*3.1. Full BRDF as a Sum of BRDF Orders*

To calculate paint visual appearance expressed via BRDF, we can use ray tracing which leads to the following expansion of the full BRDF:

$$f = f\_1 + f\_2 + f\_3 + \dotsb \tag{1}$$

where *fn* corresponds to BRDF created by ray paths with *n* flake hits. It is called the *n*-th order of BRDF. The order equals the number of ray scattering (the changes of ray direction), so only reflections are counted but the specular transmissions that do not change ray direction and only attenuate it are not counted (Figure 1). In a sense, our BRDF orders are similar to the scattering orders in [28].

**Figure 1.** Side view of ray paths which form BRDF of the first (**a**), second (**b**), third (**c**) order. Light rays are shown by red arrows, flakes are thick green dashes and the binder is shaded light gray. The number of ray transmission events is irrelevant, so a random number of them are shown.

To be precise, any ray that contributes to the BRDF must penetrate the top Fresnel boundary twice, entering the paint layer and then leaving it. Thus, formally the minimal count of ray scattering events is 2, not 0, and BRDF components should be numbered as *f*2, *f*3, . . . We, however, do not count these two constant events in the orders.

Every BRDF order is an average over all the ray paths of the corresponding type. For *f*1, this means that the ray transmits Fresnel boundary and enters the paint layer, then descends until it hits a flake. After that, it is reflected by that flake and ascends to the top boundary but not intersecting any flake. Finally, it transmits through the Fresnel boundary and leaves the paint layer (Figure 1a).

Generally, different flakes can be slightly correlated. If the flakes are well aligned and there are no forces between them, then we can neglect correlations between flakes at different depths. As a result, the flake hit event is rather independent from the obscure event (if another flake intersects the incident or reflected rays). Therefore

$$f\_1(v, u) = t\_F(v) t\_F(u) \int\_0^H a\left(v', u', z\right) \mathfrak{f}\_1\left(v', u'; z\right) dz \tag{2}$$

where *v* , *u* are direction of the refracted incident and observation rays *v* and *u*, respectively, *tF* is Fresnel transmittance of the binder-air boundary, *a*(*v* , *u* , *z*) is the product attenuation along both ray segments and f1(*v* , *u* ; *z*)*dz* is the first-order BRDF of the sublayer of thickness *dz* at depth *z*. It is independent from *z* in case of a homogeneous paint. We do not use its explicit form which is the same for both LTE and accurate approach (derivation is in [1]).

By definition, attenuation *a* is the average probability that the given fixed full path (along *v* down to *z* then upwards along *u* ) over all possible geometries of the flake ensemble which we can write as

$$a(v',u',z) = \langle \mathfrak{a}(v',z)\mathfrak{a}(u',z)\rangle$$

where a is the probability that the given one ray goes from the surface to the depth *z* (or vice versa). The first term is for the incident ray, and the second is for the reflected one.

In the continuous medium approximation, it is assumed that the fates of these two rays are statistically independent, and therefore

$$a(v',u',z) = \langle \mathfrak{a}(v',z) \rangle \langle \mathfrak{a}(u',z) \rangle.$$

Meanwhile calculation of the mean probability of ray transmitting the paint layer is simple and leads to

$$
\langle \mathfrak{a} \rangle = \mathfrak{e}^{-(1-t)DSz} \tag{3}
$$

(regardless of direction), and therefore

$$a(v', u', z) = e^{-2(1-t)\overline{D}\overline{z}}\tag{4}$$

where *t* is the flake transmittance, *S* is the mean flake's area and *D* is the total concentration of flakes as their number in unit paint volume:

$$
\overline{D} = \frac{\text{PVC}}{\overline{\text{S}}h} \tag{5}
$$

where PVC is the pigment volume concentration and *h* is flake thickness. This value determines BRDF of paint with thin planar flakes.

The formula (3) of attenuation for a single ray is very general and correct. It is true for both continuous medium approximation and ray tracing individual flakes of finite size. However, (4) is not accurate because it assumes statistical independence of the fates of the incident and reflected rays. Meanwhile, in reality they are correlated, and below we shall calculate how this affects attenuation and demonstrate large deviation from (4).
