images: increasing β decreases the intensity of sharp reflections; increasing β h only affects haziness.

Figure 9 :

 9 Figure 9: Left: the Speedshape model rendered with and without haze (we use rc = 0.04, αn = 0.03, β h = 0.1 and λ h = 5 and a purple Lambertian base). Middle: when spatial variations are applied to haziness β h (bottom), the spiral pattern appears clearly, as opposed to when we vary β (top). Right: variations of haze extent λ h (bottom) are much more noticeable compared to variations of wide roughness αw (top). The physical parameter ranges for β and αw are computed using our haze mapping for the sake of comparison.

Figure 10 :

 10 Figure 10: Detailed variations of haziness β h ∈ [0, 0.99]. Other parameters are kept fixed at rc = 0.5, g = 0, αn = 0.026 and λ h = 5.Table 3: List of parameters used in Figures 1 and 11

  Figure1demonstrates the use of our model in global illumination rendering. Each material probe is shown with and without haze on the spherical part (other parts are left unchanged). Observe how our method preserves the intensity of sharp specular reflections for either dielectrics, achromatic conductors or colored conductors. Figure11shows a more complex scene that showcases various hazy materials: dielectrics and conductors, isotropic and anisotropic. The material parameters are listed in Table3. Our model is not limited to opaque materials as shown in the left of Figure13. Rendering with Bidirectional Transmittance Distribution Functions (BTDF) actually requires no modification to our approach -another advantage of relying on existing material models through a simple haze mapping. We simply use the same physical parameters (mixture ratio β, refractive index η + iκ and roughnesses α

Figure 11 :

 11 Figure11: A 3D scene rendered with path tracing using hazy (top) and haze-free (bottom) materials. From left to right: the Fertility statue is made of a colored conductor using a wide haze extent and a pinkish edge tint; the Probe B object uses the achromatic conductor from Figure1; the Vase combines a dielectric BRDF and a dark greenish Lambertian base; the body of the Teapot is made of a colored conductor with an anisotropic halo component of vertical extent. All other materials do not use our composite BRDF model.

Figure 13 :

 13 Figure 13: Increasing haziness modifies the refractive index. Left: for a dielectric, transmitted radiance is affected as seen by comparing background distortions between the two halves (we use rc = 0.04). Right: for a conductor, edge tint is affected in the left half but is hardly visible in the right half (we use rc = [0.6, 0.52, 0.3] and g = [1, 0, 0.85]). In both cases, αn = 0.02 and λ h = 4.

Table 1 :

 1 Table of symbols and notations

	Functions
	fr f {n,w} ∈ R + ∈ R + F {r,c} ∈ [0, 1] {global, core} Fresnel terms composite BRDF {narrow, wide} BRDF terms g h ∈ R halo term (not a BRDF term)

Table 2 :

 2 Peak ratios for common BRDF models.

	BRDF models for f {n,w}	peak ratios p
	Blinn-Phong [Bli77]

c 2018 The Author(s) Computer Graphics Forum c 2018 The Eurographics Association and John Wiley & Sons Ltd.

Acknowledgements We would like to thank Laurent Belcour and the anonymous reviewers for their feedback. This project has been supported by the ANR VIDA (ANR-17-CE23-0017). We are grateful to the Aim@shape library for 3D models and Paul Debevec for environment maps. The SpeedShape model is courtesy of Jiri Filip.

Appendix

The quadratic rational Bézier curve C with control points {p 0 , p 1 , p 2 } and weights {1, w, 1} of Figure 3 is given by:

We first invert the first coordinate of C to retrieve u, yielding:

This is the unique solution that yields a parametric coordinate in the desired [0.

.1] range. To avoid numerical inacuracies when b ≈ 1 we transition to a Taylor expansion of u in a small window around the corresponding core reflectivity r ⋆ c :

The halo intensity at θ d = 0 is then directly obtained from the second coordinate of C, yielding:


Damien Bricard
email: bricard@irdes.fr

Stéphane Legleye

Changes in smoking behaviour over family transitions: evidence for anticipation and adaptation effects

Keywords: smoking, family life transitions, life cycle, longitudinal analysis, anticipation effects; adaptation effects

come

Introduction

Over the past decades, the dissemination of information on the harmful effects of tobacco and the implementation of public policies have contributed to a gradual and continuous decrease in smoking prevalence. In fact, this decline in smoking was largely attributed to a decrease in prevalence in the most privileged groups and was more contrasted for women [START_REF] Bricard | Educational inequalities in smoking over the life cycle: an analysis by cohort and gender[END_REF][START_REF] Khlat | Disadvantaged Social Groups and the Cigarette Epidemic: Limits of the Diffusion of Innovations Vision[END_REF][START_REF] Legleye | Widening inequalities in smoking initiation and cessation patterns: a cohort and gender analysis in France[END_REF] . These evolutions have led to a sharp increase in social inequalities in smoking and a convergence of male and female prevalence.

At the present time, differences between men and women remain in the age pattern of smoking for the most recent cohorts: in the most educated group, women more frequently smoke regularly than men before the age of 25, and do so less frequently after this age. These differences can probably be explained by a greater cessation of women at the approach of the their first pregnancy, especially among the most educated, and could also reflect the specificities of smoking patterns by gender and social group [START_REF] Graham | Socioeconomic lifecourse influences on women's smoking status in early adulthood[END_REF][START_REF] Graham | Lifecourse influences on women's smoking before, during and after pregnancy[END_REF][START_REF] Pampel | Life Course Changes in Smoking by Gender and Education: A Cohort Comparison Across France and the United States[END_REF] .

Exploring the dynamics of changes in smoking behaviors over the life course appears to be a promising line of research to understand the persistence as well as the evolution of smoking disparities over cohorts and over ages.

Smoking during childhood and adolescence has been studied in detail since this period of life is crucial to the understanding of the context of experimentation and transition to daily use of cigarettes. This literature has highlighted the responsiveness of smoking initiation to price and regulation factors as well as family and peer contexts explaining smoking initiation [START_REF] Chaloupka | Price, tobacco control policies and smoking among young adults[END_REF][START_REF] Francesconi | The effect of lone motherhood on the smoking behavior of young adults[END_REF][START_REF] Powell | The importance of peer effects, cigarette prices and tobacco control policies for youth smoking behavior[END_REF] . But the period of smoking initiation goes beyond adolescence and corresponds to the period of the transition into adulthood. This period of life is associated with the transitions to adult work and family roles and the related changes in health behaviors [START_REF] Oesterle | Men's and women's pathways to adulthood and their adolescent precursors[END_REF][START_REF] Staff | Substance use changes and social role transitions: Proximal developmental effects on ongoing trajectories from late adolescence through early adulthood[END_REF] , which could explain differences in smoking across social groups [START_REF] Pampel | Life course transitions in early adulthood and SES disparities in tobacco use[END_REF] . To date, the literature on the impact of family life transitions on smoking behaviors is very limited. The present study is therefore an attempt to fill this gap by focusing on the impact of family transitions on smoking behavior, especially in relation to couple formation, first childbirth and separation.

Empirical findings on smoking behaviors and family events are mostly based on contemporaneous correlations in cross-sectional data, with of a focus on the effect of family status rather than on family events. A protective effect of being in couple on smoking has been shown with a greater influence of living with a partner for men, whereas women are more affected by separation [START_REF] Cho | Marital status and smoking in Korea: the influence of gender and age[END_REF][START_REF] Nystedt | Marital life course events and smoking behaviour in Sweden 1980-2000[END_REF] which is consistent with the greater vulnerability of women to stressful events [START_REF] Mckee | Sex differences in the effects of stressful life events on changes in smoking status[END_REF] . These effects can also be explained by an influence of partner on smoking, but for that it is necessary to distinguish the effects of matching or bargaining: matching between partners would come from a tendency of individuals to choose a partner sharing the same lifestyles or preferences whereas bargaining refers to a negotiation process between spouses or partners about smoking [START_REF] Umberson | Gender, marital status and the social control of health behavior[END_REF] . Some authors have shown that the correlation of smoking between partners would rather be due to the effects of matching than bargaining [START_REF] Clark | Don't give up on me baby: Spousal correlation in smoking behaviour[END_REF] but an influence of partner on smoking have also been demonstrated [START_REF] Homish | Spousal influence on smoking behaviors in a US community sample of newly married couples[END_REF] . Concerning the birth of a child, the literature shows few changes in smoking of men in comparison with women. For women, there would be relatively little long-term effect because of relapse after pregnancy [START_REF] Fingerhut | Smoking before, during, and after pregnancy[END_REF][START_REF] Kahn | A reexamination of smoking before, during, and after pregnancy[END_REF][START_REF] Lelong | Postpartum return to smoking among usual smokers who quit during pregnancy[END_REF] . Studies of men's smoking behaviors at the time of the birth of their child are rare and show little changes in smoking behavior or only among the besteducated [START_REF] Brenner | The role of childbirth in smoking cessation[END_REF] , with the exception of changes in their habits such as not smoking in the home [START_REF] Blackburn | Smoking behaviour change among fathers of new infants[END_REF] .

One shortcoming of available studies is that they have considered family transitions as discrete events and only considered the subsequent changes. Viewing family transitions as discrete events per see is questionable, as those transitions are the outcome of a long process and their consequences develop over time. For instance, marital relations start to deteriorate long before the partners break up, and there is often a long time elapsing from the initiation of the project of having a child to the actual birth [START_REF] Nystedt | Marital life course events and smoking behaviour in Sweden 1980-2000[END_REF] . Family transitions may therefore lead to changes in behavior occurring before the happening of the event ("anticipation effects"). They may also lead to changes after the transition has taken place, and those may be temporary or permanent ("adaptation effects"). Another aspect which has hardly been addressed is that of the gendered dimension of the long-term processes. In other words, do transitions within the family similarly influence men's and women's health behaviors, and if not, what are the gender-specific patterns?

In this paper, we investigate the question of family life transitions and smoking with a perspective integrating both the time and gender dimensions. We are specifically interested in the roles of partnership, separation or childbirth as transitions triggering changes in terms of smoking uptake or cessation and in the gender-specific dynamics in this respect. Furthermore, we enlarge the observation window by considering both the three-year period preceding the actual transitions and the three-year period following them, in order to visualize the unfolding of the processes, from anticipation of impending changes to adaptation to different family life circumstances, be they positive (partnership consolidation, birth of a child) or negative (partnership breakdown), or both at the same time. To conduct this analysis, we use the Santé et Itinéraires Professionnels (SIP) survey conducted in 2006 in France, which provides both retrospective information on the timing of family life events and age of smoking initiation and eventual cessation.

Methods

Empirical strategy

The aim of this paper is to estimate the impact of family events on smoking trajectory, and particularly the changes in smoking behavior before (anticipation), during (contemporary) and after (adaptation) the transition to a different family situation.

A general approach

We favor a discrete time approach with panel econometric strategies to distinguish between individual variations and within individual variations in longitudinal data structure. This strategy is usual in multilevel analysis and estimates both within and between individual effects by including individual means of time-varying variables, called within-between estimator or Mundlak specification in the econometric literature [START_REF] Dieleman | Random-Effects, Fixed-Effects and the within-between Specification for Clustered Data in Observational Health Studies: A Simulation Study[END_REF][START_REF] Mundlak | On the pooling of time series and cross section data[END_REF] . The formulation allows to test within and between effects in the same model.

An alternative approach often used in linear model is to estimate within effects separately using fixed effects model. For the case of a binary outcome variable, such as smoking status, we favor the withinbetween strategy instead of the conditional logit fixed effects model that would have excluded individuals without longitudinal variation. This approach has the advantage to include ever-smokers and never-smokers in the analysis.

We consider a discrete-time logistic regression of smoking with the following formulation of the latent variable of smoking for the individual i at age t:

𝑆 𝑖𝑡 * =

A specific approach to measure leads and lags

The general approach presented before assumes a contemporary relationship between variables which could be very restrictive. Here, we propose a specific approach to measure leads (anticipation) and lags (adaptation) effects of family transitions on smoking in addition to contemporary effects. This approach estimates effect of life events on smoking behaviors up to three years before and after the event. It has been used in the context of the study of the effect of life events on life satisfaction [START_REF] Clark | Lags and leads in life satisfaction: a test of the baseline hypothesis*[END_REF][START_REF] Frijters | Life Satisfaction Dynamics with Quarterly Life Event Data*[END_REF][START_REF] Powdthavee | What happens to people before and after disability? Focusing effects, lead effects, and adaptation in different areas of life[END_REF] , alcohol consumption [START_REF] Tamers | The impact of stressful life events on excessive alcohol consumption in the French population: findings from the GAZEL cohort study[END_REF] and more generally in the context of treatment effect analysis of public policy for example [START_REF] Autor | Outsourcing at Will: The Contribution of Unjust Dismissal Doctrine to the Growth of Employment Outsourcing[END_REF] .

We propose to consider a three years window around life events, with dummies of leads effect (3 years before, 2 years before, on year before), the contemporary effect and dummies of lags effect (one year after, two years after, three of more years after) : We assume here a reference category for the interpretation of this dynamic, which is "being at least three years before the occurrence of the event" (including never experiencing it).

𝑆 𝑖𝑡 * =
The effect of the different transitions are estimated jointly in the same model to adjust for successive events. We estimate a model for smoking initiation and another for cessation, separating men and women in each of them. The questionnaire made it possible to retrospectively date the age of initiation of daily smoking and the age at the time of smoking cessation ("Si vous fumez actuellement tous les jours, depuis quel âge?" / "Si auparavant vous fumiez tous les jours, de quel âge à quel âge ?"). Assuming the continuity of smoking from the age the respondent started until the age s/he quit (ex-smokers) or the year of the survey (current smokers), we constructed a smoking binary indicator for every year between the age of 10 and the age at interview, whose value is one if a respondent smoked or zero otherwise.

Data

We use information collected on a biographical grid to construct the family status variables. Partnership episodes are reported and allow to date the age at union and separation periods ("Avez-vous déjà vécu en couple ? Si oui, situez la ou les périodes"). The questionnaire defined a partnership episode as a cohabitation period (not necessarily a marriage) during at least one year. Shorter episodes are reported only in the case of the birth of a child. We also used information on the age at first birth ("Avez-vous eu ou bien adopté des enfants ? Si oui, notez les années de naissance").

Adjustment was made for activity status as a time-varying variable in 5 categories: long-term employment (five years or more), short-employment (less than five years, or inactivity and unemployment of less than one year), unemployment (more than one year), inactivity (more than one year) and schooling. The following variables were also adjusted for: social class from last or current (6 categories); number of major life events during childhood (handicap or serious illness, health problems or death of a relative, family conflict or separation, violence, financial difficulties, etc.); educational level; migration status; residence in a rural municipality at the time of the survey; birth cohort, and; log of cigarette prices of the period (see Table S2 in supplementary materials for descriptive statistics of the variables).

The distributions of age at smoking initiation and cessation and of family events are presented in supplementary materials Table S1 (see Figures S1 andS2 in supplementary materials for graphical representations). This retrospective information is used to construct a person-years database of smoking status, family status and other time-varying or fixed individual variables. Given the age of the first family events, we restricted the person-year database of analysis from 17 years old to 50 years old.

Results

Anticipation and adaptation in smoking behaviors with family transitions

First, we highlight the impact of family transitions on smoking trajectories of men and women separately (for a complete table of regression results including all covariates, see supplementary materials Table S3-S4). The results originate from different models (smoking initiation and cessation) where the effects of the family events are estimated jointly. This section presents a graphical illustration of the dynamic effect of each family transition on smoking behavior through marginal effects of the timing variables.

This dynamic effect has to be interpreted in reference to the baseline level which represents at least three years before the event or never having experienced it.

The graphs in Figure 1 refer to couple formation. Living with a partner has a protective effect on smoking initiation: it delays and decreases smoking initiation after 3 years of cohabitation for men and at least after two years for women (top panel). This protective effect of living in couple occurs as a break in behavior as smoking initiation was significantly higher for women 2 years before this event and as similar but non-significant effects are observed for men.

Figure 1: Leads and lags in smoking -Living in couple (Marginal effects)

Living with a partner has also an effect on smoking cessation but this effect is mainly a contemporaneous effect which is significant for men and women at the time of the event (bottom panel). This protective effect is consistent with the hypothesis of bargaining effect between partners at the time of couple formation leading to a change in behavior or with the pleasure or ease of being in couple. There is also a long term effect of living with a partner as it increases slightly smoking cessation for men after at least 2 years from this transition (although no such effect is observed in women). Those results confirm the behavioural change following couple formation and the long-term effects of living with a partner, the effects being larger for men than for women.

The graphs in Figure 2 Further to that, there are adaptation effects related to the birth of a child. We find an increase in smoking initiation at least three years after first childbirth for women and to a lesser extent for men (top panel). This result can be interpreted as a response to the stress and constraints related to the joint exercise of the work and family roles [START_REF] Khlat | Women's health in relation with their family and work roles: France in the early 1990s[END_REF] , that would limit the protective effect of childbirth in the long term, especially for women.

The dynamics of separation is also characterized by a contrast between anticipation and adaptation effects and a strong gender pattern (Figure 3). Anticipation effects precede separation for women with an increase in smoking initiation two years and one year before the separation, while the effect is only contemporaneous for men (top panel). Additionally, there is an adaptation effect, with an increase in smoking cessation after three years of separation for men (column 1, bottom panel) and a decrease in smoking initiation after three years of separation for women (column 2, top panel).

Discussion

We studied the associations between family transitions and tobacco smoking with a method that distinguishes anticipation, contemporaneous and adaptation effects. This takes into account the fact that most of the family transitions are not precisely delimited in time nor defined with a set of criteria: there are continuums in definition and time. As stated by Nystedt (2006) "Most marital life course changes are not instant and totally unexpected since a marriage generally creaks some time before it breaks".

Living in couple or even cohabiting with the partner is the result of a continuous change in life circumstances; even the first childbirth is generally expected and anticipated and it may produce long term behavioural effects, but those might not be permanent. By comparing years before, during and after the family transitions, we shed light on the dynamic of the smoking behaviour associated with major events of the lifecycle.

Summary of the findings

Men and women tend to behave in the same way with two exceptions: the first childbirth and the couple formation. The first childbirth accelerates cessation and delays initiation among women, while it only has a positive effect on cessation for men, which is mainly contemporaneous and fades quickly.

These results are in line with previous publications regarding anticipation of pregnancy among women and reduced effects in men [START_REF] Graham | Lifecourse influences on women's smoking before, during and after pregnancy[END_REF][START_REF] Blackburn | Smoking behaviour change among fathers of new infants[END_REF] . But this beneficial effect of childbirth may also be temporary for some women who initiate smoking after the birth of their child.

Similarly, living with a partner deters initiation for both genders, while it has a long-term enhancing effect on cessation for men but not for women (for whom only a contemporaneous effect is found). This is in line with the literature that emphasises the fact that marriage is protective against smoking [START_REF] Nystedt | Marital life course events and smoking behaviour in Sweden 1980-2000[END_REF][START_REF] Pennanen | Smoking, nicotine dependence and nicotine intake by socio-economic status and marital status[END_REF] and more generally favourable in terms of health behaviours [START_REF] Umberson | Gender, marital status and the social control of health behavior[END_REF] and coherent with findings showing that being married/cohabiting is important for men's cessation success but not for women's. No anticipation effect is clearly visible. Finally a separation is accompanied by an increase in initiation the years before and an increase in cessation some years after. This accords with previous studies concluding that marital disruption is connected to increased smoking risks [START_REF] Nystedt | Marital life course events and smoking behaviour in Sweden 1980-2000[END_REF][START_REF] Mcdermott | Changes in smoking behaviour among young women over life stage transitions[END_REF] , but we further reveal that the adaptation effects lead in the long term to a return to more balanced behaviours.

Limitations

Our study is based on retrospective data which may lead to some biases and measurement errors. First, differential mortality among smokers could lead to an underestimation of the prevalence of smoking for people older than 70 as suggested by a study testing the validity of retrospective data on smoking [START_REF] Christopoulou | Dying for a smoke: How much does differential mortality of smokers affect estimated life-course smoking prevalence?[END_REF] .

This bias may be limited because only a very small part of our sample is aged more than 70. The use of retrospective data could also entail recall bias and reporting errors. It was demonstrated that these biases were lower for heavy smokers [START_REF] Kenkel | Accounting for misclassification error in retrospective smoking data[END_REF] but we cannot address this problem because of lack of information on the number of cigarettes that former smokers were smoking. These biases may vary with level of education and lead to underestimating educational inequalities in smoking in the case of under-reporting by the less educated. The reconstruction of smoking itineraries is also limited by the recall of a single smoking period per individual, not allowing for temporary quitting, which is commoner among the less educated, whose attempts to quit are more numerous and less successful [START_REF] Caraballo | Relapse among cigarette smokers: the CARDIA longitudinal study-1985-2011[END_REF][START_REF] Kotz | Explaining the social gradient in smoking cessation: it's not in the trying, but in the succeeding[END_REF] . This bias could in fact minimize the short term effect of some family events or transitions as adult ex-smokers are more likely to start smoking again. Nevertheless, the use of retrospective data on smoking avoids attrition and mortality with age which can be encountered with longitudinal studies. Furthermore, the questionnaire was based on an age-event grid, a validated technique of high reliability that allows a homogenous and coherent recall of family and occupational life events [START_REF] Antoine | Fuzzy States and Complex Trajectories: Observation, Modelization and Interpretation of Life Histories[END_REF] . In addition, the advantage of our data is that the unit of time is the year, which is shorter than what is used in the vast majority of longitudinal surveys in which subjects are surveyed every two or five years without any retrospective report of change in smoking behaviour. This allows a much more detailed analysis, especially the measurement of lags and leads effects.

It is worth noting that we focus on some key family events related to the life cycle, adjusting on the mean time passed in each configuration (couple, first childbirth and separation/single) during the whole life. Couple formation and separation may occur many times during life and it has been shown that multiple broken partnerships reduce the probability of smoking cessation in men [START_REF] Kriegbaum | Reduced probability of smoking cessation in men with increasing number of job losses and partnership breakdowns[END_REF] , while it is likely that the anticipation or adaptation effect would vary over age and repetition of the same events. And yet, the succession of the events and their timing are likely to moderate the effects on tobacco smoking: for example maternal age is strongly linked to an initiation after birth [START_REF] Higgins | Educational disadvantage and cigarette smoking during pregnancy[END_REF] . Our measure of the average effects of the family events is a strong argument for the robustness of our results.

Contrarily to most papers published on the subject, we did consider cohabitation and living with a partner rather than marital status: this definition is broader and less specific, but maybe more adapted to contemporary life. Nystedt (2006) found that the cohabitation was linked to an increase in smoking compared to marriage, for both genders, which is consistent with a long term protective effect of partnership.

Conclusions

Family life transitions influence health behaviors, leading to permanent or more temporary changes, and those changes are visible well before and sometimes well after the actual shift to a different family configuration. The anticipation effects develop during the long term process leading to those transitions, and the adaptation to this different family setting also necessitates a long period of time. This vision is a better reflection of the unfolding of health behaviour changes during adult life in response to the reconfiguration and instability of close family ties. Our approach shed light on the limitation of the traditional analytic strategy that takes only the contemporary effect of the life events or family status into account and lead to miss the real timing of the changes in smoking behaviour. Attending to the specific vulnerabilities related to partnership disruption and using forthcoming roles and responsibilities as a motivator for positive changes could be an important component of smoking prevention and cessation efforts.

Figures

  This empirical work is based on the data derived from the national representative SIP (Santé et Itinéraire Professionnel) survey, performed by the French Ministries of Labour and Health (DARES and DREES), the French Centre for Employment Studies (CEE) and the French National Institute of Statistics and Economic Studies (INSEE). This survey conducted in France in 2006-2007 is retrospective and jointlyaccounts for health events and the career paths of individuals. In 2006, households were randomly selected from the 1999 census, which was updated for new housings, and in each household, one individual aged between 20 to 74 years was randomly selected for interview. Finally, 13,648 men and women from the general French population were interviewed at home by a trained interviewer. The participation rate was 76%. The scope of the analysis is restricted to individuals aged between 25 and 74 at the time of the survey, which amounts to 4,989 men and 5,812 women.

  refer to first childbirth. Contrary to couple formation, the birth of a child has a different effect for men and women. Anticipation effects are visible among women two years before childbirth with a rise in smoking cessation up to two years before the birth as well a fall in initiation the year preceding the birth of a child (column 2, top and bottom panel). For men, the birth of a child has only a contemporaneous effect on smoking with a slight increase in smoking cessation at the time of birth (column 1, bottom panel).

Figure 2 :

 2 Figure 2 : Leads and lags in smoking -First childbirth (Marginal effects)

Figure 3 :

 3 Figure 3 : Leads and lags in smoking -Separation (Marginal effects)

Figure S1 :

 S1 Figure S1: Prevalence of smoking over ages Figure S2: Distribution of family events over ages

  𝛼. 𝑋 𝑖𝑡 + 𝛽. 𝐸 𝑖𝑡 +𝛿. 𝐸 𝑖 ̅ + 𝛾. 𝑍 𝑖 + 𝑤 𝑖 + 𝑒 𝑖𝑡 (𝐸𝑞. 1)

with i=1,…,N and t=17,…T.

where 𝑋 𝑖𝑡 is a vector of time-varying individual variables, 𝐸 𝑖𝑡 is a vector of time-varying individual family variables (i.e. living with a partner, having a first child, being separated, including leads and lags), 𝐸 𝑖 ̅ is a vector of individual means of the family variables, 𝑍 𝑖 a vector of fixed individual control variables, 𝑤 𝑖 an individual specific effect of unobserved variables and 𝑒 𝑖𝑡 an idiosyncratic errors terms.

In this model, the successive observations for a given individual from the retrospective age of 17 to the age at the time of the survey are not independent. Consequently, logistic model was estimated with heteroscedasticity adjustment on individual clusters.

Acknowledgments

This project received funding and administrative support from the French Institut National d'Etudes Démographiques (INED). Damien Bricard also benefited from a Post-doctoral fellowship from the INED.

Supplementary materials

Tables Table S1: Distributions of age at smoking initiation and cessation and at family events Table S2: Descriptive statistics Table S3: Discrete-time logistic regression of smoking initiation -Men and Women (Odds-ratio) Table S4: Discrete-time logistic regression of smoking cessation -Men and Women (Odds-ratio)

Author Contributions

Damien Bricard conceived the original idea for the analysis and wrote a first draft of the paper; Damien Bricard prepared the data and made the statistical analysis with support from Stéphane Legleye; Myriam Khlat and Stéphane Legleye provided critical feedback and revisions.

Role of Funding Source

The data collection was funded by the National Institute for Health Education and Promotion (INPES) which is a public health agency. The analysis was funded by the National Institute for Demographic Studies which is a public research centre.

Conflict of interest

The authors declare no conflict of interests.