# The visibility of rock art under paleolithic lighting: perceptual experimentation and probabilistic modeling applied to the Chauvet-Pont d'Arc cave

Guillaume Thibault, Jean-François Hullo, Kim Génuite, Gilles Tosello, Jacques Droulez, Stéphane Jaillet, Jean-Jacques Delannoy, Benjamin Sadier, Julien Monney, Laurent Dollé, et al.

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Guillaume Thibault, Jean-Francois Hullo, Kim Génuite, Gilles Tosello, Jacques Droulez, et al. The visibility of rock art under paleolithic lighting: perceptual experimentation and probabilistic modeling applied to the Chauvet-Pont d'Arc cave. 2022. hal-05234253v2

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Distributed under a Creative Commons Attribution 4.0 International License
THE VISIBILITY OF ROCK ART UNDER PALEOLITHIC LIGHTING:
PERCEPTUAL EXPERIMENTATION AND PROBABILISTIC
MODELING APPLIED TO THE CHAUVET-PONT D'ARC CAVE

Guillaume Thibault<fnref n="1" />*, Jean-François Hullo<fnref n="1" />, Kim Genuite<fnref n="2" />, Gilles Tosello<fnref n="3" />, Jacques Droulez<fnref n="4" />, Stéphane Jaillet<fnref n="2" />, Jean-Jacques Delannoy<fnref n="2" />, Benjamin Sadier<fnref n="2" />, Julien Monney<fnref n="2" />, Laurent Dollé<fnref n="1" />, Jean-Michel Geneste<fnref n="5" />

ABSTRACT

The article investigates the issue of the audience of European Paleolithic cave art. Our methods consist of an original combination of experimental archaeology and spatial modeling to better understand the role of perceptual factors, such as viewing distance, lighting, the shape and contrast of the parietal depictions, and the morphologies of the cave. A methodological framework was thus elaborated to identify the locations from which a Paleolithic light-carrier in an underground space could perceive artworks on the walls, and to quantify this visibility. We conducted a visual perception experiment to produce a probabilistic model of the two distances from which a human holding a flickering light is able to detect and identify planar parietal depictions. Two lighting types (torches and grease lamps) and three artistic techniques (drawing, engraving and painting) were considered. The resulting model is a function of the angle of approach and two features of the graphic entity: its size and contrast, the latter resulting from the combination of the technique used to create the rock art and the nature of the cave wall. This perceptual model was extrapolated to a wider probabilistic framework to include non-planar and partially hidden graphic entities. It was applied to the data corpus of the Chauvet-Pont d'Arc Cave (photographs of graphic entities, artistic techniques, meshes of the walls and graphic entities). Ray casting yielded the visibility probability of 487 graphic entities from more than 20,000 possible Paleolithic human observation points, enabling several analyses and the creation of original visibility maps at different distances. The methodology developed in this study is applicable to the datasets of other sites and paves the way for new quantitative, perceptually-based analyses of cave art.

**Keywords** Paleolithic · cave art · visibility · vision · lighting · spatial analysis · subterranean landscape · Chauvet-Pont d'Arc Cave

# 1 Introduction

For more than a century, archaeologists have devoted considerable attention to the positioning of Paleolithic artworks in open-air, rock shelter and cave sites, and have published spatial analyses [1,2,3] and studies of

<footnote n="1">¹ EDF Recherche & Développement, Palaiseau, France</footnote>

<footnote n="2">² Laboratoire Environnements, Dynamiques et Territoires de la Montagne, Centre National de la Recherche Scientifique, Chambéry, France</footnote>

<footnote n="3">³ Centre de Recherche et d'Etude pour l'Art Préhistorique, Toulouse, France</footnote>

<footnote n="4">⁴ Institut des Systèmes Intelligents et de Robotique, Centre National de la Recherche Scientifique, Paris, France</footnote>

<footnote n="5">⁵ Ministère de la Culture, Paris, France</footnote>

<footnote>*Corresponding author: guillaume.thibault@edf.fr. All authors contributed equally to this work.</footnote>
human social structures [4,5,6,7]. The seminal work of Leroi-Gourhan [8] shows that the relief and geology of cave walls may have influenced the intentional positioning of some representations or the nature of certain artistic stagings, resulting in aesthetic and formal creations that are themselves works of art [9,10,11,12].

Beyond the geological and geomorphological characteristics, questions concerning the visibility distance and the concealment or exposure of cave art, as well as their lighting and visual perception, have only recently been raised. These questions were first addressed from the perspective of lighting techniques entailing a technical type of fixed or portable light source [13,14]. [15] recently summarized the few experiments made in this domain and quantified the main characteristics of Paleolithic lighting systems based on archaeological and empirical data. Pastoors and Weniger [16] proposed a model of human behavior based on archaeological objects, imprints, the dimensions of the subterranean space, and lighting. The application of this method to three caves did not discern any behavioral patterns, mainly due to gaps in the published archaeological record. A study of the contribution of fire-lighting to the analysis of Paleolithic cave art emphasized the importance of shadows on the perception of shapes and lines [17]. Following [18], studies [19,20], applied to two major Upper Paleolithic sites in France, addressed the question of visibility versus legibility, as well as aspects of "social grouping", such as the intentional choices made when positioning and grouping artworks on the walls.

Visibility and legibility refer to two classic perceptual tasks: the detection and identification of objects [21,22,23]. There is general consensus in the scientific literature that the detection and identification of visual objects are closely related but differ in both psychophysical performances [24,25,26] and brain area activation [27]. Detection is the capacity of a human observer to detect the presence of a target that differs from the background by at least one visual feature (orientation, color, size, contrast, etc.). It is characterized as a fast, preattentive, parallel process. Identification is the capacity of the observer to precisely recognize the visual target by integrating several features into a coherent global percept. Identification is slower and requires greater attention, contrast and categorization.

The visibility calculations of [19,20] are based on the hypothesis that the decorated walls could be uniformly lit and were perceived by a distant observer not equipped with a Paleolithic lighting source. This hypothesis is tenuous in locations with walls that require complex lighting or when the artworks are very large. These two situations are encountered, for example, on the large panels in the End Chamber of the Chauvet-Pont d'Arc Cave (Ardèche, France) (abbreviated below as Chauvet cave), whose frequent undulating and folded walls with intermittent alcoves complicate efforts to achieve uniform lighting. We thus agree more with the conditions studied by [16,17], consisting of a more circumscribed Paleolithic lighting technique (a torch or grease lamp) that illuminates a smaller area. We also consider that the light source is carried by the observer and that its characteristics are thus partly determined by the geomorphological context of the site: its lighting effect differs depending on its position relative to the wall and its potential occulting folds, and the luminance it creates decreases according to the square of the distance from the wall. Using a virtual replica of the Ledge of the Horses sector in the Atxurra Cave in Spain, [15] simulated in the maximum distance from which these lighting systems can illuminate walls covered with *graphic entities* (GEs), as defined in ([28], p.90). However, the visual perceptive effect for each GE according to its production technique (engraving, painting, combination of the two) under each lighting system is not specified.

Montello [29] specified how cognitive sciences have highlighted the influence of location and scene on human psychology. Among the six perception processes he listed, three are directly related to the question of visual motor perception, which is central in our study. Recently, Opitz [30] simulated perception to monitor where the visual attention of people could have been attracted by salient distant engravings in a Neolithic cave and, therefore, their possible intentional positioning. Boudier-Blet [31] introduced a psycho-experimental protocol involving 24 participants. These individuals were asked to identify whether an animal was depicted in the stimuli consisting of photos of 40 parietal GEs realized by five techniques and using four types of lighting. The subjects were immobile and the observation distance of the displayed rock art was fixed. The bichrome GEs achieved the highest success rate with more than 90% of correct answers: this is the technical modality with which the GEs are most easily seen by the human eye, especially in natural lighting. These studies illustrate the interest of research on human perception through simulation or stimulation as a new way to study cave art from the observer's point of view.

These issues underlie the questions addressed in our study: from which locations could a Paleolithic light-carrier *perceive* parietal GEs in complete darkness, and how can we *quantify* this visibility? Our analysis of the visibility of GEs was based on two questions concerning two visual tasks: under what conditions
is an observer able to *detect* the presence of a graphic entity on the wall, and then to *identify* the figure represented? To address these questions, the performances of moving human observers were measured as the maximum distance from the wall at which each of these two tasks can be achieved under different conditions. Based on this perceptual experimentation, we then developed a model for a light-carrying human visually perceiving flat GEs. This model describes the probability distribution of two viewing distances for a GE—the detection distance and the identification distance—depending on the lighting type used, the size of the GE, the artistic technique, the nature of the wall underlying the GE, the distance between the light-carrier and the GE, and the angle of incidence with the wall. We then extrapolated this model to non-planar GEs and those partially masked by the sinuous cave walls and formations.

This model was applied to the case of the Chauvet cave, making use of its available corpus of digital data (photographs, Geographic Information System (GIS), 3D mesh). We thus quantified the visibility of hundreds of GEs from thousands of observation points and for three observation postures. Our results are illustrated with visibility maps that provide new supports for comparative analyses between the different spaces in the cave, which contains diverse artworks and variable geomorphologies.

This use of experimental perceptual factors in a probabilistic framework as a method to study a visual archaeological question, and tested on a large sample of GEs, opens new methodological avenues for the study of cave art and its spatial distribution.

# 2 Materials and methods

## 2.1 Experimental measures of the perception of flat graphic entities under Paleolithic lighting

There is a long history of visual psychophysics experiments [32,33,34] under well-controlled conditions aiming to identify various characteristics of the human visual system, such as the sensitivity to contrast as a function of spatial and temporal frequencies, the dynamics of light and dark adaptation, the capacity to discriminate colors and textures, etc. In most experiments, the participants sit quietly in front of a computer screen on which sequences of controlled stimuli are displayed. This situation differs dramatically from that of a cave visitor carrying a torch or a grease lamp and walking on an uneven surface while observing GEs painted or engraved on a 3D wall, under moving shadows.

While walking, the distance to visual stimuli, the mean luminance, the spatial and temporal frequencies as well as many other visual characteristics change. Therefore, we must be careful when extrapolating visual psychophysics data to determine the capacity of an observer to detect or identify GEs. Several discoverers of decorated caves have indeed reported their inability to perceive many Paleolithic artworks despite their modern lighting equipment [9].

As proposed, for instance, by [35], Virtual Reality is a very promising approach to study the visibility of artworks by a moving observer carrying a light source. However, there is still a lack of experimental validation for critical questions such as the simulation of flame during motion, the participant's gaze, or the low dynamic range of virtual reality displays.

For these reasons, we decided to conduct an experiment in which the participants were placed in conditions as close as possible to that of a cave visitor, in order to collect measurements on the visual capacity of humans to perceive GEs in this underground environment.

### 2.1.1 Participants

Six individuals (four men and two women, 28 to 60-years-old) participated in the experiment as light-carrying subjects. All had normal or corrected-to-normal vision. Three of them had varying degrees of prior knowledge of prehistoric decorated caves. All six subjects participated in the entire experimental procedure. This study was approved by the ethics committee of the Institut des Sciences Biologiques du Centre National de la Recherche Scientifique française. Participants gave their informed consent prior to the beginning of the experiment.
## 2.1.2 Experimental parameters

Our aim was to make experimental measurements of the visibility thresholds of GEs present on walls under conditions similar to those in decorated caves. We conducted the experiment from February 2 to 5, 2015 in an abandoned limestone quarry in the Dordogne region of France. This site was chosen to focus mainly on the factors of visibility (locomotor constraints are addressed later).

The following parameters were considered:

* two Paleolithic lighting types (vegetal torch, grease lamp),

* one GE subject (woolly rhinoceros) of two widths (~50 cm, ~120 cm),

* three GE artistic techniques (drawing with a charcoal crayon, painting with hematite [red ochre], engraving with a flint tool),

* two wall types (uniform and non-uniform),

* two angles of incidence between the locomotion direction and the normal to the mean plane of the GE (0°, 45°).

The GEs were made in totally dark areas of the quarry.

## 2.1.3 The graphic entities (GEs) analyzed in the experiment

On the quarry walls, an archaeologist of the Chauvet cave research team created GEs similar in proportion and quality to those in Chauvet cave. The variability of wall surface states and the artistic techniques (including pigment color) used there were also representative. To simplify the GE identification phase, we chose to analyze only one type of figure: woolly rhinoceros (Figure 1). This is one of the most frequent species in Chauvet cave, represented by more than 70 individuals. The prehistoric artist's manner of depicting the rhinoceroses comprises several precise graphic conventions (massive body with a rounded belly, short legs with pointed or ball-shaped extremities, a median vertical band on the body, symmetrical double-curve ears) [36]. The artists combined these elements in various ways, resulting in no two rhinoceroses that are alike, but all of which share many similarities regardless of the artistic technique used. Emblematic of Chauvet cave, this species was thus well adapted to the objectives and methods of our experiment.

Experimental stimuli are given in the S1 Appendix.

<figure><img src="image_2.png" /><figcaption>**Figure 1:** A Chauvet cave rhinoceros (0) and the six experimental graphic entities (1 to 6) arranged by technique and size (in cm). (1) Large red (120x50); (2) Small red (40x19); (3) Large engraved (100x45); (4) Small engraved (45x18); (5) Large black (100x50); (6) Small black (50x21). Red entities drawn with a narrow brush or fingertips and using a mixture of hematite. Black entities drawn with charcoal. Engravings created with the fingertips or wood fragments. Published under a CC BY 4.0 license, with permission from French Ministry of Culture / Jean-Michel Geneste (0) - Gilles Tosello & Jean-François Hullo (1 to 6).</figcaption></figure>
Three techniques were selected and each associated with a color in concordance with the data collected in Chauvet cave [37]. For the red pigment GEs, ground hematite powder was mixed with water to produce a thick mixture that could be applied with a narrow brush or fingertips. For the black drawing, Scots pine (Pinus sylvestris/nigra/mogo/unici) branches provided charcoal pieces with various sections and lengths. The engravings were made with fingertips (i.e., finger-tracings) or wood fragments, creating white or clear lines and marks.

The artistic technique determines both the color of the subject and the background. In Chauvet cave, the red and black GEs are located on light-colored walls, creating a visual contrast that facilitates their perception. The engravings, on the other hand, are located on dark-colored surfaces that are naturally covered with a brown clay or ochre film.

One final parameter, the rock texture, is important in the artistic process. In Chauvet cave, the limestone is very fine-grained and the walls are smooth with rounded relief forms. The actions of the prehistoric artists were thus fluid and met with little resistance from the support. In the quarry, we sought walls similar to those in Chauvet cave. For the engravings, these were panels with a uniform surface covered with a layer of brown infiltration clay (approx. 5 mm thick) (Figure 1, GE#3 and #4). For the red and black GEs, it was more difficult to find appropriate walls. Though the quarry has appropriate light-colored, uniform walls, they are covered with grooves made by the metal picks used to quarry the limestone (Figure 1, GE #1). There are also natural surfaces that have not been transformed by human actions, but they have an uneven relief of depressions and protrusions (Figure 1, GE #2), as well as darker alteration zones (Figure 1, GE #5 and #6).

### 2.1.4 Lighting types

Two portable Paleolithic lighting types were selected: torch and grease lamp. The 40-to-60-cm-long torches (Figure 2) were made with a softwood, Scots pine (Pinus sylvestris/nigra), which is the main species identified among the charcoal sampled on the floors and walls of Chauvet cave [38]. Many marks attributed to torch contacts have been identified on the walls and objects on the floor of the cave. It was, therefore, logical to experiment first with this type of lighting. The branches of 8 to 12 mm in diameter each were tied together at regular intervals with leather bands, forming bundles measuring about 7 to 10 cm in diameter (Figure 2).

<figure><img src="image_3.png" /><figcaption>**Figure 2:** Example of the type of torch used for the experiments. Published under a CC BY 4.0 license, with permission from French Ministry of Culture / Julien Monney.</figcaption></figure>

Portable grease lamps are mainly known from the end of the Upper Paleolithic (Magdalenian), and to a lesser extent in the Gravettian, along with a few specimens from the earlier Aurignacian period [13]. In our experiment, we used a portable stone lamp made from a 17 x 11 x 5 cm piece of urgonian limestone bearing a natural cupule (fossil) on one of its faces used as grease receptacle, 6 to 7 cm in diameter and 2 cm deep (Figure 3). The wick was made of vegetal fiber (moss). The fuel used was suidea fat.
<figure><img src="image_4.png" /><figcaption>**Figure 3: The grease lamp used during the experiments.** In the background, the wall shows the front legs of a black GE made for the experiments. Published under a CC BY 4.0 license, with permission from French Ministry of Culture / Stéphane Jaillet.</figcaption></figure>

Even if several lighting sources could have been used during a visit to the Paleolithic cave, we chose to consider only one lighting source at a time (a torch or a grease lamp): this reduces the parameters taken into account in the analysis so as not to further increase the complexity of the experiment.

The mean luminance of the wall was measured with a MINOLTA LS-100 luminance meter for the two types of lighting at distances ranging from 1 m to 10 m, in a direction orthogonal to the wall. The mean luminance for the 6 GEs at 1 m was 0.555 cd/m² with the torch and 0.225 cd/m² with the grease lamp (see Table S2 in appendix), yielding a ratio of 0.40 at 1 m from the flame. These values are coherent with those of [15, Table 1] for the torches, with a ratio of 0.20 between the average luminance for the five torches and the two grease lamps at 40 cm from the flame. We observed a slightly stronger luminance in our experiment with the grease lamps, which can be, as stated by [15], related to the nature of the fuel or the wick, as well as to the dispersion of the experimental results. For the other distances, the mean luminance decreases as expected as the inverse of the squared distance.

### 2.1.5 The experimental protocol

We selected two visibility thresholds involved in the detection and identification tasks, proceeding from the farthest to the nearest (Figure 4):

* D2, the minimum distance from the wall where the subject indicates that he/she can detect an artifact on it without perceiving its details,

* D1, the minimum distance to the wall where the subject identifies a GE with all its details.
<figure><img src="image_5.png" /><figcaption>**Figure 4:** A participant performing the experiment. Carrying a grease lamp, the participant approaches the "Large red rhinoceros" GE #1, walking perpendicularly toward the wall and looking toward it (angle of incidence of 0°). Published under a CC BY 4.0 license, with permission from French Ministry of Culture / Jean-François Hullo.</figcaption></figure>

For each of the six GEs, each participant first walked up to the wall to see all the details of the GE. They then backed away far enough to no longer see the GE, although they knew its approximate location and shape. They then moved forward again toward the wall and indicated first their *D2*, and then their *D1* distance.

In this experimental procedure, the measurements of the *D1* and *D2* thresholds were recorded for both lighting types (grease lamp and torch), the angle of incidence relative to the wall (0° and 45°), and for the three artistic techniques and two GE sizes. The average time taken to complete the trials was 2.5 hours per subject. Each subject performed three trials at a 0° angle of incidence and one at 45°, for each of the six GEs.

## 2.2 Modeling the visual perception distance of graphic entities under Paleolithic lighting

### 2.2.1 Parietal Contrast and Parietal visibility coefficient

The intuitions of [16] and [31], and the recommendations of [39] lead us to search for a relationship between *D1* (respectively *D2*), the size of the GE and the luminance contrast of the GE against the more or less regular underlying wall. This last information appears on photographs taken with a flash in the quarry. This information is exploited in several phases.

First, the values of the red (*R*<sub>pix</sub>), green (*G*<sub>pix</sub>) and blue (*B*<sub>pix</sub>) components of the pixels in the image are converted into luminance values (*Y*<sub>pix</sub>) between 0 and 255, equivalent to the perceived intensity, using the formula [40]:

$$Y_{pix} = 0.299R_{pix} + 0.587G_{pix} + 0.114B_{pix} \quad (1)$$

The pixels on the GE are then manually segmented (Figure 5).
<figure><img src="image_6.png" /><figcaption>**Figure 5:** Segmentation of two graphic entities from luminance images, with associated statistics. (A) GE #3; (B) GE #6; in green, the GEs pixels; in red, the wall pixels near the GEs, and associated histograms of the luminance of both groups of pixels with means and standard deviations. Published under a CC BY 4.0 license, with permission from French Ministry of Culture / Gilles Tosello - Jean-François Hullo.</figcaption></figure>

The analysis of this first set of pixels provides a measure of the average luminance $\mu_G$ of the pixels of the GE. We also manually segmented the pixels located on the wall near the GE. The analysis of this second set of pixels provides a measure of the average luminance $\mu_W$ of the wall pixels, and $\sigma_W$, its standard deviation. A first variable, the *parietal contrast* (PC), describes, independently of the size of the GE, a luminance contrast between the wall and the GE, including the irregular nature of the wall, which generates parasitic information. The PC is a dimensionless quantity, built on the Mahalanobis distance between the average luminance of the GE $\mu_G$ and the luminance distribution of the wall ($\mu_W, \sigma_W$) by the formula:

$$PC_G = \frac{|\mu_G - \mu_W|}{\sigma_W} \qquad (2)$$

A second variable, the GE *parietal visibility coefficient* (PVC), combines the GE parietal contrast and size via the square root of the GE area, noted $S_G$, calculated from the GE envelope (Figure 6):

$$PVC_G = PC_G \sqrt{S_G} = \frac{|\mu_G - \mu_W|}{\sigma_W} \sqrt{S_G} \qquad (3)$$

The PVC thus has the dimension of a distance, coherent with D1 and D2.

<figure><img src="image_7.png" /><figcaption>**Figure 6:** Segmentation of the envelope of two graphic entities. (A) GE #3; (B) GE #6; Published under a CC BY 4.0 license, with permission from French Ministry of Culture / Gilles Tosello - Jean-François Hullo.</figcaption></figure>
When a GE covers two geological types of walls (named facies), its PVC is simply weighted, yielding for instance with facies A and facies B:

$$PVC_G = PC_A \cdot \sqrt{S_A} + PC_B \cdot \sqrt{S_B} \quad (4)$$

where $S_A$ and $S_B$ are the GE areas on each facies and where $PC_A$ and $PC_B$ are the GE parietal contrasts on each of the two facies.

### 2.2.2 Probabilistic model of the visibility from an observation point

The experimental analyses in the quarry concerned GEs made on flat walls without occultation. Inspired by [41], we propose a first probabilistic visibility model adapted to these conditions and then extend it to the general case of non-planar GEs with partial occultation.

The solid angle at which a GE is viewed from an *observation point* (OP), where the light-carrying observer is located, quantifies the size of the retinal image of that GE for the observer,

$\mathcal{E}$ is the triangular mesh of the GE on the wall. $\mathcal{P}$ is the average plane passing as close as possible to the vertices of $\mathcal{E}$ (Figure 7).

<figure><img src="image_8.png" /><figcaption>**Figure 7:** Overhead view in the simplistic case of a graphic entity assumed to be vertical. The geometric elements used in the probabilistic model are located. Published under a CC BY 4.0 license, with permission from French Ministry of Culture / Jean-François Hullo.</figcaption></figure>

$\theta$ is the angle between the ray cast from P toward the GE barycenter $B_{\mathcal{E}}$ and the normal to the mean plane P of $\mathcal{E}$ at $B_{\mathcal{E}}$. $a_k$ is the angle between the ray cast from P toward the $B_k$ barycenter of the triangle k and the normal at this triangle in $n_k$. D is the distance between P and $B_k$. $d_k$ is the distance between P and $B_k$.

**Probability of seeing a graphic entity without occultation**

Let G be a roughly planar GE located in the average plane of the triangles. The probability of identifying the graphic entity G located at a distance D from P and at an angle $\theta$ is calculated as the complement of the cumulative distribution function:

$$P_1(G, D, \theta) = \frac{1}{\sigma_1 \sqrt{2\pi}} \int_D^\infty e^{-\frac{1}{2} \left( \frac{\chi - D1\theta}{\sigma_1} \right)^2} \cdot dx \quad (5)$$

Similarly, the probability of *detecting* a GE without being able to perceive its details is calculated as:
$$P_2(G, D, \theta) = \frac{1}{\sigma_2 \sqrt{2\pi}} \int_D^\infty e^{-\frac{1}{2} \left( \frac{x-D^2\theta}{\sigma_2} \right)^2} \cdot dx \quad \text{eq6} \quad (6)$$

$\sigma_j, j \in (1, 2)$ denotes the standard deviation associated with the identification and detection thresholds Dj.

The distance $D_{PG}$ and the angle of view $\theta_{PG}$ can be computed for a given OP P and a graphic entity G (see Figure 7). Equation 5 and Equation 6 yield two visibility scores: the identification score (j=1) based on the distance threshold $D1$ and the detection score (j=2) based on the distance threshold $D2$ of the GE:

$$Score_j(G, P) = P_j(G, D_{PG}, \theta_{PG}) \quad (7)$$

In principle, this score is inspired by the work of [42], but with an original feature: its probabilization. It should be noted that [43] probabilistic approach focuses on the uncertainty of the 3D model and not on the uncertainty of the visual perception itself, as in this study.

### Probability of seeing a graphic entity with partial occultation

This situation has not been tested experimentally. A plausible extrapolation of the previous situation is to penalize visibility scores from the OPs that allow only a partial view of the GE. This partial view is quantified by the apparent size of each of the GE triangles visible from the OP: this is the proportion of the solid angle that each triangle forms from this OP. The solid angle $\Omega_k$ of a triangle k is conventionally approximated by the formula:

$$\Omega_k = \frac{S_k \cdot \cos(a_k)}{d_k^2} \quad (8)$$

with:

* $k$, the index of the current triangle of the GE,

* $S_k$, the area of triangle k,

* $a_k$, the angle between its normal at the center of the triangle and the line passing through this same center and P,

* $d_k$, the distance between the center of the triangle and P.

The visible solid angle is obtained by summing the solid angles of the GE triangles seen from P. Equation 7 and Equation 8 are therefore adapted as follows for $D1$ (j = 1) and $D2$ (j = 2):

$$Score_j(G, P) = P_j(G, D_{PG}, \theta) \frac{\sum_{k \in \Omega_k} \Omega_{k \in \Omega_k}}{\sum_{k \in \Omega_k} \Omega_k} \quad (9)$$

where $k_{no}$ indicates that this triangle is not occulted. An occultation occurs when a ray cast from an OP to a GE is blocked by a wall triangle or by a triangle of the GE that occludes another triangle of the same GE, as may be the case under grazing angles. If all triangles are seen, the score defined by Equation 9 is equal to the probability defined by Equation 7.

### 2.2.3 Principal visibility scores

From a given OP P, the sum of the $Score_j(G, P)$ for all GEs is, by definition, the expectation (in the probabilistic sense) of the number of visible GEs from this OP, for both identification task (j=1) and detection task (j=2). $E_j(P)$ denotes these two expectations:

$$E_j(P) = \sum_{G=1}^{N_{GE}} Score_j(G, P) \quad \text{with } 0 \le E_j(P) \le N_{GE}, \text{ the number of GEs} \quad (10)$$

Following the work of [42] on the Line of Sight technique, the rays are cast in 3D from each OP to the GEs (i.e., toward the center of each triangle of each GE), as if the head was turned freely, the camera model being spherical (360° in azimuth and 180° in elevation).
Similarly, $Pmax_j(P)$ denotes the probability of the most visible entity from a given observation point P, for both identification task (j=1) and detection task (j=2):

$$Pmax_j(P) = \max_{G \in (1, N_{GE})} \{Score_j(G, P)\} \quad \text{with} \quad 0 \le Pmax_j(P) \le 1 \quad (11)$$

The calculation of $E_j(P)$ and $Pmax_j(P)$ can be restricted to subsets of GE according to different attributes recorded in the GIS such as artistic techniques (red, black, engraving, etc.) or animal species (rhinoceros, feline, bison, etc.).

## 2.3 Motricity modeling

### 2.3.1 Postures

In this study, our first task was to identify the locations in which a prehistoric human could be positioned in three different postures to potentially observe the GEs, taking into account the locomotor constraints imposed by the shapes of the cave walls and floors.

A study of the panels of dots in Chauvet cave suggests the presence of a 1.8-meter-tall individual [44]. This height is compatible with that of the fossils of the tallest *Homo sapiens* in Europe during the Upper Paleolithic, such as that of one individual from the site of Cro-Magnon, estimated at 179 cm [45]. In our study, this first standing posture is thus designated as H180. The second posture corresponds to a human in a lowered position, at 1.25 m tall (designated as H125), and the third posture to a human in a crouched position, at 0.70 m tall, designated as H70. For these three postures, the sightline (eyes) is set as 10 cm below the top of the skull.

It is also necessary to consider the body volume to identify the locations that would have been too small for a human to be positioned. For this purpose, the body in a standing position was assimilated to a simple cylinder, 20 cm in diameter. For the crouched posture, the cylinder diameter was fixed at 60 cm.

Floor zones with an altitude difference greater than 40 cm and a slope greater than 45° were excluded from this study because they imply natural locomotor difficulties that prehistoric humans probably avoided when exploring the cave. These postural parameters are summarized in Table 1.

<table><caption>Table 1: Definition of the three postures considered in the calculation of the observation zones.</caption><thead><tr><td>Posture</td><td>Height</td><td>Sight line</td><td>Corpulence (diameter)</td><td>Clearance capacity</td><td>Name</td></tr></thead><tbody><tr><td>standing</td><td>1.80 m</td><td>1.70 m</td><td>0.2 m</td><td>0.4 m, 45°</td><td>H180</td></tr><tr><td>lowered</td><td>1.25 m</td><td>1.15 m</td><td>0.2 m</td><td>0.4 m, 45°</td><td>H125</td></tr><tr><td>crouched</td><td>0.70 m</td><td>0.60 m</td><td>0.6 m</td><td>0.4 m, 45°</td><td>H70</td></tr></tbody></table>

The clearance capacity is defined by two values: the maximum obstacle height and maximum slope that can be crossed by individuals without risk.

### 2.3.2 Observation meshes

As in [42], it is necessary to determine OP locations even if the modeling of floors and walls is complex. An intermediary geometry, called an *observation mesh* (OM), is traditionally used in the field of video games, relying on navmeshing algorithms [46]. This mesh (distinct from a wall mesh or GE mesh) materializes the locations where virtual subjects can stand in a 3D scene depending on their postural characteristics and the geometry of the scene. For instance, in the Red Panels Gallery (Figure 8), one can crouch under the overhang in posture H70 (4), but not in postures H125 and H180 (5-6).
<figure><img src="image_9.png" /><figcaption>**Figure 8: Observation meshes in the Red Panels Gallery of Chauvet cave.** In this gallery (1), with complex ground and walls (2), cylinders associated with postures H70, H125 and H180 contact or not with the overhang (3a-3c) and define the boundary of the corresponding OM (4-6). Published under a CC BY 4.0 license, with permission from French Ministry of Culture / Carole Fritz (1) – Guillaume Thibault (2, 3a, 3b, 3c, 4, 5, 6).</figcaption></figure>

### 2.3.3 Locating the observation points of the observation meshes

The three OMs are regularly sampled to determine the 3D positions of the OPs that follow the floor model while taking into account the obstacles for the three postures (Figure 9), as in [47].
<figure><img src="image_10.png" /><figcaption>**Figure 9:** Observation Points for H180 posture in part of the End Chamber of Chauvet cave. Sampling of the OM is done in 50 cm steps; (A) isometric top view; (B) perspective view from the observation point located in green on the top view. The GEs meshes are shown in blue. Published under a CC BY 4.0 license, with permission from French Ministry of Culture / Guillaume Thibault - Jean-François Hullo - Adrien Nivaggioli.</figcaption></figure>

**2.4 Cartographic rendering of visibility and motricity data**

The visibility and motricity modeling offer a spatial quantification for identification and detection of the GEs under Paleolithic lighting. Visualscapes, introduced by [48], redefined and interpreted by [49] as "the spatial representation of any visual property generated by, or associated with, a spatial configuration", are powerful tools to explore results globally (at the scale of the cave).

**2.4.1 Spatial representation: cartographic projection of visibility scores**

The visibility score of each GE for the *D1* and *D2* and for the OPs of the three postures yields two *spatial representations* in the form of two vector fields, as presented in [50]: each vector at its starting point at the OP is directed toward the barycenter of a GE and has, as its magnitude, the probability of identifying the GE or detecting it.

The topography of Chauvet cave (where no floor is vertically placed over another), like that of many caves, enables cartographic illustration of the calculation results by a vertical projection. However, the position of the artworks on the walls is flattened (Figure).
<figure><img src="image_11.png" /><figcaption>**Figure 10: Projection of GEs on the ground for cartographic illustration.** Some GEs can appear beyond the cave floor (left wall) or on the floor (right wall). Published under a CC BY 4.0 license, with permission from French Ministry of Culture / Stéphane Jaillet.</figcaption></figure>

A spatial representation can be further processed as a visual property [49], a reduction of the vector field in the scalar field by this vertical projection. The visual properties chosen are $E_j$ and $Pmax_j$.

The creation of maps then proceeds as follows:

* shading the floor, encoded in grayscale. The slope can be accentuated in the rendering to highlight locomotor discontinuities.

* adding the visual property scalar field (bi-linearly interpolated to the resolution where one pixel represents 5 cm x 5 cm) on the OM considered, encoded on a color palette, with contour lines and semi-transparency.

* adding a rosette of segments directed toward the ten main GEs at each OP (for each direction, the length of the segment is equal to the visibility probability of this GE).

* Several examples of maps are given in the Results chapter.

**2.5 Spatial configuration of hotspots**

On the scalar field, it is possible to identify local maxima zones that correspond to favored GE observation zones, as in [47]. These are called visibility hotspots and correspond to the maxima of visual exposure, as defined by Llobera (2003). On a 2D field, whether $E_j$ and $Pmax_j$, these maxima can be determined by a neighborhood analysis (via a focal filter on a sliding window): a maximum is found at a point when all the neighbors of this point have a lower value than the one at the considered point. In the Results section, this processing is applied to $E_j$ with a typical grid of 5 x 5 proximal OP neighbors.

In each of the hotspots, the GEs are divided into visible and non-visible entities (for the two visibility thresholds, D1 and D2). Analyzing the subsets of visible GEs among these hotspots yields an oriented network of visual interdependence of the hotspots: one hotspot is linked to another (by a segment of the network) if, from the first one, GEs of the second are visible.
**2.5.1 Visual changes**

To quantify *visual changes* following [49], it is necessary to compare each of the GEs visible from neighboring OPs. The proposed method extends the filtering of an angular field of visibility [49]: a directional convolution filter is applied to $P_{max_j}$ and $E_j$ fields in the study area. A positive difference in the $P_{max_j}$ field by taking a step in this direction corresponds to the appearance of this GE, while a negative difference signals its disappearance. When applied to the $E_j$ field, it is possible to identify areas of visibility continuity for all GEs along a path.

**2.5.2 Interactive 3D viewer of maximum scores**

In addition to the production of maps, an interactive 3D viewer of visibility scores displays the geometries of the cave (walls, floors, GEs). It allows the user to select the posture, the score and the OP, and the visible GEs are colored in connection with the color scale (Figure 11).

<figure><img src="image_12.png" /><figcaption>**Figure 11: Interactive 3D visualization of the scores.** The spheres are the OPs. They are colored here by the $P_{max1}$ field for the H70 posture in the End Chamber of Chauvet cave. Published under a CC BY 4.0 license, with permission from French Ministry of Culture / Jean-François Hullo – Adrien Nivaggioli.</figcaption></figure>

**2.6 Chauvet cave study area and its data corpus**

The artworks in the Chauvet cave were created by anatomically Modern Humans (*Homo sapiens*). If we consider that the sensory-motor capacities of humans have remained unchanged for thousands of years, we can assume that the sight distances and their variabilities measured for humans living today were essentially the same for the prehistoric humans who observed the GEs in Chauvet cave. We can, therefore, use the data corpus of Chauvet cave to quantify the visibility of GEs for humans in the past and present by applying the method developed above to humans of the 21st century.

In the cave, we studied spaces that are located beyond the area into which daylight penetrated (direct and semi-darkness) into the cave when it was still open to the outside (Figure 12).
<figure><img src="image_13.png" /><figcaption>**Figure 12: Map of the Chauvet-Pont d'Arc cave and geographic zone studied.** Published under a CC BY 4.0 license, with permission from French Ministry of Culture / Jean-Jacques Delannoy, Kim Genuite, Julien Monney (2016).</figcaption></figure>
Starting at the end of the Chamber of the Bear Hollows (North), the path into the cave began to be enveloped in semi-darkness, even though the cave entrance could still be seen. From the Sill Gallery on, the cave was in complete darkness to the End Chamber. Within these geographic perimeters, 487 GEs were studied from the following spaces:

* Cactus Gallery (probable semi-darkness at its entrance),

* Red Panels Gallery (probable darkness),

* Sill Gallery (complete darkness),

* Hillaire Chamber (complete darkness),

* Skull Chamber (complete darkness),

* Megaloceros Gallery (complete darkness),

* End Chamber and Sacristy Chamber (complete darkness).

### 2.6.1 Photographs of the graphic entities

A photographic corpus [51-52] provided 60 suitable photographs for the quantization of the PC of the GEs in the study area.

### 2.6.2 3D survey of the cavity

Underground cavities, and decorated caves in particular, are now frequently digitized in the context of archaeological research [53,54,55,56,57]. The 3D survey of Chauvet cave, carried out by Perazio Engineering between 1998 and 2010, is based on a topographic survey made with a theodolite and a terrestrial phased array LIDAR. In 2010, 274 scan stations produced a consolidated and geo-referenced cloud of 16 billion points [58]. The resolution of the point cloud depends on the types of scanners used, the distances to the scanned objects and the objectives. For the study zone, the proximity between the walls and walkways in the cave, and the cloud-to-cloud superpositions [59], allowed by the large number of scanning stations produced a fairly homogeneous distribution of points and a sufficient resolution (> 1 point/cm², average calculated for 20 points taken at random for all scans) to achieve a high definition mesh.

### 2.6.3 Geographic Information System resources

The GIS of the cave [28] integrates into a single database all the data collected in the cave: GEs with associated information (name, artistic technique, animal species, etc.), places of measurement and detailed cartographic support (S3 Table).

## 3 Results

### 3.1 Perceptual results

#### 3.1.1 Inter-graphic entity analysis of visibility distances

Though the *D1* visibility distance from the GEs varies from one participant to another, the estimate of each average is robust, as shown in Figure 13 for torch lighting: the average *D1* has an uncertainty of ± 1 m for the 6 GEs for the first and third quartiles, i.e., about one step forward or backward.
<figure><img src="image_14.png" /></figure>

**Figure 13: D1 identification distance (in meters) for torch lighting at a 0° angle of incidence for each of the six graphic entities.** Box-and-whisker plots with median and quartiles are represented for the 108 measurements generated by the six participants. Published under a CC BY 4.0 license, with permission from French Ministry of Culture.

The same is true for the *D2* detection distance, as seen in Table 2 (detailed experimental measures of *D1* and *D2* thresholds are given in the S4 Tables for torch and grease lamp, under 0° and 45° incidence).
<table><caption>Table 2: Estimates of the D1 and D2 visibility distances for each graphic entity.</caption><thead><tr><th rowspan="2">Lighting</th><th rowspan="2">Distance (m)</th><th rowspan="2">Feature</th><th colspan="6">Graphic entity Number</th></tr><tr><th>1</th><th>2</th><th>3</th><th>4</th><th>5</th><th>6</th></tr></thead><tbody><tr><td rowspan="12">Torch</td><td rowspan="3">D1<sub>0°</sub></td><td>Mean</td><td>4,01</td><td>3,53</td><td>5,49</td><td>3,59</td><td>3,63</td><td>3,55</td></tr><tr><td>Standard deviation</td><td>1,13</td><td>0,71</td><td>1,15</td><td>1,1</td><td>0,89</td><td>1,09</td></tr><tr><td>Standard Error of the Mean</td><td>0,27</td><td>0,17</td><td>0,27</td><td>0,26</td><td>0,21</td><td>0,26</td></tr><tr><td rowspan="3">D1<sub>45°</sub></td><td>Mean</td><td>3,3</td><td>2,52</td><td>4,8</td><td>3,47</td><td>3,59</td><td>2,44</td></tr><tr><td>Standard deviation</td><td>0,98</td><td>0,62</td><td>1,22</td><td>1,91</td><td>1,05</td><td>0,88</td></tr><tr><td>Standard Error of the Mean</td><td>0,4</td><td>0,25</td><td>0,5</td><td>0,78</td><td>0,43</td><td>0,36</td></tr><tr><td rowspan="3">D2<sub>0°</sub></td><td>Mean</td><td>7,64</td><td>6,07</td><td>9,51</td><td>6,34</td><td>6,57</td><td>6,15</td></tr><tr><td>Standard deviation</td><td>1,42</td><td>0,88</td><td>1,59</td><td>1,15</td><td>1,41</td><td>1,72</td></tr><tr><td>Standard Error of the Mean</td><td>0,33</td><td>0,21</td><td>0,37</td><td>0,27</td><td>0,33</td><td>0,41</td></tr><tr><td rowspan="3">D2<sub>45°</sub></td><td>Mean</td><td>5,68</td><td>4,72</td><td>7,99</td><td>6,08</td><td>6,1</td><td>4,28</td></tr><tr><td>Standard deviation</td><td>1,25</td><td>1,33</td><td>1,87</td><td>1,88</td><td>1,49</td><td>1,28</td></tr><tr><td>Standard Error of the Mean</td><td>0,51</td><td>0,54</td><td>0,76</td><td>0,77</td><td>0,61</td><td>0,52</td></tr><tr><td rowspan="12">Grease lamp</td><td rowspan="3">D1<sub>0°</sub></td><td>Mean</td><td>3,09</td><td>2,59</td><td>3,89</td><td>2,65</td><td>2,61</td><td>2,39</td></tr><tr><td>Standard deviation</td><td>0,68</td><td>0,71</td><td>0,87</td><td>0,76</td><td>0,55</td><td>0,5</td></tr><tr><td>Standard Error of the Mean</td><td>0,16</td><td>0,17</td><td>0,2</td><td>0,18</td><td>0,13</td><td>0,12</td></tr><tr><td rowspan="3">D1<sub>45°</sub></td><td>Mean</td><td>2,75</td><td>1,89</td><td>3,39</td><td>2,54</td><td>2,68</td><td>2,41</td></tr><tr><td>Standard deviation</td><td>0,33</td><td>0,33</td><td>0,44</td><td>0,75</td><td>0,77</td><td>0,63</td></tr><tr><td>Standard Error of the Mean</td><td>0,13</td><td>0,13</td><td>0,18</td><td>0,31</td><td>0,31</td><td>0,26</td></tr><tr><td rowspan="3">D2<sub>0°</sub></td><td>Mean</td><td>5,64</td><td>4,25</td><td>6,88</td><td>4,81</td><td>4,75</td><td>4,02</td></tr><tr><td>Standard deviation</td><td>0,79</td><td>0,9</td><td>1,31</td><td>0,71</td><td>0,79</td><td>0,8</td></tr><tr><td>Standard Error of the Mean</td><td>0,19</td><td>0,21</td><td>0,31</td><td>0,17</td><td>0,19</td><td>0,19</td></tr><tr><td rowspan="3">D2<sub>45°</sub></td><td>Mean</td><td>5,73</td><td>3,41</td><td>6,12</td><td>4,57</td><td>4,91</td><td>4,48</td></tr><tr><td>Standard deviation</td><td>1,1</td><td>0,49</td><td>0,67</td><td>0,66</td><td>1,24</td><td>1,9</td></tr><tr><td>Standard Error of the Mean</td><td>0,45</td><td>0,2</td><td>0,27</td><td>0,27</td><td>0,51</td><td>0,78</td></tr></tbody></table>

D1 with a 0° angle of incidence is noted as D1<sub>0°</sub>, and D1 with a 45° angle of incidence is noted as D1<sub>45°</sub>. The same is true for D2<sub>0°</sub> and D2<sub>45°</sub>. The standard deviations and the standard error of the mean are given, for torch and grease lamp lighting conditions and incidence angles of 0° (18 measurements) and 45° (6 measurements).

Figure 14 shows the parallelism of the values of D1<sub>0°</sub> for the two lighting types and the same for D2<sub>0°</sub>.
<figure><img src="image_15.png" /><figcaption>**Figure 14:** Mean and Standard Error of the Mean distance measurements *D1*₀° and *D2*₀°. The two lighting conditions are considered for the six GEs at the 0° incidence angle. Published under a CC BY 4.0 license, with permission from French Ministry of Culture.</figcaption></figure>

It is observed that the large, engraved rhino (GE #3) is clearly more visible than the other 5 GEs, in particular reds and blacks, whether for the distance thresholds *D2* and *D1*, for the 0° and 45° incidences. One-way ANOVA confirms these observations. For instance, for the torch, at 0° incidence, one-way ANOVA on *D1* for the factor GE gives a global p-value < 0.001. The assumptions for the ANOVA have been verified using Shapiro-Wilk (normality) and Bartlett (homoscedasticity) tests. Post-hoc Dunnett test provides evidence of the superiority of GE #3 relative to the other ones: p-values < 0.001 for the five pairs of GEs (#3 vs #1, #3 vs #2, etc.). The same is established for *D2* (global p-value is < 0.001, and paired p-values are lower than 0.001 too). Same findings are obtained for the grease lamp, for *D1* and *D2*, for 0° incidence: all p-values stay under 0.003. These observations are weaker for the incidence 45° (for torch and grease lamp, for *D1* and *D2*) given the small number of measurements which reduces the power of pair analysis.

Figure 15 shows the effect of GE size on the *D1*₀° and *D2*₀° values: the larger the GE, the bigger *D1*₀° and *D2*₀°.
<figure><img src="image_16.png" /><figcaption>**Figure 15:** Mean threshold values of D1₀° and D2₀° (meters) for the six graphic entities in the two size groups. The three large GEs are in one group, the three small GEs in another one, at a 0° incidence angle, with torch lighting. The error bars represent the standard error of the mean. Published under a CC BY 4.0 license, with permission from French Ministry of Culture.</figcaption></figure>

### 3.1.2 Effect of the angle of incidence

We study the effect of the angle of incidence (Table 3) via several ratios: $D2_0°/D1_0°$, $D2_{45°}/D2_{45°}$, $D1_{45°}/D1_0°$ and $D2_{45°}/D2_0°$.

<table><caption>**Table 3:** Average ratios for each of the six graphic entities and each of the two lighting types.</caption><thead><tr><th rowspan="2">Ratio</th><th rowspan="2">Lighting</th><th colspan="6">Number of the Graphic Entity</th></tr><tr><th>1</th><th>2</th><th>3</th><th>4</th><th>5</th><th>6</th><th>Mean</th></tr></thead><tbody><tr><td rowspan="2"><i>D2</i><sub>0°</sub>/<i>D1</i><sub>0°</sub></td><td>Torch</td><td>1.90</td><td>1.72</td><td>1.73</td><td>1.77</td><td>1.81</td><td>1.73</td><td>1.78</td></tr><tr><td>Grease lamp</td><td>1.83</td><td>1.64</td><td>1.77</td><td>1.81</td><td>1.82</td><td>1.68</td><td>1.76</td></tr><tr><td rowspan="2"><i>D2</i><sub>45°</sub>/<i>D1</i><sub>45°</sub></td><td>Torch</td><td>1.72</td><td>1.87</td><td>1.67</td><td>1.75</td><td>1.70</td><td>1.75</td><td>1.75</td></tr><tr><td>Grease lamp</td><td>2.08</td><td>1.80</td><td>1.81</td><td>1.80</td><td>1.84</td><td>1.86</td><td>1.86</td></tr><tr><td rowspan="2"><i>D1</i><sub>45°</sub>/<i>D1</i><sub>0°</sub></td><td>Torch</td><td>0.82</td><td>0.71</td><td>0.87</td><td>0.97</td><td>0.99</td><td>0.69</td><td>0.84</td></tr><tr><td>Grease lamp</td><td>0.89</td><td>0.73</td><td>0.87</td><td>0.96</td><td>1.02</td><td>1.01</td><td>0.91</td></tr><tr><td rowspan="2"><i>D2</i><sub>45°</sub>/<i>D2</i><sub>0°</sub></td><td>Torch</td><td>0.74</td><td>0.78</td><td>0.84</td><td>0.96</td><td>0.93</td><td>0.70</td><td>0.82</td></tr><tr><td>Grease lamp</td><td>1.02</td><td>0.80</td><td>0.89</td><td>0.95</td><td>1.03</td><td>1.11</td><td>0.97</td></tr></tbody></table>

$D2_0°/D1_0°$ ratio is very homogeneous. These observations are almost identical for the $D2_{45°}/D1_{45°}$ ratio. For torch lighting, $D1_{45°}/D1_0°$ and $D2_{45°}/D2_0°$ are very close to 0.84. For grease lamp lighting, there is a greater difference between the ratios.

These observations are consistent with Lambert's cosine law in which the diffuse component of surface illumination is proportional to $cos(\theta)$, where $\theta$ is the angle between the normal at the illuminated surface and the direction of the light source. As the observer is the light-carrier in the experiment, $\theta$ is also the angle between the surface normal and the observer's viewing direction. In addition, the amount of light received per unit of illuminated area is inversely proportional to the square of the distance between the area and the
light source. Finally, the distance threshold at an angular deviation $\theta$ from the 0°, $D1_\theta$, is deduced from $D1_{0\circ}$ according to the following equation:

$$D1_{\theta} = D1_{0\circ} \cos(\theta) \qquad (12)$$

The formula also applies to the $D2$ threshold:

$$D2_{\theta} = D2_{0\circ} \cos(\theta) \qquad (13)$$

These formulas accurately model the $D1_{45\circ}/D1_{0\circ}$ and $D2_{45\circ}/D2_{0\circ}$ ratios of distances measured experimentally for the 45° incidence angle relative to the 0° incidence (~0.84), which matches $\cos(45°)$. Above all, these equations enable us to extrapolate the $D1_\theta$ and $D2_\theta$ distances for any $\theta$ angle of incidence, which will be used for the GEs of Chauvet cave.

### 3.1.3 Affine model

The adjustment of the measurements of $D1_{0\circ}$ (resp. $D2_{0\circ}$) by linear regression (S5) reveals an affine relationship between an objective quantity, the *parietal visibility coefficient* (PVC), and a percept, the visibility distance $D1_{0\circ}$ (resp. $D2_{0\circ}$):

$$D1_{0\circ} = 1.41 \text{ PVC} + 2.6 \qquad (14)$$

$$D2_{0\circ} = 2.46 \text{ PVC} + 4.67 \qquad (15)$$

The graphic representations of each of the two functions are provided in Figure 16.

<figure><img src="image_17.png" /><figcaption>Figure 16: Values of the sight distances relative to the parietal visibility coefficient of the six graphic entities, with regression lines. $D1_{0\circ}$ blue dots and $D2_{0\circ}$ orange dots are given for the 0° incidence and torch lighting. Published under a CC BY 4.0 license, with permission from French Ministry of Culture.</figcaption></figure>

This result shows that for the GE in the experiment, the sight distances $D1_{0\circ}$ and $D2_{0\circ}$ under a 0° incidence angle can be reasonably predicted from a photograph by segmenting the area of the GE and separating
its artistic representation from the surrounding wall. This model is robust for variable segmentations (tests conducted with two different analysts on several GEs resulted in a similar affine model). The rightmost points on each of the two straight lines in Figure 16 correspond to the large engraved rhino (GE #3): these values of the sight distances, much higher than those associated with the other GEs (see also Figure 13 and Figure 14), are explained by the high value of the PVC of this GE.

### 3.1.4 Lighting type analysis

Figure 14) and Table 3 show that between torch lighting and grease lamp lighting, the *D1* and *D2* values vary significantly, but always in the same *D1/D2* ratio. The difference in values between these two lighting types is explained mainly by the ratio of the intensities of the two sources. Based on the mean of the measured luminance of the walls, the average luminance of them for *D1* at 0° was 0.040 cd/m² with the torch and 0.031 cd/m² with the grease lamp (S6 Tables). Similarly, for *D2* at 0°, the average luminance was 0.013 cd/m² with the torch and 0.010 cd/m² with the grease lamp. These values correspond well to the range of mesopic vision. In this domain, contrast sensitivity and color discrimination are known to strongly depend on the luminance level [60-61]. We will use only torch lighting, whose use is known in Chauvet cave, in the following calculations. The results obtained with the grease lamp can be used for other caves where evidence for this type of lighting has been found.

## 3.2 Grouping approach to process the large number of graphic entities in Chauvet cave

For each of the 487 GE of the study area, we need to estimate the distributions of *D1* and *D2* based on its surface and PC. The PC calculation requires photographic documentation that is adequate in terms of resolution, lighting and camera angle. Faced with this large-scale work, a grouping approach was chosen, using 60 photographs, covering more than 80% of the possible combinations of line types and geological facies, to quantify the PCs, using the following approach:

* Grouping into profiles the GEs displaying the same artistic technique (provided by the GIS) on the same type of wall (known via the segmentation of wall meshes).

* Identification of a representative GE for each group for which a photograph is available.

* Calculation of the PC of this GE using this photograph and Equation rereq:eq2

* Attribution of this PC to all GEs of the group.

* Calculation of the PVC of each GE of the group by injecting, via Equation 3 the surface of the GE obtained via 3D segmentation.

* Use of affine relationships (Equation 14 and Equation 15) to calculate the GE-specific thresholds for *D1* and *D2*, with standard deviations σ₁ and σ₂ attributed to them.

Each step is detailed below.

The notion of facies and the facies types are detailed in subsection Segmentation of the walls and GE meshes by facies. In the study area, ten wall facies are decorated, among which four facies are associated with 95% of the GE, as shown in Table 4.

The artistic techniques used to create the GEs are designated as follows:

* B: Black

* R: Red

* Y: Yellow

* f: finger-tracing

* e: engraving (without wall scraping)

* s: wall scraping
<table><caption>Table 4: Distribution of graphic entities per facies studied, by descending order.</caption><tr><td>Facies index</td><td>1b</td><td>1</td><td>4</td><td>5</td><td>3</td><td>6</td><td>2</td><td>9</td><td>7</td><td>8</td></tr><tr><td>Number of GE</td><td>210</td><td>121</td><td>100</td><td>72</td><td>17</td><td>10</td><td>3</td><td>3</td><td>1</td><td>1</td></tr></table>

Some artistic techniques (blowing, impression) are not currently included in the study.

A single GE is sometimes created using a combination of techniques and/or across several facies. We designate a technique and facies pair as a profile. Table 5 presents the 55 profiles that correspond to all the GEs in the study zone.

<table><caption>Table 5: Distribution of graphic entities per profile, in descending order.</caption><thead><tr><th>Profile</th><th>N</th><th>Profile</th><th>N</th><th>Profile</th><th>N</th><th>Profile</th><th>N</th><th>Profile</th><th>N</th></tr></thead><tbody><tr><td>1b B</td><td>115</td><td>1b Bs</td><td>7</td><td>2 R</td><td>3</td><td>5 Bes</td><td>2</td><td>4 es</td><td>1</td></tr><tr><td>1 R</td><td>89</td><td>3 B</td><td>7</td><td>4 sf</td><td>3</td><td>5 sf</td><td>2</td><td>5 Bs</td><td>1</td></tr><tr><td>4 f</td><td>49</td><td>5 Be</td><td>7</td><td>6 e</td><td>3</td><td>1 es</td><td>1</td><td>5 R</td><td>1</td></tr><tr><td>1b Be</td><td>35</td><td>5 f</td><td>7</td><td>6 B</td><td>3</td><td>1 RY</td><td>1</td><td>5 Res</td><td>1</td></tr><tr><td>4 e</td><td>28</td><td>5 es</td><td>6</td><td>9 R</td><td>3</td><td>1 Y</td><td>1</td><td>5 s</td><td>1</td></tr><tr><td>1b e</td><td>22</td><td>1 Be</td><td>5</td><td>1 BR</td><td>2</td><td>3 et</td><td>1</td><td>6 Bes</td><td>1</td></tr><tr><td>5 B</td><td>22</td><td>1 e</td><td>4</td><td>1b BR</td><td>2</td><td>3 Bes</td><td>1</td><td>6 BR</td><td>1</td></tr><tr><td>5 e</td><td>20</td><td>1b es</td><td>4</td><td>1b BRe</td><td>2</td><td>3 Bs</td><td>1</td><td>6 Bs</td><td>1</td></tr><tr><td>1 B</td><td>18</td><td>1b Bes</td><td>4</td><td>4 ef</td><td>2</td><td>3 R</td><td>1</td><td>6 R</td><td>1</td></tr><tr><td>1b R</td><td>16</td><td>3 e</td><td>4</td><td>4 B</td><td>2</td><td>3 f</td><td>1</td><td>7 BR</td><td>1</td></tr><tr><td>4 s</td><td>15</td><td>1b s</td><td>3</td><td>5 ef</td><td>2</td><td>3 s</td><td>1</td><td>8 R</td><td>1</td></tr></tbody></table>

The facies identifier originates from the 3D segmentation. The GIS provides the artistic technique. The PC of each profile was calculated based on the 60 photographs. Nine samples are given in Table 6.
<table><caption>**Table 6: Examples of the photographs used to estimate the parietal contrast of graphic entity profiles.**</caption><tr><td>Profile 2R, GE 113<br>PC = 4</td><td>Profile 5B, GE 193 and 195<br>PC = 3.5</td><td>Profile 9R, GE 329 and 330<br>PC = 3</td></tr><tr><td>Profile 1B, GE 196<br>PC = 2</td><td>Profile 1bR, GE 353<br>PC = 2</td><td>Profile 5f, GE 4<br>PC = 2</td></tr><tr><td>Profile 4f, GE 24<br>PC = 2</td><td>Profile 5E, GE 201<br>PC = 1.5</td><td>Profile 4E, GE 495<br>PC = 1</td></tr></table>

<figure><img src="image_18.png" /></figure>

<figure><img src="image_19.png" /></figure>

<figure><img src="image_20.png" /></figure>

<figure><img src="image_21.png" /></figure>

<figure><img src="image_22.png" /></figure>

<figure><img src="image_23.png" /></figure>

<figure><img src="image_24.png" /></figure>

<figure><img src="image_25.png" /></figure>

<figure><img src="image_26.png" /></figure>

Samples are given in descending order of their PC value.

The segmentation was carried out as described in Figure 5.

For a few of the profiles (~15% of GE), resulting from a combination of artistic techniques (1bBES, 6BR, etc.), the value of the marginal profile with the highest parietal contrast was used. Therefore, profile 1bBES inherits the PC of profile 1bB, etc. Finally, some “orphan” profiles that did not have adequate documentation (~2% of GE) received the average of the parietal contrasts of the other profiles with the same artistic technique. Due to inter-profile variability, contrast values are rounded to 0.5 (Table 7).
<table><caption>Table 7: Parietal contrasts for the different profiles.</caption><thead><tr><th rowspan="2">Technique</th><th colspan="5">Pigments only</th><th colspan="5">Engravings only</th><th colspan="5">Mixed</th></tr><tr><th>B</th><th>R</th><th>Y</th><th>BR</th><th>RY</th><th>e</th><th>f</th><th>s</th><th>es</th><th>sf</th><th>ef</th><th>BRe</th><th>Be</th><th>Bs</th><th>Bes</th><th>Res</th></tr></thead><tbody><tr><td>1b</td><td>3</td><td>2</td><td></td><td>3</td><td></td><td>1,5</td><td></td><td>1</td><td>1,5</td><td></td><td></td><td>3</td><td>3</td><td>3</td><td>3</td><td></td></tr><tr><td>1</td><td>2</td><td>1,5</td><td>1,5</td><td>2</td><td>2</td><td>1,5</td><td></td><td></td><td>1,5</td><td></td><td></td><td></td><td>2</td><td></td><td></td><td></td></tr><tr><td>4</td><td>3</td><td></td><td></td><td></td><td></td><td></td><td>1</td><td>2</td><td>1,5</td><td>1,5</td><td>2</td><td>2</td><td></td><td></td><td></td><td></td></tr><tr><td>5</td><td>3,5</td><td>2,5</td><td></td><td></td><td></td><td>1,5</td><td>2</td><td>1,5</td><td>1</td><td>2</td><td>2</td><td></td><td>3,5</td><td>3,5</td><td>2,5</td><td>2,5</td></tr><tr><td>3</td><td>3</td><td>2,5</td><td></td><td></td><td></td><td>1,5</td><td>1,5</td><td>2</td><td></td><td></td><td>1,5</td><td></td><td></td><td>3</td><td>2,5</td><td></td></tr><tr><td>6</td><td>3,5</td><td>2,5</td><td></td><td>3,5</td><td></td><td>2</td><td></td><td></td><td></td><td></td><td></td><td></td><td>3,5</td><td>2,5</td><td></td><td></td></tr><tr><td>2</td><td></td><td>4</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>9</td><td></td><td>2,5</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>7</td><td></td><td></td><td></td><td>2,5</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>8</td><td></td><td>1,5</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr></tbody></table>

Based on these profiles, the PVC can be calculated for each GE of the considered profile using Equation 3 by entering the value of $S_{GE}$, the GE area (provided by the segmentation of the 3D mesh of this GE according to the envelope principle illustrated in Figure 6).

The uncertainty associated with each profile remains to be quantified. The main profiles are assigned the experimental standard deviations of all GEs, for all subjects, under the 0° incidence, for torch lighting, i.e., $\sigma_{10°} = 1.0$ m and $\sigma_{20°} = 1.4$ m (Table 2). The standard deviations for 45° are derived from the experiment by observing that these values for 45° are very close to those estimated for 0° (Table 2): $\sigma_{145°} = 1.03 \approx \sigma_{10°}$ and $\sigma_{245°} = 1.45 \approx \sigma_{20°}$. Given the small number of trials under the 45° incidence, differences with the 0° incidence angle are neglected. The standard deviations of inherited (inh) and orphaned (orp) profiles are weighted as follows (for 0° and 45° incidences):

$$ \sigma_{1inh} = 1.5 \sigma_{10°} \qquad \sigma_{2inh} = 1.5 \sigma_{20°} \qquad (16) $$

$$ \sigma_{1orp} = 2 \sigma_{10°} \qquad \sigma_{2orp} = 2 \sigma_{20°} $$

When a GE covers two facies or more, the maximum value of the $\sigma_{10°}$ and $\sigma_{20°}$ of the facies on which the GE is made is used.

## 3.3 3D modeling of the Chauvet cave walls and graphic entities

### 3.3.1 Objective

It is necessary to know the surface area of each GE, per facies, to determine its PVC, as well as the 3D spatial extent of each GE to determine if it is fully or partially visible from each OP. We thus meshed the geometry of the walls and GEs via the 3D survey, recreated the geomorphology of the cave in prehistoric times where needed, and segmented the mesh of each GE, per facies.

### 3.3.2 Cloud meshing

The point cloud of the walls was meshed in the form of a two-manifold triangular 3D mesh [62] using the Delaunay 3D triangulation method [63]. The final resolution of the model is set at 3 cm (average triangle based on [20] independent measurements on the points and final mesh sizes). The objective was to obtain, from a cloud of variable resolution, a continuous model with no holes, whose resolution is compatible with the point cloud and the geometric complexity of the wall to be analyzed [53-62] (Figure 17 left column).
<figure><img src="image_27.png" /><figcaption>**Figure 17:** Meshing process of the point cloud acquired on the walls (left column) and the segmentation process (right column). Published under a CC BY 4.0 license, with permission from French Ministry of Culture / Stéphane Jaillet - Kim Genuite.</figcaption></figure>

### 3.3.3 Recreating the geomorphology of the cave in prehistoric times

A virtual reconstruction of the cave during the Paleolithic period to ensure that the visibility calculations are based on the morphology of the walls, their conditions, and the topography of the GEs at the time of their creation. This work is based on a geomorphological study of the cave walls and floors, the intercalation of the wall and floor deposits, their dating, and the displacement of archaeological and paleontological remains on the floors [64-65]. Based on a speleogenetical analysis, it is thus possible to propose different representations of the cave, before, during and after the prehistoric occupations. To accurately calculate the OMs and place the OPs on them, it was essential to reconstruct the topography of the subterranean
landscapes contemporary with Paleolithic uses. In some sectors, based on geomorphological criteria, we thus filled in the solution-collapses that formed after the cave entrance was sealed (Figure 16 A-B). It was also necessary to remove the Holocene concretions, recent calcite veils on the walls, and carbonate crusts that formed on the floors after the Paleolithic occupations (Figure 18 C-D).

<figure><img src="image_28.png" /><figcaption>**Figure 18: Reconstruction of the morphologies and conditions of the cave walls at the time of Paleolithic occupations.** Current underground landscapes in the Hillaire Chamber (A) and Cactus Gallery (C). (B-D) Morphological reconstructions of the cave during the Paleolithic period based on geomorphological evidence (restitution of the clayey nature of the floors and removal of the Holocene concretions). Published under a CC BY 4.0 license, with permission from French Ministry of Culture / Benjamin Sadier - Stéphane Jaillet.</figcaption></figure>

These calcite deposits modify the visibility of the GEs, the reflectance of the walls and the absorption of light by the floors. In the Cactus Gallery, for example, the curtain of white calcite columns that now hides the second section of the gallery was formed after the Paleolithic; to recreate the morphology and visual field of the time, it was thus necessary to remove the mesh of these speleothems.

### 3.3.4 3D segmentation of the graphic entity meshes

In the study zone, we manually segmented 487 GEs (Figure 17, right column): this segmentation consisted of cutting the envelope of each GE in the 3D mesh. This work was guided by the point cloud laser reflectance information (the cloud and mesh being positioned in the same coordinate system, Figure 19).
<figure><img src="image_29.png" /><figcaption>**Figure 19: Point cloud with laser reflectance information in grayscale.** This intensity recorded by the scanner makes it possible to distinguish the GEs (light lines) on the walls. Published under a CC BY 4.0 license, with permission from French Ministry of Culture / Kim Genuite.</figcaption></figure>

The measurement of a 3D point for each GE (performed in the field using a tacheometric survey [28]) facilitated their location in the point cloud. However, since reflectance is partly a function of the color of the wall elements touched by the laser spot, some GEs, especially red paintings, are difficult to read in the point cloud. To resolve this difficulty, the GE segmentation was carried out jointly by a parietal art specialist and a 3D operator. A pair of specialists working together thus observed the point cloud, mesh, photographs, manual surveys and GIS tables.

Because the finesse of the contours of the GEs on the cave walls is greater than the average size of the mesh triangles, the segmentation generates a spatial uncertainty of ± one triangle [66] relative to the ideal segmentation specified in Figure 6. The mesh being composed of relatively regular triangles with an average edge of 3 cm, this source of error is considered as low relative to the resolution of the calculations by ray casting.

A division per elementary GE, rather than per panel (cf. 4 Discussion), was privileged in coherence with our experimentation and to facilitate the interpretation of the results.

The wall and the GE meshes were simplified into a Triangle Irregulated Network mesh (TIN) (Figure 17, right column) to moderate the calculation times [62, 67]. These optimizations nearly preserve the contour geometry of each GE generated during the facies/GE cuts, while significantly reducing the number of triangles.

The surface area of a GE, necessary to calculate the PVC of the GE, was simply computed by summing the area of each triangle of the GE mesh.

### 3.3.5 Segmentation of the walls and graphic entity meshes by facies

The same facies as those identified for the replication of the Chauvet-Pont d'Arc Cave [68] were selected. The term facies covers the lithological nature of the bedrock that outcrops on the wall, the state of the rock (altered or unaltered) and the state of the surface (covered or not with clay or calcite). Each facies is described on an identity sheet indicating the nature and state of the wall (Figure 20 A). The spatial distribution of these different facies within the cave served as a guide for cutting the wall meshes [68] as shown in (Figure 20 B-C).

Nevertheless, to obtain the values required for this study, additional geomorphological surveys were completed in the cave, drawing the boundaries between the different facies directly on the point cloud. Twelve main facies were selected, and each was assigned a color code and a number (Figure 20 A).
A)

<figure><img src="image_30.png" /></figure>

B)

<figure><img src="image_31.png" /></figure>

C)

<figure><img src="image_32.png" /></figure>

**Figure 20:** The twelve main facies in Chauvet cave and two examples of the segmentation of the wall and floor meshes according to facies. (A) The main facies; (B-C) Segmentations of the meshes in the Hillaire Chamber and the End Chamber. Published under a CC BY 4.0 license, with permission from French Ministry of Culture / Jean-Jacques Delannoy.

Several GEs are located on two, or even three, facies (Figure 21).
<figure><img src="image_33.png" /><figcaption>**Figure 21: Segmentation of a wall into distinct facies (one color per facies) and segmentation of the graphic entities on top of the facies.** Example of the Megaloceros Gallery. Published under a CC BY 4.0 license, with permission from French Ministry of Culture.</figcaption></figure>

The mesh of such GEs was separated into as many single-facies meshes, considering that some were
created across two facies. Then Equation 4 was used.

3.3.6 Naming the graphic entity meshes taken from the GIS tables

The designations of the GE meshes are composed of their GE identifier in the GIS, suffixed with the name of
the corresponding facies (e.g., 242_facies01b in the figure below). The purpose of this nomenclature is to
automate the data processing: when a ray falls on a GE, the calculation code accesses not only the distance
measurement D (Figure 7) but also the name of the facies (necessary for the calculation of the PVC), as well
as the GE identifier. This enables access, via the GIS tables, to the attributes of the GE, such as its artistic
technique and its associated animal species, etc. (Figure 22).

<figure><img src="image_34.png" /><figcaption>**Figure 22: Identification of a GE in connection with the GIS and facies.** In the example of the End Chamber (a), GE 242 mesh name includes facies 1b information (b) and refers to the GIS data (c). Published under a CC BY 4.0 license, with permission from French Ministry of Culture.</figcaption></figure>

The GE scores can then be filtered according to the attribute considered. This enabled us to create the maps
shown below.

3.4 Calculations

3.4.1 Calculating the observation meshes and points for the three postures

To correctly process the narrow passages (S7 Appendix for the observation meshes data), the OM resolution
was chosen so that the edges of the triangles have a length smaller than the diameter of the subjects (20 cm,
60 cm). There are few differences between the three OMs in the spaces studied, except in the Red Panels Gallery and the End Chamber (Figure 23).

<figure><img src="image_35.png" /><figcaption>**Figure 23:** Observation mesh for each posture. From left to right: H70, H125 and H180. Published under a CC BY 4.0 license, with permission from French Ministry of Culture / Jean-François Hullo.</figcaption></figure>

Table 8 summarizes the number of OPs considered for each posture (S8 Appendix for the observation points data).

<table><caption>**Table 8:** Number of Observation Points per posture and the total for the study area, for a 50 cm step.</caption><thead><tr><td>Posture</td><td>Number of observation points</td></tr></thead><tbody><tr><td>H70</td><td>8,977</td></tr><tr><td>H125</td><td>8,843</td></tr><tr><td>H180</td><td>7,249</td></tr><tr><td>Total</td><td>25,069</td></tr></tbody></table>

### 3.4.2 Calculating the visibility scores

The model equations were applied to the data of Chauvet cave, thus providing, for each OP, the visibility score of each GE for *D1* and *D2*, as well as *E*<sub>j</sub> and *P*<sub>maxj</sub> (S9 Appendix).

In total, the calculation provided 12,258,741 scores from the 25,069 OPs for 489 GEs. Nine hours of calculation were required (with an Intel Core i7 processor at 2.6 GHz) to obtain the result files for a 0.5 m spacing between the OPs.

## 3.5 Cartographic renderings

The method presented in the previous section enables us to quantify the visibility for any posture. For the sake of simplicity, only the H180 posture will be discussed in this section.
Below, we discuss several archaeological questions with reference to several visibility maps.

**3.5.1 In which areas can a light-carrying person see at least one graphic entity?**

As a light-carrier approaches some of the decorated walls, some initial GEs can be discerned even if their details are not perceptible, which is the definition of the *D2* threshold. Determining the GE best seen from each OP, e.g., by exploiting the *Pmax*₂ field yielded the maps in Figure 24, showing the zones in which a light-carrier can perceive the GEs.

<figure><img src="image_36.png" /><figcaption>**Figure 24:** Map of visibility zones from which one can detect the graphic entities without perceiving their details. The *Pmax*₂ field is encoded on a scale from blue to yellow. (A) Rendering for the entire study area; (B) Zoom in the Red Panels Gallery (located by the blue rectangle in (A)). The white lines (rosettes) starting from each OP indicate the directions in which the GEs are visible. Published under a CC BY 4.0 license, with permission from French Ministry of Culture / Jean-François Hullo.</figcaption></figure>

**3.5.2 In which zones can a light-carrying person see at least one graphic entity in detail?**

The light-carrier must approach the walls up to a certain threshold from which they can see all the details of the GE. This is the definition of the *D1* threshold. At each OP, the score of the most visible GE according to *D1*, namely *Pmax*₁, locates these visibility zones from which details can be seen.

Figure 25 illustrates this field of visibility.
<figure><img src="image_37.png" /><figcaption>**Figure 25:** Map of visibility zones from which one can perceive the GEs with their details. The *Pmax*₁ field is encoded on a scale from black to yellow to orange. (A) Rendering for the entire study area; (B) Zoom in the Red Panels Gallery. Published under a CC BY 4.0 license, with permission from French Ministry of Culture / Jean-François Hullo.</figcaption></figure>

### 3.5.3 In which areas can a light-carrying person identify several graphic entities, discerning their artistic technique and animal species?

When the light-carrier enters the visibility zones, they can, in many locations, observe several GEs in detail. The expectation of the number of visible GEs at the *D*1 threshold, *E*₁, enables us to locate the visibility zones of groups of GEs, as shown in Figure 26.
<figure><img src="image_38.png" /><figcaption>**Figure 26:** Map of the expectation of detailed visibility of the graphic entities. The *E*₁ field is encoded on a gray-green-yellow-red-brown color palette. (A) Rendering for the entire study area; (B) Zoom in the Red Panels Gallery. Published under a CC BY 4.0 license, with permission from French Ministry of Culture / Jean-François Hullo.</figcaption></figure>

The rosette of directions in each OP (Figure 26 A) shows where the observer benefits from a more panoramic viewpoint, where the GEs cannot be taken in with a single gaze (the rosette is then fan-shaped). Conversely, a rosette narrowed in a single direction characterizes an OP where the visibility is concentrated on a few GEs that are close to each other or even a single GE. This vector rosette provides more information on the angular distribution of visible GEs than a tensor does if it is illustrated as an ellipsis field, as proposed by [69].

A map can be produced per artistic technique by sorting the scalar fields according to the attribute attached to the GE, extracted from the GIS, as shown in Figure 27.
<figure><img src="image_39.png" /><figcaption>**Figure 27:** Maps of the expectation of detailed visibility of the GEs according to the artistic technique. The *E*₁ field is encoded in a gray-green-yellow-red-brown color palette. (A) GE in red; (B) GE in black; (C) GE without pigment, produced by finger-tracing, scraping or engraving. Published under a CC BY 4.0 license, with permission from French Ministry of Culture / Jean-François Hullo.</figcaption></figure>

Another rendering enables an analysis of the GE observation zones according to the animal species depicted,
by sorting per species in the GIS attribute field (Figure 28).
<figure><img src="image_40.png" /><figcaption>**Figure 28:** Maps of the locations from which the GEs could be identified according to the animal species. (A) for felines; (B) for mammoths, with respect to *Pmax*₁. It thus appears that mammoths are generally not co-visible, while felines are almost always arranged in groups of co-visible GEs. Published under a CC BY 4.0 license, with permission from French Ministry of Culture / Jean-François Hullo.</figcaption></figure>

**3.5.4 Which locations offer the maximal visibility of the graphic entities? What are the inter-visibility relationships between these locations?**

Similar questions on visibility maxima were addressed by [47]. In the case of Chauvet cave, in the GE visibility zones, the light-carrier was able to find the precise location combined with the best posture to see a maximum number of GEs. From these exceptional OPs, each movement to another OP reduces the number of GEs perceived. These locations, defined as hotspots in the previous chapter, correspond to the local maxima of the *E*₁ field. Twenty-five hotspots, numbered by their *D*1 expectation, were highlighted in the study zone (Figure 29). From these detailed visibility hotspots, the light-carrier can detect (in the sense of the *D*2 threshold) at least one of the GEs visible in detail from other hotspots. This inter-visibility is represented by an arrow between two hotspots, the base of which corresponds to the hotspot from which the artworks of the second hotspot (arrow tip) are seen. A double arrow indicates that from both hotspots the observer can see some of the artworks of the other one. Six independent networks can be identified.
<figure><img src="image_41.png" /><figcaption>**Figure 29:** Map of the visibility hotspots for *E*₁ and their inter-visibilities. The hotspots are numbered in descending order of their expectation. Published under a CC BY 4.0 license, with permission from French Ministry of Culture / Jean-François Hullo.</figcaption></figure>
### 3.5.5 Which areas have high visibility variation in the Skull Chamber?

Following the Visual changes definition (cf. previous chapter), directional filtering of GE subsets seen per OP allows GE visibility changes to be recorded as one moves from OP to OP. This directional filtering is applied here in the northwest direction of the Skull Chamber (Figure 30).

It locates strong discontinuities in visibility resulting from the effect of the cave wall folds.

<figure><img src="image_42.png" /><figcaption>**Figure 30: Study of visual landscape changes per difference in visibility, GE by GE.** (A) Photographic view of the panels of the Skull Chamber; (B) Top view of the field of expectation that reveals a hotspot; (C) Top view of the difference field, GE by GE, as the observer takes a step (1 m) in the northwest direction (white arrows). Published under a CC BY 4.0 license, with permission from French Ministry of Culture / Stéphane Jaillet - Jean-François Hullo.</figcaption></figure>
# 4 Discussion

## 4.1 The visibility model

### 4.1.1 About the experimental protocol

In our experiment, only a small number of subjects was tested: six observers and six GEs. Compared to typical psychophysical laboratory experiments, the protocol we used, in real outdoor conditions, is far more constraining for the participants, who had to walk in darkness while carrying a torch or a grease lamp in their hand. Thus, for practical reasons, we could neither ask more people to participate in the experiment nor ask the participants to do more tests. Considering the additional time required for the realization of GEs, the preparation of light sources and the measurements, we could not have guaranteed the stability of the experimental conditions with more participants.

Despite this limitation, the foundations of the proposed probabilistic model are already consistent between the *D1* and *D2* measurements, for both the 0° and 45° angles of incidence, and for both the torch and grease lamp lighting techniques. It is striking to observe that the *D2/D1* ratio is nearly identical for the six GEs, both lighting techniques and both incidence angles (Table 2). This constancy of the *D2/D1* ratio is consistent with a wider experimental result obtained by [22-23]: there exists a fixed ratio between the detection and identification of visual patterns relative to the visual eccentricity. In our experiment, the subjects were free to visually explore the GE and the environment, thus reducing the effect of eccentricity. In addition, the size of the six GEs and the perception distances were such that only the foveal and parafoveal visions were stimulated (for example, the eccentricity for the GE #1 at *D1* lies under 8°). In such conditions, [23] provides values of 0.06 for detection and 0.1 for identification at a threshold level of 75% of correct answers, yielding a detection/identification ratio of 1.7, which is very close to our values (Table 2). Similar experimental results were obtained by [22] with a mean ratio of the probability of identification divided by the probability of detection for the same stimulus close to 0.65 in foveal and parafoveal vision. These results, though obtained in different experimental conditions, confirm our evaluation of the performance in a detection task (*D2*) compared to an identification task (*D1*).

We have no reason to believe that the visual system of *Homo sapiens* has evolved significantly since the time of the paintings of Chauvet or Lascaux. There is, of course, currently no experimental measurement of visual acuity to confirm our working hypothesis, except for the exquisite quality of certain Paleolithic paintings and engravings.

[29] points out that there are cognitive effects of a *priori* knowledge that modify the threshold at which shapes are perceived. While these memory effects are difficult to characterize, the protocol that we developed partially reduces this difficulty. The observer does not need to concentrate his/her efforts on memorizing the GE since at the beginning of the trial they move forward to see the GE and then move backward along a simple path: the observer thus knows what and where the GE is, at least approximately. This allows them to focus on the visual detection and identification of the GE, which contributes to the quality of *D1* and *D2* measurements.

### 4.1.2 About the perceptual measures

The domain of mesopic vision [61] ranges between approximately 0.001 and 5 cd/m², depending on the eccentricity and measurement method [70]. In this region, the color vision and the visual acuity increases smoothly from the scotopic range (below 0.001 cd/m²) to the photopic domain (> 5 cd/m²) where color vision and visual acuity are optimal. The average D20° for the 6 GEs (S6 Tables) yields an average luminance of 0.013 cd/m² for the torch (respect. 0.01 cd/m² for the grease lamp) at this distance threshold, which shows that the experiment was entirely conducted within the domain of mesopic vision.

In this mesopic domain, the color vision is not optimal. In particular, the wavelength corresponding to the maximal sensitivity tends to shift progressively from 555 nm (photopic vision) toward 500 nm (scotopic vision). We might therefore address the question of the choice of pigments in terms of visibility. The possible dominant use of red and black pigments selected by the artist based on their capacity to be revealed by the predominantly red color of fire-light [15,17] is not observed in the experiment. The distance values proposed by [16] for the visibility of red artworks by a single candle (2.24 m and 1.29 m) are weaker than the statistics of *D2* and *D1* obtained experimentally for a grease lamp, for the #1 and #2 red GEs.
4.1.3 About the perceptual model

This part of our contribution is a psychophysical simulation of visibility based on a computational model of
perception (namely the visual detection and identification of GEs). One can draw a parallel with the work
of [30] who also resorted to the simulation, to study another perception (visual attention) using another
computational model (dedicated to saliency in her case).

Several papers emphasize the importance of the segmentation of a figure relative to the background
[71,72,73], as did we (Figure 5). These studies focused both on the role of luminance versus chrominance
in this task and the statistical differences in the distribution of pixel values. First, the perceptual ability to
segment shapes based only on luminance (with achromatic stimuli) or only chrominance (with isoluminant
stimuli), [72] for example, showed that with chrominance alone, it is possible to discriminate two shapes,
even if they have the same luminance as that of the background. This work corroborated earlier studies that
compared the detection thresholds of luminance vs. chrominance, e.g., [74]. By comparing the detection
thresholds of luminance contrast and chrominance contrast, these authors demonstrated that the thresholds
are similar when the background noise level is high (> 0.1), but the luminance contrast threshold is lower than
that of chrominance when the background noise levels are low (< 0.1). [75] confirms this observation for an
orientation discrimination task. These studies support the proposal to build the PC based only on luminance
(Equation 1). Second, the use of statistical differences in the distribution of pixel values refers to the signal
detection theory [76] broadly used in psychophysics and experimental psychology. In a simple case, this
theory considers that the signal to be detected and the noise (whatever the origin, environmental, sensory,
neuronal, etc.) are modeled by Gaussian distributions of the same variance but with different means. The
ability to detect the signal then relies on a quantity called the *discrimination index*, which is the ratio between
the difference of means and the noise standard deviation (taken as the square root of the variance). This
theory was applied with success to a large number of experimental situations. For instance, [75] quantified
the ability to discern different orientations in luminance and chrominance with an index which coincides with
our definition of the PC (Equation 2). Since the pixels of the GE are much less numerous than the pixels of
the wall, we assume that the variance of the signal, *i.e.*, the spatial variance of the luminance of the GE, is
equal to the spatial variance of the background. This variance is mainly due to the non-uniform texture of the
stone. Concisely, our definition of the PC, as a discrimination index based only on the pixel luminance, is
thus supported by studies using the standard signal detection theory, widely recognized as explanatory of
visual psychophysics experiments.

[16] elaborated distance values for the visibility of red artworks in the light of a single candle, in an otherwise
lightless environment. However, their approach relied on illuminance, not converted into luminance, the only
quantity that matters when studying human perception (See the Appendixes for the definition of illuminance
and luminance). Consider an illuminated wall, as proposed by [19-20]. It is the amount of light entering
through the pupil of the observer and processed by the visual system, i.e., the luminance, which has a
perceptive impact. When one moves away from a lit wall, without carrying a light with us, the portion of the
surface seen increases, which can lead to an increase (if it brings in a more reflective surface) or decrease
(if it makes entering a darker area) in the amount of luminance. But, if the surface is uniformly illuminated
and uniformly reflective, the luminance remains invariant. This is because the surface seen increases with
the square of the distance, but the solid angle corresponding to the pupil also decreases with the square of
the distance. Therefore, for a uniformly illuminated surface, the luminance does not depend on the distance
but only on the average albedo of the surface and its emitted illumination. In opposition to [19-20], we
consider the case of a light-carrying observer where the luminance decreases quadratically with increasing
distance from the wall, as pinpointed by [16]. To top off the relationship between luminance, contrast and
distance, [77] quantified the fact that when one moves away from the wall, due to luminance decreasing,
it is increasingly difficult to perceive the contrast, which is the parietal contrast in our model. Extending
the studies of [19-20], in which the lighting of the wall is presumed to be uniform, with the visibility being
impacted only by the concealment caused by the walls, we integrate an attenuation of the visibility score with
the distance between the light-carrying human and the parietal GEs, leading to a detailed perceptual model.

In the proposed model, for a given GE and a given OP, when the ratio between the solid angle of the seen
triangles and that of all the triangles of the GE is 50%, for example, the resulting score is 50%, and we then
consider that the GE was seen at 50%. However, research in the field of visual cognition on the organization
of the objects one perceives, Gestalt theory in particular [78-79], shows that observers complete the percept
if they are familiar with the entire shape. Our study thus addresses the question of visibility (in the dark, with
a light source carried by the observer, in panoramic vision) from a physical and physiological angle—"where
do I see/not see the GE?"—and responds to the question—"am I in the right place to detect and identify
it well?". The same question when the GE is partly concealed could be investigated with an experiment extending the one of [80] with occluding walls.

### 4.1.4 About the extrapolation of the experimental model

Some limitations arise when extrapolating the experimental model to a data corpus such as the one derived from the Chauvet cave. First, no measurements are available for non-planar GEs, or GEs that are much larger than the experimental ones, or under incidence angles greater than 45°. For example, the felines located on the left wall at the entrance to the End Chamber are much larger than the rhinos used in the experiment, raising uncertainty as to the validity of extrapolating our model to large surfaces (meaning to PVCs with high values). Conversely, the very small GEs present in various locations can also lead to aberrations of the model. Furthermore, the rhinoceroses made for the experiment have a compact shape while in Chauvet cave there are GEs with more complex shapes, such as the very filiform GEs on the Panther Panel. In some cases, the lines are almost completely folded back onto themselves and the GE thus takes the form of a disk, while a tiny gap in such a line makes the GE homeomorphic to a ring. These two nearly similar cases have nevertheless a large difference in surface area, and therefore on the PVC. Including all these cases in the study, as well as more extreme parietal contrasts (e.g., low wall/GE contrasts), would certainly improve our model, and it can be achieved in the future through new dedicated experiments.

We experimentally determined the perception distances by considering whole GEs, and not the multiple entangled GE portions that often make up the panels in Chauvet cave. At this stage of research, the aim of our experimental procedure was not to measure the perception of panels composed of several GEs since the visual perception of a single GE on a wall, separated from any other anthropogenic marks, is probably different from the perception of a group of assembled GEs.

Furthermore, to generalize to large panels such as those in Chauvet cave, whose surface cannot be uniformly illuminated from a single lighting position, and whose surface extends beyond the simple visual field, it would have been necessary to define and record a more complex trajectory than the one selected for the current protocol. Another approach could have been to install several lighting sources and to position the observers some distance from them, in which case the observers would no longer be the light-carriers. An archaeological situation such as this is realistic as fireplaces and ground combustion areas are present in Chauvet cave [81-82]. They can be considered as fixed points of light. Some of them are constructed, while others consist only of large piles of charcoal. In other areas of the cave, rubified (reddened) surfaces and traces of thermal shock are clearly visible on the walls and ceilings of some of the narrowest galleries. In addition, the Paleolithic observers in Chauvet cave could have used several light sources held by several light-carriers. In the present study, however, to focus on the dynamic aspects of the lighting used by the visitors of Chauvet cave, and to avoid multiplying the experimental conditions, only portable light sources were considered.

For two reasons, the implemented method makes it possible to study a specific type of engravings in Chauvet cave, those often made with fingers on soft limestone, but not the thin engravings made with a tool on a hard wall surface. Firstly, thin engravings require a specific type of lighting (lateral grazing) to make them visible, which is not treated in our study. Secondly, the surface material of the fine engravings is not distinguished from the original wall material, in contrast to the finger engravings on soft limestone, which remove a significant portion of the wall surface material, exposing a lighter color underneath, which contrasts with the darker color of the original surface. Thin engravings, therefore, do not produce the same segmentation of the form and the background that is exploited for the PC. Given the omnipresence of engravings on soft limestone in Chauvet cave, and the great rarity of fine-grained hard wall engravings, especially in the study area [52], the finely-engraved rhinoceros were intentionally excluded from the experimental stimuli, to retain only the rhinos engraved on soft panels (engraved line width from 0.5 to 2.5 cm).

More globally, in the present study of visibility, certain behavioral elements that are associated with all Paleolithic rock art cannot be ignored. Firstly, there is the possibility that some artworks were not meant to be seen. Beyond the line itself (cf. thin engravings), the study of visibility zones can therefore implicitly contribute to the detection of this intention. Secondly, the visibility of an engraving depends on its alteration, and thus on its age. Fortunately, the red and black lines, as well as those engraved on soft clay surfaces are very well preserved in Chauvet cave. This excellent state of preservation, especially of some of the GEs carefully chosen for the determination of profiles, enables us to consider that we are working in a hypothetical framework that is close to the Paleolithic conditions of the cave.
### 4.1.5 Probabilistic modeling

Numerous publications address the application of Bayesian approaches to perception, especially visual perception research [83, 85, 87, 89]. It is thus very common in visual perception research to construct a probabilistic model to analyze so-called “natural images,” using different statistics (luminance, contrast, etc.), taking care to integrate the different sources of uncertainty. In our case, for example, the six GEs in the quarry are similar to those in Chauvet cave without being exact copies. The uncertainty concerning the statistical quantities *D*1 and *D*2, namely *σ*1 and *σ*2, is introduced via the probabilistic formulation (Equation 5 to Equation 11) for a legitimate extrapolation to Chauvet cave.

The choice of the Gaussian model is justified by the fact that only two observables are available under each condition: the mean of *D*1 and its variance *σ*1 (resp. *D*2 and *σ*2). In this case, the Gaussian distribution is the one that accounts for these two observables, while maximizing entropy. It thus responds to the principle of simplicity or "Occam's razor."

### 4.1.6 Differentiated influences of the parameters of the experimental model

The probabilities of visual detection and identification are influenced by various parameters depending on the OP and GE considered.

For a given GE, Equations 1, 5, 6 and 7, together with Equations 8, 11, 12 and Figure 7) show that the two leading parameters are the surface of the GE and its PC. First, the larger the GE, the greater is its visibility from a distance (variation according to the square root of the surface). Second, the higher the PC of a GE, the greater is its visibility from a distance (linear variation). These two influences are conjugated through Dj.

For a given OP, the three leading parameters are D, *θ* and the percentage of non-occultation. First, D is a factor whose influence is mainly around the mean of Dj, with a span dictated by the standard deviation *σ*<sub>j</sub>. In the large chambers of Chauvet cave that were studied (Hillaire and Skull chambers), this effect is crucial for the many OPs located far from the GEs, without obstruction: without this probabilistic model of attenuation, all the GEs would be falsely considered as visible. Second, Equation 12 influences mainly grazing targets. The probability of visibility in OPs very close to the walls is thus affected by this parameter. In Hillaire and Skull chambers, as in the Red Panels Gallery, this factor gives a systematic advantage to the lowered posture because of the slight inclination of the panels toward the ground (Figure 31).

<figure><img src="image_43.png" /><figcaption>**Figure 31:** Histogram of the angular deviation between the normal vector of each triangle of each studied graphic entity and the vertical direction of the cave. Published under a CC BY 4.0 license, with permission from French Ministry of Culture.</figcaption></figure>

Most of this distribution has a negative value, which is related to the overhang of the lower part of the walls.

The percentage of non-occultation (the ratio that weights the probability in Equation 9 is influential in galleries or places with a complex geometry because a small shift from an OP to its neighbor can lead to total concealment. If the leading parameters of the set of equations are effective at any OP and any GE, then the probabilities of detection and identification depend on both the locations at which the observers stood and the positions of the entities. In addition, the surface of the GE plays a major role in all locations. Close to the
walls, the angle of incidence and the concealment play a strong role. In areas with a wide vantage point, such as in the Hillaire and the Skull Chambers, the PC leads the visibility probability.

### 4.1.7 Comparative use of probabilistic variables

By definition, visibility expectation is a sum (Equation 10). It is therefore suitable for locating the visibility areas of groups of GEs, but less suitable for detecting areas where isolated GEs are visible. Conversely, the maximum probability at *D*1 is, by nature, a variable more sensitive to the presence of a GE, even if isolated. The comparison of the maps produced by these two operators highlights the spatial differences in the visibility of grouped and isolated GEs (Figure 32).

<figure><img src="image_44.png" /><figcaption>**Figure 32: Visibility expectation map (I) and maximum visibility map (II) at D1 for the Hillaire and Skull Chambers.** Published under a CC BY 4.0 license, with permission from French Ministry of Culture / Jean-François Hullo.</figcaption></figure>

The probabilistic operators emphasize different elements on maps. For example, in zone B of Figure 32 I, *E*₁ shows a high visibility of GE clusters, while *P*max₁ does not (Figure 32 II). In zone A of Figure 32 II, *P*max₁ indicates isolated but still visible entities, which are obliterated by *E*₁ (Figure 32, I). These differences illustrate the very definition of these operators.

It is therefore advisable to use the appropriate probabilistic variable to reach archaeological interpretations depending on whether they are based on single entities, one-by-one, or based on a cluster of entities.

### 4.1.8 Significance of the visibility zones

Following [47], the Getis-Ord *gi*\* enables comparisons of the hotspots calculated above and the null hypothesis (i.e. hotspots that would be produced by a random spatial process). This algorithm is designed to eliminate the bias associated with the generally similar values of neighboring points. Figure 33 gives the results for the Red Panels and Cactus Galleries.
<figure><img src="image_45.png" /><figcaption>**Figure 33: Comparison of the hotspot map (A) and the significance map (B) from the Getis-Ord gi* algorithm in the Red Panels Gallery and the Cactus Gallery.** Published under a CC BY 4.0 license, with permission from French Ministry of Culture / Jean-François Hullo – Guillaume Thibault.</figcaption></figure>

The method indicates that in the green and red areas one can be sure at ±3σ that the values (of the visibility expectation in *D*1 in this case) are not due to a random phenomenon. Outside these areas, the significance decreases from ±2σ to ≈ 0σ. The concordance with the hotspot map reinforces its validity.

**4.2 Factors concerning the data corpus**

**4.2.1 Calculating the parietal contrast of each graphic entity from the photographic collection**

First, 3D cutting of the GEs enables us to know the facies of each one (the wall meshes themselves being named by facies). Second, the GIS provides the artistic technique of each GE. We then group the GE-facies pairs into 55 profiles, characterizing a GE-facies pair from a photograph considered representative of the profile, i.e., valid for all GEs of the profile, modulating by conservatively chosen standard deviations (Equation 16). However, each GE-facies pair of each profile was not analyzed using a dedicated photograph. The determination of the parietal contrast also presupposes that the photograph used to do so offers a sufficient exposure dynamic: an overexposure or underexposure reduces the quality of the determination of the PC and, therefore, the estimates of *D*1 and *D*2. The 60 photographs selected for the PC calculations were sufficiently dynamic. An additional effort is still necessary to determine the PC of each GE-facies based on more photographs taken orthogonally to the wall and with an adapted lighting dynamic.
### 4.2.2 About the 3D segmentation of the graphic entities

This task was difficult due to the complex shapes of the walls and GEs, as well as the large number of GEs. We thus segmented the GEs either by closely following their contours, or in a slightly more enveloping manner. In the case of superimposed figures, the triangles were either assigned uniquely to one GE (southern Hillaire Chamber) or duplicated them for several superimposed GEs (northern Hillaire Chamber). These modulations nonetheless have little impact on the results given the small size of the triangles and the probabilistic nature of the models.

### 4.2.3 Planar and non-planar graphic entities

Of the 487 GEs in the study zone, 474 are nearly flat: their differential mean curvature is less than 10% of the highest mean curvature observed for all GEs (the GE meshes were sampled by points randomly positioned on the mesh triangles; the mean curvature was calculated in a spherical neighborhood of 1 m around the current 3D point). Each of these GEs was legitimately assimilated to its average plane and Equation 9 applies without difficulty. The other thirteen GEs have a more pronounced concavity or convexity. In this case, the use of Equation 9 becomes questionable. In particular, GEs 267 and 268, located on the Pendant of the Woman, are almost dihedral due to their strong high convexity. An observer thus cannot see them from a single viewpoint and must thus move around the pendant for a more complete view. We thus segmented these GEs into two parts, each being homologous to a plane, and then summed the score associated with each part (*D*1 and *D*2 being adapted to the new areas).

### 4.2.4 About the postures considered and the observation meshes

The proposed method allows the visibility of a GE to be calculated at any point on the cave floor where an observer can stand, depending on his/her posture (similar to a cylinder), and obstacle clearance ability. While three main postures were considered, the maps in the Results section concern only the H180 posture. Figure 34 shows the possibility of comparing the maps of the three postures.

<figure><img src="image_46.png" /><figcaption>**Figure 34: Maps of the expectation in detail visibility E1 in the Red Panels Gallery for three postures.**
(A) H70; (B) H125; (C) H180. Published under a CC BY 4.0 license, with permission from French Ministry of Culture / Jean-François Hullo.</figcaption></figure>

Beyond the three studied postures, this method allows an analysis of GE visibility in the cave for any human configuration, as described by [39], even for that of an observer lying on his/her back to view the ceiling.

The obstacle crossing capacity, however (defined for Chauvet cave as a difference in height of less than 40 cm and a slope of less than 45°), will probably need to be adapted for use in other caves. This can be done with the navmeshing tools.
**5 Conclusion**

The methodological approach we present here offers a new procedure to quantify the visibility of parietal GEs (artworks on cave walls) in cave chambers and galleries without natural light. In the tested scenario, the light source is carried by a Paleolithic human, thus creating a situation in which the lighting varies with the distance to the GEs. This approach highlights the importance of luminance contrast between the artworks and the background for the different facies and the three artistic techniques tested (drawing, engraving, painting). Our study is based on two components: 1) the development of a probabilistic model to determine the visual detection and identification of parietal GEs, and 2) the application of this model to the digital data corpus of the Chauvet-Pont d'Arc Cave.

The modeling component, based on an active vision experiment in a limestone quarry, addresses the visibility of planar GEs depending on their lighting, artistic technique, wall type, distance and angle of incidence. Our analysis focused on two distances: the distance from which a light-carrying human is able to detect the presence of an anthropogenic modification to the wall (D2), and the distance from which he/she is able to identify a GE on the wall (D1). An affine formula, fit to the measurements, binds the visibility distance (D1 and D2) to a single parameter—the parietal visibility coefficient—which combines three factors: the technique(s) used to produce the GE, the type of wall, and the surface area of the GE. The experimental data further suggest that the D1 and D2 distances vary according to the angle of incidence between the direction of observation and the normal to the mean plane of the GE. The model formulated by these equations is robust, as shown by the consistency of the D2/D1 ratio for the two lighting types tested and the observation incidences. This model is generalized in a probabilistic framework to take into account the various sources of variability and uncertainty. Epistemologically, our modeling approach to the experimental measures, notably for the parietal contrast, was based on the classic signal detection theory.

We then applied this modeling procedure to a large digital data corpus of Chauvet cave, consisting of photographs, Geographic Information System data, and a 3D model representing the geometry of the cave at the time of Paleolithic human presence, the walls and GEs of which are segmented by facies. This allowed us to determine the parietal visibility coefficients of 487 GEs in four chambers and four galleries. The visibility probability was calculated by ray casting, taking into account the shapes of the cavity and the GEs. Several probabilistic operators were developed for D1 and D2 (GE of maximum probability at the current OP, expectation at the current OP, hotspots, variation of GE identifiers along a direction), illustrated by complementary visibility maps between them. These maps provide a new tool that can improve the current ability to analyze the artworks in Chauvet cave since, to ensure the preservation of the cave, researchers are not allowed to leave the walkways installed in the cave or use fire (torches or grease lamps) to study the GE visibility conditions.

For example, the results obtained in the Cactus Gallery concerning the visibility of GE groups are useful because they are concrete. They provide the archaeologists, who currently know the GEs of this space only from the vantage point of the walkway at the gallery entrance, with a new wealth of vantage points (Figure 35).
<figure><img src="image_47.png" /><figcaption>**Figure 35:** Map of the expectation of detailed visibility *E*₁ with the walkways (in purple) ending at the entrance of the Cactus Gallery. The Gallery is located on the right. Published under a CC BY 4.0 license, with permission from French Ministry of Culture / Jean-François Hullo – Guillaume Thibault.</figcaption></figure>

In addition to the first maps produced, this study offers new perspectives to the archaeologists working on Chauvet cave. Using the probabilistic visibility model, they can now produce visibility and inter-visibility maps of several chambers according to the artistic technique, representation type (e.g., animal species), or the size of the observer, and conduct comparative analyses of them.

Beyond the study of these elementary factors, the present method enables researchers to easily explore the complete range of combined factors, comparing species and artistic techniques, for example. Investigations lacking a digital approach are very time-consuming, not only for the analysis of each animal species and each technique but also for the co-visibility of species (by couple or triplet of species).

It is also possible to produce these maps as of the moment of the last GE created by the prehistoric artists: if the GEs themselves are dated in the future, the archaeologists could again use the model to produce maps of the dated GEs, and thus of the different states of the cave wall decorations through time, enabling a study of the evolution of their visibility.

Research on Paleolithic scenography [18-20] faces the difficulty of considering a very large number of factors, such as the 49 factors included in the study by [20], thus raising questions concerning the possible dependence between factors, their relative weight, and, in short, the interpretability of the statistical model. With our approach, a new perspective emerges: Bayesian inference would make it possible to integrate the elementary variables (detection probability, identification probability, visibility expectation), along with their uncertainties, or even the derived variables (inter-visibility and their hotspots, directional visibility) in a framework expanded to include even more variables to refine the ins and outs of grouping artworks by area of visibility.
Consequently, beyond Chauvet cave, all the digital processing techniques of this method can be applied to other caves with similar digital data. Our model thus enables researchers to study various features of cave art (artistic techniques, animal species, etc.) using probabilistic operators and their cartographic rendering to contribute to interpretations of the visibility of GE modalities. This probabilistic visibility modeling based on psycho-experimental data and a corpus of digital data (photographic, GIS and 3D) thus offers a new analytical tandem that can significantly contribute to scientific research in the context of the increasing inaccessibility of decorated caves due to preservation issues.

## Acknowledgments

Marc Drouet, Assistant Director of Archaeology at the Ministry of Culture and Communication, provided strong support to the TIPTOP project due to its original multidisciplinary nature. We thank G. Pinçon and S. Konik from the French National Prehistory Center for their continuous support. We thank Adrien Nivaggioli from the Ecole Polytechnique for developing the visualization tool for the calculation results during his internship at EDF R&D, as well as Fabien Dory, Christian Boucheny, Stéphane Ploix, Nicolas Filliard, Stéphane Gougeon, Miranda Coninx and Olivier Morvant from EDF R&D. We are grateful to Jean-Pierre Chadelle and Mathilde Régeard of the Departmental Archaeology Service of the Dordogne for access to the quarry and their participation in the experimentation. We thank Magen O'Farrell for the great care given to the English translation.

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representation in an image or display) distinguishable. There are several formal definitions of contrast, the
two most used being:

* Weber (or local) contrast $C = \frac{L_f - L_b}{L_f}$ with $L_f$ and $L_b$ representing the luminance of the features and the background, respectively.

* Michelson (or global) contrast $C = \frac{L_{max} - L_{min}}{L_{max} + L_{min}}$ with $L_{max}$ and $L_{min}$ representing the highest and lowest luminance.

## 5.1 Statistical methods

Within-subjects design was selected for the experimental protocol: each participant tested all the conditions.

No outlier was removed.

R statistical open-source software (www.r-project.org) was used to compute mean, standard deviations and standard errors of the mean of the experimental measures (Tables 2 and 3), PC and PVC values of the GEs, and then the affine model (Equation 14 and 15). R also provided the tools to verify that the large engraved rhino (GE #3) is more visible than the other 5 GEs, for both distance thresholds, for both incidences (Figure 16): one-way ANOVA for the factor GE gave the global p-values. The assumptions for the ANOVA have been verified using Shapiro-Wilk (normality) and Bartlett (homoscedasticity) tests from R. Post-hoc Dunnett tests (many-to-one comparisons) were given by R. The visibility hotspots and visual changes were computed and cartographically rendered using the raster package of R.

CloudCompare open-source software (cloudcompare.org) was used to compute the histogram of the slope of the GEs (Figure 31) by converting their normal vector into standard Dip value and Dip direction. CloudCompare also provided the distribution of their mean curvature (cf. Planar and non-planar GEs section of the Discussion chapter).

The Getis-Ord gi* algorithm enables comparisons of the hotspots and the null hypothesis (i.e., hotspots that would be produced by a random spatial process). This algorithm is designed to eliminate the bias associated with the generally similar values of neighboring points. The Getis-Ord gi* analysis was performed using ArcGIS (www.arcgis.com) and led to Figure 33.

## 5.2 Other non-statistical computations

Conversion of the photos following Equation 1 was done using ImageJ open-source software (https://imagej.nih.gov/ij/). Adobe Photoshop (https://www.adobe.com/fr/products/photoshop.html) allowed the segmentation of each GE (Figure 5) and its envelope (Figure 6). The wall point cloud was meshed (Figure 17) using 3DReshaper software (www.3dreshaper.com). This software provides a multi-step meshing algorithm to gradually refine the model by reusing the point cloud to improve the shape and resolution of the mesh at each new step to obtain a surface close to that of the cavity [87]. Several other functions of 3DReshaper were also used: cutting and filling hole algorithms for clearly visible concretions and sculpting to smooth some light calcitic formations. The facies cutting was carried out on a tablet inside the cave, using the Meshlab open-source software (www.meshlab.net). The GEs were segmented using 3DReshaper, which was also used to simplify these meshes with the additional use of the ProOptimiser function of the 3dsmax software (https://www.autodesk.fr/products/3ds-max/overview).

Meshes of walls and GEs were then imported into Unity3D (unity.com) without losing the GIS facies identification information, origin and scale. The three OMs (Fig 23) were then calculated using the Unity3D navmeshing tool. Unity3D offered ray casting from each OP toward each triangle of each GE.

In the calculation of scores, in addition to the classic analysis calculations (solid angle, etc.), the complement of the cumulative distribution function was evaluated using the Math.Net library (www.mathdotnet.com) called from Unity3D.

Using CloudCompare open-source software (cloudcompare.org) we rasterized the 3D OMs to create background maps.