Astrid Jourdan
email: astrid.jourdan@eisti.fr

Jessica Franco
email: jessica.franco@total.com

A new criterion based on Kullback-Leibler information for space filling designs

Keywords:

Experimental designs are tools which can drastically reduce the number of simulations required by timeconsuming computer codes. One strategy for selecting the values of the inputs, whose response is to be observed, is to choose these values so that they are spread evenly throughout the experimental region, according to "space filling designs". In this article, we suggest a new criterion based on the Kullback-Leibler information for design construction. The aim is to minimize the difference between the empirical distribution of the design points and the uniform distribution which is equivalent to maximizing the Shannon entropy. The entropy is estimated by a Monte Carlo method, where the density function is replaced with its kernel density estimator or by using the nearest neighbor distances.

Introduction

For many scientific phenomena, physical experimentation is very expensive, time-consuming, or impossible. Engineers and scientists have been in the forefront of developing mathematical models and numerical solutions to describe physical systems. As models become more sophisticated, running times increase rapidly, and computer experiments are necessary to characterize physical phenomena. values of the governing input parameters (input factors) of the computer model. The outputs, y(x 1 ),...,y(x n ), at a given set of inputs x 1 ,...,x n , are used to provide a predictor of the simulated response. In this paper, we suppose that no information is available about the relationship between the response and the input factors (exploratory phase). Hence, the designs should allow one to adapt a variety of statistical models (kriging models, neural network models, ...). One strategy consists of selecting the set of inputs in order to fill up the experimental region in a uniform fashion. These kinds of designs are called "space filling designs" and have been widely investigated in the past decade (e.g. [START_REF] Koehler | Computer Experiments[END_REF] and [START_REF] Franco | Planification d'expériences numériques en phase exploratoire pour des codes de calculs simulant des phénomènes complexes[END_REF]). They provide information about all parts of the experimental region and enable one to spot possible irregularities of the computer response within the experimental region.

In this paper, we propose a new criterion based on the Kullback-Leibler information (KL information) for design construction. As with the discrepancy method, the KL information measures the difference between the empirical distribution of the design points and the uniform distribution. The idea is to minimize this difference by using an exchange algorithm.

In section 2, the properties of the KL information are presented, especially the link with the Shannon entropy in the case of the uniform distribution. In sections 3 and 4, two methods of the entropy estimation are investigated. The first one is a Monte Carlo method where the density function is replaced with its Gaussian kernel estimate. The second one avoids the density estimation by using the nearest neighbor distances. In section 4, we compare the new designs with the most commonly used space filling designs.

Kullback-Leibler information

Suppose that the design points X 1 ,...,X n , are n independent observations of the random vector X=(X 1 ,...,X d ) with absolutely continuous density function f concentrated on the unit cube [0,1] d . The aim is to select the design points in such way as to have the density function "close" to the uniform density function. The Kullback-Leibler (KL) information measures the difference between two density functions f and g supported by E,

∫       = E dx ) x ( ) x ( ln ) x ( ) , ( I g f f g f .
Note that the KL information is not a metric function since the triangle inequality is not satisfied and the information is not symmetrical. Hence, the density functions can not be exchanged and f is the "design" density and g is the "target" density, i.e the uniform density. However, the KL information is

always non-negative, 0 ) , ( I ≥ g f , with 0 ) , ( I = g f
if and only if f=g. The aim is then to minimize the KL information so as to make f converge towards the uniform density.

If g is the uniform density function, then the KL information becomes

( ) [ ] f f f f H dx ) x ( ln ) x ( ) ( I - = = ∫
, where H[f] denotes the Shannon entropy. Therefore, minimizing the KL information amounts to maximizing the entropy.

We recognize the definition of the maximum entropy sampling commonly used in computer experiments [START_REF] Shewry | Maximum Entropy Sampling[END_REF], [START_REF] Currin | A bayesian approach to the design and analysis of computer experiments[END_REF]). The difference lies in the fact that the entropy used in this paper is independent of any metamodel. The aim of maximizing the entropy is not the gain in information on the model parameters. On the other hand, for f concentrated on [0,1] d , one always has H(f)≤0 and the maximum value of H(f), zero, being uniquely attained by the uniform density. This latter property confirms our choice to minimize the KL information. Note that [START_REF] Dudewicz | Entropy-Based tests of uniformity[END_REF] proposed an entropy-based test of uniformity but only for d=1. In order to avoid any confusion with the maximum entropy designs mentioned above, the designs constructed in this paper are called «minimal KL information designs ».

Finally, we use an exchange algorithm that maximizes the entropy for the design construction. The main point of the method is the estimation of the Shannon entropy. [START_REF] Beirlant | Nonparametric entropy estimation : an overview[END_REF] gave an overview of several techniques of estimation. Two methods have been selected. The first one is a Monte Carlo method where the density function is replaced with its Gaussian kernel estimate. The second one is based on the nearest neighbor distances.

Estimate of entropy based on the Monte Carlo method (MC)

Given that the entropy can be written as follows,

( ) [ ] ) X ( ln E ) X ( H f - =
where E denotes the expectation for f, the Monte Carlo method provides an unbiased and consistent estimate of the entropy, [START_REF] Ahmad | A nonparametric estimation of the entropy for absolutely continous distributions[END_REF] proposed to replace the unknown density function f with its kernel density estimate [START_REF] Silverman | Density estimation for statistics and data analysis[END_REF][START_REF] Scott | Multivariate Density Estimation : Theory, practice and visualization[END_REF],

∑ = - = n 1 i i ) X ( ln n 1 ) X ( H ˆf .
∑ =         - = n 1 i i d h X x nh 1 ) x ( f ˆK , ∀x∈[0,1] d
The quality of a kernel estimate depends essentially on the value of its bandwidth h (smoothing parameter). In our application, the bandwidth is chosen using Scott's rule where the standard deviation estimates are replaced with the standard deviation of the uniform distribution on [0,1],

) 4 d /( 1 n 1 12 1 h ˆ+ =
The choice of kernel function K is much less important for the behavior of the estimate than the choice of h. Most of the common kernels (uniform, Epancehnikov, triangle, ...) have a bounded support (unit sphere), so that, in our application, the probability that the kernel values are not zero is extremely small. So, the chosen kernel function K is the multivariate Gaussian distribution,

      - π = - 2 2 d 2 / d z s 2 1 exp s ) 2 ( ) z ( K ,
where s 2 is the variance of the uniform distribution on [0,1] multiplied by the dimension d, 12 d s 2 = .

Joe (1989) obtained asymptotic bias and variance terms for the estimator described above. The bias depends naturally on the size n and the dimension d but also on the bandwidth h (fixed during the exchange algorithm). Moreover, he pointed out that the sample size needed for good estimates increases rapidly with the dimension (fig 1a). Hence, in the context of experimental design, the entropy estimation will be approximate since the size n is small. However, the objective is not to compute an accurate approximation of the entropy. And, despite the low accuracy, the exchange algorithm converges rapidly (fig 1b) toward designs with the expected properties (filling up the space) (fig. 3).

Estimate of entropy based on nearest neighbor distances (NN)

The performance of the Monte Carlo methods generally reduces as dimensionality increases.

Moreover, the density function needs to be estimated for each exchange in the algorithm. Hence, the design construction with the previous method becomes time-consuming as the sample size increases.

Kozachenko and [START_REF] Kosachenko | Sample estimate of entropy of a random vector[END_REF] proposed estimating the entropy by

) 1 n ln( C ) 1 2 / d ( ln ln n d ) ( H ˆE 2 / d n 1 i i - + +       + Γ π + ρ = ∑ = f
where C E ≈0.5772 is the Euler constant, Γ is the Gamma function and ρ i is the nearest neighbor distance of X i and the other X j ,

j i n j 1 , i j i X X min - = ρ ≤ ≤ ≠ .
They proved the asymptotic unbiasedness and the consistency of the estimator. Figure 2a illustrates the consistency and the bias since the estimated values are positive whereas the entropy is negative.

Figure 2b shows that the exchange algorithm converges more slowly than the previous one. Finally, even if the density function estimation is avoided, this method of construction requires nearly the same running-time.

Design comparison

The aim of this section is to study the properties of the designs constructed by the methods described above, and to compare them with the usual space filling designs.

In an exchange algorithm, the design depends more or less on the initial setting. Consequently, several initializations are run, and the best design is selected so as to reduce the dependence. However, in this section, several designs are built from only one initialization in order to study this dependence.

Improvement of the initial setting

The aim of the construction methods is to fill up the parameter space in a uniform fashion. It can be seen on figure 6 that the objective is reached for the two methods. Whatever the initialization, the algorithm converges toward designs with the same characteristics : • The points lie on the edge of the factor space but also in the interior like a scrambled regular grid (quasi-periodical distribution). Such distribution assures the points are spread evenly in the unit cube.

• Contrary to a regular grid, many levels are tested for each parameter.

• For small size n, the minimal KL designs will generally lie on the exterior of [0,1] d and fill in the interior as n becomes large. Koelher & Owen (1996) pointed out the same behavior for maximin designs.

The MST criterion ( § 5.3.) shows that the designs keep these characteristics in dimension d≥2.

Comparison with traditional space filling designs

Criteria

It is impossible to study visually the point distribution when d≥2. The analysis requires criteria to assess the uniformity and/or the distribution throughout the unit cube. The criteria selected in this paper are intrinsic, that is, they are independent of any kind of metamodel.

• The cover measure (Cov) calculates the difference between the design and a regular grid [START_REF] Gunzburger | Uniformity measures for point sample in hypercubes[END_REF]). The goal is to minimize the criterion in order to be close to a regular grid and, in so doing, to guarantee that the points fill up the space. [START_REF] Johnson | Minimax and maximin distance design[END_REF] proposed the maximin and minimax distances to spread the points evenly in the unit cube. The maximin criterion (Mindist) maximizes the minimal distance between the points.

• Contrary to the previous criterion, the discrepancy is not based on the distance between the design points. This criterion measures the difference between the empirical cumulative distribution and the uniform cumulative distribution [START_REF] Niederreiter | Point sets and sequences with small discrepancy[END_REF]). Two kinds of discrepancy are selected, the L2-discrepancy (DL2) and the centered L2-discrepancy (DC2).

Traditional space filling designs

The designs selected for the comparison are the most common designs used for computer experiments. [START_REF] Koehler | Computer Experiments[END_REF] and [START_REF] Franco | Planification d'expériences numériques en phase exploratoire pour des codes de calculs simulant des phénomènes complexes[END_REF] proposed an overview of space filling designs.

• Random : designs built by simple applications of the random function.

• LH : latin hypercube designs (without optimization).

• Discr : sequences with small discrepancy (Sobol, Niederreiter, Hammersley, Halton) [START_REF] Niederreiter | Point sets and sequences with small discrepancy[END_REF]).

• Dmax : designs which maximize the determinant of a covariance matrix [START_REF] Shewry | Maximum Entropy Sampling[END_REF], [START_REF] Currin | A bayesian approach to the design and analysis of computer experiments[END_REF]). These designs are commonly used when the metamodel is a kriging surface.

• Strauss : designs created using a Strauss procedure, which considers the repulsion between two points, in such a way as to maximize filling the space of the parameters [START_REF] Franco | Planification d'expériences numériques en phase exploratoire pour des codes de calculs simulant des phénomènes complexes[END_REF].

• Maximin : optimal designs based on the maximin distance.

• MC : minimal KL information designs constructed by the Monte Carlo method.

• NN : minimal KL information designs constructed by the nearest neighbor method.

The minimal KL designs provide the best results except for the discrepancy (fig. 4 and fig. 5). This conclusion is surprising since the two criteria, discrepancy and KL information, minimize the difference between the point distribution and the uniform distribution, whereas the cover measure and the maximin criteria are based on the distance between the design points. This result may be explained by the difficulty in evaluating the discrepancy.

Figure 5 shows that the design qualities do not decrease when the dimension d increases. The minimal KL information designs compete with maximin designs (even concerning the maximin criteria) which are commonly used in the exploratory phase.

MST criterion

The MST (Minimal Spanning Tree) criterion proposed by [START_REF] Franco | Planification d'expériences numériques en phase exploratoire pour des codes de calculs simulant des phénomènes complexes[END_REF] (following the studies of [START_REF] Wallet | Comparison of spatial point patterns and processes characterization methods[END_REF] classifies the designs according to their structure: grid, cluster,… . The classification depends on the mean and standard deviation of the edge lengths of the MST associated with the design under study (fig. 6). The designs of the region of quasi-periodicity represent the best compromise between regular grid (space filling) and uncertain distribution (uniformity). The designs of minimal KL information also give the best results in this case (figures 7 and 8) with a clear demarcation in dimension 3. This result confirms the remarks on the characteristics of the designs of dimension 2 ( §5.1). The designs constructed using the Monte Carlo method with Gaussian kernels are still higher performing than those constructed by nearest neighbor method.

Mean

Standard deviation

Quasi-periodical distribution

Cluster models

Gradient distribution

Random distribution

Points ordered Mean

Standard deviation

Quasi-periodical distribution

Quasi-periodical distribution

Cluster models

Cluster models

Gradient distribution

Random distribution

Points ordered

Fig. 6. Empirical distribution created using MST criteria

The classification depends on the mean and standard deviation of the edge lengths of the MST associated with the design under study. The designs of the region of quasi-periodicity represent the best compromise between regular grid (space filling) and uncertain distribution (uniformity). The designs of minimal KL information also give the best results in this case (fig. 7 and fig. 8) with a clear demarcation in dimension 3. This result confirms the remarks on the characteristics of the designs of dimension 2 ( §5.1). The designs constructed using the Monte Carlo method with Gaussian kernels are still higher performing than those constructed by nearest neighbor method.

Conclusion

The aim of the criteria presented in this paper is to minimize the difference between the distribution of the design points and the uniform one by using the KL information. The criterion is equivalent to maximizing the Shannon entropy, and the design construction is then based on the entropy estimation.

Two methods have been investigated, and they provide roughly the same results. The first one is a Monte

Carlo method where the density function is replaced with its Gaussian kernel estimation. The second one avoids the density estimation by using the nearest neighbor distances. The tests show that minimal KL information designs achieve the main goal, which is to spread the design points evenly throughout the experimental region. They are indisputably the best designs with regard to the usual criteria, even in high dimensions. They compete with maximin designs which are widely used in the exploratory phase.

Despite the context of experimental designs, which is far-removed from asymptotic conditions, a criterion based on the Shannon entropy estimation provides very good results. However, the Monte Carlo method performs less well as the dimension increases and the design size decreases. Moreover, the Monte Carlo method requires running time, since the density has to be estimated for each exchange. Leonenko et al. pointed out that nearest-neighbor methods show promise when the dimension is large, but the tests show a slow convergence of the exchange algorithm. One idea consists of replacing the Shannon entropy by another one, for instance, the Rényi entropy (estimated by minimal spanning trees, [START_REF] Hero | Applications of entropic spanning graphs[END_REF], or the Tsallis entropy (analytic expression when the density is replaced with its kernel estimate, [START_REF] Bettinger | Design of experiments for response diversity[END_REF].

Fig. 1 .

 1 Fig. 1. Convergence of the Monte Carlo method (d=3)

Fig. 2 .

 2 Fig. 2. Convergence of the nearest neighbor method (d=3)

Fig. 3 .

 3 Fig. 3. Results of the exchange algorithm for designs of size 20 in dimension 2

Fig. 4 .

 4 Fig. 4. Criterion for 20 designs of size 30 with dimension 3

Fig. 7 .Fig. 8 .

 78 Fig. 7. Representation of the MST criterion calculated on 20 designs with 30 points in dimension 3

Acknowledgment

This work was supported by Total. We are grateful for the encouragements from Bernard Corre.


Chao Gao

Chang-Bing Tang

Yan Zhang

Lin Wang
email: lin.wang@pasteur.fr

Zhao Zhang
email: zhaozhang@zjnu.cn

What can AI learn from bionic algorithms?

Keywords: Physarum-based algorithms, Artificial intelligence, Optimization, Bionic algorithms

. Surprisingly, this is a self-organized process without centralized control. Gao et al. [11] reviewed the latest progress of Physarum-based models and computations. Through a systematic review of publications in the Web of Science, they constructed a network of scientific citations to overview the hot research areas. Their major interests lie in the computational models inspired by the two fundamental features of Physarum's foraging behavior, i.e., extension and retraction, which are applied as the morphology, taxis and positive feedback dynamics in the top-down and bottom-up modeling techniques. They also surveyed some real-world applications based on the core features of Physarum for solving difficult computational problems. Furthermore, they outlined recent advancements in bionic algorithms that are grounded in the bio-intelligence of Physarum polycephalum.

. Recent decades, AI has accumulated substantial achievements in

supervised learning via synthesis with neuroscience, psychology and cognitive science [START_REF] Russell | Artificial intelligence: a modern approach[END_REF]. However, unsupervised learning, as the main direction towards the "real" AI, is still a hard problem. Bionic algorithms may inspire useful tools to solving such problems.

For example, social insects such as ants and bees are typical organisms with distributed intelligence [START_REF] Bonabeau | Self-organization in social insects[END_REF]. Their population survival relies on the collective predation, the observation of which leads to the establishment of Ant Colony Algorithms (ACA) that was first proposed by Marco Dorigo [START_REF] Dorigo | Learning and Natural Algorithms[END_REF]. For path optimization problems such as "finding a shortest path between two points on a polyhedron", the strategy of ACA is to continuously release "ants" from one of these two points and let them do random walks on the polyhedron. Similar to the real ants that excrete pheromones for triggering social response, the simulated "ants" record their positions to guide others in subsequent simulation iterations. In fact, the development of swarm-intelligence algorithms such as the ACA echoes the progress in modern statistical and machine learning (e.g. Markov chain Monte Carlo and Hamiltonian Monte Carlo [START_REF] Wang | Characterizing the dynamics underlying global spread of epidemics[END_REF][START_REF] Li | Reconstruction of stochastic temporal networks through diffusive arrival times[END_REF][START_REF] Duane | Hybrid Monte Carlo[END_REF][START_REF] Brooks | Handbook of Markov Chain Monte Carlo[END_REF]).

Let us revisit Physarum polycephalum reviewed by Gao et al. [START_REF] Gao | Does being multi-headed make you better at solving problems? A survey of Physarum-based models and computations[END_REF]. An interesting case study [START_REF] Liu | A Physarum-inspired vacant-particle model with shrinkage for transport network design[END_REF] has shown the potential of such unicellular organism in designing planar transportation networks. For example, if food sources are placed according to the spatial location of multiple cities on the map of a country, Physarum can connect the food resources by reshaping its body as a network of growth routes, which resembles the real-world roads and railways networks of that country. This is because of the strong deformability of Physarum, i.e. the dramatic local deformation of cell membrane in response to the chemical source concentration gradient of food resources. Physarum keeps on growing along the directions at which it perceives a higher food chemical concentration after deforming, while its body remains the same or retracts along the other directions with lower perceived concentration. This bio-intelligent strategy is conceptually similar to many other likelihood-based searching algorithms.

Although AI-based research and development have been highly successful in supervised learning, unsupervised learning is still running on a bumpy and winding road [START_REF] Ganguli | The intertwined quest for understanding biological intelligence and creating artificial intelligence[END_REF]. Organisms such as Physarum polycephalum and social insects have shown the advantages in self-organization, adaptive learning, distributed and dynamical optimization and control, etc. Bionic algorithms built by deciphering the mechanisms underlying biological intelligence will provide cues for designing unsupervised learning algorithms.

Acknowledgement

This work was partly supported by the Key Projects of National Natural Science Foundation of China (No. 61751303), the National Natural Science Foundation of China (No. 11475003, 71871233, 11771013, 11531011) and the Zhejiang Provincial Natural Science Foundation of China (Nos. LY16F030002, LD19A010001). The related research was conducted in part using the research computing facilities and advisory services offered by Information Technology Services, The University of Hong Kong.


Y Le Cozler

C Allain

L Depuille

A Caillot

J M Delouard

L Delattre

T Luginbuhl

P Faverdin

Volume and surface area of Holstein dairy cows calculated from complete 3D shapes acquired using a high-precision scanning system: interest for body weight estimation

Keywords: volume, area, cows, sensors, 3D images, weight estimation

Three-dimensional (3D) imaging is a solution for monitoring morphology and growth of dairy cows, but it can also estimate indicators such as body volume, surface area and body weight. A 3D full-body scanning device was used to scan 64 lactating Holstein cows from March-June 2018. The cows were individually and automatically weighed at a static weighing station (mean ± standard deviation = 673 ± 65 kg).

These measured weights were compared to those predicted from regression models based on volume, area or morphological traits determined from 177 3D images.

Since some images were truncated due to cow movement or technical problems, we developed additional regression models to reconstruct total volume or area. The accuracy of volume and area measurements was first tested on an inert cylindrical form (coefficients of variation (CVs) < 0.72%). The CVs for repeatability and reproducibility of the method of calculating volume and area from truncated images

Introduction

New technologies based on image analysis are successfully used to improve management of most types of animal production. They have been developed to detect lameness in cows [START_REF] Van Hertem | Automatic lameness detection based on consecutive 3D-video recordings[END_REF]Zhao et al., 2018) or to measure body parameters such as body condition score (BCS; [START_REF] Halachmi | Cow body shape and automation of condition scoring[END_REF][START_REF] Fischer | Rear shape in 3 dimensions summarized by principal component analysis is a good predictor of body condition score in Holstein dairy cows[END_REF]Sploliansky et al., 2016). Two-dimensional (2D) image approaches used in the past [START_REF] Marchant | Extending the snake image processing algorithm for outlining pigs in scenes[END_REF]Schofield et al., 1998) were less effective due to the lack of a third dimension, distortion problems, the complexity of calibration and the need for multiple cameras and three-dimensional (3D) reconstruction models.

Inexpensive 3D cameras (< 300 €) are now available, which increases the interest in 3D approaches. They have been used to analyze BCS in dairy cattle, using either a fixed [START_REF] Fischer | Rear shape in 3 dimensions summarized by principal component analysis is a good predictor of body condition score in Holstein dairy cows[END_REF] or mobile [START_REF] Kuzuhara | A preliminarily study for predicting body weight and milk properties in lactating Holstein cows using a three-dimensional camera system[END_REF] device. Other authors [START_REF] Negretti | Determination of live weight and body condition score in lactating Mediterranean buffalo by Visual Image Analysis[END_REF][START_REF] Buranakarl | Estimation of body weight and body surface area in swamp buffaloes using visual image analysis[END_REF][START_REF] Guo | LSSA_CAU: an interactive 3d point clouds analysis software for body measurement of livestock with similar forms of cows and pigs[END_REF]Pezzuolo et al., 2018) have also developed and used 3D imaging technologies for a wider variety of livestock. In most cases, animals were measured under laboratory conditions (heavy equipment and landmarks on animals to guide computer measurements), which provided images of the entire body of restrained animals or portions of the body of unrestrained animals.

Equipment can often be adapted to farm conditions and recent literature describes devices used to estimate body weight (BW) of pigs in barns (Pezzuolo et al., 2018;Wang et al., 2018) and broilers in houses [START_REF] Mortensen | Weight prediction of broiler chickens using 3D computer vision[END_REF]. Inexpensive and portable equipment based on the Microsoft Kinect® v1 sensor have often been used; however, authors concluded that most methods needed additional technical development to acquire and extract data automatically. Using 3D image technology to estimate animal BW is also of interest, since it reduces risky situations for both animals and humans and can provide frequent records. This technology is also suitable for collecting information about animal volume and area. For example, to examine heat stress in an increasingly warm environment, it would be useful to focus on area, since evaporative heat loss in most mammals occurs via perspiration from the skin and respiration (Berman, 2011).

The Morpho3D scanning device was developed to record and monitor morphological traits [START_REF] Cozler | High precision scanning system for complete 3D cow body shape imaging and analyzing morphological traits[END_REF]. In the present study, we tested the hypothesis that it could also determine volume and area of dairy cows accurately and thus their BW.

Materials and methods

2.1. Animals: Data were obtained from the INRA-UMR PEGASE experimental dairy station at Méjusseaume, Le Rheu, in western France (48°11' N; 1°71' W; elevation 35 m). The study involved 64 Holstein dairy cows with a mean BW of 673 kg (standard deviation (SD) = 65 kg) and parity ranging from 1-5. After each milking (twice a day), cows were individually and automatically weighed at a static weighing station (DeLaval France, Elancourt, France) at the milking parlor's exit. Mean BCS, based on the French scoring scale of 0-5 [START_REF] Bazin | Grille de notation de l'état d'engraissement des vaches pie-noires[END_REF], was 2.05 (± 0.25). One month, after weighing, cows were scanned with the Morpho3D device (section 2.3). Data were collected from March-June 2018, yielding 289 3D images. However, due to abnormal behavior of the cows or excessive light in May (which generated artifacts on images), cloud points were considered of too low quality to be used for 3D images reconstruction (Fig. 1). As a result, only 177 3D images were used to estimate morphological traits, volume and area. To limit image quality problems due to animal movements and excess light, the Morpho3D device was then partially covered with an opaque tarpaulin and the installation of a feed fence limited the cow movement.

Cylinder model:

A cylinder model, on which measurements could be performed easily and precisely, was scanned to determine the accuracy of measurements and calculations of volumes and areas from 3D images. The cylinder model was considered the reference object, and its volume and area were measured manually and calculated.

Morphological acquisition system:

3D images of the cows were acquired automatically using Morpho3D, a sliding acquisition system, located near the weighing station (Fig. 2). Briefly, the system had five cameras, each paired with a laser projector. The VGA image resolution of each camera was 640 × 480 pixels. Two cameras were fixed at 0.40 and 1.77 m above ground level, respectively, on each side of the portal. The fifth camera was fixed to the middle of the top of the portal (3.00 m above ground level). The portal moved at a mean speed of 0.5 m.s -1 from back to front (phase 1) and returned to its initial position at a mean speed of 0.3 m.s -1 (phase 2). Each camera took 80 images per second only during phase 1, yielding a total of 2,000 images. The cameras were attached to the sliding portal (l = 5.00 m; w = 2.58 m; h = 3.00 m; Fig. 2). See Le [START_REF] Cozler | High precision scanning system for complete 3D cow body shape imaging and analyzing morphological traits[END_REF] for additional details about Morpho3D. Reconstructing an animal in 3D is a generalization of laser triangulation. Each Morpho3D laser generates a vertical plane, whose intersection with the object appears as a stripe in each image, yielding more points per image. Knowing the equation of the plane in the camera frame allows the 3D position of the points in each stripe in the camera frame to be determined by intersecting the plane with the ray passing through the origin of the camera and the points observed in the image plane.

Sliding the system's portal horizontally scans the laser plane over the entire object, yielding a point cloud consisting of several "slices" of 3D points (1 slice = 1 image).

These point slices are aggregated based on the location of the system's portal. The cameras were calibrated using a black-and-white checkerboard placed in different locations in the Morpho3D device. Each camera-laser pair was calibrated individually and then two-by-two to calibrate all five cameras collectively. Images of the laser stripes projected onto the cow were captured by their corresponding camera and sent to a computer to reconstruct the cows' 3D information. First, images from each camera were processed to build separate point clouds using calibration information and the speed of the portal. A 3D reconstruction of the entire cow was generated by recording and merging the multiple 3D point clouds from the five camera-laser pairs. This resulted in a single point cloud representation of the entire cow.

Two camera filters were used during image capture: a physical filter centered on the laser wavelength to reduce ambient light and increase the contrast of laser stripes in each image and a software filter to prevent recording of undesirable points too far from the camera. Undesired objects were deleted during a cleaning process using Metrux2α® software (3D Ouest, Lannion, France). This step ensured that the point cloud is a sampling of a smooth surface on which surface normal vectors can be estimated. Finally, surface normals were estimated from the point cloud, and a screened Poisson surface-reconstruction algorithm was applied to build a triangulated mesh [START_REF] Kazhdan | Screened Poisson surface reconstruction[END_REF] using Meshlab® open-source software [START_REF] Cignoni | MeshLab: an Open-Source Mesh Processing Tool[END_REF].

Calculating volume and area

Principles of calculating volume and area

Metrux2α® was used to perform linear measurements and estimate morphological traits, volume and area. Le [START_REF] Cozler | High precision scanning system for complete 3D cow body shape imaging and analyzing morphological traits[END_REF] describe the linear measurements and their validation. Volume and area were automatically calculated by algorithms integrated into Metrux2α®, similar to the method used for morphological traits. Area was calculated from a triangular 3D mesh created from a list of interconnected 3D vertices. Each vertex shared by several triangles was then indexed to identify its position in relation to the other vertices to create a list of triplets of vertices that form the triangles. From this list, the area of each triangle was calculated. The total area of the cow equaled the sum of the areas of the triangles of the 3D mesh.

Volume was calculated using the method of [START_REF] Mirtich | Fast and accurate computation of polyhedral mass properties[END_REF], based on divergence theorem and Green's theorem. In it, volume equals the volumetric integral of the characteristic function of the object, but this integral cannot be calculated directly for a complex volume. Instead, faces and points, from which calculations can be made, must be introduced. The divergence theorem transforms an integral of the volume into an integral of the area, as follows

∇. = .
where v is the volume, δv is the surface around the volume, ∇ is the nabla operator which characterizes the divergence, F is the vector field of the volume and n is the normal to the surface oriented towards the outside.

This divergence theorem is used on a vector field whose divergence is equal to 1 in order to find the characteristic function of the object. In a second step, the surface around the volume is calculated using Green's theorem, which transforms the integral of the area of the triangles of the 3D mesh into an integral of the segments, themselves calculated from the 2D coordinates of the vertices. This step requires transforming the 3D coordinates into local 2D coordinates. The volume and area of the 3D image are then calculated from the vertices of the triangular 3D mesh.

Application to the cylinder model

We first applied the method to calculate the volume and area of the cylinder model.

After scanning, image acquisition and cleaning, a single point-cloud representation of the cylinder was obtained and a triangulated mesh was built (Fig. 3). This process was repeated 10 times to acquire 10 images. We then compared the volumes and areas calculated by Metrux2α® to those calculated by Meshlab®, since the latter is widely used for this purpose. We also compared the volume and area measured manually to those calculated by Metrux2α®.

Application to living animals

Due to animal movement, not all images could be used in full. Although the cow's body was digitized properly, its head usually moved, which sometimes distorted its To decrease cow movement without affecting the lasers, we provided concentrate feed in a rectangular bowl, either opaque plastic or transparent glass. The presence of the opaque plastic bowl changed the shape of the cow's volume (Fig. 5). The depth of the cow's head in the bowl at the time of acquisition varied, but the addition of the bowl to the cow's head added a maximum volume of 5 dm 3 and area of ca. 0.2 m² (Fig. 5a). In some images, the bowl truncated the head, removing a mean volume of 3 dm 3 and a mean area of 0.1 m² (Fig. 5b). To avoid this bias, the transparent glass bowl was used, which did not distort the cow's head, allowing volume and area of the cow to be calculated accurately (Fig. 5c). Given the need to restrain some cows, we assessed the ability to estimate total volume and area from images of animals cut off at shoulder level. Only high-quality images (i.e. with the head scanned correctly) were selected. Two volumes and areas were determined on from the images: total volume and area, calculated using the automatic algorithm, and truncated volume and area, measured by placing one plane at the tip of the shoulder blades (similar to that used by [START_REF] Minagawa | Surface area, volume, and projected area of Japanese-shorthorn cattle measured by stereo photogrammetry using non-metric cameras[END_REF]) and

another at the rump (Fig. 6). The two planes were placed using 3D image-processing software Metrux2α® after placing three points on the image (Fig. 5). The Metrux2α® was used to estimate the volume and area between the two planes and to measure six morphological traits from the images: heart girth, chest depth, wither height, hip width, backside width and ischial width. Details of these traits are available in Le [START_REF] Cozler | High precision scanning system for complete 3D cow body shape imaging and analyzing morphological traits[END_REF].

Area and volume could be influenced by surface noise and some apparent roughness. Cleaning the resulting points cloud before creating the image required then patience and precision. Nevertheless, for some images, the reconstruction was sometimes not totally satisfactory (e. g. tail stuck to the animal, which was not the case on points cloud). But these errors were finally negligible, since the validation tests, carried out on different images of the same animals showed extremely small variations (see section 3.3)."

Repeatability and reproducibility analyses

As done for morphological traits (Le [START_REF] Cozler | High precision scanning system for complete 3D cow body shape imaging and analyzing morphological traits[END_REF], we used the Morpho3D device to estimate the repeatability and reproducibility of calculating volume and area.

Repeatability and reproducibility of plane placement

Since total volume and area were calculated automatically, their repeatability and reproducibility did not require assessment (i.e. no variation among measurements due to the operator). For truncated images, however, the repeatability of plane placement and point identification was analyzed to determine whether, for a given image, the placement of planes was reliable and whether the same volume or area could be calculated. Five images of complete cows were used, on which the same operator placed the cutting planes five times. Truncated volume and area were thus calculated five times from each of five different images.

For reproducibility, we assessed the ability of different operators to identify visually where to place the cutting planes. On images of five cows scanned only once, two operators repeated the placements five times each. In total, we assessed 50 measurements (25 per operator) on five different images. Truncated volume and area were then calculated.

Reproducibility of volume and area of the Mopho3D device in a changing environment

Analysis of reproducibility aimed to verify whether the method was able to calculate the same values from different image acquisitions in a changing environment caused by variability in cow position, image reconstruction and operator plane placements.

Reproducibility was assessed using images of nine cows, each undergoing the entire procedure, from initial acquisition to calculation of volume and area, five times. The same operator performed all of the manipulations. In this procedure, the reproducibility combined the repeatability error of volume and area calculation with that of plane placement.

Calculation

Variability in volume and area calculated from the 3D images was corrected for the effect of individual cows by extracting residuals from an analysis of variance (ANOVA) model. Repeatability and reproducibility were estimated using the SDs of placement repeatability (σrp), inter-operator reproducibility (σRo) and reproducibility in a changing environment (σRi). Values of σrp and σRi were calculated from the residues of the single-factor ANOVA of the individual-cow effect, while σRo was calculated from the mean of the reproducibility SDs of an ANOVA of the operator effect. Existing variations among cows were then excluded from the analysis. The coefficients of variation of repeatability (CVrp) and those of inter-operator and changing environment reproducibility (CVRo and CVRi, respectively) were then estimated from their respective means (µrp, µRo and µRi, respectively) and SDs (σrp, σRo and σRi, respectively). The more repeatable (or reproducible) the 3D measurement, the smaller was its CVrp (or CVRo or CVRi).

Estimation of body weight

Cow BW was measured twice a day. BW was also predicted using several regression models based on morphological traits, volume and area calculated with the Morpho3D device. Predictions of these models were then compared to the BW measured with the weighing system. From the measurements performed on complete images, a correlation matrix of Pearson coefficients was calculated to identify relationships between traits. Volume, area and morphological traits were measured or/ calculated from 3D images, while other traits, such as BCS and BW, were available in the database of the herd management system.

Several linear regression models to predict BW from traits measured or calculated from the 3D images were tested. The models were developed using the Akaike information criterion (AIC) with "backward elimination" variable selection. This method begins with all variables and then removes one or more variables at each iteration until it obtains the minimum value of the AIC, defined as 2 × (n-2) × log(L),

where n is the number of variables in the model and L is the maximum probability of the likelihood function. Each final model was cross-validated to estimate its prediction error. Cross-validation was preferred due to the small size of the dataset, which precluded separating it into sufficiently large calibration and validation datasets. The dataset was thus randomly divided into 2 groups, one of them containing 90%of datasets. This large group was used to develop BW models, which were then validated on data from the remaining group. Ten iterations of each model were crossvalidated 100 times using different random draws to obtain a large number of variations and calculate the root mean square error of prediction (RMSEP).

Coefficient of determination (R²) and RMSEP of the models were calculated to determine the quality of the linear regression and quantify the mean prediction error made by the model during cross-validation, respectively.

From truncated to complete images

Total volume and area were predicted using regression models based on truncated images. As before, models were developed using AIC with "backward elimination" variable selection. The same variables, except for total volume and area, were used in the models. Two models for each variable (total volume and total area) were developed: one with multiple variables that minimized RMSEP and another with only truncated area or volume. As before, 10 iterations of each model were crossvalidated 100 times using different random draws to obtain a large number of variations and calculate RMSEP.

Statistical analysis

All statistical analyses (i.e. ANOVA, repeatability, reproducibility, correlation analysis)

were performed using R software (R Core Team, 2016).

Results

Volume and area of the cylinder model

The error in determining both volume and area of the cylinder model was 0.0005% for Metrux2α® and 0.0001% for Meshlab®. Differences between Metrux2α®

estimates and manual measurements were less than 1%, regardless of the trait (Table 1).

Volume and area of cows

Morphological traits, volume and area were measured or calculated for the 64

Holstein dairy cows (Table 2). Mean (± 1 SD) volume and area of cows was 0.76 (± 0.07) m 3 and 6.84 (± 0.45) m 2 , respectively. Truncated volume and area equaled 90% and 81% of total volume and area, respectively. Density of the cows (i.e. BW/total volume) was 0.89 kg dm -3 , but ranged 0.79 to 0.95 kg dm -3 .

Reproducibility and repeatability

Repeatability estimated the error related to an operator's placement of planes on truncated images. Placement of the planes varied little (σrp for volume and area = 0.17% and 0.32%, respectively). Inter-operator reproducibility had larger error, σRo for volume and area 1.00% and 1.80%, respectively. Reproducibility of total and truncated values in a changing environment (σRi) had even higher errors (CVs for both = ca. 2-3%) (Table 3). In calculations of total volume and area, CVs expressed measurement error related to image acquisition and processing. In calculations of truncated volume and area, CVs combined image acquisition and processing error with plane-placement error.

Estimating body weight from 3D images

Traits measured from 3D images were strongly correlated with each other and with BW (Table 4). BCS was not correlated with other variables (maximum r = 0.22). BW and volume (total or truncated) were strongly correlated (r = 0.93 and 0.92, respectively). Volume and area (total or truncated) were also strongly correlated (r = 0.84-0.87). Since correlations between BW and the main traits were linear, linear regression models (1) and (2) were developed with the maximum number of variables from complete images using AIC (Table 5). After testing these more complex models, we included only 1-3 variables in models ( 3) and (4). (3) = 620 × + 379 × ! "#$% &% ℎ + 2870 × ℎ%) &% ℎ -174 0.88 22.5 3.36%

(4) = 827.5 × + 45.8 0.85 24.9 3.72%

(5) BW = 102.3 × total area -30.33 0.49 45.2 8.75%

For the selected traits, see corresponding units in tables 3 and4 Model (1) was the most accurate (RMSEP = 2.72%, R 2 = 0.93); it included six of nine possible traits (BCS not included because of the lack of correlation) previously mentioned): total volume, total area, hip width, backside width, wither height and chest depth. However, the coefficient of collinearity between total volume and total area was greater than 4 (not presented). Other models, free of strong collinearity and therefore more generalizable, were tested, such as including area but not volume (model (2)), or volume but not area (model (3)), but they were less accurate (RMSEP = 3.36% and 4.70%, respectively). A 4 th model, which considered only volume, was only moderately accurate (R 2 = 0.85, RMSEP = 3.72%), which corresponded to an error of 25 kg in predicted BW. In a similar way, a final model ( 5), which considered only area, indicated a poor accuracy (R 2 = 0.49, RMSEP = 6.75%), which corresponded to an error of 45 kg in predicted BW.

Predicting total volume and area from truncated images

Since it was not always possible to acquire complete images, total cow volume and area were predicted from truncated images ( For the selected traits, see corresponding units in tables 3 and 4

Discussion

Scanning an inert cylinder showed that the scanner could accurately acquire and calculate volume, area, length and circumference with less than 1% error. The validation process assessed image acquisition and processing and thus validated the overall accuracy of the method. This method can calculate volume and area with high reproducibility (CV = 2.24% and 2.43%, respectively). On living animals, the reproducibility coefficients for calculating volume and area were slightly higher due to manual manipulation (placement of planes) and were quantified by the repeatability of the placement of planes (0.17% and 0.32% for volume and area, respectively). A future study could calculate the repeatability and reproducibility of each step of the method to identify when the method becomes less accurate.

To assess the interest of developing this tool to render measurements automatic or semi-automatic, inter-operator reproducibility of plane placement was considered high (CVs for volume and area measurement = 1.00% and 1.80%, respectively).

According to [START_REF] Fischer | Rear shape in 3 dimensions summarized by principal component analysis is a good predictor of body condition score in Holstein dairy cows[END_REF], a method is repeatable and reproducible when its CV is less than 3-5%. Unfortunately, repeatability and reproducibility tests are not widely available in the literature, even though they are essential for determining the relevance of tools and methods (Marinello et al., 2015). It is therefore difficult to compare the method we developed to other tools. BW 0.46 ). When we applied these models to the recorded BW in our database, some Errors resulting from the image-reconstruction process or low-quality images may occur with the Morpho3D device. Other methods that estimate area, however, also have limits and can be difficult to use (e.g. the model of Mitchell (1928)). Future studies that accurately estimate area are needed with other methods than with Morpho3D device, but access to a large number of animals could be challenging, as well as determining arear with such methods Like for body area, little information is available on cow volume. [START_REF] Minagawa | Surface area, volume, and projected area of Japanese-shorthorn cattle measured by stereo photogrammetry using non-metric cameras[END_REF] estimated the volume of the neck and head of beef cows, and the results indicated that the rest of the body's volume was similar to our truncated volume. By applying our model (5b, Table 6) to this truncated volume, we predicted less than 1.5% difference in the total volume for four of Minagawa's five cows. The difference in total area was less than 6%.

Validating the methods used to estimate morphological traits, volume and area from 3D images enabled us to use these data to estimate cow BW, which is useful in breeding and feeding programs, commercial transactions and in the search for new indicators to estimate food-use efficiency in dairy animals. Estimation of BW, particularly during the rearing period, has been based for decades on development indicators [START_REF] Heinrichs | Predicting body weight and wither height in Holstein heifers using body measurements[END_REF]. It is indeed traditionally and commonly used in livestock production to follow growth of animals and/or then, adapt their feeding regimes. It is also used for commercialisation purposes for examples. Most feeding regimes for ruminants, pigs or poultry are based on animal body weight, which is rarely measured on-farms. As a result, recommendations are often based on visual estimation of BW. In addition, when performed so far, BW measurements are done manually, using weighing systems, which are also time consuming, sometimes costly, risky for human and animals health. Methods to estimate BW based on image technologies are then of increasing interest for many research groups, breeding organization, farmers and advisors. Estimating animal BW from measurements made from 3D images is a recent possibility [START_REF] Anglart | Automatic estimation of body weight and body condition score in dairy cows using 3D imaging technique[END_REF][START_REF] Buranakarl | Estimation of body weight and body surface area in swamp buffaloes using visual image analysis[END_REF][START_REF] Kuzuhara | A preliminarily study for predicting body weight and milk properties in lactating Holstein cows using a three-dimensional camera system[END_REF]. [START_REF] Anglart | Automatic estimation of body weight and body condition score in dairy cows using 3D imaging technique[END_REF] applied this approach to 3D acquisition of the backs of Holstein dairy cows using a time-of-flight camera, obtaining a R² for BW of 0.87. Of the 6224 acquisitions, however, 30% of the BW estimated from the images had an error of more than 30 kg. [START_REF] Buranakarl | Estimation of body weight and body surface area in swamp buffaloes using visual image analysis[END_REF] acquired full-body 3D images of buffaloes and developed 12 BW prediction equations (four each for all buffaloes, females and males) by changing the number of parameters considered.

For females, they obtained an R² for BW of 0.89 using four parameters (wither height, shoulder width, ischial width, length from ischia to shoulders). [START_REF] Kuzuhara | A preliminarily study for predicting body weight and milk properties in lactating Holstein cows using a three-dimensional camera system[END_REF] predicted BW from seven traits (RMSEP = 42 kg). The models used in the present study were therefore satisfactory (R² = 0.82-0.93, RMSEP from 2.72% (18.2 kg) to 4.38% (29.3 kg)) compared to those previously published. Since only 16% of the BW had an error greater than 30 kg (model ( 3)), the scanner and the method used appear reliable. Unlike previous studies that used only linear traits as parameters, 3D

imaging can create new models using volume and area. Ultimately, 3D imaging enables accurate and simplified estimation of BW from the traits measured, especially because it allows calculation of volume, which is strongly correlated with BW. Nevertheless, more automation (e.g. image preparation and measurement) is required to fully benefit from this tool.

Conclusion and future studies

The scanning technology described in this study provides new perspectives for assessing animal morphology and can be used to calculate volume and area of dairy cows. Analyses indicated that it was possible to estimate total volume and area from truncated images; however, all images had the same degree of truncation. Future studies are needed to determine whether other cutting planes (i.e. types of other types of truncation) could be useful. This suggests that truncated images from a less complicated or portable device could be used in the future. In addition, the device can also be used to estimate BW which is useful information, without investing in a weighing system. Preliminary results also indicated that the device can estimate changes in volume in the short term (rumen content), medium term (embryo development) and long term (growth).

The Morpho3D device allowed the acquisition of new phenotypes, not accessible until now, such as area and volume, at high-flow rates. There were no plans to implement this device on a large scale in commercial farms. For this purpose, a new version, not depending on animal movements and not sensitive to ambient light, is being tested. The quality of the 3D-images is lower, but sufficient to estimate most parameters previously presented, that will be compared to values obtained from Morpho3D device, considered as "Gold Standards".

Figure 1 .

 1 Figure 1. Examples of poor quality cloud points due to (a) excessive light or (b) animal movement.

Figure 2 .

 2 Figure 2. Design of the Morpho3D scanner

Figure 3 .

 3 Figure 3. Data acquisition from (a) the reference cylinder model (b) the raw point cloud after cleaning and (c) the final 3D image after Poisson surface-reconstruction.

  volume and area. It was sometimes necessary to restrain younger and/or nervous animals in a feed fence during image acquisition. As a result, the image was cut off at shoulder level (Fig.4).

Figure 4 .

 4 Figure 4. Images obtained when a cow was (a) restricted in a feeding fence (truncated image) or (b) its head moved too much during image acquisition.

Figure 5 .

 5 Figure 5. Differences in cow head reconstruction with (a) an opaque plastic bowl, (b) an opaque plastic bowl that truncates the head or (c) a transparent glass bowl.

Figure 6 .

 6 Figure 6. Position of the cutting planes used to estimate truncated volume and area when the head was missing from an image (i.e. cow restrained by a feed fence)

Figure 6 .

 6 Figure 6. Surface area of cows predicted by published models vs. those calculated using the Morpho3D device.

Table 1 .

 1 Mean and standard deviations (SD) of traits of the cylinder model estimated by the

	Morpho3D device (10 images) and absolute and relative differences between them and
	manual measurements.
				Difference
	Trait	Mean	SD	Absolute Relative
	Length (h), m	1.552	0.010	-0.010	0.62%
	Circumference (d x π), m	1.880	0.005	-0.008	0.24%
	Surface area (π x d x (h + d/2)), m 2	3.490	0.015	-0.028	0.44%
	Volume (π x (d/2) 2 x h), m 3	0.440	0.003	-0.003	0.72%

Table 2 .

 2 Mean, standard deviation (SD), and minimum and maximum values of body weight, body condition score (French scale of 0-5[START_REF] Bazin | Grille de notation de l'état d'engraissement des vaches pie-noires[END_REF]), morphology, volume and area of 64 dairy Holstein cows measured or calculated from 177 images. Truncated volume and area were calculated from the tip of the shoulder blades to the rump by placing one plane at

	each location.
	Body trait	Mean SD	Min.	Max.
	Body weight, kg	673	65	539	871
	Body condition score	2.05 0.25	1.50	2.88
	Heart girth, cm	228	10	210	256
	Chest depth, cm	85.9	3.2	78.8	95.3
	Wither height, cm	146	5	135	160
	Hip width, cm	58.8	3.4	50.7	66.7
	Backside width, cm	53.8	2.7	44.3	63.5
	Total volume, m 3	0.76 0.07	0.61	0.96
	Truncated volume, m 3	0.69 0.07	0.56	0.84
	Total area, m 2	6.84 0.45	5.80	8.32
	Truncated area, m 2	5.53 0.39	4.60	7.17
	Density, kg dm -3	0.89 0.03	0.79	0.95

Table 3 .

 3 Reproducibility of total and truncated volume and area in a changing environment assessed by standard deviation (SD) and coefficient of variation (CV). Truncated volume and area were calculated from the tip of the shoulder blades to the rump by placing one plane at

	each location.
	Repeatability	Mean	SD	CV
	Total volume, m 3	0.78	0.02	2.24%
	Truncated volume, m 3	0.72	0.02	2.43%
	Total area, m 2	6.94	0.02	2.85%
	Truncated area, m 2	5.59	0.1	3.12%

Table 4 .

 4 Correlation matrix between traits measured and/or estimated from 3D images. Truncated volume and area were calculated from the tip of the shoulder blades to the rump by placing one plane at each location. (Under the diagonal: Pearson correlation coefficient; Above the diagonal: p-value of the correlation). BCS: body condition score, BW: body weight.

		Total	Truncated	Total	Truncated	Hip	Wither	Chest	Heart	Backside
	Trait										BCS	BW
		volume	volume	area	area	width	height	depth	girth	width
	Total volume	-	<0.001	<0.001	<0.001	<0.001	0.012	0.296	0.502	0.008	0.020 <0.001
	Truncated volume	0.99	-	<0.001	<0.001	<0.001	0.012	0.298	0.535	0.008	0.018 <0.001
	Total area	0.87	0.86	-	<0.001	0.004	0.017	0.407	0.948	0.025	0.007	0.003
	Truncated area	0.84	0.86	0.94	-	0.005	0.019	0.485	0.871	0.027	0.010	0.004
	Hip width	0.76	0.76	0.64	0.62	-	0.010	0.221	0.537	0.002	0.002 <0.001
	Wither height	0.66	0.66	0.62	0.61	0.61	-	0.008	0.435	0.191	0.020	0.007
	Chest depth	0.56	0.56	0.50	0.45	0.52	0.67	-	0.083	0.934	0.318	0.152
	Heart girth	0.54	0.53	0.40	0.31	0.44	0.43	0.56	-	0.824	0.718	0.334
	Backside width	0.61	0.61	0.53	0.50	0.61	0.38	0.30	0.26	-	0.001	0.004
	BCS	0.15	0.14	0.04	0.01	-0.03	0.05	0.20	0.22	-0.12	-	0.031
	BW	0.93	0.92	0.72	0.71	0.80	0.70	0.63	0.57	0.66	0.19	-

Table 5 .

 5 Models predicting body weight (BW, in kg) of cows as a function of selected traits measured from complete images from the Morpho3D device. Model quality was assessed with a coefficient of determination (R 2 ) and root mean square error of prediction (RMSEP).

		Model	R²	BW RMSEP
	(1)	= 812.1 × ! "#$% &% ℎ + 273.8 × ℎ%) &% ℎ + 208.8 × -81.4 × + 343.8 × ℎ +% ℎ + 113.7 × &% ℎ ℎ %+ℎ -280.7	0.93	18.2	2.72%
	(2)	= 31,7 × ! "#$% &% ℎ + 257.2 × &% ℎ ℎ %+ℎ + 152.8 × + 608.8 × ℎ%) &% ℎ + 593.4 × ℎ +% ℎ -905.3	0.82	29.3	4.38%

Table 6

 6

	). A model using only truncated

Table 6 .

 6 Models estimating total volume or total area of cows from truncated images. Model quality was assessed with a coefficient of determination (R 2 ) and root mean square error of prediction (RMSEP). Truncated volume and area were calculated from the tip of the shoulder blades to the rump by placing one plane at each location.

				Model			R²	RMSEP
	(5a)	: 0.0165 × ℎ	= 1.0595 × +% ℎ -0.0149	"	+	0.99 0.008	1.02%
	(5b) :	= 1.0704 ×	+ 0.0154		0.98 0.008	1.05%
	(6a)	: ! "#$% &% ℎ + 0.518 × ℎ = 0.990 × "	+ 1.039 × +% ℎ -0.365	0.90 0.146	2.14%
	(6b) :	= 1.07 ×	"	+ 0.94		0.85 0.152	2.22%

Acknowledgments

The authors wish to thank everyone involved in the Morpho3D project, especially technicians at the Méjusseaume experimental station, who took excellent care of the animals. The Morpho3D project is supported by the National Fund CASDAR, which supports innovation in agriculture (RFP "Recherche Technologique" 2015, no. 005), special funds from the INRA Animal Physiology and Livestock Systems division for innovative projects and the collaborative ANR -APIS-GENE project DEFFILAIT.