**2 Simulation of front-lighting in fog **

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**#### 2.1 Light scattering

Fog usually occurs by radiational or advectional cooling of moist air, resulting in a high concentration of microscopic airborne water droplets. When visible light interacts with one of these droplets, it is scattered in different directions with different intensities, following the so-called phase function, which depends on wavelength and droplet size according to Mie’s theory. Hence, light propagating along a path through fog undergoes multiple scattering, and consequently loses energy as an exponential function of distance, according to Beer-Lambert law. These optical phenomena were thoroughly described by Middleton [1].

**2.2 Backscattered luminance **

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**Scattered light does not simply disappear as if it was absorbed by smoke or dust. It simply propagates along different paths, some of which end up into the driver’s eyes. It results in an added luminance which impairs visibility by reducing contrasts. Paradoxically, the driver’s own headlamps, which are designed to increase visual range in night-time, may actually add to this problem by illuminating the fog ahead of the vehicle, causing a veil of backscattered luminance. The only way to minimize this phenomenon is to work on the beam pattern of the headlamps, in order to avoid stray light above the cut-off line [2].

**2.3 Simulation techniques **

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**Simulation is a precious tool for designing and testing headlamps, specially for foggy weather conditions, which can hardly be experimented on the field. But the simulation needs to be photometrically accurate in order to build efficient beam patterns. When the proper optical data is available, and with enough computation time, global illumination techniques may be satisfactory in terms of accuracy. But when it comes to real-time driving simulation, some approximations are needed to achieve interactive frame rates.

To the best of our knowledge, there are two techniques to simulate vehicle headlamps in night-time fog in real-time. The technique developed by Lecocq et al [3] focuses on the study of front-lighting systems. It uses a real-time ray tracing technique to account for the interactions of the pattern emitted by the headlamps with the scattering medium (Figure 1a). The basic approximations it uses are far field photometry and single scattering. The technique developed by Dumont et al [4] rather focuses on the study of road vision. It is based on a photometric model of the visual effects of fog. The backscattered luminance distribution is pre-computed – possibly without the previous approximations – and stored as a texture to be added in real time onto the simulated image of the foggy road scene (Figure 1b). The approximation comes from the fact that volume shadows cannot be rendered, since the backscattered veil is precomputed.

(a) Real-time ray tracing [3]. (b) Pre-computed texture [4]. Figure 1: Interactive rendering techniques for the backscattered veil.

**3 Computational assessment of the approximations **

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**#### 3.1 Assessment method

In order to assess the previously listed approximations, backscattered luminance distributions were computed with a software called PROF (for Photometric Rendering Of Fog), developed by the Laboratoire Central des Ponts et Chaussées of Paris, France. PROF is based on a Monte Carlo light tracing technique which simulates the multiple anisotropic scattering of light in fog [5]. The computations were made for radiation- and advection- types of fog (with droplet diameters respectively around 2 and 12 micrometers), for optical ranges between 10 and 150 meters, and for one pair of low-beam halogen headlamps for which the near-field photometry was available. The backscattered luminance distribution were computed in a 50° horizontal field of vision, and the average backscattered luminance inside a 20° cone of vision was used as a quantitative criterion of comparison (Figure 2).

Figure 2: Backscattered luminance distribution in a 50° horizontal field of vision.

**3.2 Far field photometry **

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**Computations of the backscattered luminance were first conducted with extended light sources, using the near-field photometric data available for the headlamps. Then they were conducted with point light sources, using hemispheric far field beam patterns computed from the previous near field data. In the sample results presented in Figure 3, it can be seen that the only significant difference introduced by the far field approximation is located right where the light exits the headlamps, at a position in the field of vision which is normally HIDden from the driver by the hood of the vehicle.

Figure 3: Influence of the far field approximation on the backscattered luminance distribution (for radiation fog and 50 m optical range).

Diagram 1 clearly shows the influence of droplet size on the average backscattered luminance (forward scattering is stronger with bigger droplets). On the other hand, the relative error introduced by the far field approximation is low, with less than 10% underestimation.

Diagram 1: Influence of the far field approximation on the average back-scattered luminance.

**3.3 Single scattering **

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**In order to assess the single scattering approximation, simulations were once again conducted with PROF, only this time light paths were traced until their first interaction with the scattering medium and then discarded. In the sample results presented in Figure 4, it can be seen that discarding multiply scattered luminous energy leads to an underestimation of the backscattered veil in the upper parts of the field of view, above the cut-off line of the low beam.

Figure 4: Influence of the single scattering approximation on the

backscattered luminance distribution (for radiation fog and 50 m optical range).

Diagram 2 shows that the single

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