Pierre Lebreton

Alexander Raake

Marcus Barkowsky

Patrick Le Callet

PERCEPTUAL PREFERENCE OF S3D OVER 2D FOR HDTV IN DEPENDENCE OF VIDEO QUALITY AND DEPTH

Keywords: 3D video, pair comparison, preference, quality of experience, picture quality, depth, coding

3D video quality of experience (QoE) is a multidimensional problem and many factors contribute to the global experience by the user. Due to this multidimensionality, this paper evaluates the integral 3D video QoE and relates it with image quality and depth. Subjective tests have been conducted using paired comparison to evaluate 3D QoE and the preference of 3D over 2D with different combinations of coding conditions. Depth scores were available from previous work and were used to check their relation with 3DQoE; the difference between 2D and 3D QoE is found to be a function of the picture quality, and the desired preference of 3D presentation over 2D can be reached when pictorial quality is high enough (VQM score lower than 0.24). A factor ranging from 0.08 to 0.76 with a mean of 0.71 between pictorial quality and preference of 3D was also found.

INTRODUCTION

3D video Quality of Experience (QoE) is a multidimensional problem; many factors contribute to the global rating like image quality, depth perception, depth quality, visual discomfort, sense of presence, naturalness, etc. Evaluating QoE taking into consideration all these factors is no easy task. The simplest approach, namely to instruct observers in subjective experiments to judge the "Quality of Experience", does not necessarily provide the expected result, but is rather often based on only one factor of QoE, the picture quality [START_REF] Benoit | Quality assessment of stereoscopic images[END_REF]. This type of results is due to the internal reference of the observer: when observers are asked to rate a 3D image/video, they base their judgment on their expectations. In a test with different coding conditions, the expectations of the observers are mainly driven by picture quality. Consequently, especially with an absolute, single stimulus rating paradigm, people have difficulties to consider factors other than image quality of 3D when they are asked for 3D video QoE. Hence, it is hard to quantify the added value of 3D as compared to 2D, as can be seen in [START_REF] Brunnström | 2D no-reference video quality model development and 3D video transmission quality[END_REF] where 3D is not rated better than 2D in an absolute category rating (ACR) test, and no improvement of QoE due to the 3D effect can be measured.

Alternative evaluation schemes have been considered by Seuntiëns by addressing other dimensions like naturalness or immersion, or by investigating specific factors like depth perception or eye-strain [START_REF] Seuntiens | Visual experience of 3D TV[END_REF], which provides some insight on isolated factors of the general QoE, but not the overall QoE. In the present study, as global measure of QoE, the subjective preference has been considered. It is believed that when subjects are asked for preference between two videos, they may considers all factors (picture quality, both depth quantity and depth quality, visual discomfort and probably other factors) to take the decision which of two versions of a sequence they prefer. This way, the entire multidimensionality of 3D QoE is considered. Missing factors of 2D video QoE when evaluating it using ACR were shown by Belmudez [START_REF]Comparison of the ACR and PC evaluation methods concerning the effects of video resolution and size on visual subjective ratings[END_REF], where another multidimensional question was studied. Here, image size and image resolution were compared in terms of quality ratings, one using ACR, one using paired comparison (PC). Results showed that the two test methods do not provide the same results: using ACR, observers give higher QoE ratings for images at their native resolution; using PC, observers prefer larger images obtained after upscaling. The results are different, and show that using the ACR methodology observers only judge image quality, but with paired comparison they extend their rating to other dimensions, including the image size. PC however has an important drawback: its cost and time consumption. To obtain scale value quality scores from PC data, two models exist: the Bradley-Terry model or the Thurstone-Mosteller model [START_REF] Handley | Comparative analysis of Bradley-Terry and Thurstone-Mosteller model of paired comparisons for image quality assessment[END_REF]. Both need a full PC matrix: each condition has to be compared to another. However, several efficient approaches have been developed in the literature to reduce the number of required comparisons [START_REF] Li | Analysis and improvement of a paired comparison method in the application of 3DTV subjective experiment[END_REF], [START_REF] Lee | Paired comparison-based subjective quality assessment of stereoscopic images[END_REF]. In [START_REF] Lee | Paired comparison-based subjective quality assessment of stereoscopic images[END_REF] six video sequences were recorded. Each of these videos were captured at six inter-camera distances (10 cm to 60 cm). The 36 video sequences were then compared through paired comparison, and the Bradley-Terry scores of each condition were determined. Results show that the Bradley-Terry scores reveal quality fluctuation due to the different depth and comfort. The relation between inter-camera distance and QoE was found highly content dependent. In [START_REF] Barkowsky | Influence of depth rendering on the quality of experience for an autostereoscopic display[END_REF], 3D was compared to 2D using a PC approach on an autostereoscopic display. 3D was produced internally by the display based on a texture and a depth map. The texture was used at four different qualities levels (three encodings and a reference). Results show that 3D was rejected in 70% of the cases and for the lowest quality rejected at 56%. However, the results may be influenced by the technology used at the time of the experiment and the quality of 3D rendering of the 3D display as mentioned by the authors.

In this study, paired comparison will be used to evaluate 3D and 2D video sequences to show the quality improvement/decrease due to 3D. In addition, coding conditions were created and included in the experiment. This enables the comparison between known artefacts such as coding and effect of content characteristics such as depth on 3D-QoE. This information was missing in [START_REF] Lee | Paired comparison-based subjective quality assessment of stereoscopic images[END_REF]. An increase the number of the number of contents enables also a content-based analysis Section 2 provides information on the subjective experiment: the selection of source sequences, the generation of processed video se- Fig. 1: Depth, visual comfort, bitrate of source sequences quences (PVS), and the evaluation process. Section 3 links pictorial quality, depth and preference of 3D 2D. Section 4 extends the analysis to quantifying the "3D added value" and shows to which extent depth contributes to QoE. Section 5 concludes the paper.

EXPERIMENTS

Definition of test conditions

Selection of Source sequences

The selection of the source contents (SRC) is based on three databases. All sequences were full HD stereoscopic videos; each view had a resolution of 1920x1080, with a frame rate of 25 images per second and was of 10s length. Seven SRCs come from a first database composed of 64 source reference signals (SRCs) [START_REF] Lebreton | Evaluating depth perception of 3D stereoscopic videos[END_REF]. The SRCs were used at the highest quality available, and contained various types of scenes. They were rated on three different scales: overall quality of experience, depth and visual comfort. The methodology used was Absolute Category Rating (ACR). Perceived depth was rated on a five-point scale with labels: "very high", "high", "medium", "low" or "very low". Using this general depth scale, the observers rated their general impression of the depth, which takes into account both depth layout perceptions and depth quality. The comfort was evaluated on an absolute manner by asking subjects, if the 3D sequence is "much more", "more", "as", "less", "much less" -"comfortable than watching 2D video". Based on this data, the seven SRCs were chosen to cover the entire range of depth ratings. To ensure the reproducibility of our results, it was decided to include five open video materials [START_REF] Urvoy | NAMA3DS1-COSPAD1 : Subjective video quality assessment database on coding conditions introducing freely available high quality 3D stereoscopic sequences[END_REF]. These sequences include "Tree Branches", "Hall", "Umbrella" and two new sequences designed to reach our depth effect requirements: one with low and high depth quantity, respectively, "Timelaps" and "Drone". A third source of SRC coming from Blu-Ray disk where three other non-open sequences named "Alice" were added to the test. These last sequences were not available at the time of the previous test [START_REF] Lebreton | Evaluating depth perception of 3D stereoscopic videos[END_REF], our depth perception model developed in [START_REF] Lebreton | Evaluating depth perception of 3D stereoscopic videos[END_REF] was used to have an objective value of the perceived depth for these sequences. The top scatter plot of Figure 1 shows how the selected sources cover the depth scale, based on subjective data [START_REF] Lebreton | Evaluating depth perception of 3D stereoscopic videos[END_REF]. The second scatter plot shows the results of the depth score estimated from the depth model described in [START_REF] Lebreton | Evaluating depth perception of 3D stereoscopic videos[END_REF]. The third scatter plot shows the available subjective data regarding visual discomfort. This data has been added, however not used in this paper for content selection, and is shown to let the reader have a view on the principal characteristics of the 3D sequences.

Selection of coding conditions

Coding was performed using a Harmonic Electra 8000 H.264 encoder at constant bitrate. Since it was planned to use a polarized display with horizontal interlacing, the 3D sequences were in the Top/Bottom frame compatible format, this choice limiting the loss of resolution. Each full-HD view was downscaled to half the vertical resolution using a lanczos filter. No further interpolation was done for optimizing resolution but have result in half a line of vertical parallax. Each sequence is then encoded to four different "quality levels" by using four different values of bitrate. The four bitrate values were defined for each source sequence individually, since each of them had different spatial and temporal complexity. In previous experiments, it was revealed that VQM (ITU-R Rec. J.144) performs sufficiently well in estimating picture quality of 3D sequences [START_REF] Lebreton | A subjective and objective evaluation of a realistic 3D IPTV transmission chain[END_REF] (Pearson correlation of 0.89, and RMSE of 5.4). As a consequence, the procedure for determining the bitrate values corresponding to the "quality levels" is based on quality estimations obtained from the VQM general model. The four adequate quality levels have been determined as 0.1, 0.2, 0.3, and 0.4 on the VQM scale. These values correspond to the quality score of the most complex sequence of the test, "TreeBranches", encoded respectively at a quantization parameter QP of 26, 32, 38, and 44 using the reference H.264 encoder JM 18.2. The range of bitrates used in the test is illustrated in Figure 1 and noted for example 2DQ1, for 2D at quality level 1. Quality level 0 being the reference.

At the end of the selection process, the 15 SRCs were encoded at four individually chosen bitrates leading to VQM scores close to 0.1, 0.2, 0.3 and 0.4 as described above. These sequences are used in two versions: 2D and 3D. The 2D sequences were encoded at the same bitrate as their respective counterpart sequences in 3D. In addition, a 3D reference with no compression was added to the test. This results in 15 • (2 • 4 + 1) = 135 video sequences to be evaluated.

Evaluation of 3D QoE using paired comparison

The global QoE of the video sequences was evaluated in a paired comparison experiment. 35 Observers participated in this test. The laboratory environment was in accordance with ITU-R Recommendation BT.500. The observers' vision was screened in terms of acuity, color vision (Ishihara test), and stereovision (Randot stereo test). For the test, two polarized 23" Hyundai displays (ViewSonic V3D231) with horizontal interlacing were used. The displays were calibrated using a display calibration device (X-Rite i1 Pro) to make the rendering as similar as possible between the two displays. The observers were facing two distinct displays, and were instructed to give their preference between the two presentations they could see on the displays. Considering the number of possible presentations (3D or 2D, 4 quality levels in 2D and 3D, and a 3D reference), a full PC matrix approach would have required 9 × (9 -1)/2 = 36 comparisons per SRC, hence 540 comparisons for evaluating all video sequences. This high number of comparisons is impracticable for a subjective experiment [START_REF] Li | Analysis and improvement of a paired comparison method in the application of 3DTV subjective experiment[END_REF]. For more efficient testing, the square design matrix was employed. Based on this approach it was possible to reduce the number of comparisons to 18 comparisons per SRC, hence 15 × 18 = 270 comparisons. The comparisons made in the test can be found in listing 1. The sequence pairs were randomized such that in case of comparison A vs. B, both orders A vs. B and B vs. A were seen by the observers. This avoids any dependency of the preference ratings on the display and possible default answers by observers (right vs. left). The test was split into two sessions

| 3DQ4 vs 3DQ0 | 2DQ3 vs 2DQ1 | | 3DQ4 vs 2DQ4 | 2DQ3 vs 3DQ2 | | 3DQ0 vs 2DQ4 | 2DQ1 vs 3DQ2 | | 3DQ3 vs 3DQ1 | 3DQ4 vs 3DQ3 | | 3DQ3 vs 2DQ2 | 3DQ4 vs 2DQ3 | | 3DQ1 vs 2DQ2 | 3DQ3 vs 2DQ3 | | 3DQ0 vs 3DQ1 | 3DQ0 vs 2DQ1 | | 3DQ1 vs 2DQ1 | 2DQ4 vs 2DQ2 | | 2DQ4 vs 3DQ2 | 2DQ4 vs 2DQ4 |

PREFERENCE OF 3D OVER 2D AND PICTORIAL QUALITY

The main goal of the paired comparison test was to analyze how the preference of 3D over 2D would depend on the respective "pictorial quality". As outlined in Section 2, pictorial quality was varied at four different bitrates, i.e. quality levels Q1-Q4. Figure 2a illustrates for one SRC, how observers answered. It is visible that when the bitrate increases, the preference of the 3D presentation over the 2D version increases. This is however not true for one of the contents, "SkydiversInsideGroup". This content was found to be less preferred when the quality increased. In Figure 1, it can be seen that this content is the least comfortable sequence of the database. The blurring added by coding may have contributed in such way that this content was perceived as more comfortable. This would be in agreement with [START_REF] Roumes | The effects of spatial frequency on binocular fusion: from elementary to complex images[END_REF], where binocular fusion was found to be dependent on the retinal disparities and spatial frequencies within images. In the paired comparison tests, some video sequences were found to always be preferred in 2D "SkydiversAlignment", "SkydiversInsid-eGroup", and "Waterfall". These sequences correspond to the least comfortable sequences of the test. In turn, one sequence was always preferred in 3D: "CarRace3". Based on the test results, the bitrate at which 2D and 3D were equally preferred can be determined, as well as the respective VQM scores. In the following, these points are referred to in terms of isopreference. The VQM scores at isopreference has been estimated using linear regression between two known points in the 2D domain spanned by "preference percentage" and "VQM scores". On average, it has been found that the isopreference at the same bitrate between 2D and 3D is reached when the picture quality of the 3D sequence measured by VQM is at least equal to 0.24. The relation between the VQM scores at isopreference and the depth score rating (subjective depth score when available, and objective if not, see Section 2) was considered. However, no simple relation can be found between these two factors, and other factors have to be taken into account. These other factors may include monocular depth cues such as blur from defocus, linear perspective, texture gradient, motion parallax and also visual discomfort.

QUANTITATIVE ANALYSIS OF THE "3D ADDED VALUE"

Thanks to the test design it was possible to use the Bradley-Terry (BT) model. In Figure 2b, results of the model are depicted for one SRC. The BT-scores provide the continuous perceptual scale which quantifies the difference between 2D and 3D QoE. It is then possible to evaluate the "added value of 3D" by measuring the difference between the BT-scores at conditions where the bitrate is the same. As only pairs for the same source content were evaluated in our test, the BT-scores cannot be used to compare preferences across contents. For example, it is not possible to compare the content "Alice1" in 3D at quality level 3 to "SkydiversInsideGroup" in 2D at quality level 2. This inter-content comparison was not targeted, and instead the goal was to determine 3D preference thresholds as a function of pictorial quality for different degrees of depth information. Making inter-contents comparisons would have added individual judgement of the observer regarding his preference of one type of scene compared to another, which would have made the data noisy and hard to interpret. As a consequence, it is not possible to compare one BT-score from one SRC to another score from another SRC since it exists an unknown offset between these two scores. However, since the scale remains the same between SRCs, it is then possible to compare inter-SRC differences of BT-score. Let the "3D added value" be the difference of BT-score between two similar coding conditions reflecting the score fluctuation due to the presence of depth (see Figure 2b). At least two factors are of influence on the "3D added value" (3DAV) scores, one covers the 3D characteristics of the video sequences including depth, comfort, naturalness, immersion, etc. And the other one covers the pictorial quality of the video. Figure 3 depicts the relation between the quality factor measured through VQM scores of the 3D video sequences and the 3DAV. These two factors show a Pearson correlation of -0.65 and a Spearman correlation of -0.67. Using a N-Way Analysis of Variance (NANOVA) analyzing the 3DAV based on the factors "QualityLevels" as defined in Section 2 and "DepthLevels" (grouping the SRCs in five class of depth effect) shows that there is a strong influence of quality on the 3DAV (F = 13.5, p < 0.001) and that there is also a significant influence of the "DepthLevel" on the 3DAV (F = 3.98, p = 0.0069). Considering the rather small amount of data (four 3DAV values per SRC) no significant influence can be observed on a per-content analysis. Let BT3D(k) be the BT-score of the condition 3DQk as listed in listing 1. The 3D-QoE fluctuation due to coding can be obtained by: ∀k ∈ [START_REF] Benoit | Quality assessment of stereoscopic images[END_REF][START_REF]Comparison of the ACR and PC evaluation methods concerning the effects of video resolution and size on visual subjective ratings[END_REF], ∆BT3D(k) = BT3D(k) -BT3D(0). The ∆BT3D(k) provide a subjective value of how coding affects 3DQoE. This includes loss in pictorial quality, loss in depth [START_REF] Yamagishi | Subjective characteristics for stereoscopic high definition video[END_REF] and the impact on comfort. The relation between ∆BT3D(k) and 3DAV is depicted in Figure 4. In the latter scatter plot, it should be reminded that both 3DAV and ∆BT3D(k) have the same scale. To compare content's specificities it is proposed to perform a regression of the 3D added value (3DAV) as a function of ∆BT3D(k) (see Table 1). The slope values (α) between the two factors range from 0.02 to 0.76. The result on the overall data is a slope of 0.71. This shows that on average, a quality variation of X will impact by 0.71 • X the added value of 3D over 2D. However, there are high fluctuations due to contents specificities (depth and comfort) which need to be studied further. The values of β provides information on the added value of the content when available at the highest quality possible and then its suitability to be presented in 3D. Considering the high variation inter-content of α and β a characterization of the scenes appears to be needed for the development of 3D-QoE models. This is part of on-going work.

CONCLUSION

In this paper, 3D QoE was evaluated by using paired comparison. This way, preference of 3D could be investigated as a function of picture quality. Results show that increasing picture quality increases the probability of preference of 3D over 2D. On average, a VQM score of 0.24 was found to be required to ensure preference of 3D over 2D. Bradley-Terry scores were estimated, and the "3D added value" was determined. The results show that, on average, there is a factor of 0.71 between variation of pictorial quality and "3D added value". There is however lots of variation between contents, which will be investigated in further studies.

Fig. 2 :

 2 Fig. 2: Illustration of preference results for one source content (Hall).

Fig. 3 :

 3 Fig. 3: 3D added value as a function of VQM scores.

Listing 1 :

 1 List of sequence pairs compared by observers

Fig. 4 :

 4 Fig. 4: 3D added value as a function of ∆BT3D.

Table 1 :

 1 Relationship between added value of 3D and difference of BT-score between coding and reference condition.

		3DAV = α • ∆BT3D + β
	Content	α	β	Content	α	β
	Timelaps	0.54 -0.10 Alice7	0.33 -0.07
	Sky.Alignment 0.021 -0.94 Alice4	0.58	0.38
	Waterfall	0.08 -0.67 Hall	0.68	1.05
	Alice1	0.40	0.77	Sky.InsideGroup 0.38 -0.72
	TreeBranche	0.57	0.23	PauseOnARock	0.76	0.13
	Umbrella	0.76	0.90	Firework	0.38	0.65
	LampBlowUp	0.41	0.53	CarRace3	0.51	1.33
	Drone	0.71	1.15	overall	0.71	1.15


Marcus Barkowsky
email: marcus.barkowsky@univ-nantes.fr

Nicolas Staelens
email: nicolas.staelens@intec.ugent.be

Lucjan Janowski
email: janowski@kt.agh.edu.pl

Open collaboration on hybrid video quality models -VQEG Joint Effort Group Hybrid

Open collaboration on hybrid video quality models -VQEG joint effort group hybrid

I. INTRODUCTION

Since many decades it is well-known that Peak Signal to Noise Ratio (PSNR) has severe limitations. This has been shown both theoretically as well as experimentally [START_REF] Chikkerur | Objective Video Quality Assessment Methods: A Classification, Review, and Performance Comparison[END_REF], [START_REF] Wang | Image Quality Assessment: From Error Visibility to Structural Similarity[END_REF]. Nevertheless it is still the most often used measurement tool. The list of alternatives is endless and outside the scope of this paper. For more information on objective video quality assessment, the interested reader is referred to [START_REF] Chikkerur | Objective Video Quality Assessment Methods: A Classification, Review, and Performance Comparison[END_REF]. The development of each such alternative objective quality metric requires time and effort. For most of them their authors showed and eventually verified that they performed better than PSNR. Few of them were validated. None of them has replaced PSNR so far.

In scientific literature, objective video quality metrics usually have a rather limited scope and applicability.

Often, specific use cases are envisaged and some specific test conditions are considered for creating a number of impaired video sequences. These sequences with their limited scope are then used to set up a subjective experiment and train and verify new objective video quality metrics. As such, the validity of these metrics is usually restricted to the considered use case. Furthermore, the (impaired) video sequences and corresponding subjective quality ratings are too often kept secret. For example, the use of copyrighted video materials might prohibit the redistribution of particular video sequences.

In industry, execution speed and ease of use are often considered paramount features. Under these conditions, simple measurements as PSNR are welcomed. Several companies offer more complex solutions to the industry which have a higher prediction performance. The International Telecommunication Union (ITU) has recommended several methods for different scopes that represent typical applications such as mobile transmission or Full-HD IPTV. These methods have often been validated in an open competition approach within the Video Quality Experts Group (VQEQ) showing that they outperformed PSNR in a statistical sense.

With the start of VQEG's Joint Effort Group (JEG), a collaborative approach is now followed towards constructing novel objective video quality metrics [START_REF]Project Homepage of VQEG Joint Effort Group -Hybrid[END_REF]. As opposed to the competitive approach traditionally used within VQEG, JEG encourages and facilitates a free, open, and joint collaboration in subjective and objective video quality assessment. Within JEG, different members from universities, model developing industries, and video service providers join forces. By combining all available know-how during every stage of objective video quality metric design and development, more in-depth and profound video quality research can be conducted as a whole.

The paper provides an overview of recent standards by the ITU and their validation process in Section 2. Advantages and limitations are highlighted. In Section 3, the newly proposed joint approach is summarized, including current and future activities. Section 4 concludes the paper.

II. ITU STANDARDIZATION FOR VIDEO QUALITY MEASUREMENT

Standardisation is considered important because of the reproducibility and the traceability of algorithms and their performance. Notably in ITU standardisation important fairness rules apply. ITU Recommendations have therefore often an important impact on the industry producing or using technology tackled in standardisation. A prominent example is video coding which allows for the efficient exchange of media between various entities regardless of manufacturer or country. The manufacturers are ascertained that they can provide the technology on equal terms with their competitors.

In subjective video quality assessment, most of the standards have been created in a collaborative manner. Discussions have led to refining standards such as ITU-R BT.500 or ITU-T P.910 to name two examples [START_REF]Subjective video quality assessment methods for multimedia applications[END_REF].

A. Competitive validation procedure for objective video quality models

For objective video quality, companies underwent a competition phase in order to determine the best algorithm. In most cases, several algorithms showed similar performance and were therefore standardized.

The competition phase often took place within VQEG in the following way:

1. The proponents developed independent objective models for a given application area. 2. A common document (test plan) was created containing the limits of the application area such as technical constraints on the degradations, the subjective assessment method for creating the validation databases, the procedure for validation including the statistical analysis, and the way in which the models would be ordered.

Proponents submitted executable programs

and eventually encrypted source code, so called frozen models, to a member of the Independent Laboratory Group (ILG) 4. Processed Video Sequences (PVS) were generated obeying the restrictions given in the test plan on known and secret content 5. Subjective assessments were performed creating the Mean Opinion Scores (MOS) for validation 6. Based on the statistical analysis of each model's prediction performance the models were ordered 7. Standardisation in the ITU was proposed if the best model outperformed PSNR, which was eventually adapted to the requirements of the application area 8. The ITU recommended in most cases all models which performed statistically equivalent to the best performing model In this way, ITU-T J.144 [START_REF]Objective perceptual video quality measurement techniques for digital cable television in the presence of a full reference[END_REF] was created for Standard Definition (SD) television at 50Hz and 60Hz including interlaced and progressive, ITU-T J.246 [START_REF]Objective perceptual video quality measurement techniques for digital cable television in the presence of a full reference[END_REF] and J.247 [START_REF]Objective perceptual multimedia video quality measurement in the presence of a full reference[END_REF] for multimedia signals in VGA (640x480 pixels, Video Graphics Array), CIF (352x288, Common Intermediate Format), and QCIF (176x144, Quarter CIF) format including severe coding artefacts, transmission degradations, and pausing and skipping. J.247 requires access to the source video signal and is therefore called Full-Reference (FR) while J.246 requires only access to a signal that was extracted from the source video signal which may be transmitted as side information, a socalled Reduced Reference (RR) algorithm. ITU-T J.341 [START_REF]Objective perceptual multimedia video quality measurement of HDTV for digital cable television in the presence of a full reference[END_REF] allows for measuring Full-HD video signals in 1080i25, 1080i30, 1080p25, and 1080p30. J.341 and J.246 contain a single algorithm while J.144 and J.247 contain four different algorithms each.

In the process of this standardisation, various subjective databases have been created, notably the two well-known VQEG Phase 1 SDTV databases, two hidden databases for SDTV Phase 2, 14 QCIF, 14 CIF, and 13 VGA databases, and 6 databases for HDTV of which 5 are freely available via the Consumer Digital Library (CDVL) [START_REF]Consumer Digital Video Library website[END_REF].

Currently, VQEG evaluates objective measurement algorithms that do not require access to the source video sequence. Instead, information in the decoded video and the bitstream as transmitted over the network is exploited. These models are called Hybrid-No-Reference algorithms.

B. Limitations

The current approach for standardising objective video quality prediction algorithms is well established, the procedure is sound, and allows for fair conditions and equal terms. There are some drawbacks which should be considered, particularly when comparing to other standardisation activities such as video coding:

-Delay: Due to the validation procedure, the typical delay between the submission of the model and the standardisation is two to three years. -Test conditions: Although a large number of test conditions have been evaluated in subjective experiments conducted for validation as described above, they may not suffice to cover the entire application scope and to test for robustness of the models -Secret content: Validation requires content that is not known prior to model submission. This content usually needs to be shot particularly for each validation data set. Shooting video sequences that are balanced in terms of visual features is a time consuming and costly task. -Exploitation: The results are often only used for the standardisation. This applies both to the analysis results because of their usage restrictions, and to the video sequences that become available only after the standardisation has finished. -Critical model performance analysis: Evaluation of the model's performance in different parts of the targeted scope is often difficult. As an example, it may be seen that some of the models for multimedia sequences in J.247 do not measure explicitly frame rate or pauses and skips within the analysed video sequence. It has not been analysed whether this impacts on the model's performance for these particular conditions. -Missing continuity: The standardized models do not provide a basis for future developments by the standardization group. Each proponent improves his own model for the next competition. Splitting the analysis into building blocks and comparing the performance of the algorithms' internal indicators may show advantages and weaknesses that can be exploited for the next version of the standard. -Reproducibility: A rigorous test whether the standards contain all required information for implementing the algorithms has not been undertaken. Despite the enormous and time consuming effort of each proponent to document their algorithm, information may be missing.

III. COLLABORATIVE APPROACH

Most of the above mentioned limitations are a consequence of the competitive approach. Therefore, a collaborative effort was started in order to continuously work on improvements of objective video quality measurement. This includes the creation of source video databases, processed video sequences and associated hardware and software production, the development of additional tools, and the storage of side information.

A. Overview and Advantages

A graphical overview of the complexity of the approach is provided in Figure 1. The diagram mostly focuses on the engineering of an objective model. In particular it does not show the required prerequisites such as psychophysical studies. The diagram shows the steps for creating reasonable source content on the top, degrading those videos in the middle, and at the bottom the development of an objective model. Each part contains an iteration loop and the complete process requires iterations as well.

In a joint effort approach, priorities are significantly different from the competitive approach. In particular, a large database that is well balanced may reduce the requirements for independent validation as the model parameters are known. This allows, for example, avoiding overtraining by consequent analysis of fitted parameters. Following this approach, the creation of a large scale database that allows testing for robustness becomes an important factor. Therefore, the joint effort started with the creation of this database. Several research questions have since then been posed such as the selection of different balanced subsets from a large video source set. The measurement of the uniform coverage of sequence characteristics has previously been mostly limited to simple measures such as Temporal perceptual Indicator (TI) and Spatial perceptual Indicator (SI), defined in ITU-T P.910. The notion of art work and the question of the perceived difference between professionally shot content, such as in cinema or TV broadcast, and user generated content, such as vacation videos or informal videos found on internet platforms, was tackled. The annotation of source content with meta information such as visual attention may be considered, bringing up the question whether video quality assessment as a task is realistic for watching videos of different type, i.e. entertainment or educational. The induced degradations are no longer restricted by rules available in a test plan document. Again the measurement of uniform coverage of the scope becomes important, introducing notably requirements for the definition of distinguishable perceptual degradations as different technical parameters may lead to similar perceived degradations. A toolchain was developed that allows the automated creation of large datasets of degraded videos. Annotating the videos with video quality scores may be performed in different ways, ranging from FR measurement algorithms, over crowd sourcing strategies to formal subjective experiments in lab conditions. Combinations of several objective measurements and subjective experiments are possible and important progress is expected to be achieved by the application of data mining and machine learning algorithms.

Model construction requires the development of isolated indicators which are backed up by psychophysical studies and experimental validation on relevant parts of the large scale database. It also requires establishing a common framework for integrating the different measurements. This is particularly important because the structure of the model determines the way in which the human visual system can be modelled. Evaluating the performance of different strategies known to the perception and psychophysical community on a large amount of annotated databases may provide insight into the processing of visual information by the human observer.

B. Organisation

The joint effort has been started within the VQEG. VQEG's JEG is free and open to everyone. No subscription fees are involved for joining VQEG JEG. Contributions can be made concerning every step involved in subjective and objective video quality assessment. Furthermore, JEG encourages contributions from both academia and private industries.

Since its start, several important links have been established via Liaison Statements to other organisations and networks. An important example is the European Network on Quality of Experience in Multimedia Systems and Services, Qualinet, COST IC2003. Continuous exchange between the organisations led to several innovative publications.

C. Current status and future activities

Currently, JEG focuses on the following topics:

-Creation, analysis, and complementarity of source video sequences for a large scale database to be used freely by the research community -Providing an easy-to-use tool chain enabling the creation of video sequences containing compression and/or transmission artefacts -Creation of a large scale database containing degraded videos with various features -Enabling the development of coding standard independent Hybrid Video Quality metrics by converting the encoded bit-stream data into parsed information stored in XML text files -Researching the feasibility of measuring the video quality on more than ten thousand videos -Developing new statistical methods for evaluating and validating metrics In the near future, JEG plans to launch call for proposals for hybrid measurement methods evaluated on the large scale database. This will include calls for evidence to both, the industry as to the academic world.

D. Research topics and cooperation

JEG-Hybrid's preliminary results discovered new scientific questions which should be addressed by future research. We would also like to encourage the scientific community to use tools and databases already created within JEG Hybrid. For each of the following research questions, a point of contact is provided on the VQEG JEG-Hybrid homepage [START_REF]Project Homepage of VQEG Joint Effort Group -Hybrid[END_REF]. If you have any question and you would like to contribute, please get in contact.

Existing data base We created a database of more than 10000 sequences. Those sequences cover numerous different compression parameters. In addition some full and non reference metrics were computed for all sequences within the database.

Combining multiple subjective and/or objective scores A large database calls for specific model estimation. Machine learning algorithms or data mining algorithms may be applied. For some of the sequences, subjective ground truth data obtained in a formal experiment is available.

Single FR measurement scope and prediction accuracy FR measurements computed for a large number of sequences does not necessary result in a similar quality estimation. A methodology for selecting a correct answer in case of a FR metrics disagreement is needed.

Parsed bitstream data in XML format One of JEG Hybrid's goal is to provide a standardized XML data format combining all transmission and compression information available in the transmitted bitstream for currently wide-spread coding standards, i.e. blockbased H.264, H.265. This greatly facilitates the construction of Hybrid Models applicable across protocols and coding standards. Initial tools for parsing network bitstreams and in RTP format and for parsing H.264 Annex-B files exist. Volunteers are welcome for the consideration of further transmission and multiplex protocols and H.265 video coding.

Including new databases We are very much interested in including more subjective experiments' results. Even if your experiment covers a small part of the hybrid methodology (for example compression only) your results are very helpful and welcome.

Including new metrics We are very much interested in including more metrics independent of their methodology FR, RR, NR or Hybrid. Please join the effort and test your code on the databases.

IV. CONCLUSIONS

Future developments of objective video quality measurement require algorithms with increasing complexity that need to be evaluated on large datasets in order to prove their performance and robustness in real world scenarios. While simple algorithms may provide some rapid answers, their performance is far lower than the performance of human evaluation of video quality. A collaborative approach allows for combining different research domains in order to achieve an optimized algorithm which continuously approaches the prediction performance of a group of human observers in a subjective assessment. Advantages of the joint effort have been detailed and possible future developments have been traced in this paper. Current and future VQEG-JEG activities have been summarized.

Figure 1 :

 1 Figure 1: Overview of processing steps to develop, train and verify a video quality assessment model

ACKNOWLEDGEMENTS

Lucjan Janowski's work was financed by The National Centre for Research and Development (NCBiR) as part of project no. SP/I/1/77065/10.


Guido Bacciagaluppi
email: g.bacciagaluppi@abdn.ac.uk

Non-equilibrium in Stochastic Mechanics

The notion of non-equilibrium, in the sense of a particle distribution other than ρ = |ψ| 2 , is imported into Nelson's stochastic mechanics, and described in terms of effective wavefunctions obeying non-linear equations. These techniques are applied to the discussion of non-locality in non-linear Schrödinger equations.

Introduction

The ideal of quantum mechanics as an emergent theory is well represented by Nelson's stochastic mechanics, which aims at recovering quantum mechanics from an underlying stochastic process in configuration space. More precisely, as we sketch in Section 2, Nelson starts from a time-reversible description of a diffusion process in configuration space, and then introduces some (time-symmetric) dynamical conditions on the process, leading to the Madelung equations for two real functions R and S, which are implied by the Schrödinger equation for ψ = Re iS/ .

The resulting theory has a number of similarities with de Broglie and Bohm's pilot-wave theory (although in the latter ψ is interpreted as a fundamental quantity). Indeed, particle trajectories in Nelson's stochastic mechanics can be intuitively thought of as de Broglie-Bohm trajectories with a superimposed white noise; and several concepts and techniques from pilot-wave theory can be easily and usefully imported into stochastic mechanics (although this is not usually discussed explicitly). One of these concepts, that of non-equilibrium, is the topic of the present paper.

Non-equilibrium in pilot-wave theory is defined as a situation in which the particle distribution is not equal to ρ = R 2 = |ψ| 2 . In Section 3 we discuss non-equilibrium in pilot-wave theory and how it makes sense even in stochastic mechanics.

1

In Section 4 we develop a formalism for describing non-equilibrium in stochastic mechanics in terms of effective wavefunctions and effective non-linear Schrödinger equations related to but different from the ones describing equilibrium.

In Section 5, this formalism is applied to the discussion of non-locality in the context of non-linear Schrödinger equations. Specifically, we shall find a class of entangled solutions of the equation

i ∂ψ ∂t = - 2 2m ∆ψ + V ψ + 2 m ∆|ψ| |ψ| ψ (1)
(with no interaction terms in V ), possessing a local hidden variables model. Some open questions conclude the paper (Section 6). One should note in particular that the theory of non-equilibrium in stochastic mechanics is potentially richer that the corresponding one in pilot-wave theory. The techniques developed in this paper, however, do not yet allow to discuss the genuinely novel cases (which, among other things, might be useful for the study of causally symmetric models of quantum mechanics).

Nelson's stochastic mechanics

In this section, we give a brief summary of Nelson's theory (largely after [START_REF] Nelson | Derivation of the Schrödinger equation from Newtonian mechanics[END_REF][START_REF] Davidson | A generalization of the Fényes-Nelson stochastic model of quantum mechanics[END_REF], with particular reference to aspects we shall need later. We consider only one particle for simplicity (the general case is analogous). 1Nelson (1966) considers the following stochastic differential equations,

dx(t) = b(x(t), t)dt + dw(t) , (2)
dx(t) = b * (x(t), t)dt + dw * (t) , (3)
where b(x(t), t) is the mean forward velocity and b * (x(t), t) the mean backward velocity of the particle, and w(t) and w(t) * are suitable Wiener processes with mean square fixed by the diffusion coefficient ν > 0.

These equations provide a time-symmetric kinematics for describing diffusion processes. In pilot-wave terminology they can be thought of as forward and backward stochastic 'guidance equations' for the particle.

One has

Dx = b and D * x = b * , (4)
with the forward and backward stochastic derivatives D and D * given by Dx(t)

x(t)=x = lim ε→0 + E t x(t + ε) -x(t) ε x(t) = x (5)
and

D * x(t) x(t)=x = lim ε→0 + E t x(t -ε) -x(t) -ε x(t) = x . ( 6
) (E t [ . |x(t) = x]
is the expectation value at time t conditional on the value of the process being x.2 ) Now take an arbitrary solution ρ of the forward Fokker-Planck (FP) equation

∂ρ ∂t = -div(bρ) + ν∆ρ . (7)
If we define an osmotic velocity

u ρ := ν ∇ρ ρ , (8)
and current velocity

v ρ := b -u ρ , (9)
ρ satisfies a continuity equation:

∂ρ ∂t = -div v ρ + ν ∇ρ ρ ρ + ν∆ρ = -div(v ρ ρ) . ( 10
)
If it is further the case that

b * = v ρ -u ρ , (11)
the same ρ satisfies also the analogous backward equations:

∂ρ ∂t = -div(b * ρ) -ν∆ρ = -div v ρ + ν ∇ρ ρ ρ -ν∆ρ = -div(v ρ ρ) . (12)
Note that if only (2) is given, the time reversal (3) is not unique. From any solution ρ of the forward FP equation, one could define a backward mean velocity b * = v ρ -u ρ and construct a time reversal of (2). (We shall use this in Section 4.) But if b * is given, the single-time distribution ρ of the process will need to solve both FP equations and will thus be unique, because the corresponding current and osmotic velocities are fixed by

1 2 (b + b * ) = v ρ and 1 2 (b -b * ) = u ρ = ν ∇ρ ρ , (13)
and ρ is normalised. As yet, b and b * (which define the dynamics of the process) are left unspecified. Nelson's aim is to find natural constraints that will yield the Madelung equations for R and S, with R 2 defining the distribution of the process and with

v = 1 m ∇S . ( 14
)
The latter can be justified for instance from the variational approach of [START_REF] Guerra | Quantization of dynamical systems and stochastic control theory[END_REF], further motivated in [START_REF] Nelson | Quantum Fluctuations[END_REF]. Alternatively, note that v being a gradient implies

Db = D * b * . (15)
This was pointed out already by de la Peña and Cetto (1982, eq. ( 16)).3 Thus we can also justify setting v equal to a gradient as a simple way of enforcing the time-symmetry condition (15) -which we now see is an identity in Nelson's stochastic mechanics. Given ( 14) and writing R 2 := ρ we obtain the first Madelung equation (the continuity equation),

∂R 2 ∂t = -div 1 m (∇S)R 2 = - 1 m (∆S)R 2 - 1 m ∇S∇R 2 . ( 16
)
By defining further the mean stochastic acceleration as

a := 1 2 (Db * + D * b) , (17)
and imposing 'Newton's law',

ma = -∇V , (18)
Nelson obtains also the Hamilton-Jacobi-Madelung (HJM) equation

∂S ∂t = - 1 2m (∇S) 2 -V + 2 2m ∆R R . (19)
As originally shown by Madelung (1926a,b), equations ( 16) and ( 19) are implied by Schrödinger's equation for ψ = Re iS/ . However, as pointed out by [START_REF] Wallstrom | Inequivalence between the Schrödinger equation and the Madelung hydrodynamic equations[END_REF], the converse is not true unless S has the right multivaluedness behaviour. 4More generally [START_REF] Davidson | A generalization of the Fényes-Nelson stochastic model of quantum mechanics[END_REF], we can define the mean acceleration as

α 2 (Db * + D * b) + β 2 (Db + D * b * ) , (20)
with α, β ≥ 0, α + β = 1. Imposing Newton's law with the acceleration defined by ( 20), 5 we now obtain

∂S ∂t = - 1 2m (∇S) 2 -V + 2mν 2 (α -β) ∆R R . (21)
As is easy to see, this HJM equation is implied by a generally non-linear Schrödinger equation:

i ∂ψ ∂t = - 2 2m ∆ψ + V ψ + 2 2m -2mν 2 (α -β) ∆|ψ| |ψ| ψ . (22)
We can distinguish three canonical cases:

(1) ν = 0 and α > β: one obtains the linear Schrödinger equation by choosing

ν = 2m √ α -β ; ( 23
) (2a) ν = 0 (deterministic case), or (2b) α = β: one has i ∂ψ ∂t = - 2 2m ∆ψ + V ψ + 2 2m ∆|ψ| |ψ| ψ , (24)
corresponding to the classical Hamilton-Jacobi equation and thus called the 'Schrödinger equation of classical mechanics' (Holland 1993, Sect. 2.6 and references therein);6

(3) ν = 0 and α < β: one can choose

ν = 2m √ β -α (25)
to obtain the non-linear equation

i ∂ψ ∂t = - 2 2m ∆ψ + V ψ + 2 m ∆|ψ| |ψ| ψ . ( 26
)
3 Pilot-wave theory and non-equilibrium

The pilot-wave theory by de [START_REF] De Broglie | La nouvelle dynamique des quanta Électrons et Photons: Rapports et Discussions du Cinquième Conseil de Physique Solvay[END_REF] and [START_REF] Bohm | A suggested interpretation of the quantum theory in terms of 'hidden' variables, I. and II[END_REF] describes deterministic particle trajectories in configuration space, where the particle velocity at any time is given by the 'guidance equation'

v = 1 m ∇S (27)
(or v i = 1 m ∇ i S for several particles), S being the phase of Schrödinger's wavefunction. If one considers an ensemble of particles guided by identical wavefunctions and distributed according to ρ = |ψ| 2 , then this relation will be preserved by the Schrödinger equation (a fact already known to de Broglie). Thus the theory describes very easily any diffraction and interference phenomena with material particles.

In order to account for measurements more general than simple detections, in particular of observables other than position, one has to include also the measuring apparatus into the description. The wavefunction of the combined system wlil develop into a superposition of components that are and -provided no macroscopic reinterference takes place -remain (approximately) nonoverlapping in the combined configuration space, and the system and apparatus will thus be trapped inside one of these components, the others being 'empty waves'. This component will be solely responsible for guiding the future motion of the combined system. Thus, the theory recovers an effective collapse of the wavefunction.

This analysis of measurement was Bohm's decisive contribution, and can be extended to more general cases of decoherence by the environment, thus arguably allowing pilot-wave theory to reproduce the classical regime of quantum mechanics as understood in the theory of decoherence (see the discussion in [START_REF] Bacciagaluppi | The role of decoherence in quantum theory The Stanford Encyclopedia of Philosophy[END_REF].

Since the current velocity v in stochastic mechanics has the same form as the particle velocity in de Broglie-Bohm theory, many of the results and techniques developed in the context of pilot-wave theory can be straightforwardly imported into stochastic mechanics, even though in the latter theory the wavefunction is not considered a fundamental quantity. 7 We shall now consider the notion of (quantum, or sub-quantum) non-equilibrium as discussed in pilot-wave theory, and suggest it should also be imported into stochastic mechanics.

In pilot-wave theory the (independently postulated) wavefunction ψ has two roles: it guides the particle via v = 1 m ∇S, and it defines the particle distribution via ρ = |ψ| 2 . At first, it may seem puzzling how the wavefunction can play these two roles simultaneously. Indeed, if one thinks of the wavefunction principally as defining the particle distribution (in a maybe hypothetical ensemble), it will be puzzling to say it also guides the motion of the (actual) particle. 8 If, however, one thinks of the wavefunction principally in its dynamical role as guiding the particle's motion, then it is perfectly natural to expect the particle distribution to depend on the dynamics -namely, if one understands the distribution as an equilibrium distribution. By the same token, non-equilibrium distributions, in which the statistical distribution in the ensemble is not equal to |ψ| 2 , become perfectly intelligible.

7 This is true in particular of the notion of effective collapse. This is an obvious move, but usually not considered explicitly in treatments of stochastic mechanics. I conjecture it would, among other things, resolve the puzzle of two-time correlations raised by [START_REF] Nelson | Afterword Diffusion[END_REF].

8 Indeed, this was precisely the reason for Schrödinger's dismissal of Bohm's 1952 theory, as expressed in a letter to Einstein (Schrödinger to Einstein, [after 18 but before 31 January] 1953, Archive for the History of Quantum Physics, microfilm no. 37, sect. 005-012 (manuscript) and 005-013 (carbon copy) (in German)) reproduced in the collection of Schrödinger's correspondence edited by von Meyenn (2011, vol. 2., pp. 673-675).

The notion of non-equilibrium in pilot-wave theory has been discussed by several authors over the years. Immediately following his initial papers, [START_REF] Bohm | Proof that probability density approaches |ψ| 2 in causal interpretation of the quantum theory[END_REF] discussed relaxation of the particle distribution from initial non-equilibrium towards |ψ| 2 via random external influences. The next year, [START_REF] Bohm | Model of the causal interpretation of quantum theory in terms of a fluid with irregular fluctuations[END_REF] hypothesised relaxation through an intrinsic random noise. This theory was discussed in more detail by Bohm and Hiley (1993, Chap. 9), and leads to trajectories identical to those of stochastic mechanics (!). The differences between this stochasic variant of pilot-wave theory and stochastic mechanics proper are, first and obviously, that Bohm and co-workers take the wavefunction to be fundamental, and, second, that they disregard the backward equations and thus the time-symmetric formulation of the theory. As a matter of fact, they are explicitly concerned with the relaxation of non-equilibrium distributions towards (future) equilibrium, and thus with much the same situation as we shall discuss in the next section, of particles evolving under Nelson's stochastic guidance equation (say, the forward one), but subject to a (say, initial) non-equilibrium constraint.

Further important work has focused on the analogy between |ψ| 2 in pilotwave theory and equilibrium in classical statistical mechanics. Major examples are Valentini's (1991a,b) sub-quantum H-theorem, the analysis of equilibrium and of how it relates to uncertainty by [START_REF] Dürr | Quantum equilibrium and the origin of absolute uncertainty[END_REF], various studies of relaxation behaviour, e.g. the recent work by [START_REF] Towler | Timescales for dynamical relaxation to the Born rule[END_REF] (see also references therein), and various studies of the consequences, signatures and possible residues of non-equilibrium by Valentini (see e.g. [START_REF] Valentini | Astrophysical and cosmological tests of quantum theory[END_REF][START_REF] Valentini | Inflationary cosmology as a probe of primordial quantum mechanics[END_REF]. The last type of investigations is particularly exciting, because it opens up the possibility of new empirical predictions.9

The situation is now perfectly analogous in Nelson's theory. The (independently postulated) vector fields b and b * have two roles: they guide the particle via v = 1 2 (b + b * ) and u = 1 2 (b -b * ), and they define the particle distribution via u = ν ∇ρ ρ . The guidance equations are law-like, and so is ρ, but this is unsurprising if we interpret it as the equilibrium distribution of the process, while the actual distribution ρ is contingent and might be different from ρ. Perhaps, however, I am labouring an obvious point. A stochastic process (as in Nelson's theory) is a probability measure over a space of trajectories, and actual frequencies neither are nor need always match the probabilities (see the more extensive discussion in Bacciagaluppi 2010).

Description of non-equilibrium

Suppose we impose a constraint on the particle distribution at some time t 0 ,

ρ(t 0 ) = |ψ(t 0 )| 2 . ( 28
)
We shall now describe the constrained process for t ≥ t 0 . (The same result can be extended to all times t.10 ) In this case, the time development of the distribution still satisfies the forward FP equation for t > t 0 ,

∂ ρ ∂t = -div(bρ) + ν∆ρ , (29)
but (always for t > t 0 ) it no longer satifies the backward equation,

∂ ρ ∂t = -div(b * ρ) -ν∆ρ . ( 30
)
Since ρ does satisfy the forward equation, however, we can define:

ũ := u ρ = ν ∇ρ ρ , (31) ṽ
:= b -ũ = v + u -ũ , (32) and b
* := ṽ -ũ = v + u -2ũ . ( 33
)
As discussed in Section 2, ρ will now satisfy a new effective backward FP equation:

∂ ρ ∂t = -div( b * ρ) -ν∆ρ . (34)
(Note that if v is a gradient, given that u and ũ are also gradients, so is ṽ.)

We can now define R := ρ ,

and

S := S + νm(log R 2 -log R2 ) = S + 2νm(log R -log R) , ( 36
) so that 1 m ∇ S = 1 m ∇S + ν ∇R 2 R 2 -ν ∇ R2 R2 , ( 37
)
and a new effective wavefunction ψ := Re i S/ . (Note also that S and S have the same multi-valuedness behaviour.) By construction, R2 satisfies the continuity equation with current velocity 1 m ∇ S:

∂ R2 ∂t = -div 1 m (∇ S) R2 = - 1 m (∆ S) R2 - 1 m ∇ S∇ R2 . ( 38
)
We shall now obtain also an effective HJM equation. Partial differentiation of (36) gives

∂ S ∂t = ∂S ∂t + νm 1 R 2 ∂R 2 ∂t - 1 R2 ∂ R2 ∂t . ( 39
)
And substituting ( 21), ( 16) and ( 38) into (39) yields

∂ S ∂t = - 1 2m (∇S) 2 -V + 2mν 2 (α -β) ∆R R -ν∆S -ν∇S ∇R 2 R 2 + ν∆ S + ν∇ S ∇ R2 R2 . ( 40
)
Using further the fact that

∆S = ∆ S -2νm ∆R R + 2νm (∇R) 2 R 2 + 2νm ∆ R R -2νm (∇ R) 2 R2 , (41)
and

∇S = ∇ S -νm ∇R 2 R 2 + νm ∇ R2 R2 , (42)
and hence also

(∇S) 2 =(∇ S) 2 -2νm∇ S ∇R 2 R 2 + 2νm∇ S ∇ R2 R2 + ν 2 m 2 ∇R 2 R 2 2 -2ν 2 m 2 ∇R 2 R 2 ∇ R2 R2 + ν 2 m 2 ∇ R2 R2 2 , ( 43
)
after suitable simplification we finally obtain

∂ S ∂t = - 1 2m (∇ S) 2 -V + 2mν 2 (α -β + 1) ∆R R -2mν 2 ∆ R R . ( 44
)
We see that the equation for S is not the same as that for S, but has acquired the extra term

2mν 2 (α -β + 1) ∆R R - ∆ R R . ( 45
)
And the corresponding effective Schrödinger equation for ψ is

i ∂ ψ ∂t = - 2 2m ∆ ψ+V ψ+2mν 2 (α-β+1) ∆|ψ| |ψ| ψ+ 2 2m + 2mν 2 ∆| ψ| | ψ| ψ . (46)
Thus, the non-equilibrium process can be equally well described as an equilibrium process with a different wavefunction and Schrödinger equation. In general this equation is not the same as the one satisfied by ψ (and it is not an autonomous equation for ψ, because of the presence of the 'old' quantum potential). In special cases, however, the extra term in the HJM equation vanishes, and ψ and ψ will both satisfy the same (autonomous) equation. This happens iff 2mν 2 (αβ + 1) = 0, i.e. Case (b) corresponds to the non-linear Schrödinger equation

i ∂ψ ∂t = - 2 2m ∆ψ + V ψ + 2 2m + 2mν 2 ∆|ψ| |ψ| ψ , (47)
and a non-equilibrium distribution in this case is simply an equilibrium distribution corresponding to a different wavefunction. For this special case thus, there is no privileged equilibrium distribution (and a case to be made that the wavefunction is to a large extent epistemic).

Non-linearity and non-locality

We now apply the results of the previous section to the discussion of non-locality in the context of theories with non-linear Schrödinger equations.

Rigorous results are available on this issue, specifically the well-known paper by [START_REF] Gisin | Stochastic quantum dynamics and relativity[END_REF], to which we shall return below. What these results, however, do not show is that one automatically has superluminal signalling whenever one has entangled states obeying non-linear equations. Indeed, we shall now construct a counterexample to this proposition.

First of all, note that for product wavefunctions, and only for product wavefunctions,

S(x 1 , x 2 ) = S 1 (x 1 ) + S 2 (x 2 ) (48) and R(x 1 , x 2 ) = R 1 (x 1 )R 2 (x 2 ) . (49)
In this case, and only in this case,

v i = 1 m i ∇ i S(x 1 , x 2 ) = 1 m i ∇ i S i (x i ) (50)
(i = 1, 2) and

u i = ν∇ i log R 2 (x 1 , x 2 ) = ν∇ i log R 2 i (x i ) . (51)
In particular, if the wavefunction has product form, the Nelsonian dynamics decomposes into independent dynamics for the two particles, and is thus local. Now take any Schrödinger equation of the form (22), i.e.

i ∂ψ ∂t = - 2 2m ∆ψ + V ψ + 2 2m -2mν 2 (α -β) ∆|ψ| |ψ| ψ . (52)
If there are no interaction terms in the potential, this equation preserves product wavefunctions. We can see this by noting that, if the potential decomposes as V 1 (x 1 ) + V 2 (x 2 ), the linear Schrödinger equation preserves products. But the quantum potential

2 2m -2mν 2 (α -β) ∆R R (53)
just is a potential that decomposes in this way:

∆R R = (∆ 1 + ∆ 2 )R 1 R 2 R 1 R 2 = ∆ 1 (R 1 R 2 ) R 1 R 2 + ∆ 2 (R 1 R 2 ) R 1 R 2 = ∆ 1 R 1 R 1 + ∆ 2 R 2 R 2 . ( 54
)
To simplify the construction of our counterexample, we now specialise to one of the invariant non-linear Schrödinger equations derived above, e.g.

i ∂ψ ∂t = - 2 2m ∆ψ + V ψ + 2 m ∆|ψ| |ψ| ψ (55) (case of α = 0, β = 1, ν = 2m ). 11
We take an initial product wavefunction ψ(x 1 , x 2 , t 0 ). From the above, it follows that the corresponding stochastic mechanics for this system is local (and similarly if the two particles separately interact locally with measurement apparatuses).

Now we impose classical correlations on the initial distribution:

ρ(t 0 ) = ρ1 (t 0 )ρ 2 (t 0 ) . ( 56
)
The resulting effective wavefunction ψ is entangled, because the functions

R(t 0 ) = ρ(t 0 ) (57)
and

S(t 0 ) = S(t 0 ) + log R(t 0 ) -log R(t 0 ) (58)
fail to decompose appropriately. The constrained process is thus described by an entangled ψ satisfying a non-linear Schrödinger equation. But, by construction, it is a local process with added classical correlations. This refutes the 'folk' claim that entanglement and non-linearity together imply superluminal signalling. What about Gisin's result, however? What [START_REF] Gisin | Stochastic quantum dynamics and relativity[END_REF] has shown (illustrated even more strikingly in [START_REF] Gisin | Weinberg's non-linear quantum mechanics and supraluminal communications[END_REF]) is that if a theory with a non-linear Schrödinger equation is such that it reproduces the usual phenomenology of collapse, then one can use non-linearity in conjunction with entanglement to obtain signalling.12 As Gisin is very well aware (private communicaton, Arolla, Switzerland, July 2009), this is a substantive assumption, and his theorem is not meant to apply more generally. Our counterexample thus shows (perhaps unsurprisingly) that the family of non-linear equations we have considered are not good candidates for reproducing the standard collapse phenomenology of quantum mechanics!

Open questions

To conclude, let us briefly return to the comparison between stochastic mechanics and pilot-wave theory, and to a few open questions that suggest themselves.

Specifically in the context of non-linear Schrödinger equations, we have seen that an apparent threat of non-locality arises from the entanglement of the effective wavefunction describing the non-equilibrium situation, i.e. from the fact that ṽ and ũ do not decompose in terms of the velocities of the single particles. Locality, however, is preserved because their sum b = v + u = ṽ + ũ decomposes appropriately. On the other hand, in a pilot-wave model of the same situation (wavefunction ψ satisfying the given non-linear equation) the motion of the particles depends only on ṽ. Thus, despite the fact that the particle distribution in both models would be the same, it would appear that the de Broglie-Bohm model of the situation would be more non-local than the Nelsonian model. 13Further, whether one considers linear or non-linear Schrödinger equations, non-equilibrium stochastic mechanics is a potentially richer theory (including a potentially richer range of novel phenomena) than non-equilibrium pilot-wave theory. This is so because the theory is stochastic. Indeed, in a deterministic theory such as pilot-wave theory one can only consider constraints at a single time: the distribution of trajectories is fixed by the particle distribution at a single instant. By contrast, in a stochastic theory any additional constraint on the distribution of trajectories merely subselects further from the ensemble of all possible trajectories that defines the stochastic process; thus, independent multi-time constraints make perfect sense. In particular, one can impose initial and final constraints on a process and might thereby obtain some qualitatively new phenomena (for instance genuine non-locality even if the unconstrained process is local). Such time-symmetrically constrained processes could provide good test cases for the ideas put forward in particular by [START_REF] Price | Time's Arrow and Archimedes' Point: New Directions for the Physics of Time[END_REF] about retrocausal hidden variables models for quantum mechanics. The techniques developed in this paper, however, cannot be straightforwardly extended to the case of multi-time constraints.

Pavia, Sydney and Utrecht and at Perimeter Institute. Finally, a warm thanks goes to the organisers of the Vienna conference, especially to Gerhard Grössing.

Appendix

We now show explicitly that requiring that v be a gradient is a simple sufficient condition for the time-symmetry condition (15), i.e.

Db = D * b * . ( 59
)
Note first of all that applying the stochastic derivatives D or D * to an arbitrary random variable f (x(t), t) one obtains Df (x(t), t)

x(t)=x = ∂ ∂t + b(x, t) • ∇ + ν∆ f (x, t) (60)
and

D * f (x(t), t) x(t)=x = ∂ ∂t + b * (x, t) • ∇ -ν∆ f (x, t) . (61)
By ( 60) and ( 61), condition ( 59) is equivalent to

∂u ∂t + (v • ∇)u + (u • ∇)v + ν∆v = 0 . (62)
Now, by the definition of u, we have

∂u ∂t = ∂ ∂t ν∇ ln ρ = ν∇ 1 ρ ∂ρ ∂t . ( 63
)
And using the continuity equation ( 10),

∂u ∂t = -ν∇ 1 ρ ∇ • (vρ) = -ν∇ 1 ρ (∇ • v)ρ + 1 ρ v • ∇ρ = = -ν∇ ∇ • v + v • ∇ ln ρ = -ν∇(∇ • v) -∇(v • u) . (64)
Therefore, (62) becomes

ν ∇(∇ • v) -∆v + ∇(v • u) -(v • ∇)u -(u • ∇)v = 0 . ( 65
)
From the identity

∇(a • b) = (a • ∇)b + (b • ∇)a + a × (∇ × b) + b × (∇ × a) , (66)
and the fact that

∇(∇ • v) = ∇ × (∇ × v) + ∆v , (67)
we can further transform (65) to

ν∇ × (∇ × v) + u × (∇ × v) + v × (∇ × u) = 0 . (68)
Finally, since u is a gradient, we obtain the equivalence of (59) and

ν∇ × (∇ × v) + u × (∇ × v) = 0 , (69)
for which v being a gradient is obviously a sufficient condition.

If we further impose (18) or some similar dynamical law, v being a gradient is presumably also a necessary condition for (59), since all time derivatives of (69) need to vanish, too, and they will generally depend on the arbitrary external potential V .14

  (a) if ν = 0 (but not if α = β, although in both cases the Schrödinger equation is the 'Schrödinger equation of classical mechanics'), or (b) if β = 1, α = 0 and ν is arbitrary.

For a much more comprehensive overview, see[START_REF] Nelson | Quantum Fluctuations[END_REF]. An introduction with an eye to conceptual questions is given by[START_REF] Bacciagaluppi | A conceptual introduction to Nelson's mechanics Endophysics[END_REF].

The definition of Dx(t) thus involves the forward transition probabilities from time t to times t + ε, while the definition of D * x(t) involves the backward transition probabilities from time t to times tε.

We give an explicit proof in the Appendix.

Possibly the most promising strategy for overcoming Wallstrom's objection is the suggestion by L.Smolin (personal communication, Waterloo, Ontario, July 2005) to exploit the fact that quantum mechanical ground states will be typically nodeless under appropriate conditions. Instead, the line of argument suggested in Smolin (2006, Sect. IV) does not go through; seeValentini (2010, footnote 3) andSchmelzer (2011, Sect. 2.5). For two recent approaches to overcoming Wallstrom's objection from the point of view of slightly different theories, see de la[START_REF] De La Peña | Origin and meaning of quantum nonlocality[END_REF]

Sect. 5.3) and the detailed proposal bySchmelzer (2011). 5 This is equivalent to the procedure by de la Peña (equation (25) in his (1969)) and de la[START_REF] De La Peña | Stochastic theory for classical and quantum mechanical systems Found[END_REF]. Note that in their treatment the case β = 1 is shown to correspond to (classical) Brownian motion.

Note that the description of ensemble motions provided in the case ν = 0 corresponds to the special case of classical mechanics in which an initial momentum field is fixed by the choice of the initial Hamilton-Jacobi function. For a careful discussion of these and related issues, seeHolland (1993, Chap. 2, esp. Sects. 2.5 and 2.6). The case α = β instead is physically very different, as we shall note again in Section 4.

Dürr and co-workers instead take it as an advantage of pilot-wave theory over classical statistical mechanics that there is no obvious non-equilibrium to be observed that would require explanation.

This can be done by carrying out the analogous proof for t ≤ t 0 , or -more economically -by exploiting the time-symmetry properties of the resulting equations.

With slight modifications, the argument will cover all the non-linear Schrödinger equations (22).

Imagine for simplicity an EPR setup with a singlet state. If Alice by measuring spin in direction x or y can create on Bob's side equal up-down mixtures of spin eigenstates in direction x or y, respectively, Bob can then use the non-linearity of the evolution to discriminate between these two different mixtures. Thus, Alice can signal arbitrarily fast to Bob using an ensemble of sufficiently separated EPR pairs.

This also suggests the question of whether entangled wavefunctions with decomposable1 m ∇S + ν ∇R 2R 2 are somehow less intrinsically non-local than generic ones.

The condition Db * = D * b is similarly equivalent to ∂u ∂t + (v • ∇)u -(u • ∇)v -ν∆v = 0, which assuming (59) simplifies to ν∆v + (u • ∇)v = 0.

Acknowledgements

I would like to thank Elise Crull, Thomas Durt, Chris Timpson, Sonja Smets, Rob Spekkens and Antony Valentini for correspondence, discussion and other precious help, and Rafael Sorkin for very perceptive comments on a previous talk I gave on this material. I would also like to thank Mark Davidson, Edward Nelson, Ángel Sanz and other members of the audience at Vienna for questions, comments and correspondence, in particular Ana Maria Cetto. Previous versions of this talk were given at the Universities of Aberdeen, Firenze, Oxford,