Yana Nehmé
email: yana.nehme@insa-lyon.fr

Jean-Philippe Farrugia
email: jean-philippe.farrugia@univ-lyon1.fr

Florent Dupont
email: florent.dupont@liris.cnrs.fr

Patrick Le Callet
email: patrick.le-callet@univ-nantes.fr

Guillaume Lavoué
email: glavoue@liris.cnrs.fr

Comparison of subjective methods, with and without explicit reference, for quality assessment of 3D graphics

Keywords: Computing methodologies → Appearance and texture representations, Perception, Mesh models, Virtual reality Visual quality assessment, 3D graphics, subjective methodologies, single stimulus, double stimulus

Numerous methodologies for subjective quality assessment exist in the field of image processing. In particular, the Absolute Category Rating with Hidden Reference (ACR-HR) and the Double Stimulus Impairment Scale (DSIS) are considered two of the most prominent methods for assessing the visual quality of 2D images and videos. Are these methods valid/accurate to evaluate the perceived quality of 3D graphics data? Is the presence of an explicit reference necessary, due to the lack of human prior knowledge on 3D graphics data compared to natural images/videos? To answer these questions, we compare these two subjective methods (ACR-HR and DSIS) on a dataset of high-quality colored 3D models, impaired with various distortions. These subjective experiments were conducted in a virtual reality (VR) environment. Our results show differences in the performance of the methods depending on the 3D contents and the types of distortions. We show that DSIS outperforms ACR-HR in term of accuracy and points out a stable performance. Results also yield interesting conclusions on the importance of a reference for judging the quality of 3D graphics. We finally provide recommendations regarding the influence of the number of observers on the accuracy.

INTRODUCTION

Nowadays, three-dimensional (3D) graphics are widely used in many applications such as digital entertainment, architecture and scientific simulation. These data are increasingly rich and detailed; as a complex 3D scene may contain millions of geometric primitives, enriched with various appearance attributes such as texture maps designed to produce a realistic material appearance. These huge data tend to be visualized on various devices (e.g., smartphone, head mounted display) and possibly via the network. Therefore, to avoid latency or rendering issues, there is a critical need for the compression and simplification of these high quality 3D models. These processing operations may impact the visual quality of the 3D models and thus the quality of user experience (QoE). Thus to evaluate the visual quality as perceived by human observers, it is fundamental to resort to subjective quality assessment tests. In these tests, a group of human subjects is invited to judge the quality of a set of images subject to some predefined distortions. Such subjective experiments are also the most convenient way to create a ground-truth for judging the performance of objective quality metrics. However, selecting the best subjective methodology is not a trivial task since we should ensure that such method give valid and reliable results.

In the past years, several methodological guidelines have been defined for 2D image and video quality assessment by the International Telecommunication Union (ITU) [BT.500-13 2012;P.910 2009]. In the field of computer graphics, previous subjective experiments were carried out to evaluate the visual quality of still and animated 3D models [START_REF] Corsini | Watermarked 3-D mesh quality assessment[END_REF][START_REF] Guo | Subjective and Objective Visual Quality Assessment of Textured 3D Meshes[END_REF][START_REF] Lavoué | Quality assessment in computer graphics[END_REF]. However, no comparison of subjective methodologies have been made for such 3D data. So, there is no consensus about the best methodology to adopt for quality assessment of 3D models. In this work, we propose to compare two of the most prominent methods: Absolute Category Rating with Hidden Reference (ACR-HR) and Double Stimulus Impairment Scale (DSIS). We also assess whether or not the presence of an explicit reference is necessary for evaluating the quality of 3D graphics. We chose to make the experiment in Virtual Reality (VR) using the HTC Vive Pro headset because VR is becoming a popular way of consuming and visualizing 3D content.

Our psycho-visual experiment is detailed in section 3. In section 4, we present the results and analysis. Finally, concluding remarks are outlined in section 5.

RELATED WORK

In this section we first review popular methodologies for subjective quality assessment of (natural) images and videos, and then focus on existing subjective tests conducted with 3D graphics. We finally discuss previous work that compares subjective methodologies. The reader is referred to [START_REF] Lavoué | Quality assessment in computer graphics[END_REF] for a comprehensive survey of subjective quality assessment in computer graphics.

Methodologies for subjective quality assessment of images and videos

Several methodologies for 2D video/image quality assessment exist in the literature and have been standardized by the International Telecommunication Union [BT. 500-13 2012]. Four subjective quality assessment methodologies are notably used nowadays: Absolute Category Rating (ACR), Double Stimulus Impairment Scale (DSIS), Subjective Assessment Methodology for Video Quality (SAMVIQ) and pairwise comparison (PC). The ACR method consists of presenting each impaired sequence individually to the observer and then asking him/her to rate its quality on a quality scale. In the DSIS method, the reference video is presented first, followed by the same video impaired. The observer is asked to rate, on an impairment scale, the degradation of the second video compared to its reference. These methods are categorical rating since they use a 5-level discrete scale [START_REF] Rafał | Comparison of Four Subjective Methods for Image Quality Assessment[END_REF]. They are dominant in video subjective quality tests [BT.500-13 2012;P.910 2009]. Furthermore, ACR with hidden reference (detailed in section 3) is notably used by the Video Quality Experts Group (VQEG) [VQEG 2007]. The pairwise comparison method (PC) is an alternative method in which two distorted videos are displayed, side by side, and the observer has to choose the one having the highest quality. The fourth method is SAMVIQ. It differs form the others in several aspects. SAMVIQ uses a multi-stimuli with random access approach [START_REF] Huynh | Examination of the SAMVIQ methodology for the subjective assessment of multimedia quality[END_REF]. The test sequences are presented one at a time but the observer is able to review each video and modify the quality score multiple times. In addition, it uses a continuous quality scale (0-100).

Note that for graphics applications requiring localized information on the distortion visibility, the methodology based on the local marking of visible distortions is commonly used [START_REF] Piórkowski | Automatic Detection of Game Engine Artifacts Using Full Reference Image Quality Metrics[END_REF][START_REF] Wolski | Dataset and Metrics for Predicting Local Visible Differences[END_REF]]. In such subjective experiments, observers manually mark the visible local artifacts in the impaired images.

Subjective quality assessment of 3D graphical models

When it comes to subjective tests involving 3D models, no specific standard or recommendation exist. Researchers have adapted existing image/video protocols, while considering different ways to display the 3D models to the observers (e.g., 2D still images, animated videos, interactive scenes). Lavoué et al. [START_REF] Lavoue | Perceptually driven 3D distance metrics with application to watermarking[END_REF]] and [START_REF] Corsini | Watermarked 3-D mesh quality assessment[END_REF]] considered single stimulus protocols, derived from ACR, to assess the quality of impaired 3D meshes. The observers were able to freely interact with the 3D models and then had to rate the visibility of the distortions between 0 (invisible) and 10. Despite these two works, in the majority of existing experiments a double stimulus protocol (derived from DSIS) is used, with diverse modalities of display. Watson [START_REF] Watson | Measuring and predicting visual fidelity[END_REF]] used still screenshots to evaluate mesh simplification distortions while Lavoué [START_REF] Lavoué | A local roughness measure for 3D meshes and its application to visual masking[END_REF]] considered free-viewpoint interactions for evaluating 3D meshes subject to smoothing and noise addition. Pan et al. [START_REF] Pan | Quality metric for approximating subjective evaluation of 3-D objects[END_REF]] and [START_REF] Luis | Point cloud quality evaluation : Towards a definition for test conditions[END_REF]] considered animations (e.g. low speed rotations) for assessing the quality of resp. textured meshes and colored point clouds. It seems that most researchers intuitively felt that rating the absolute quality of a 3D graphical model (i.e., without the reference nearby) might be a difficult task for a naive observer (i.e. non-expert). Note that several recent works used pairwise comparison methods [START_REF] Guo | Subjective and Objective Visual Quality Assessment of Textured 3D Meshes[END_REF][START_REF] Vanhoey | Visual quality assessment of 3D models: On the influence of light-material interaction[END_REF]].

Comparison of subjective methodologies

Several works evaluate and compare the performance of the methodologies described above (mostly for natural image or video content). 2018] recommend to use at least 24 observers. Nevertheless, the SAMVIQ method is considerably more time-consuming than an ACR (or DSIS) method.

Moving to double stimulus methods, the main difference between DSIS and ACR is the presence of explicit references. According to the ITU [P.910 2009], DSIS ratings are less biased compared to ACR ratings. Indeed due to the presence of the references, subjects are able to detect shape and color impairments that they may miss with the ACR method. In addition, in DSIS, the scores are not influenced by the subjects opinion of the content. Surprisingly, [START_REF] Rafał | Comparison of Four Subjective Methods for Image Quality Assessment[END_REF] denote that for the experimental procedures, images and distortions used in their study, there was "no evidence that the double stimulus method is more accurate than the single stimulus method". They demonstrated that since the PC methodology is straightforward, it tends to be the most accurate from the 4 tested methods (single stimulus, double stimulus, forced choice pairwise comparison, and similarity judgments methods). However, despite the simplicity of the task of this method, it may become tedious if all sequences need to be tested (PC requires n(n-1) 2 trials to assess n sequences while ACR requires n +1 trials and DSIS requires n trials). Recently, Singla et al. [START_REF] Singla | Comparison of Subjective Quality Evaluation Methods for Omnidirectional Videos with DSIS and Modified ACR[END_REF]] investigated the performance of the DSIS and a Modified Absolute Category Rating (M-ACR) method for omnidirectional (360°) videos using an Oculus Rift. They denoted that M-ACR is statistically slightly more reliable than DSIS since DSIS gave larger confidence intervals.

To conclude, while many methodological guidelines have been defined for natural video/image quality assessment (using a screen), no similar standards exist for quality evaluation of 3D graphics. No consensus has emerged toward the best methodology for such data, especially in a virtual or mixed reality environment. One particular open question is whether or not a reference is necessary. In this context, we compare the performance of the Double-Stimulus Impairment Scale (DSIS) and the Absolute Category Rating with Hidden Reference (ACR-HR) methodologies for assessing the quality of 3D graphics. We consider a VR context using the HTC Vive Pro headset, a high-end virtual reality headset1 . The present work attempts to make a first step toward standardizing a methodology for assessing the quality of 3D graphics.

SUBJECTIVE EXPERIMENT

We compare the effectiveness and the accuracy of two subjective quality assessment methods for 3D graphic objects in an immersive virtual environment. For this purpose, we conducted a psychovisual experiment using the HTC Vive Pro, in the fixed position mode. This section provides the details of our subjective study.

Experimental methodologies

Several methodologies exist in the literature of subjective quality assessment. In this paper, we investigate two categorical rating methods: a single and a double stimulus methods. The selected methodologies are presented below and illustrated in Figure 1.

• Absolute Category Rating with Hidden Reference (ACR-HR): also known as single stimulus categorical rating, in which the impaired stimuli are presented one at a time in addition to the original unimpaired stimuli (references), without informing the subjects of their presence. The observers are asked to evaluate the quality of the stimulus shown using a Likert-type scale ranged from 1 to 5 (or five-level scale), where the discrete levels correspond to bad, poor, fair, good, and excellent. Note that some methods favour continuous rather than categorical scales to avoid quantizing errors [BT. 500-13 2012]. According to the ITU-T recommendations, the presentation time for the stimulus should be 10s. It may be reduced or increased according to the content of the test sequence [P.910 2009]. In our pilot study (pretests), we found that 6s presentation is sufficient to assess the quality of the presented 3D model. • Double Stimulus Impairment Scale (DSIS): also called Degradation Category Rating (DCR), in which the viewer sees an unimpaired reference model, then the same model impaired. Following that, the subject is asked to rate the impairment of the second stimulus in relation to the reference [P.910 2009] using the following five-level impairment scale: Imperceptible(5), Perceptible but not annoying(4), Slightly annoying(3), Annoying(2),Very annoying (1). Similarly to ACR, 10s presentation time is recommended per stimulus (≈20s/pair). However, this methodology slows-down the experiment too much since it requires at least twice as much time as ACR method. The overall length of the experiment affects the efficiency of the experimental method especially in virtual reality where most of the subjects are not used to the VR headset and tend to exhibit symptoms of cybersickness both during and after the VE experience [LaViola 2000]. To avoid these issues, we chose to display the reference and the test stimulus simultaneously side by side in the same scene. In this way, the number of presentations is halved. In addition, using simultaneous presentation makes the evaluation of the differences between the stimuli easier for the subjects [P. For this methodology, we increased the presentation time to 10s, since, comparing to ACR-HR, 6s is not sufficient to observe the 2 stimuli displayed in the scene, compare them and assess their impairments.

Experiment design

The objective of our study is to compare the accuracy of the ACR-HR and DSIS methods. For this purpose, observers were divided into two groups and were asked to rate the quality of a set of 80 distorted models (from 5 references), using both methodologies. ACR-HR and DSIS tests were made in different order according to the groups. Details about our study are described below.

3.2.1 Stimuli Generation. We selected five triangle meshes of high resolution, each having a vertex color map: "Aix", "Ari", "Chameleon", "Fish", "Samurai" (see Figure 2). These 3D models are considered to be "good" or "excellent" quality. The number of vertices of the five models ranges from 250000 to 600000. They belong to very different semantic categories (human statues, animal, Art) and have different shapes and colors (Monocolor, warm cool and dull colors) (Figure 2). These reference models have been corrupted by 4 types of distortions, each applied with four different strengths:

• Uniform Geometric quantization (QGeo): applied on the geometry. • Uniform LAB color quantization (QCol): applied on the vertex colors.

• "Color-ignorant" simplification (SGeo): mesh simplification algorithm that takes into account the geometry only [START_REF] Garland | Surface simplification using quadric error metrics[END_REF]].

• "Color-aware" simplification (SCol): mesh simplification algorithm that takes into account both geometry and color [START_REF] Lee | Rate-distortion optimization for progressive compression of 3D mesh with color attributes[END_REF]].

The strength of these distortions was adjusted manually in order to span the whole range of visual quality from imperceptible levels to high levels of impairment. For this task, a large set of distortions was generated and viewed by the authors, and a sub-set of them spanning the desired visual quality (i.e. "Excellent, " "Good, " "Fair, " and "Poor") was chosen to be included in the database (as in [START_REF] Guo | Subjective and Objective Visual Quality Assessment of Textured 3D Meshes[END_REF][START_REF] Seshadrinathan | Study of subjective and objective quality assessment of video[END_REF]). Thus, we generated 80 distorted models (5 reference models×4 distortion types × 4 strengths). Figure 2 illustrates some visual examples.

Rendering parameters.

In designing our subjective experiment, we had to choose whether we select static or dynamic scenes. In fact, deciding the way the 3D models are displayed to the observers is a crucial problem. No standardized procedures exist for subjective evalution of the quality of 3D objects and current studies show a lack of generalization in the methodology that should be used [START_REF] Corsini | Watermarked 3-D mesh quality assessment[END_REF][START_REF] Guo | Subjective and Objective Visual Quality Assessment of Textured 3D Meshes[END_REF][START_REF] Rogowitz | Are image quality metrics adequate to evaluate the quality of geometric objects?[END_REF]. [START_REF] Rogowitz | Are image quality metrics adequate to evaluate the quality of geometric objects?[END_REF] proved that the perceived degradation of still images may not be adequate to evaluate the perceived degradation of the equivalent 3D model. Indeed, still images may mask both artifacts and the effect of light and shading. Following this approach, Corsini et al. [START_REF] Corsini | Watermarked 3-D mesh quality assessment[END_REF]] allowed the subject to interact with the model by rotating and zooming it. While it is important for the observer to have access to different viewpoints of the 3D object, the problem of allowing free interaction is the cognitive overload which may alter the results. Hence, we decided to control the interaction between the subject and the stimulus displayed on the scene. So, based on the principle of pseudo-videos and as in Guo et al. [START_REF] Guo | Subjective and Objective Visual Quality Assessment of Textured 3D Meshes[END_REF], we used animations. For each object in our database, we selected the viewpoint that covers most of the shape. We then applied a slow rotation of 15 degrees around the vertical axis in clockwise and then in counterclockwise directions (i.e. total rotation of 30 degrees). These dynamic stimuli are shown in the scene at a viewing distance fixed to 3 meters from the observer. Their size is approximately 36.87 degrees of visual angle. They are visualized in a neutral room (light gray walls) under a directional light (all the vertices are illuminated as if the light is always from the same direction. It simulates the sun). We aimed to design a neutral room so that the experimental environment does not influence the users (quality-) perception of the stimulus. The default color calibration of the HTC Vive Pro was used.

3.2.3 Experimental procedure. The goal of this experiment is to evaluate the impact of the test methodologies (ACR-HR, DSIS) on the user quality assessment. For this purpose, we divided our experiment into 2 sessions, one for each methodology i.e. one session consisted of presenting the stimuli using ACR-HR and the other session presented them using DSIS. In addition, in order to study whether a methodology has an influence over the other and if the order of the methodologies matters, we divided the subjects into 2 groups (G1 and G2). G1 refers to the participants who completed the ACR-HR test before DSIS and G2 refers to those who passed the DSIS session first then the ACR-HR session. None of these sessions took place on the same day in order to reduce the learning effect between stimuli. Thus, these two sessions occurred at least two days apart. In each session, the stimuli were displayed in a random order (3D models, distortions types and levels all mixed) to each observer. Each stimulus (for ACR-HR) or pair of stimuli (for DSIS) was presented once; the observer was not able to replay/review the objects.

Rating interface. We opted to ensure in our test a user experience and quality of experience (QoE) in fully immersive virtual environment (VE). So, we integrated a rating billboard in the VE of our experiment (see Figure 1). This board is adapted to each methodology and is displayed after the presentation time of each stimulus. There is no time limit to vote and the stimulus to rate is not shown during that time. The same neutral room (light gray walls) utilized to show the stimulus was used in the rating environment. To vote, the subject selects and saves the score using the trigger of the HTC Vive controller. As in [START_REF] Regal | VRate: A Unity3D Asset for integrating Subjective Assessment Questionnaires in Virtual Environments[END_REF], to facilitate the interaction with the rating panel, we attached a raycast beam to the controller.

Training. As recommended in the ITU-R500 [BT.500-13 2012], both sessions started with a training in which observers could familiarize themselves with the virtual environment and the task. We selected a training 3D model not included in our original test set: "Dancing Drummer" (see Figure 2) and generated 11 distorted models that span the whole range of distortions. At the beginning of each session, the training models are shown in the same manner (single or pairwise) and with the same time (6s or 10s) adopted in the upcoming session. After each stimulus, the rating panel, with the corresponding scale, is displayed for 5s. The score attributed/assigned to this distortion is highlighted. We added a practice trials stage at the end of the training: we displayed 2 extra stimuli and asked the subject to rate the quality or the impairment, according to the session. The results of these stimuli were not recorded. This stage was used to allow the observer to familiarize with the experimentation, to focus appropriately and to ensure that observers fully understand the task of the experiment.

Duration. No session took longer than 30 minutes to avoid fatigue and boredom: the total time was 18 minutes for the ACR-HR session (informed consent/instructions + 11 training stimuli × (6s display + 5s Rating) + 85 Test stimuli × (6s display + ≈4s rating)) and 23 minutes for DSIS session (informed consent/instructions + 11 training stimuli × (10s display + 5s Rating) + 80 Test stimuli × (10s display + ≈4s rating)).

The whole experience was developed in Unity3D using c# scripting. Snapshots of the experimental environment are provided in the supplementary material.

Participants.

As mentioned in the previous section, the stimuli were rated by 30 subjects divided into 2 groups of 15. The participants were students and professionals at the University of Lyon and LIRIS laboratory. 27 males and 3 females, aged between 19 and 45, they were naive about the purpose of the experiments. All observers had a normal or corrected to normal vision. In order to avoid the effect of the temporal sequencing factor, the order of stimuli was randomly generated so that each participant views the stimuli in a different order.

RESULTS AND ANALYSIS

The following sections analyze and discuss the results of our experiment. The purpose is to compare the two methodologies ACR-HR and DSIS and to determine which method is the best to assess the quality of 3D graphic models.

Screening observers

Before starting any analysis, participants were screened using the ITU-R-BT-500.11 recommendation [BT. 500-13 2012]. Applying this procedure on our data, we did not find any outlier participant from group 1 (G1). However, one subject from group 2 was rejected (G2) by reason of reporting implausible scores in the DSIS session (the first session for G2).

Computing the mean ratings MOS/DMOS

The first step of the analysis of the results is the calculation of the mean score for each of the stimuli [ITU-R BT.500-13]. For ACR-HR, it is advised to compute the difference scores between hidden reference and test stimuli instead of using directly the raw rating results. Indeed, studies [START_REF] Rafał | Comparison of Four Subjective Methods for Image Quality Assessment[END_REF][START_REF] Van Dijk | Quality asessment of coded images using numerical category scaling[END_REF] show that subjects tend to assign a different quality scale for each object. It is influenced by their opinion of the content (whether they like or dislike the object). Therefore, assessing differences in quality allows to take into account this variability in the use of the rating scale:

d i j = s iref(j) -s i j (1)
s i j refers to the score assigned by observer i to the stimulus j. ref(j) is the reference of stimulus j. The difference scores for the reference stimuli (d iref(j) = 0) are removed from the collected data in the ACR-HR session for/of both groups G1 and G2. Finally, we computed the Difference Mean Opinion Score (DMOS) of each stimulus for both groups:

DMOS j = 1 N N i=1 d i j (2)
N denotes the remaining subjects after screening observers i.e, N =15 for G1 and N =14 for G2.

For DSIS, we don't need to compute the DMOS since DSIS is based on the comparison between the reference and test models. Hence, we can directly use the rating results and compute the MOS.

MOS j = 1 N N i=1 s i j (3)

Resulting MOS/DMOS

Figure 3 shows the results of ACR-HR and DSIS tests for all stimuli, averaged over all screened observers. For a better readability in the interpretation of results, we show the MOS (instead of the DMOS) for ACR-HR. Note that DMOS are used in the statistical tests presented in section 4.4. A comparison of the G1's and G2's DMOS and confidence intervals of the computed DMOS/MOS are provided in the supplementary material.

As expected the MOS decrease as the distortion strengths increase. For the DSIS method (Figure 3.b), we can notice a strong consistency between the two groups and a good use of the entire rating scale. Indeed the observers of both groups showed almost the same behavior for each stimulus and the rating scores reach the scale limits.

For the ACR-HR method (Figure 3.a), we can notice some differences between the rating scores of the two groups. In fact, observers of G1 tend to downrate the reference stimuli, i.e. the rating scores given by G2 observers to almost all the references, except the Chameleon, exceed those of G1 observers. As a consequence, the amplitude of the rating scale is reduced. The specificity of the Chameleon model will be discussed in the next subsection. Moreover, we note that G2 observers were able to detect some distortions that G1 observers missed, notably the color distortions: e.g. QCol distortion with high strength (strengths ≥ 3) for Aix, Ari and Samurai (row 2 col 1,2,5) obtained better scores in G1 than in G2. These first results reveal several differences in the performance and behavior of the methodologies. In the next section, we assess whether these differences are statistically significant and we attempt to provide explanations for their causes.

Quantitative analysis

In this section, we analyze and compare quantitatively the results of both methodologies. In particular, we evaluate if the orders of the ACR/DSIS sessions have an impact on their results and why, and we demonstrate which methodology provides the most accurate results.

4.4.1 Normality and dependency analysis / preliminary tests. The statistical analysis is affected by the dependencies between the samples. In our experiment, two groups of observers (G1 and G2) rated the same stimuli. The only difference for the two groups was the order of the ACR-HR/DSIS sessions. We aimed to test whether there are differences in scores between the two groups so we could evaluate whether a methodology has an influence over the other. Hence, for the analysis, the raw rating scores are independent and thus we could have used unpaired two-sample t-tests. However before using a parametric test, it is important to make sure that the data follow a normal distribution. We applied several normality tests, on the rating scores; such as Shapiro-Wilk's test, Lilliefors's test, Anderson-Darling's test. All these tests ascertained that the distribution of our data is not-normal (p-value ≪ 0.05). Hence, for our data analysis, we have opted for the unpaired two-samples Wilcoxon test (also known as Wilcoxon rank-sum test or Mann-Whitney test). It is a non-parametric alternative to the unpaired two-samples t-test.

4.4.2 Consistency across the groups. To assess whether, for a given methodology, there are significant differences in rating scores between the two groups of observers, we conducted for each stimulus the unpaired two-samples Wilcoxon test on the scores s i jд (for DSIS) or the differential scores d i jд (for ACR-HR) of the 2 groups. The null hypothesis (H0) is that, for a given stimulus, the rating scores of the G1's observers are equal to those of the G2's observers at the 95% confidence level. The alternative hypothesis (H1) is that the scores of G1 are greater (or lesser) than the scores of G2. The p-values are presented in Figure 4. The red boxes (p-value<0.05) indicate that the corresponding stimuli have been rated significantly different by the two groups of subjects.

Figure 4: p-values computed between the rating scores of the two groups for all stimuli of both methodologies (the red color indicates a significant difference between the scores of G1 and G2).

For the ACR-HR method, we noticed that the scores of the two groups are not consistent (i.e. differ significantly) for 12 stimuli, out of 80; especially for the LAB quantization of all the models excluding the Chameleon. This is coherent with the results observed in section 4.3. Our hypothesis is that this is due to the absence of explicit references. Indeed for G1's observers, as they did the ACR-HR test first, the assessment was absolute. Thus, it was difficult for them to detect the distortions of some models especially the color impairments, notably for Samurai and Ari (10 red boxes out of 12). The reason is that, for statues like Ari and Samurai, they have no

(a) (b) (c) (d)
Figure 5: Variation of the accuracy according to the number of subjects for both methodologies and both groups (G1's subjects did the ACR-HR session 1 st followed by the DSIS session, while G2's subjects did the DSIS session 1 st and then the ACR-HR session). The accuracy (y-axis) is defined as the percentage of pairs of stimuli whose qualities were assessed as statistically different. Curves represent mean values of these percentages and areas around curves represent 2.5th -97.5th percentiles.

prior knowledge of the exact color of the model. This is not the case for G2's observers since they had already seen the references during the DSIS session. Hence, they were able to detect the distortions (even the color distortions) that G1 observers might miss. For the Chameleon, there is no significant difference between the 2 groups. We believe that this is related to the fact that people have strong prior knowledge about this model: the chameleon/Iguana is an animal known worldwide and everyone has an idea of its shape, color and geometry characteristics. We observe, for certain models, a better consistency/agreement among the subjects of the two groups, for the DSIS method. This confirms the fact that the presence of the reference makes the DSIS methodology more consistent across the groups and independent of the sessions order. The absence of reference in the ACR-HR method makes it difficult for observers to asses certain distortions, especially when they do not have prior knowledge about the models. This result is inconsistent with comparative studies conducted with images and videos, including omnidirectional videos [START_REF] Singla | Comparison of Subjective Quality Evaluation Methods for Omnidirectional Videos with DSIS and Modified ACR[END_REF], in which M-ACR was slightly more reliable than DSIS. This is because people have more prior knowledge about the quality of (natural) images/videos than 3D graphics, and therefore, the presence of references is not necessary to assess the quality of these data.

4.4.3 Accuracy of the quality scores. As stated by [START_REF] Rafał | Comparison of Four Subjective Methods for Image Quality Assessment[END_REF]]: « A more accurate method should reduce randomness in answers, making the pair of compared conditions more distinctive. A more accurate method should result in more pairs of images whose quality can be said to be different under a statistical test. ». To assess the accuracy of the methodologies, we thus computed the number of pairs of stimuli rated significantly different by G1 and G2 subjects. For this task, we conducted unpaired two-samples Wilcoxon tests between rating scores of each possible pairs of stimuli. We conducted 80 x 79/2 = 3160 tests. The α levels used here is 0.05.

In order to study the behavior of this accuracy according to the number of subjects, we made these tests for different numbers of subjects and assessed the evolution of the number of pairs of stimuli significantly different. For each number N of subjects, we considered all possible combinations (without repetition) (with 3 ≤ N ≤ 15 for G1 and 3 ≤ N ≤ 14 for G2) and averaged the number of pairs significantly different over all these combinations of observers. Results are shown in Figure 5. The numbers of pairs in y-axis are given in percentages of the total number (i.e., 3160).

From Figure 5.a, it can be noticed that, for the DSIS method, the accuracy do not evolve much from G1 to G2. Hence, double stimulus methodology seems, once again, stable and independent of the sessions order. However, this is not the case of the ACR-HR method since the accuracy undergoes a large increase for G2 compared to G1 (Figure 5.b). This demonstrates anew that the method without explicit reference is not consistent across the groups. G2's subjects -who completed the ACR-HR test in the 2 nd session-were more familiar with the stimuli than G1's subjects since they had already seen the models and their references in the 1 st session (the DSIS test). Therefore, they are capable of distinguishing/detecting the degradations/loss in the visual quality of the stimuli more easily than the G1's observers. Beyond this better consistency observed for DSIS, Figures 5.c and 5.d clearly show that the DSIS method is more accurate than the ACR-HR method. This is valid even for G2, in which ACR-HR was conducted after DSIS.

In Figure 6, we determined the number of subjects required in both methodologies to obtain the same accuracy. As can be seen in the figure, ACR-HR requires almost twice as many subjects as DSIS for G1. For instance, for a given number Figure 7: Width of confidence intervals (CI) for both ACR-HR (red curves) and DSIS (blue curves) methodologies as a function of the number of observers involved in both groups (G1's subjects did the ACR-HR session 1 st followed by the DSIS session, while G2's subjects did the DSIS session 1 st and then the ACR-HR session). For (c), the turquoise and violet curves refer to CI of G1 and G2 respectively.

of observers unfamiliar with the test stimuli, ACR-HR requires minimum 14 observers to get a discrimination with an overall level of 53% while DSIS requires only 7 observers.

Confidence intervals.

Another way to evaluate the accuracy of the methodologies is to compute the 95% confidence intervals of the obtained MOS/DMOS. We thus computed these 95% confidence intervals (CI) for both groups and methodologies, in order to determine the "true" mean score (i.e. the interval in which the MOS/DMOS will reside if we have an ∞ number of observers) [BT. 500-13 2012]. We evaluated the evolution of the width of these intervals for both methodologies according to the number of subjects. The curves of Figure 7 were obtained by averaging the width of CI over all the possible combinations of subjects. Note that for a given reference model and type of distortion, we average the widths of the CI over the four strengths of the distortion. We can observe that width of CI increases as the sample size decreases. For G1 (see Figure 7.a), we notice that, for most stimuli, the CI of the ACR-HR experiment are much larger than the CI given by the DSIS experiment, implying a strong dispersion of the ACR-HR scores across the G1's subjects. This disagreement is due to the fact that the references of the models are unknown for G1's subjects. This disagreement is not so apparent for G2 where the widths of CI given by the ACR-HR method are closer to the CI of the DSIS method (see Figure 7.b). These results confirm that DSIS is more accurate than ACR-HR, regardless the group. We illustrate, in the supplementary material, that there is almost no difference between the CI of G1 and G2 involved in DSIS, while the CI of G1's ACR-HR test are always superior to those of G2, except color quantization distortions of the Chameleon. Figure 7.c illustrates the confidence intervals of the Chameleon. As explained in section 4.4.2, the strong prior knowledge of observers on the color of this animal increases their accuracy, even without the presence of the explicit reference.

Recommended number of observers for the DSIS methodology

As previously demonstrated, DSIS is more stable and more accurate than ACR-HR method. In this section we attempt to make recommendations about the required number of observers for this methodology. For this purpose, we aggregate the DSIS test's scores of the 2 groups (G1 and G2) and thus obtain 30 subjects. This aggregation is possible since we demonstrated in previous subsections that DSIS scores are consistent among the two groups. We then recompute the accuracy (as in section 4.4.3) according to the number of observers. The accuracy being defined as the percentage of pairs of stimuli statistically distinguishable by their rating scores.

From Figure 8, we observe that at least 20 test subjects are required to be able to discriminate 70% of all possible pairs of stimuli. With 15 observers, the recommended number by the ITU, we obtain an accuracy of 66%. However, with 25 subjects the discrimination increases to 73% and reaches 75% with 30 subjects. As a conclusion, and with regard to the shape of the curve, 24 subjects seem to be a good compromise.

CONCLUSION

In this study, we designed a psycho-visual experiment that compares two of the most prominent subjective methodologies, with and without explicit reference (ACR-HR and DSIS). We compare these methods for the quality assessment of 3D graphics in a VR environment. Results assert that DSIS is consistent among the two groups of subjects (stable performance). It is also more accurate than the ACR-HR method and thus requires less subjects. In fact, in ACR-HR, the assessment is absolute (absence of explicit references) and therefore observers, who had never seen the reference models before, are not able to detect all the distortions, especially the color impairments. Thus, they tend to be less discriminating than those who are familiar with the test stimuli. These conclusions are not consistent with recent comparative studies conducted with images and videos. This is due to the fact that people have less prior knowledge of 3D graphic quality than of (natural) images.

This study makes the first step toward standardizing a methodology for assessing the quality of 3D graphics. In our future work, we will compare DSIS with both paired-comparison (PC) and SAMVIQ methods. Furthermore, we will work towards evaluating the impact of the display devices (2D screen, VR/MR headset) on the perceived quality of 3D graphics.

Figure 1 :

 1 Figure 1: Illustration and timeline of the two subjective quality assessment methods explored in this study.

Figure 2 :

 2 Figure 2: Illustration of the 3D graphic reference models (Left) and some examples of distorted models (Right). Acronyms for distorted models refer to Type_Strength.

  Figure 3: Comparison of the G1's and G2's mean scores of the ACR-HR and DSIS experiments for all stimulus (the blue and orange dots refer to the MOS of G1 and G2 respectively).

Figure 6 :

 6 Figure 6: Number of observers required to obtain the same accuracy with ACR-HR and DSIS methods.As can be seen in the figure, ACR-HR requires almost twice as many subjects as DSIS for G1. For instance, for a given number

Figure 8 :

 8 Figure 8: Accuracy of the DSIS method according to the number of subjects.

https://www.vive.com

ACKNOWLEDGMENTS

This work was supported by French National Research Agency as part of ANR-PISCo project (ANR-17-CE33-0005).


Susanna Zimmermann
email: susanna.zimmermann@univ-angers.fr

CONTINUOUS AUTOMORPHISMS OF CREMONA GROUPS

Keywords:

We show that if a group automorphism of a Cremona group of arbitrary rank is also a homeomorphism with respect to either the Zariski or the Euclidean topology, then it is inner up to a field automorphism of the base-field. Moreover, we show that a similar result holds if we consider groups of polynomial automorphisms of affine spaces instead of Cremona groups.

Introduction

The Cremona group in d-variables over a field k is the group of birational transformations of the projective d-space P d k over k. Equivalently, it can be seen as the group of k-automorphisms of the field k(x 1 , . . . , x d ). Cremona groups have been the object of research for over 150 years and considerable progress in the understanding of their structure has been achieved recently. This article concerns group automorphisms of Cr d (k).

An important subgroup of Cr d (k) is the automorphism group Aut(P d k ), which is isomorphic to PGL d+1 (k). Every field automorphism α of k naturally induces a group automorphism on both, Cr d (k) and PGL d (k), which we denote by g → α g. The group automorphisms of PGL d (k) are well-known: every automorphism of PGL d+1 (k) is the composition of an inner automorphism with an automorphism of the form g → α g or g → α g ∨ , where α is a field automorphism of k and g ∨ denotes the inverse of the transpose of g. Note that this result implies in particular, that all group automorphisms of PGL d+1 (k) are Zariski continuous.

In [START_REF] Déserti | Sur les automorphismes du groupe de Cremona[END_REF], Déserti showed that every group automorphism of Cr 2 (C) is inner up to automorphisms induced by a field automorphism of C. It is a natural question, whether the theorem of Déserti holds in all dimensions and for all fields. A difficulty in studying group automorphisms of Cr d (k) for d > 0 is that no easy to handle set of generators of the group is known. However, Cr d (k) can be endowed with the Zariski topology, a topology that extends the Zariski topology of PGL d+1 (k) (see Section 2.1). We generalize Déserti's theorem to arbitrary dimensions and arbitrary fields of chracteristic 0 under the additional assumption that the group automorphisms are also homeomorphisms with respect to the Zariski topology.

Theorem 1.1. Let k be a field of characteristic 0 and let ϕ : Cr d (k) -→ Cr d (k) be a group automorphism that is a homeomorphism with respect to the Zariski topology, where d ≥ 2. Then there exists a field automorphism α of k and an element f ∈ Cr d (k) such that ϕ(g) = f ( α g)f -1 for all g ∈ Cr d (k).

If k = R or k = C, then Cr d (k) can moreover be equipped with the Euclidean topology -a topology that makes Cr d (k) a Hausdorff topological group and which restricts to the standard Euclidean topology on Aut(P d k ) (see Section 2.2). In this setting, Theorem 1.1 also holds if we consider the Euclidean topology instead of the Zariski topology: Theorem 1.2. Let k = C or k = R and let ϕ : Cr d (k) -→ Cr d (k) be a group automorphism that is a homeomorphism with respect to the Euclidean topology, where d ≥ 2. Then there exists a field-automorphism α of k that is Euclidean continuous and an element f

∈ Cr d (k) such that ϕ(g) = f ( α g)f -1 for all g ∈ Cr d (k).
The main ingredients for the proof of Theorem 1.1 and Theorem 1.2 are a result of Cantat and Xie about embeddings of finite index subgroups of SL d+1 (Z), the classification of automorphisms of PGL d+1 (k), and a topological deformation argument.

Déserti's theorem implies that every group automorphism of Cr 2 (C) is Zariski continuous. It is therefore a natural question whether every group automorphism of Cr d (k) is Zariski continuous.

Observe that Theorem 1.1 implies in particular, that if k doesn't have any nontrivial field automorphisms (for instance, if k = Q or k = R), then every continuous isomorphism of Cr d (k) is inner. In this setting, Theorem 1.2 can be seen as an algebraic analogue of [START_REF] Filipkiewicz | Isomorphisms between diffeomorphism groups[END_REF], where it is shown that all group automorphisms of diffeomorphism groups are inner. Following this theme, we look in Section 6 at continuous automorphisms of the group of diffeomorphisms of P d R (R) that are induced by birational transformations.

In Section 5, we consider instead of Cr d (k) its sister, the group of polynomial automorphisms Aut(A d k ), and obtain the following result:

Theorem 1.3. Let k be an infinite perfect field and let ϕ : Aut(A d k ) -→ Aut(A d k ) be a group automorphism that is a homeomorphism with respect to the Zariski topology. Then there exists a field automorphism α of k and an element

f ∈ Aut(A d k ) such that ϕ(g) = f ( α g)f -1 for all g ∈ Aut(A d k ).
The group Aut(A d k ) has the additional structure of an ind-group (see [START_REF] Furter | On the geometry of the automorphism groups of affine varieties[END_REF] for details). Theorem 1.3 implies in particular that every ind-group automorphism of Aut(A d k ) is inner, if k is an infinite perfect field. In the case where k is of characteristic zero and algebraically closed, this fact was proven by Kanel-Belov, Yu, and Elishev [START_REF] Kanel-Belov | On the augmentation topology of automorphism groups of affine spaces and algebras[END_REF]. Again, one can ask the question, whether all group automorphisms of Aut(A d k ) are Zariski continuous. In dimension 2, the question has a positive answer if k is uncountable [START_REF] Déserti | Sur le groupe des automorphismes polynomiaux du plan affine[END_REF]. A partial generalization of [START_REF] Déserti | Sur le groupe des automorphismes polynomiaux du plan affine[END_REF] to higher dimensions has been obtained in [START_REF] Kraft | On automorphisms of the affine Cremona group[END_REF], [START_REF] Stampfli | A note on automorphisms of the affine Cremona group[END_REF], [START_REF] Urech | On automorphisms of the affine cremona group[END_REF].

It could be interesting to generalize Déserti's theorem as well as the results of this paper to more general fields, in particular to finite fields. Another natural question is in as far similar results hold if we drop either the condition of surjectivity or injectivity of the group endomorphisms; these questions have been raised in [START_REF] Déserti | Le groupe de Cremona est hopfien[END_REF], [START_REF] Popov | Three plots about Cremona groups[END_REF], and also [START_REF] Blanc | Quotients of higher dimensional Cremona groups[END_REF].

Topologies on the Cremona groups

Cremona groups carry the so-called Zariski topology, which was introduced by M. Demazure [START_REF] Demazure | Sous-groupes algébriques de rang maximum du groupe de Cremona[END_REF]. Over local fields, this topology can be refined to the Euclidean topology -a construction due to J. Blanc and J.-P. Furter [START_REF] Blanc | Topologies and structures of the Cremona groups[END_REF]. Both topologies restrict to the usual Zariski or Euclidean topology on the group Aut(P d k ). In this section we will briefly recall the definitions and main properties of those topologies.

2.1. The Zariski topology. Let k be a field and X and A irreducible algebraic varieties defined over k. Consider a birational map f : A×X A×X inducing an isomorphism between open dense subsets U and V , where U, V ⊂ A × X have the property that the restriction of the first projection to U and V induces a surjective morphism onto A. Every k-point a ∈ A(k) induces a birational transformation f a of X by mapping x ∈ X to p 2 (f (a, x)), where p 2 : A × X -→ X is the second projection. The map from A(k) to Bir(X) given by a → f a is called a morphism (or k-morphism) from A to Bir(X), and is denoted by A -→ Bir(X).

The Zariski topology is now the finest topology such that the preimages of closed subsets by morphisms are closed: Definition 2.1. A subset F ⊂ Bir(X) is closed in the Zariski topology if for any algebraic k-variety A and any k-morphism A -→ Bir(X) the preimage of F is closed in A(k).

Recall that, once we choose coordinates of P d k , a Cremona transformation is given by [x 0 :

• • • : x d ] [f 0 : • • • : f d ],
where the f i ∈ k[x 0 , . . . , x d ] are homogeneous polynomials of the same degree without a non-constant common factor. The degree of f is then defined to be the degree of the f i . We denote by Cr d (k) ≤n the set of Cremona transformations of degree ≤ n and by Cr d (k) n the set of those of degree n.

J. Blanc and J.-P. Furter give an equivalent construction of the Zariski topology on the Cremona groups [3, §2] by defining a topology on Cr d (k) ≤n and extending it to Cr d (k) as the inductive limit topology. The sets Cr d (k) n can be equipped canonically with the structure of a k-variety such that the elements in Cr d (k) n are exactly its k-points. However, the sets Cr d (k) ≤n , while being closed, can not be given in any reasonable way the structure of an algebraic variety. This is part of the obstruction that Cr d (k) can not be equipped with the structure of an ind-variety when endowed with the Zariski topology [START_REF] Blanc | Topologies and structures of the Cremona groups[END_REF]Theorem 2].

Let G be an algebraic group and G -→ Cr d (k) a morphism which is also a group homomorphism from the group Proof. Assume, for a contradiction, that ϕ(A(k)) is of unbounded degree. Then there exists an infinite sequence of elements {f n } n∈Z>0 ⊂ ϕ(A(k)) such that all the f i are of different degree. This implies that the induced topology on {f n } n∈Z>0 is discrete and so in particular not noetherian. But this yields that the induced topology on ϕ -1 ({f n } n∈Z>0 ) is not noetherian, which is a contradiction, since a subspace of a noetherian space is noetherian.

G(k) of k-points of G to Cr d (k).
2.2. The Euclidean topology. We now introduce the Euclidean topology on Cr n (k) as defined in [3, §5].

Let k[x 0 , . . . , x d ] n be the k-vector space of homogeneous polynomials of degree n and consider the projectivisation

W n (k) = P((k[x 0 , . . . , x d ] n ) d+1 ), which is the set of k-rational points of a projective space W n of dimension r = (d + 1) d+n d -1. An element h = [h 0 : • • • : h d ] ∈ W n (k) induces the rational map of P d k given by ψ h : [x 0 : • • • : x d ] [h 0 (x 0 , . . . , x d ) : • • • : h d (x 0 , . . . , x d )]
, and the set

H n (k) ⊂ W n (k) of elements inducing a birational map of P d k is locally closed in W n (k).
There exists a locally closed subvariety

H n ⊂ W n such that H n (k) is exactly the set of k-points of H n [3, Lemma 2.4(2)]. The map π n : H n (k) -→ Cr d (k) ≤n , h → ψ h corresponds to a morphism H n -→ Cr d (k). It

A homomorphism of Cr d (k) is determined by its restriction to

Aut(P n k ) In this section, we show that for infinite fields a surjective Zariski or Euclidean continuous homomorphism of Cr d (k) is determined by its restriction to Aut(P d k ). The main ingredients for this result are continuous deformations of arbitrary Cremona transformations to linear maps. Consider the following example: Example 3.1. Let k be a field and pick f ∈ Cr d (k) of degree e. With respect to affine coordinates on the chart x 0 = 0 we can write

f : (x 1 , . . . , x d ) (F 1 , . . . , F d ) ,
where

F i = P i0 + P i1 (x) + • • • + P ie (x) Q i0 + Q i1 (x) + • • • + Q ie (x) ,
for some homogeneous

P ij , Q ij ∈ k[x 1 , . . . , x d ] of degree j.
For t ∈ k \ {0}, we define β t ∈ Aut(P d k ) by

β t : (x 1 , . . . , x n ) → (tx 1 , . . . , tx d ).
The map t → β -1 t f β t defines a k-morphism ρ : A 1 k \{0} -→ Cr d (k) whose image consists of conjugates of f by linear maps. Lemma 3.2. Let k be an infinite field. The k-morphism ρ :

A 1 k \{0} -→ Cr d (k) from Example 3.1 extends to a Zariski continuous map ρ : A 1 k (k) -→ Cr d (k) with ρ0 ∈ Aut(P n k ) if and only if f is a local isomorphism at the point p = [1 : 0 : • • • : 0] and f (p) = p.
The same statement holds if k a local field and if we consider the Euclidean instead of the Zariski topology.

Proof. With respect to affine coordinates x = (x 1 , . . . , x d ) on the chart x 0 = 0 we can write

ρ t (x) = β -1 t f β t (x 1 , . . . , x d ) = F t 1 , . . . , F t d ,
where Now, suppose that ρ extends according to our hypothesis and ρ0 ∈ Aut(P d k ). In particular, the limit of F t i for t -→ 0 is a rational map for all i = 1, . . . , d, which implies that P i0 = 0 and Q i0 = 0. Hence f fixes [1 : 0 : • • • : 0], and ρ0 corresponds to the derivative of f at [1 : 0 : • • • : 0], so f is a local isomorphism at p. Proposition 3.3. Let k be an infinite field and ϕ : Cr

F t i = t -1 P i0 + P i1 (x) + tP i2 (x) + • • • t e-1 P ie (x) Q i0 + tQ i1 (x) + • • • t e Q ie (x) .
d (k) -→ Cr d (k) a group homomorphism which is Zariski continuous. If ϕ| Aut(P d k ) = id Aut(P d k ) , then ϕ = id Cr d (k) .
The same statement holds for any local field k and ϕ Euclidean continuous.

Proof. Let f ∈ Cr d (k) and p ∈ P d k (k) any point such that both f and ϕ(f ) are local isomorphisms at p. We will prove that ϕ(f

)(p) = f (p) or ϕ(f )(p) = p, which implies that ϕ(f ) = f or ϕ(id), since k is infinite.
From this we will then deduce the claim.

We denote q := f (p) ∈ P d k (k). There exists an automorphism α ∈ Aut(P d k ) whose only fixed points in P d k (k) are p and q. Let us observe that the birational transformation α -1 f -1 αf fixes p and is a local isomorphism at p. Denote by ρ : A 1 k \{0} -→ Cr d (k) the morphism defined in Example 3.1 with respect to the map α -1 f -1 αf , i.e. ρ t = β -1 t (α -1 f -1 αf )β t . By Lemma 3.2, ρ extends to a morphism ρ : A 1 -→ Cr d (k) and ρ0 is the derivative of (α -1 f -1 αf ) at p. Lemma 2.4 yields that the corresponding map

A 1 k (k) -→ Cr d (k) is also Euclidean continuous.
Since ϕ is Zariski (respectively Euclidean) continuous, the map ϕ • ρ : A 1 (k) -→ Cr d (k) is also Zariski (respectively Euclidean) continuous. By hypothesis, the restriction of ϕ to Aut(P d k ) is the identity, and so

ϕ(ρ t ) = β -1 t ϕ(α -1 f -1 αf )β t = β -1 t α -1 ϕ(f ) -1 αϕ(f )β t , t = 0 ϕ(ρ 0 ) = ρ0 ∈ Aut(P d k )
Lemma 3.2 implies that the map (α -1 ϕ(f ) -1 αϕ(f )) fixes p and is a local isomorphism at p. In particular, since ϕ(f ) is a local isomorphism at p, we have

α(ϕ(f )(p)) = ϕ(f )(α(p)) = ϕ(f )(p).
By hypothesis, p and q = f (p) are the only fixed points of α, hence

ϕ(f )(p) = p or ϕ(f )(p) = f (p).
This holds for any p in an open dense subset of P d k (k). As k is infinite, ϕ(f ) coincides thus with id P d k or with f . This holds for any f ∈ Cr d (k), so in particular for γf , for any non-trivial γ ∈ Aut(P d k ). Suppose that ϕ(f ) = id. Then id = γ = ϕ(γ) = ϕ(γf ). Since we have by the same argument again that either ϕ(γf ) = γf or ϕ(γf ) = id, we obtain that ϕ(γf

) = γf , which implies f = id. It follows that ϕ(f ) = f for any f ∈ Cr d (k).
We can repeat the proof of Proposition 3.

Proofs of the main theorems

Group automorphisms of classical groups are well-understood. In [START_REF] Dieudonné | La géométrie des groupes classiques[END_REF] J. Dieudonné gives a complete classification in the case of fields, and we need the classification of group automorphisms of PGL d+1 (k). Let G be PGL d+1 (k) or GL d+1 (k). For any group automorphism ϕ : G -→ G there exists an element h ∈ G and a field automorphism α of k such that ϕ is of the form

ϕ(g) = h( α g ∨ )h -1 or ϕ(g) = h( α g)h -1
where g ∨ = t (g -1 ) is the inverse of the transpose of g. Not all automorphisms of PGL d+1 (k) can be extended to an automorphism of the Cremona group: Lemma 4.2. [24, Corollary A.12] For any field k, the automorphism of PGL d+1 (k) given by g → α g ∨ does not extend to an automorphism of Cr d (k). Theorem 4.3 ([6, Theorem A, Corollary 8.5]). Let d ≥ 1 and Γ ⊂ SL d+1 (Z) a finite index subgroup. Let X C be an irreducible complex quasi-projective variety of dimension dim(X) = n.

(1) If there is an injective group homomorphism ϕ : Γ -→ Aut(X C ), then n ≥ d.

If n = d then there is an isomorphism f :

X C -→ P d C such that f ϕ(Γ)f -1 ⊂ Aut(P d C ). ( 2
)
If there is an injective group homomorphism ψ : Γ -→ Bir(X C ), then n ≥ d.

If n = d then there is a birational transformation f :

X C P d C such that f ϕ(Γ)f -1 ⊂ Aut(P d C )
. Let p be an odd prime. We denote by Γ p ⊂ SL d+1 (Z) the congruence subgroup mod p, i.e. the kernel of the homomorphism ρ : SL d+1 (Z) -→ SL d+1 (F p ) given by reduction modulo p. Since p is odd by assumption, Γ p intersects the center of SL d+1 (Z) only in the identity and hence, the restriction of the quotient homomorphism GL d+1 (k) -→ PGL d+1 (k) to Γ p is injective and we can consider Γ p as a subgroup of PGL d+1 (k). We can now apply Theorem 4.3 to Γ p and obtain the following: Let p be an odd prime and Γ p ⊂ SL d+1 (Z) the congruence subgroup mod p. We consider Γ p as a subgroup of PGL d+1 (k). In particular, Γ p contains the subgroup of elements of the form

[x 0 : • • • : x d ] → [x 0 + kpx 1 : x 1 : • • • : x d ], k ∈ Z .
For i, j = 0, . . . , d, i = j, define the elementary subgroups

U ij := {[x 0 : • • • : x d ] → [x 0 : • • • : x i + cx j : x i+1 : • • • : x d ] | c ∈ k} ⊂ PGL d+1 (k).
Consider the Euclidean compact subset

E 01 := {[x 0 : • • • : x d ] → [x 0 + cx 1 : x 1 : • • • : x d ] | ||c|| ≤ p} ⊂ PGL d+1 (k)
where || • || denotes the Euclidean norm on k. Every element in the subgroup U 01 can be written as a product of an element in Γ p and an element in E 01 . Since E 01 is compact, ϕ(E 01 ) is of bounded degree [START_REF] Blanc | Topologies and structures of the Cremona groups[END_REF]Lemma 5.13]. By Corollary 4.4, the image ϕ(Γ p ) is of bounded degree as well and we conclude that ϕ(U 00 ) is of bounded degree. Since all the elementary subgroups U ij are conjugate to U 00 in PGL d+1 (k) it follows that ϕ(U ij ) is of bounded degree for any i, j.

Using the Gauss-Algorithm one can see that there exists an integer K ≥ 1 only depending on d such that every element in PSL d+1 (k) can be written as the product of at most K elements contained in some the subgroups U ij . Therefore, ϕ(PSL d+1 (k)) is of bounded degree and hence is regularisable, i.e. there exists a quasi-projective k-variety X k and a birational transformation . Since ϕ is continuous, Proposition 3.3 implies that ϕ = id Cr d (k) , which yields the claim. It can be shown that if k is algebraically closed, then Aut(A d k ) can be naturally equipped with the additional structure of a so-called ind-group, which induces the ind-topology on Aut(A d k ). We refer to [START_REF] Furter | On the geometry of the automorphism groups of affine varieties[END_REF] for a definition and details about this notion. We will only need that an ind-closed subgroup of bounded degree of Aut(A d k ) has the structure of an affine algebraic group such that the induced action on A d

P d k X k that conjugates ϕ(PSL d+1 (k)) to a subgroup of Aut(X k ) [

Polynomial automorphisms

Denote by

∈ Aut(A d k ) such that f ϕ(T d (k))f -1 ⊂ T d (k) is a dense subgroup. Proof. Let D k be the closure of ϕ(T d (k)) in Aut(A d k )
with respect to the indtopology. Since ϕ(T d (k)) and hence D k are of bounded degree, we obtain that D k is an affine algebraic group acting algebraically on A d k . By the structure theorem for commutative affine algebraic groups over algebraically closed fields (see [

1, Théorème XVII.7.2.1]), D k is isomorphic over k to T r (k) × U s (k)
, where T r is a split torus of dimension r ≥ 0 and U s a unipotent group of dimension s ≥ 0.

First, suppose that char(k) = 2. The image ϕ(T d (k)) contains 2 d elements of order two, hence D k contains at least 2 d elements of order two, and it follows that r ≥ d. Thus D k contains a maximal torus in Aut(A d k ) and as such coincides with its centraliser. So we obtain that s = 0. Now, if char(k) = 2, as ϕ is a homeomorphism with respect to the Zariski topology by hypothesis, and since We leave it as an exercise to the reader to verify that for an infinite field k the centralizer of C(GL

T d (k) 2 is Zariski dense in T d (k), we have that ϕ(T d (k)) 2 is dense in ϕ(T d (k))
d (k)) in Aut(A d k ) is GL d (k), that Norm Aut(A d k ) (T d (k)) = T d (k) S d and C(T d (k) S d )) = C(GL d (k)).
) (ϕ(T d (k))) = Norm Aut(A d k ) (T d (k)) = T d (k) S d .
As the image of the normaliser is contained in the normaliser of the image, this implies that ϕ(S d ) ⊂ T d (k) S d . Consider the natural projection π :

T d (k) S d -→ S d . Its kernel is equal to T d (k), which is equal to Cent Aut(A d k ) (T d (k)). Then ϕ(S d ) ∩ T d (k) = ϕ(S d ) ∩ Cent Aut(A d k ) (T d (k)) = ϕ(S d ∩ T d (k)) = {id} implies that up to an automorphism of S d the restriction ϕ| S d is a section of π. Therefore, Cent T d (k) S d (ϕ(S d )) = C(GL d (k)). It follows that ϕ(C(GL d (k)) = ϕ Cent T d (k) S d (S d ) ⊂ Cent T d (k) S d (ϕ(S d )) = C(GL d (k)).
As k is infinite and ϕ is injective, the image ϕ(C(GL

d (k))) is dense in C(GL d (k)). It follows that ϕ(GL d (k)) = ϕ Cent Aut(A d k ) (C(GL d (k))) ⊆ Cent Aut(A d k ) (ϕ(C(GL d (k)))) = Cent Aut(A d k ) (C(GL d (k))) = GL d (k).
It remains to show that ϕ(GL d (k)) is dense in GL d (k). For i, j = 1, . . . , d, i = j, we define the elementary subgroups

U ij (k) := (x 1 , . . . , x d ) → (x 1 , . . . , x i + cx j , x i+1 , . . . , x d ) | c ∈ k ⊂ GL d (k). and U ij (k) E ij : (x 1 , . . . , x d ) → (x 1 , . . . , x i + x j , x i+1 , . . . , x d ) Observe that {tE ij t -1 | t ∈ ϕ(T d (k))} ⊆ {tE ij t -1 | t ∈ T d (k)} = U ij (k)
is a dense subset which is also contained in ϕ(GL d (k)). Recall that there exists an integer N ≥ 1 and a surjective morphism

V 1 × • • • × V N -→ GL d (k), (v 1 , . . . , v N ) → v 1 v 2 • • • v N , where each V l is equal to T d (k) or to U ij (k) for some i, j. Since V l := V l ∩ ϕ(GL d (k)) is dense in V l as remarked above, this implies that ϕ(GL d (k)) is dense in GL d (k).
Let G be a group acting on a group A by σ : a → σ a for σ ∈ G, a ∈ A. Recall that the first Galois cohomology H 1 (G, A) is defined as the set of cocycles, i.e. maps ν : G -→ A satisfying ν(στ ) = ν(σ) σ ν(τ ) for all σ, τ ∈ G, up to the following equivalence relation: two cocycles ν and η from G to A are equivalent if there exists an element a ∈ A such that η(σ) = a -1 ν(σ) σ a for all σ ∈ G. Lemma 5.3. Let k be a perfect field and ϕ : GL

d (k) -→ Aut(A d k ) an injec- tive homomorphism. Assume that there is an element f ∈ Aut(A d k ) such that f ϕ(GL d (k))f -1 ⊂ GL d (k) as a dense subgroup, then there exists an element g ∈ Aut(A d k ) such that gϕ(GL d (k))g -1 ⊂ GL d (k). Proof. Let L be the finite field extension such that f is defined over L. In particular, H = f ϕ(GL d (k))f -1 ⊂ GL d (L) ⊂ Aut(A d L ) as a dense subgroup. Let σ ∈ G = Gal(L/k) be any element of the Galois group of L over k. Then σ H = σ (f ϕ(GL d (k))f -1 ) = σ f (ϕ(GL d (k)) σ f -1 = σ f f -1 Hf ( σ f ) -1 .
Using that H and hence also σ H is dense in GL d (L), we may conclude that σ f f -1 normalizes GL d (L) and therefore that σ f f -1 ∈ GL d (L). In other words, there exists an element g σ ∈ GL d (L) such that σ f = g σ f .

We define now the map µ : G -→ GL d (L) by µ(σ) := g -1 σ . It is straightforward to check that µ satisfies the cocycle condition with respect to the Galois action of G on GL d (L). By Hilbert 90 (see for example [20, Lemma 1, Chapter III.1.1]) the first cohomology H 1 (G, GL d (k)) is trivial, i.e. every cocycle is equivalent to the cocycle that is given by mapping every element in G to the identity in GL d (L). In other words, there exists an element a ∈ GL d (L) satisfying a -1 µ(σ) σ a = id for all σ ∈ G. Consider now the polynomial automorphism h = a -1 f . Clearly we have hϕ

(GL d (L))h -1 ⊂ GL d (L). It remains to show that h is defined over k. Since k is perfect, it is equivalent to prove σ h = h for all σ ∈ G. For any σ ∈ G one calculates σ h = σ a -1σ f = a -1 µ(σ) σ f = a -1 µ(σ)µ(σ) -1 f = h. As h and ϕ(GL d (k))
are both defined over k, the group hϕ(GL d (k))h -1 is defined over k as well, so hϕ(GL d (k))h -1 ⊂ GL d (k). Proof. From Lemma 5.1, Lemma 5.2 and Lemma 5.3 we obtain that there exists f ∈ Aut(A d k ) such that f ϕ(GL d (k))f -1 ⊆ GL d (k). Since, by assumption, ϕ is a homeomorphism, Lemma 2.2 implies that ϕ -1 (f -1 GL d (k)f ) is of bounded degree and closed, and is hence an affine algebraic group containing GL d (k). The Krulldimension is preserved by a homeomorphism, thus GL d (k) = ϕ -1 (f -1 GL d (k)f ). If char(k) = 2, then f 2 = id and hence ϕ(f ) 2 = id, which implies that b = 1. Now, suppose that char(k) = 2 and consider t : (x 1 , . . . , x d ) → (2x 1 , x 2 , . . . , x d ). We have tf t -1 = f 2 , so also tϕ(f )t -1 = ϕ(f ) 2 , which again implies b = 1.

After conjugating ϕ with a suitable element of the center of GL d (k), we obtain ϕ(f ) = f . The group GL d (k) acts transitively by conjugation on the set of translations different from id, therefore ϕ| Aff d (k) = id Aff d (k) .

Proof of Theorem 1.3. The theorem follows from Lemma 5.4, Lemma 5.5 and Proposition 3.4.

Lemma 2 . 2 .

 22 Let A be an algebraic variety over a field k and ϕ : A -→ Cr d (k) a Zariski continuous map. Then ϕ(A(k)) is of bounded degree.

Definition 2 . 3 .

 23 is surjective, continuous and closed with respect to the Zariski topology, and is thus a topological quotient [3, Corollary 2.9]. It turns out that the Zariski topology on Cr d (k) is the inductive limit topology given by the closed sets Cr d (k) ≤d endowed with the quotient topology [3, Proposition 2.20]. If k is a local field, we endow the W n (k) and H n (k) with the Euclidean topology and define the Euclidean topology on Cr d (k) to be the inductive limit topology given by the Cr d (k) ≤n endowed with the quotient topology π n : H n (k) -→ Cr d (k) ≤n . Endowed with the Euclidean topology, Cr d (k) is a Hausdorff topological group which is not metrisable, and any compact subset is of bounded degree [3, Theorem 3, Lemma 5.16, Lemma 5.13]. By definition, the Euclidean topology is finer than the Zariski topology and we have the following property: Lemma 2.4 ([5, Lemma 2.11]). Let A be an algebraic variety over a local field k and ρ : A -→ Cr d (k) a k-morphism. Then the corresponding map A(k) -→ Cr d (k) is Euclidean continuous.

First, suppose that

  f fixes p = [1 : 0 : • • • : 0] and is a local isomorphism at p. Then P i0 = 0 and Q i0 = 0 for i = 1, . . . , d and hence ρ 0 corresponds to the derivative (the linear part) of f at [1 : 0 : • • • : 0]. Thus ρ extends to a k-morphism ρ : A 1 -→ Cr d with ρ0 ∈ Aut(P d k ), which is in particular Zariski continuous. If k is a local field then ρ is Euclidean continuous by Lemma 2.4.

  3 word by word by considering Aut(A d k ) instead of Cr d (k) and its subgroup Aff d (k) instead of PGL d+1 (k). This way, we obtain the following statement: Proposition 3.4. Let k be an infinite field and ϕ : Aut(A d k ) -→ Aut(A d k ) a surjective homomorphism which is Zariski continuous. If ϕ| Aff d (k) = id Aff d (k) , then ϕ = id Aut(A d k ) .

Theorem 4 . 1 (

 41 [12, IV. §1.I-III, p.85-89 and IV. §6, p.98]). Let d ≥ 1 and k a field.

Corollary 4 . 4 .

 44 Let d ≥ 2, k a field of characteristic 0 and ϕ : Γ p -→ Cr d (k) an injective group homomorphism. Then the degree of the elements ϕ(Γ p ) is bounded. Proof. Let F ⊂ k be the smallest subfield over which ϕ(Γ p ) is defined. We can consider ϕ(G) as a subgroup of Cr d (F ) ⊂ Cr d (k). Since Γ p is finitely generated, F is a finitely generated extension of Q and as such embeds into C and we may consider ϕ(G) as a subgroup of Cr d (F ) ⊂ Cr d (C). By Theorem 4.3, ϕ(G) is conjugate in Cr d (C) to a subgroup of Aut(P d C ). In particular, ϕ(Γ p ) is of bounded degree. Now, we study continuous embeddings of PGL d+1 (k) into Cr d (k).

Lemma 4 . 5 .

 45 Let k be either the field of real numbers R or the field of complex numbers C and d ≥ 1. Let ϕ : PSL d+1 (k) -→ Cr d (k) be an injective group homomorphism that is Euclidean continuous. Then there exists an element f ∈ Cr d (k) such that the group f ϕ(PSL d+1 (k))f -1 is contained in Aut(P d k ). Proof. The statement is trivial for d = 1, hence we can assume that d ≥ 2.

  Aut(A d k ) the group of polynomial automorphisms of the affine d-space A d k over a field k. The goal of this section is to prove Theorem 1.3. If we choose an embedding of A d k into P d k , we can naturally consider Aut(A d k ) as a subgroup of Cr d (k). The Zariski topology on Aut(A d k ) is the induced topology of the Zariski topology on Cr d (k).

k

  is algebraic[START_REF] Stampfli | Contributions to automorphisms of affine spaces[END_REF] Proposition 3.7].1 In fact, we apply[START_REF] Rosenlicht | Some basic theorems on algebraic groups[END_REF] Theorem 1] to the Zariski closure of ϕ(PSL d+1 (k)) in Cr d (k), which is an algebraic subgroup of Cr d (k).

Lemma 5 . 1 .

 51 Denote by T d (k) ⊂ Aut(A d k ) the diagonal linear transformations. Let k be the algebraic closure of a field k. In what follows, we always consider Aut(A d k ) as a subgroup of Aut(A d k ) through the obvious inclusion. Let k be an infinite field and ϕ : Aut(A d k ) -→ Aut(A d k ) a group automorphism that is a homeomorphism with respect to the Zariski topology. Then there exists f

  and hence in ϕ(T d (k)). This implies in particular that ϕ(T d (k))2 is dense in ϕ(T d (k)), i.e. (D k ) 2 is dense in D k . Now, at the same time, char(k) = 2 implies that (D k ) 2T r (k), and we obtain s = 0. Since ϕ is a homeomorphism, it preserves the Krull dimension, therefore ϕ(T d (k)) and hence also D k has Krull dimension d, it follows also that d = r.So we have that D k is a torus of rank d. From [2, Theorem 2] it follows that D k is conjugate in Aut(A d k ) to the standard torus T d (k) in Aut(A d k ). In particular, there exists an elementf ∈ Aut(A d k ) such that f ϕ(T d (k))f -1 ⊂ T d (k). Since ϕ(T d (k)) is dense in D k , we obtain that f ϕ(T d (k))f -1 is dense in T d (k).Let us denote by GL d (k) ⊂ Aut(A d ) the subgroup of linear automorphisms, by Aff d (k) the subgroup of affine transformations, and by S d ⊂ GL d (k) the subgroup of coordinate permutations. For a group G we denote by C(G) the center of G, and for a set A ⊂ G we denote by Cent G (A) and by Norm G (A) the centraliser and normaliser of A in G, respectively.

Lemma 5 . 2 .

 52 Let k be an infinite field and ϕ :Aut(A d k ) -→ Aut(A d k ) an injective group endomorphism. Assume that ϕ(T d (k)) is contained in T d (k) as a dense subgroup. Then ϕ(GL d (k)) is contained in GL d (k).Moreover, this image is dense. Proof. Since ϕ(T d (k)) is dense in T d (k) by assumption, we have Norm Aut(A d k

Lemma 5 . 4 .

 54 Let k be an infinite perfect field and ϕ :Aut(A d k ) -→ Aut(A d k )a group automorphism that is a homeomorphism with respect to the Zariski topology. Then there exists an elementf ∈ Aut(A d k ) such that f ϕ(GL d (k))f -1 = GL d (k).

Lemma 5 . 5 .

 55 Let k be an infinite perfect field and ϕ : Aut(A d k ) -→ Aut(A d k ) a group automorphism and suppose that ϕ(GLd (k)) = GL d (k). Then ϕ(Aff d (k)) = Aff d (k)and there exists a field automorphism α of k and an element h ∈ GL d (k) such that ϕ(g) = h( α g)h -1 for all g ∈ Aff d (k).Proof. By Theorem 4.1, up to conjugation in GL d (k) and up to a field automorphism of k, we can assume that ϕ| GL d (k) is the identity on GL d (k) or given by g → g ∨ .Letf : (x 1 , . . . , x d ) → (x 1 + a, x 2 , . . . , x d ) for some a ∈ k * . Consider GL d-1 (k) as a subgroup of GL d (k) embedded as g → [(x 1 , . . . , x d ) → (x 1 , g(x 2 , . . . , x d ))]. We write ϕ(f ) : x → (p 1 (x), . . . , p d (x)) for x = (x 1 , . . . , x d ) and polynomials p 1 , . . . , p d ∈ k[x 1 , . . . , x d ]. Observe that f commutes with every element of GL d-1 (k). As ϕ(GL d-1 (k)) = GL d-1 (k), it follows that the same is true for ϕ(f ), and hencep i (x) = x i b for some b ∈ k * for i = 2, . . . , d. Moreover, p 1 (x) is of the form cx 1 + d for some c, d ∈ k. Note that d = 0 because ϕ is injective. It follows in particular, that ϕ(Aff d (k)) = Aff d (k). Consider h : (x 1 , . . . , x d ) → (x 1 + x 2 , x 2 , x 3 , . . . , x d ).Then f commutes with h but ϕ(f ) does not commute with h ∨ . It follows that ϕ| GL d (k) is the identity on GL d (k). Then ϕ(f )h = hϕ(f ), which implies that d = b.

  Proposition 4.6. Let k be a field of characteristic 0 and let ϕ : PGL d+1 (k) -→ Cr d (k) be an injective group homomorphism.If either ϕ is Zariski continuous, or if k = R or k = C and ϕ is Euclidean continuous, then ϕ(PGL d+1 (k)) is conjugate in Cr d (k) to a subgroup of Aut(P d k ). Proof. If ϕ is Zariski continuous, then ϕ(PGL d+1 (k)) is of bounded degree by Lemma 2.2. If k = R or k = Cand ϕ is Euclidean continuous, then Lemma 4.5 implies that ϕ(PSL d+1 (k)) has bounded degree in Cr d (k). Since PSL d+1 (k) has finite index in PGL d+1 (k), hence also ϕ(PGL d+1 (k)) is of bounded degree.In either case there exists a quasi-projective k-variety X k and a birational mapP d k X k which conjugates ϕ(PGL d+1 (k)) to a subgroup of Aut(X k ) [19, Theorem 1] 1 . By Theorem 4.3, X k P d k ,where k is the algebraic closure of k, and hence, by Châtelet's theorem, X k P d Since ϕ is continuous, Proposition 4.6 implies that up to conjugation in Cr d (k), we may assume that ϕ(Aut(P d k )) ⊂ Aut(P d k ). As by assumption ϕ -1 is a continuous group homomorphism as well, we argue analogously that there exists f ∈ Cr d (k) such that f ϕ -1 (Aut(P d

		19, Theorem 1] 1 . By Theo-
	rem 4.3, we have X C	P d C and hence, by Châtelet's theorem, X k	P d k [7, §IV.I,
	p.283].
	d k )) = k ). With Theorem 4.1 and Lemma 4.2 we conclude that, up to conjuation in Aut(P d Aut(P d

k [7, §IV.I, p.283]. Proof of Theorem 1.1 and Theorem 1.2. The group automorphism ϕ : Cr d (k) -→ Cr d (k) induces an injective group homomorphism ϕ| Aut(P d k ) : Aut(P d k ) → Cr d (k). k ))f -1 ⊂ Aut(P d k ). In particular, we have f Aut(P d k )f -1 ⊂ Aut(P d k ) and so f normalizes Aut(P d k ) and hence f ∈ Aut(P d k ), i.e. ϕ -1 (Aut(P k ) and up to a field automorphism of k, we have ϕ| Aut(P d k ) = id Aut(P d k )

Acknowledgements: The authors would like to thank Jean-Philippe Furter, Hanspeter Kraft, and Immanuel van Santen (né Stampfli) for interesting discussions related to this project.

During this project, the first author was supported by the Swiss National Science Foundation project P2BSP2 175008, the second author was supported by the ANR Project FIBALGA ANR-18-CE40-0003-01. Both authors received funding from Projet PEPS 2019 "JC/JC" and were supported through the programme "Research in Pairs" by the Mathematisches Forschungs Institut Oberwolfach 2019.

Birational diffeomorphisms

Let X R be a real projective rational variety, such that the set of R-points X R (R) is non-empty. We can now consider the subgroup BirDiff(X R (R)) ⊂ Bir(X) of birational diffeomorphisms, which is the subgroup of birational transformations without any real indeterminacy points. In [START_REF] Kollár | Cremona transformations and diffeomorphisms of surfaces[END_REF] it is shown that BirDiff(P 2 R (R)) and BirDiff(P 1 R (R) × P 1 R (R)) are dense subgroups of the diffeomorphism groups

) be a group automorphism that is a homeomorphism with respect to the Zariski topology on the group BirDiff(P d R Then either ϕ is an inner automorphism, or there exists f ∈ BirDiff(P d R (R)) such that ϕ(g) = f g ∨ f -1 for all g ∈ PGL d+1 (R) and this defines ϕ uniquely.

Proof. By Proposition 4.6, there exists f ∈ Cr d (R) such that f ϕ(PGL d+1 (R))f -1 ⊂ PGL d+1 (R). Since PGL d+1 (R) acts transitively on P d R (R), the maps f and f -1 have no base-points in P d R (R), and hence f ∈ BirDiff(P d R (R)). So, we can assume that ϕ(PGL d+1 (R)) = PGL d+1 (R). By Theorem 4.1, up to conjugation in PGL d+1 (R), we have that ϕ| PGL d+1 (R) = id PGL d+1 (R) or ϕ(g) = g ∨ for all g ∈ PGL d+1 (R). By Remark 6.1, ϕ is uniquely determined by its restriction on PGL d+1 (R).

Despite best efforts, the authors were not able to determine whether the automorphism PGL d+1 (R) given by g → α g ∨ extends to an endomorphism of BirDiff(P d R (R)). Remark 6.3. For any field k, we can define the group Bireg(P d k (k)) as the group of elements f ∈ Cr d (k) such that f and f -1 have no base-points in P d k (k). If k is of characteristic zero, the analogous statement of Proposition 6.2 for Bireg(P d k (k)) holds with the same proof as for BirDiff(P d R (R)).