*3.3. BRDF for Specular Flakes*

The expansion in scattering order (1) works in the accurate approach and in the LTE. In most cases, the first-order term is the principal one and the latter ones are secondary corrections to it. Therefore, if we take the first-order calculated using the accurate approach and the second and other orders calculated using the LTE, we shall give a decent approximation

$$\begin{array}{rcl} f\_1 &=& f\_1 + f\_2^{(LTE)} + f\_3^{(LTE)} + \cdots \\ &=& f\_1 + \left( f^{(LTE)} - f\_1^{(LTE)} \right) \end{array}$$

For vertically homogeneous paint, the equation for the first-order BRDF (2) becomes

$$f\_1(v, u) = t\_F(v) t\_F(u) \\ f\_1(v', u') \int\_0^H a(v', u', z) \, dz'$$

It is valid for both the LTE and the accurate approach. The pre-integral factor is the same for both approaches. The attenuation is different: (4) for the LTE and (8) for the accurate approach. So

$$f\_1(v, u) = f\_1^{(LTE)}(v, u) \frac{A|\_{\text{accumate}}}{A|\_{LTE}} \tag{14}$$

where

$$A \equiv \int\_0^H a\left(v', u', z\right) dz$$

Therefore, the final approximation to the full (all orders) BRDF becomes

$$f \approx f^{(LTE)} + \left(\frac{A|\_{\text{acurate}}}{A|\_{LTE}} - 1\right) f\_1^{(LTE)}(v, u) \tag{15}$$

For the LTE method, the integral of attenuation (4) is

$$A\_{LTE} = \frac{1 - e^{-2DSH}}{2DS} \tag{16}$$

While for the accurate model, according to (8),

$$\begin{array}{rcl} A & \equiv & \int\_0^H a(v', u', z) dz \\ & \approx & \frac{2}{c} \int\_0^{\frac{\epsilon M}{2}} e^{-2c^{-1}(1-t)(2\overline{\text{D}}\overline{\text{S}}Z - (1-t)a(Z))} dZ \\ & = & 2\rho H \int\_0^{\frac{1}{2\rho}} e^{-4(1-t)\rho F(\zeta - \frac{(1-t)}{\overline{\pi}}\widetilde{a}(\zeta))} d\zeta \end{array} \tag{17}$$

where <sup>∼</sup>

$$\begin{array}{rcl} \mathfrak{a}(\zeta) & \equiv & \frac{\pi}{2c\rho F}\mathfrak{a}(Z) \\ \zeta & \equiv & \frac{Z}{cH\rho} = \frac{Z}{R} \\ \rho & \equiv & c^{-1}r \\ r & \equiv & \frac{\overline{R}}{H} \\ \overline{R} & \equiv & \sqrt{\overline{S}/\pi} \\ F & \equiv & \overline{D}\overline{S}H \end{array}$$

The *F* is a sort of a concentration parameter like PVC. It can be named PAC (pigment area concentration) because by definition *F* is the total area of flakes in the paint layer of the unit area. *R* is the effective mean radius, and *r* ≡ *R*/*H* is the relative effective radius. *A* is determined just by the three parameters *ρ*, *F* and *H* and depends on the directions *v* and *u* only through *ρ* which is inversely proportional to *c* = *c*(*v* , *u* ).

The integral (17) can be calculated only numerically.

The functions *γ<sup>m</sup>* and *α* are determined solely by the distribution of flake area. Thus, they can be pre-calculated in advance and re-used for all color channels and all BRDF points (i.e., for all combinations of the directions of incidence and illumination) because it is only *c* that depends on these directions and *t* which depends on the color channel.
