Xavier Roynard

Jean-Emmanuel Deschaud

François Goulette

Classification of Point Cloud for Road Scene Understanding with Multiscale Voxel Deep Network

In this article we describe a new convolutional neural network (CNN) to classify 3D point clouds of urban scenes. Solutions are given to the problems encountered working on scene point clouds, and a network is described that allows for point classification using only the position of points in a multiscale neighborhood. This network enables the classification of 3D point clouds of road scenes necessary for the creation of maps for autonomous vehicles such as HD-Maps.

On the reduced-8 Semantic3D benchmark [1], this network, ranked second, beats the state of the art of point classification methods (those not using an additional regularization step as CRF). Our network has also been tested on a new dataset of labeled urban 3D point clouds for semantic segmentation.

I. INTRODUCTION

The majority of autonomous vehicles use 3D maps of the world for localization, perception and navigation tasks. As these maps improve the robustness of autonomous systems, we believe that almost all roads will be scanned in the future, representing for example 8 million km in North America and 4 million in Europe. Moreover, they must be updated regularly to take into account changes in the network. This is why it is important to set up the most automated processes possible to create and update these maps.

These point clouds must be processed after acquisition to extract the relevant information for autonomous driving: moving objects and parked vehicles must be removed, traffic signs, traffic lights and drivable areas detected. For example, for the localization task, the classified point cloud can be used as a map and moving objects can be detected with differences between the map and the current lidar frame as shown in figure 1.

To do so, the automatic classification of the data is necessary and is still challenging, regards to the number of objects present in an urban scene.

For the object classification task, deep-learning methods work very well on 2D images. The easiest way to transfer these methods to 3D is to use 3D grids. It works well when the data is just one single object [START_REF] Wu | 3d shapenets: A deep representation for volumetric shapes[END_REF].

But it is much more complicated for the task of point classification of a complete scene (e. g. an urban cloud) made up of many objects of very different sizes and potentially interwoven with each other (e. g. a lamppost passing through vegetation). Moreover, in this kind of scene, there are classes more represented (floor and buildings) than others (pedestrians, traffic signs ...).

This article proposes both a training method that balances the number of points per class during each epoch, and to 1 All authors are with Mines ParisTech, PSL Research University, Centre for Robotics. xavier.roynard@mines-paristech.fr

II. STATE OF THE ART

The focus here is on the semantic segmentation methods applied to dense registered point clouds used to create maps as HD-Maps, unlike the very sparse KITTI dataset clouds which require real-time processing methods.

A. Shallow and Multi-Scale Learning for 3D point cloud classification

There is a great variety of work for classifying 3D point cloud scenes by shallow learning methods or without learning. Methods can generally be classified into one of the two approaches: classify each point, then group them into objects, or conversely, divide the cloud into objects and classify each object.

The first approach is followed by [START_REF] Hackel | Fast semantic segmentation of 3d point clouds with strongly varying density[END_REF] which classifies each point by calculating multi-scale features, computing the same kind of features at different scales to capture both context and local shape around the point. After classifying each point, the points can be grouped into objects by CRF [START_REF] Zhang | Sensor fusion for semantic segmentation of urban scenes[END_REF] or by regularization methods [START_REF] Landrieu | A structured regularization framework for spatially smoothing semantic labelings of 3d point clouds[END_REF].

The segmentation step of the second approach is usually heuristic-based and contains no learning. [START_REF] Aijazi | Segmentation based classification of 3d urban point clouds: A super-voxel based approach with evaluation[END_REF] segments the cloud using super-voxels, [START_REF] Serna | Detection, segmentation and classification of 3d urban objects using mathematical morphology and supervised learning[END_REF] uses mathematical morphology operators and [START_REF] Roynard | Fast and robust segmentation and classification for change detection in urban point clouds[END_REF] makes a region growth to extract the soil, then groups the points by connected components. After segmentation, objects are classified by computing global descriptors that can be simple geometrical descriptors [START_REF] Serna | Detection, segmentation and classification of 3d urban objects using mathematical morphology and supervised learning[END_REF], or mixture of bag-of-words [START_REF] Behley | Laser-based segment classification using a mixture of bag-of-words[END_REF].

B. Deep-Learning for 3D point cloud classification

Over the past three years, there has been a growing body of work that attempts to adapt deep learning methods or introduces new "deep" approaches to classifying 3D point clouds.

This is well illustrated by the ShapeNet Core55 challenge [START_REF] Yi | Large-scale 3d shape reconstruction and segmentation from shapenet core55[END_REF], which involved 10 research teams and resulted in the design of new network architectures on both voxel grids and point cloud. The best architectures have beaten the state of the art on the two proposed tasks: part-level segmentation of 3D shapes and 3D reconstruction from single view image.

1) on 2D Views of the cloud:

The most direct approach is to apply 2D networks to images obtained from the point cloud. Among other things, we can think of the following projections:

• RGB image rendered from a virtual camera, • depth-map, from a virtual camera,

• range image, directly from the sensor, • panorama image [START_REF] Sfikas | Ensemble of panoramabased convolutional neural networks for 3d model classification and retrieval[END_REF],

• elevation-map. These methods can be improved by taking multiple views of the same object or scene, and then voting or fusing the results [START_REF] Boulch | Unstructured point cloud semantic labeling using deep segmentation networks[END_REF] (ranked 5th on reduced-8 Semantic benchmark). In addition, these methods greatly benefit from existing 2D expertise and pre-trained networks on image datasets [START_REF] Deng | Imagenet: A large-scale hierarchical image database[END_REF], [START_REF] Lin | Microsoft coco: Common objects in context[END_REF] that contain much more data than point cloud datasets.

2) on Voxel Grid:

The first deep networks used to classify 3D point clouds date from 2015 with VoxNet [START_REF] Maturana | Voxnet: A 3d convolutional neural network for real-time object recognition[END_REF], this network transforms an object instance by filling in an occupancy or density grid and then applies a Convolutional Neural Network (CNN). Later [START_REF] Huang | Point cloud labeling using 3d convolutional neural network[END_REF] applied the same type of network to classify urban point clouds, the network then predicts the class of a point from the occupancy grid of its neighborhood. However, we cannot compare with this architecture because the experimental data has not been published. Best results on ModelNet benchmarks are obtained using deeper CNNs [START_REF] Brock | Generative and discriminative voxel modeling with convolutional neural networks[END_REF] based on the architecture of Inception-ResNet [START_REF] Szegedy | Inception-v4, inception-resnet and the impact of residual connections on learning[END_REF] and voting on multiple 3D view of objects.

There are also significantly different approaches on voxel grids. OctNet [START_REF] Riegler | Octnet: Learning deep 3d representations at high resolutions[END_REF] uses a hybrid Grid-Octree structure that allows CNNs to be used on resolved grids of higher reolution. VoxelNet [START_REF] Zhou | Voxelnet: End-to-end learning for point cloud based 3d object detection[END_REF] instead of increasing grid resolution, increases the size of voxels and the information contained in each voxel through a network similar to PointNet [START_REF] Qi | Pointnet: Deep learning on point sets for 3d classification and segmentation[END_REF] (called Voxel Feature Encoding).

3) on Graph: Another approach is to use graphs, indeed the raw point cloud having no structure, it is very difficult to derive general information from it. Whereas a graph gives relations of neighborhoods and distances between points and allows for example to make convolutions as in SPGraph [START_REF] Landrieu | Large-scale point cloud semantic segmentation with superpoint graphs[END_REF] or to apply graph-cut methods on CRF as in SEGCloud [START_REF] Tchapmi | Segcloud: Semantic segmentation of 3d point clouds[END_REF].

4) on Point Cloud:

For the time being, there are still quite a few methods that take the point cloud directly as input. These methods have the advantage of working as close as possible to the raw data, so we can imagine that they will be the most efficient in the future. The first method of this type is PointNet [START_REF] Qi | Pointnet: Deep learning on point sets for 3d classification and segmentation[END_REF] which gets fairly good results on ModelNet for object instance classification. PointNet is based on the observation that a point cloud is a set and therefore verifies some symmetries (point switching, point addition already in the set...) and is therefore based on the use of operators respecting these symmetries like the global Pooling, but these architectures lose the hierarchical aspect of the calculations that make the strength of the CNN. This gap has been filled with PointNet++ [START_REF] Qi | Pointnet++: Deep hierarchical feature learning on point sets in a metric space[END_REF] which extracts neighborhoods in the cloud, applies PoinNet and groups the points hierarchically to gradually aggregate the information as in a CNN. Two other approaches are proposed by [START_REF] Engelmann | Exploring spatial context for 3d semantic segmentation of point clouds[END_REF] to further account for the context. The first uses PointNet on multiscale neighborhoods, the second uses PointNet on clouds extracted from a 2D grid and uses recurrent networks to share information between grid boxes.

III. APPROACH A. Learning on fully annotated registered point clouds

Training on scenes point cloud leads to some difficulties not faced when the point cloud is a single object. For the point classification task, each point is a sample, so the number of samples per class is very unbalanced (from thousands of points for the class "pedestrian" to tens of millions for the class "ground"). The classic training method by epoch would be to go through all the points of the training cloud at each epoch, making the classes with few samples anecdotal for the network.

We propose a training method that solves this problem. We randomly select N (for example N = 1000) points in each class, then we train on these points shuffled randomly between classes, and we repeat this process at the beginning of each Epoch.

Once a point p to classify is chosen, we compute a grid of voxels given to the convolutional network by building an occupancy grid centered on p whose empty voxels contain 0 and occupied voxels contain 1. We only use n × n × n cubic grids where n is even, and we only use isotropic space discretization steps ∆. To reduce neighborhood search time, we can also sub-sample point clouds from the training set with a scale less than ∆.

B. Data Augmentation and Training

Some classic data augmentation steps are performed before projecting the 3D point clouds into the voxels grid:

• Flip x and y axis, with probability 0.5 • Random rotation around z-axis • Random scale, between 95% and 105% • Random occlusions (randomly removing points), up to 5% • Random artefacts (randomly inserting points), up to 5% • Random noise in position of points, the noise follows a normal distribution centered in 0 with standard deviation 0.01m

The cost function used is cross-entropy, and the optimizer used is ADAM [START_REF] Kingma | Adam: A method for stochastic optimization[END_REF] with a learning rate of 0.001 and ε = 10 -8 , which are the default settings in most deeplearning libraries. To reduce neighborhood search time, we can also sub-sample point clouds from the training set with a scale less than ∆. In our experiments, all point clouds are subsampled at 2cm. No study has been carried out on the influence of subsampling on classification quality, but it is estimated that as long as the subsampling is performed at a scale below the discretization step of the voxel grid, the impact is negligible.

C. Test

To label a complete point cloud scene, the naive method is to go through all the points of the cloud, and for each point:

• look for all the neighboring points that fit into the occupation grid, • create this grid,

• infer the class of the point via the pre-trained network. However, two points very close to each other will have the same neighborhood occupancy and therefore the network will predict the same class. A faster test method is therefore to sub-sample the cloud to be tested. This has two beneficial effects: reduce the number of inferences and neighborhood searches, and each neighborhood search takes less time. To infer the point class of the initial cloud, we give each point the class of the nearest point in the subsampled cloud, which can be done efficiently if the subsampling method used retains the correct information.

IV. NETWORK ARCHITECTURE

The choosen network architecture is inspired from [START_REF] Simonyan | Very Deep Convolutional Networks for Large-Scale Image Recognition[END_REF] that works well in 2D. Our network follows the architecture: Conv(32, 3, 1, 0) → Conv(32, 3, 1, 0) → MaxPool(2) → Conv(64, 3, 1, 0) → Conv(64, 3, 1, 0) → MaxPool(2) → FC(1024) → FC(N c ) 1 where N c is the number of classes, and each Convolutionnal (Conv) and Fully-Connected (FC) layer is followed by a Batch Normalization, a Parametric ReLU and a Squeeze-and-Excitation block [START_REF] Hu | Squeeze-and-Excitation Networks[END_REF] except the last FC layer that is followed by a So f tMax layer. This network takes as input a 3D occupancy grid of size 32×32×32, where each voxel of the grid contains 0 (empty) or 1 (occupied) and has a size of 10cm × 10cm × 10cm.

This type of method is very dependent on the space discretization step ∆ selected. Indeed, a small ∆ allows to understand the object finely around the point and its texture (for example to differentiate the natural ground from the ground made by man) but a large ∆ allows to understand the context of the object (for example if it is locally flat and horizontal around the point there can be ambiguity between the ground and the ceiling, but there is no more ambiguity if we add context).

Since a 3D scene contains objects at several scales, this type of network can have difficulty classifying certain objects. So we also propose a multiscale version of our network called MSK_DeepVoxScene for the K-scales version (or abbreviated in MSK_DVS). We take several versions of the previous network without the fully-connected layer. The input of each version is given a grid of the same size 32 × 32 × 32, but with different sizes of voxels (for example 5 cm, 10 cm and 15 cm). We then retrieve a vector of 1024 characteristics from each version, which we concatenate before giving to a fully-connected classifier layer. See figure 2 for a graphical representation of MS3_DeepVoxScene.

V. EXPERIMENTS

A. Datasets

To carry out our experiments we have chosen the 2 datasets of 3D scenes which seem to us the most relevant to train methods of deep-learning, Paris-Lille-3D [START_REF] Roynard | Paris-Lille-3D: a large and high-quality ground truth urban point cloud dataset for automatic segmentation and classification[END_REF] and Semantic3D [START_REF] Hackel | Semantic3d.net: A new large-scale point cloud cassification benchmark[END_REF]. Among the 3D point cloud scenes datasets, these are those with the most area covered and the most variability (see table I). The covered area is obtained by projecting each cloud on an horizontal plane in pixels of size 10cm × 10cm, then summing the area of all occupied pixels.

1) Paris-Lille-3D:

The Paris-Lille-3D dataset consists of 2 km of 3D point clouds acquired by Mobile Laser Scanning using with a Velodyne HDL-32e mounted on a van. Clouds are georeferenced using IMU and GPS-RTK only, no registration or SLAM methods are used, resulting in a slight noise. Because the scene is scanned at approximately constant speed, the point density is roughly uniform. The dataset consists of 3 files, one acquired in Paris and two acquired in Lille including Lille1.ply much larger than Lille2.ply. To validate our architectures by K-fold method, we cut spatially Lille1.ply into two folds containing the same number of points. Cross-validation is thus performed on 4 folds of similar sizes In addition, this dataset contains 50 classes, some of which only appear in some folds and with very few points. We therefore decide to delete and group together some classes to keep only 9 coarser classes: ground buildings poles bollards trash cans barriers pedestrians cars natural Some qualitative results on Paris-Lille-3D dataset are shown in figure 4. We can observe that some trunks of trees are classified as poles. It may means that the context is not sufficiently taken into account (even so the 15 cm grid is 4.8 m large) In addition, the ground around objects (except cars) is classified as belonging to the object. One can imagine that cars are not affected by this phenomenon because this class is very present in the dataset.

2) Semantic3D:

The Semantic3D dataset was acquired by static laser scanners, it is therefore more dense than a dataset acquired by MLS as Paris-Lille-3D, but the density of points varies considerably depending on the distance to the sensor. And there are occlusions due to the fact that sensors do not turn around the objects. Even by registering several clouds acquired from different viewpoints, there are still a lot of occlusions. To minimize the problem of very variable density, we subsample the training clouds at 2 cm. This results in a more uniform density at least close to the sensor and avoids redundant points. After subsampling, the dataset contains 79.5M points. The training set contains 15 point clouds which after sub-sampling are of similar sizes, each cloud is used as a separate fold for cross-validation. Some qualitative results on Semantic3D dataset are shown in Figure 3.

B. Evaluation Protocol

To confirm the interest of multi-scale CNNs, we compare the performance of our two architectures on these three datasets. And on Semantic3D we compare our results with those of the literature. The metrics used to evaluate performance are the following: Except for Semantic3D benchmark, all results are obtained by cross-validation by training on all folds except one and testing on the remaining fold. All our networks are trained for 100 epochs with 1000 points per class on each fold. No validation sets are used.

C. Comparison with the state of the art

For a comparison with the state-of-the-art methods on reduced-8 Semantic3D benchmark see table II. For MS1_DeepVoxScene several resolutions have been tested, and by cross-validation on the Semantic3D training set the 10 cm resolution is the one that maximizes validation accuracy. DeepVoxScene's choice of MS3_DeepVoxScene resolution results from this observation, we keep a resolution that obtains good performance in general, and we add a finer resolution of 5 cm to better capture the local surface near the point, and a coarser resolution of 15 cm to better understand the context of the object to which the point belongs. Our method achieves better results than all methods that classify cloud by points (i. e. without regularization).The inference time of the 23.5 million points of the reduced8 test set subsampled at 2 cm is approximately 32 h. And the propagation of classes to the nearest points on the original cloud (not subsampled) takes approximately an hour.

D. Study of the different architectures

To evaluate our architecture choices, we tested this classification task by one of the first 3D convolutional networks: VoxNet [START_REF] Maturana | Voxnet: A 3d convolutional neural network for real-time object recognition[END_REF]. This allows us both to validate the choices made for the generic architecture of the MS1_DeepVoxScene network and to validate the interest of the multi-scale network. We reimplemented VoxNet using the deep-learning library Pytorch. See table III for a comparison between VoxNet [START_REF] Maturana | Voxnet: A 3d convolutional neural network for real-time object recognition[END_REF], MS1_DeepVoxScene and MS3_DeepVoxScene on the 3 datasets.

See table IV for a comparison per class between MS1_DeepVoxScene and MS3_DeepVoxScene on Paris-Lille-3D dataset. This shows that the use of multi-scale networks improves the results on some classes, in particular the buildings, barriers and pedestrians classes are greatly improved (especially in Recall), while the car class loses a lot of Precision.

VI. CONCLUSIONS

We have proposed both a training method that balances the number of points per class seen during each epoch, as well as a multi-scale CNN that is capable of learning to classify point cloud scenes. This is achieved by both focusing on the local shape of the object around a point and by taking into account the context of the object in a multi-scale fashion.

We validated the use of our multi-scale network for 3D scene classification by ranking second on Semantic3D benchmark and by ranking significantly better than state-of-the-art point classification methods (those without regularization).

Fig. 1 .

 1 Fig. 1. Application of our classified point cloud for map based localization of an autonomous vehicle (in white, the map point cloud, in color the current velodyne frame of the autonomous vehicle from blue/close to the map to red/far to the map)

Fig. 2 .

 2 Fig. 2. Our Multi-Scale Voxel Network architecture: MS3_DeepVoxScene (all tensors are represented as 2D tensors instead of 3D for simplicity).

Fig. 3 .

 3 Fig. 3. Example of classified point cloud on Semantic3D test set (blue: man-made terrain, cerulean blue: natural terrain, green: high vegetation, light green: low vegetation, chartreuse green: buildings, yellow: hard scape, orange: scanning artefacts, red: cars).

  F1 c = 2T P c 2T P c + FP c + FN c IoU c = T P c T P c + FP c + FN c Where F1 c and IoU c represent respectively F1-score and Intersection-over-Union score of class c. And T P c , T N c , FP c and FN c are respectively the number of True-Positives, True-Negatives, False-Positives and False-Negatives in class c.

Fig. 4 .

 4 Fig. 4. Example of classified point cloud on Paris-Lille-3D dataset. Left: classified with MS3_DVS, right: ground truth (blue: ground, cerulean blue: buildings, dark green: poles, green: bollards, light green: trash cans, yellow: barriers, dark yellow: pedestrians, orange: cars, red: natural).

TABLE I COMPARISON

 I OF 3D POINT CLOUD SCENES DATASETS. PARIS-LILLE-3D CONTAINS 50 CLASSES BUT FOR OUR EXPERIMENTATIONS WE KEEP ONLY 9 COARSER CLASSES. IN BRACKETS IS INDICATED THE NUMBER OF POINTS AFTER SUBSAMPLING AT 2 cm.

	Name	LiDAR type	Covered Area	Number of points (subsampled)	Number of classes
	Paris-Lille-3D [27]	multi-fiber MLS	55000m 2	143.1M (44.0M)	9
	Semantic3D [1]	static LiDAR	110000m 2	1660M (79.5M)	8

TABLE II TOP

 II -5 RESULTS ON SEMANTIC3D REDUCED-8 TESTING SET. MS3_DVS IS OUR MS3_DEEPVOXSCENE WITH VOXEL SIZES OF 5 cm, 10 cm AND 15 cm AND MS1_DVS IS OUR MS1_DEEPVOXSCENE WITH VOXEL SIZE OF 10 cm (ADDED FOR COMPARISON WITH NON MULTI-SCALE DEEP NETWORK).

TABLE IV PER

 IV CLASS PRECISION AND RECALL AVERAGED ON THE 4 FOLDS OF PARIS-LILLE-3D DATASET.

we denote Conv(n, k, s, p) a convolutional layer that transforms feature maps from previous layer into n new feature maps, with a kernel of size k × k × k and stride s and pads p on each side of the grid.


Corinne Robert
email: robert@bcgn.grignon.inra.fr

Marie-Odile Bancal

Christian Lannou

Wheat Leaf Rust Uredospore Production and Carbon and Nitrogen Export in Relation to Lesion Size and Density

Keywords: Brown rust, Triticum aestivum, Yield reduction

To develop mechanistic yield loss models for biotrophic fungi, we need better account for the export of dry matter, carbon and nitrogen from the leaf into the spores. Three experiments in controlled environment chambers were performed to study the dynamics of uredospores production of Puccinia triticina on seedling leaves of wheat in relation to time, lesion density and sporulating surface area. The detrimental effect of lesion density on the sporulation capacity of brown rust lesions was confirmed : when lesion density increased, spores production per lesion strongly decreased. However, our results showed that increasing lesion density also greatly reduces lesion size. A model was developed to summarize these relationships. The main conclusions of our work is that the density effect on spore production per lesion was accounted for by lesion size : when sporulation was related to the sporulating surface area, it became independent of density. As well, carbon and nitrogen contents of the spores were independent of lesion density. Our data suggest that when nitrogen available in the host is limiting, spore production is reduced but nitrogen content of spores tend to remain stable.

INTRODUCTION

The development of integrated pest management systems requires mechanistic models that can simulate the effect of disease on yield. But these models must be firmly rooted in a clear understanding of the biological mechanisms involved. Such models require quantitative information on the effects of a disease on physiological processes. In the case of leaf rust caused by Puccinia triticina of wheat (Triticum aestivum), the effects of the disease on photosynthesis, respiration and foliar transpiration have been documented [START_REF] Grath | Alteration of physiological processes in wheat flag leaves caused by stem rust and leaf rust[END_REF][START_REF] Shtienberg | Effects of foliar diseases on gas exchange processes : A comparative study[END_REF]. However, for brown rust as well as for many other biotrophic fungi, spore production results in a significant loss of biomass by the diseased crops that is rarely taken into account in current models [START_REF] Van Roermund | Simulation of yield reduction by leaf rust in winter wheat, applied to the analysis of genetic variation in partial resistance[END_REF]. Bastiaans (3) quantified the effect of rice leaf blast (caused by Magnaporthe grisea) on leaf photosynthesis and respiration in order to model the consequences of the disease on rice dry matter production. He found that the changes in photosynthesis and respiration were insufficient to fully explain the observed reduction in shoot dry weight, and suggested that the dry matter exported in the spores, which was neglected in the model, accounted for the overestimated diseased shoot dry weight in the simulations. For leaf rusts, Roermund and Spitters [START_REF] Van Roermund | Simulation of yield reduction by leaf rust in winter wheat, applied to the analysis of genetic variation in partial resistance[END_REF] also considered the uptake of assimilates for spore production as a cause of damage. Their field studies indicated that leaf rusts can consume up to 230 to 570 kg CH 2 O ha -1 for spore production. Therefore, any model designed to describe the effect of brown rust on crop growth and yield must accurately describe the loss of dry matter resulting from spore production. Spore production must also be quantified throughout the lifetime of a rust lesion, as damage depends greatly on the crop growth stage [START_REF] Madden | Modeling crop losses at the field scale[END_REF].

Other authors have described brown rust spore production per lesion at different disease severities [START_REF] Mehta | Uredospore production and sporulation period of Puccinia recondita f. sp. triticina on primary leaves of wheat[END_REF][START_REF] Sache | Effect of density and age of lesions on sporulation capacity and infection efficiency in wheat leaf rust[END_REF]. Spore production per lesion increases to a maximum and then gradually decreases to zero, so that the shape of the spore production curve is an asymmetrical bell-shaped curve. Density greatly affects the sporulation capacity of brown rust lesions [START_REF] Sache | Effect of density and age of lesions on sporulation capacity and infection efficiency in wheat leaf rust[END_REF]. When lesion density increases, spore production per lesion strongly decreases and reaches its maximum value faster. In these studies, spore production was quantified as the number of spores per lesion, rather than by the export of dry matter. The latter parameter is needed, however, to relate spore production to damage resulting in crop loss. These studies also linked spore production to lesion density (number of lesions per leaf), whereas damage-analysis models usually consider diseased area as an impact variable [START_REF] Bassanezi | Accounting for photosynthetic efficiency of bean leaves with rust, angular spot and anthracnose to assess crop damage[END_REF][START_REF] Gary | Effects of Aschochyta blight on the decrease in photosynthetizing leaf area and the reduction of photosynthetic efficiency by green leaf area of dried-pea[END_REF][START_REF] Lim | Leaf area as a factor in disease assessment[END_REF][START_REF] Madeira | Growth and light interception in field bean (Vicia faba) infected by Ascochyta fabae[END_REF][START_REF] Savary | Dynamic simulation of groundnut rust: A preliminary model[END_REF]. For instance, Shtienberg [START_REF] Shtienberg | Effects of foliar diseases on gas exchange processes : A comparative study[END_REF] worked on brown rust and showed that the decrease in photosynthesis is proportional to the diseased leaf area. The present study was to estimate the export of dry matter into spores and to relate spore production to the sporulating area at different lesion densities. We also measured the export of carbon and nitrogen into the spores in relation to lesion density.

MATERIALS AND METHODS

Seedlings of wheat cv. Soissons were grown in commercial compost (peat substrate, Gebr.

Brill Substrate Gmbh & Co., Germany) in 7x7 cm square pots (8 seedlings per pot) in a growth chamber with a 16-h light period (250 µE.m -2 .s -1 ) at 18°C and an 8-h dark period at 15°C. Isolate B9384-1C1 of P. triticina was used in all experiments. The primary leaves of seedlings were inoculated in a settling tower (5) 10 days after sowing, before the emergence of the second leaf.

Different lesion densities were obtained by inoculating the seedlings with a mixture of brown rust spores and talc in which the spore contents were 6, 3, 1.5, 0.75 and 0.375 mg. Inoculated plants were incubated for 16 hours under 100% relative humidity in plastic bags at 15°C and then placed in a growth chamber until the end of the experiment. When chlorosis were visible, two inoculated leaves per pot were selected according to lesion density. Spore production was estimated by collecting and weighting spores for each inoculated leaf at regular intervals after onset of sporulation as described below. The experiment was repeated 3 times (experiments 1, 2 and 3). Spore production was modeled as a function of time and lesion density using the data of experiment 1 (32 leaves). Data of experiment 2 were used to validate the spore production model (10 leaves) and to model changes in lesion size with time and lesion density (22 leaves). Spore production was not assessed on these 22 leaves because, when shooting the leaves with camera for lesion size measurement, spores would have been lost. Although the experimental design was the same for all experiments, the appearance of the leaves in experiment 3 (27 leaves) suggested that seed germination and first leaf development were not optimal. The reason for this was probably asphyxiation of the roots due to overwatering. Experiment 3 was therefore not used to estimate parameters.

Lesion density.

The number of lesions per leaf was counted twice, 8 days and 10 days after inoculation. Lesions started to sporulate 9 days after inoculation. Leaves were then selected so that lesion densities were 0.1 to 40 lesions per cm² of leaf. No secondary infection occurred during the experiment and the lesion number was considered constant after day 10. However, rings of secondary lesions appeared around very low-density primary lesions about 1 week after onset of sporulation.

Leaf area (S, in cm²) was estimated by linear regression from leaf length (L, in cm) on 40 primary leaves grown as described above (S = 0.44 x L -0.12 , R² = 0.92, P<10 -4 ). The length of each leaf was measured and lesion density was calculated by dividing the number of lesions by the estimated leaf area (S). Spore collection. Spores were collected every 2 or 3 days from day 3 after initial sporulation until the end of sporulation. Before onset of sporulation, leaves were placed horizontally in 3 cmdiameter glass tubes. Plants were placed on a turntable in the growth chamber to ensure uniform light and temperature. This caused the air to flow into the tubes and prevented water condensation. Spores were collected by shaking the leaves so that the spores fell onto the glass tubes, from which they were collected and weighed. The tubes were replaced until the next collection. Secondary leaves were cut off after each spore collection.

Spore production model. The data from experiment 1, obtained from 32 infected leaves, were used to model spore production. The relationship between cumulative spore production per lesion (Y, in mg), lesion age (t = number of days after onset of sporulation, with t = 1 for the first sporulation day) and lesion density (d = number of lesions per cm² of leaf) was established in a twostep procedure. Cumulative spore production per lesion at each collection date t was first related to lesion density d:

ln [Y t (d)] = a t -b t x ln(d) [ 1 ]
Then, parameters a(t) and b(t) were related to lesion age t by :

a(t) = m a -exp (c a -f a x t) [2.1] b(t) = m b -exp (c b -f b x t) [2.2]
where

)] = c a -f a x t [3.1] ln [b(t) -b(27)] = c b -f b x t [3.2]
Combining Equation 2.1 and 2.2 with Equation 1 gave an estimate of the cumulative spore production per lesion at day t after onset of sporulation for a lesion density d :

Y(t,d) = exp {m a -exp (c a -f a x t) -[m b -exp (c b -f b x t)] x ln(d)} [4]
From day 4 of sporulation, daily spore production (P, in mg) per lesion for lesion age t and lesion density d was then estimated by:

P(t,d) = Y(t,d) -Y(t-1,d) [5]
This model was validated with data obtained from 10 infected leaves in experiment 2. Spore production between t = 0 and t = 3 was assumed to increase linearly for any density d, according to Sache [START_REF] Sache | Comparison of the wheat brown and yellow rusts for monocyclic sporulation and infection processes, and their polyciclic consequences[END_REF].

Lesion size. In experiment 2, lesion size was measured on 22 leaves. We did not measure spore production on these leaves. Lesion size was defined as the lesion sporulating surface area.

Digital pictures were taken at each spore collection date until 18 days after start of sporulation to accurately estimate the sporulating area by image analysis (Optimas, Media Cybernetics, Silver Spring MD 20910, USA). Tissue senescence after day 18 made it more difficult to assess sporulating area, but lesion size could be considered constant after this date.

Lesion growth model. Lesion growth was modeled using the same procedure as spore production. For homogeneity with the spore production model, only data from days 4 to 18 were used to estimate parameters, although lesion size was measured from day 1. Lesion size was assumed to increase linearly with time before day 4. The relationship between lesion size S, (mm² of sporulating area), lesion age t (number of days after onset of sporulation) and lesion density d, was established in a two-step procedure. First, sporulating area was related to lesion density d at each date t by:

ln [S t (d)] = u t -v t x ln (d) [ 6 ]
The estimated values of u(t) and v(t) were then plotted against lesion age t and the following equations were fitted to the data for days 4, 6, 8, 11, 14, and 18 :

u(t) = m u -exp (d u -e u x t) [7.1] v(t) = m v -exp (d v -e v x t
S(t,d) = exp {m u -exp (d u -e u x t) -[m v -exp (d v -e v x t)] x ln(d)} [8]
Nitrogen and carbon contents of leaves and uredospores. To estimate the export of nitrogen and carbon into uredospores, the nitrogen and carbon contents of the spores produced by each infected leaf were measured at the end of experiments 1, 2, and 3. To characterize the initial status of the leaves and to estimate whether the nitrogen content of the leaves affects spore production, we measured the N and C levels in the primary leaves of check plants the day before inoculation for experiments 1 and 3. The C and N contents of both spores and primary leaves were measured by the method of Dumas (4).

RESULTS

Uredospore production. About 90% of the lesions on infected leaves started to sporulate 10 days after inoculation. This was considered to be the first sporulation day (t = 1). In experiments 1 and 2, overall spore production per cm² of leaf (Fig. 1) increased with lesion density from 0.7 mg/cm² of leaf for a lesion density of 1.5 lesion/cm² to a maximum of around 4.5 mg/cm² of leaf for lesion densities greater than 15 lesions/cm². Uredospore production per cm² of leaf was more than 70% lower in experiment 3 than in experiment 1 for all lesion densities and the maximum spore production was around 1 mg/cm² of leaf at a lesion density greater than 15 lesions/cm².

Cumulative uredospore production per lesion. Cumulative spore production per lesion was higher at lower lesion densities from day 6 after start of sporulation in experiments 1 and 2, and this density effect on spore production increased with time (Fig. 2). Cumulative spore production per lesion at the end of the sporulation period (t = 27) was around 1.5 mg/lesion for the lowest lesion densities (<1 lesion/cm²) and 0.10 mg/lesion for the highest lesion densities (>20 lesions/cm²). There was little density effect in experiment 3.

Daily uredospore production per lesion. Spore production per lesion was measured every 3 days. Hence daily spore production was estimated by dividing the amount of spores collected by the number of days between two successive collection dates. Spore production per lesion increased to a maximum and then decreased to zero at all lesion densities (Fig. 3). At low lesion densities, the length of the sporulation period and the maximum daily spore production both increased, and the date at which the maximum was reached was delayed. In experiment 1, the maximum spore production per lesion decreased from 0.11 mg/lesion/day at a lesion density of 0.6 lesion/cm² to 0.01 mg/lesion/day at a lesion density of 23.4 lesions/cm². Spore production per lesion was maximal between 3 days after onset of sporulation, at the highest lesion densities (>20 lesion/cm²), and 13 days after onset of sporulation, at the lowest lesion densities (<1 lesion/cm²). Sporulation lasted from 20 days at the highest lesion densities (>20 lesion/cm²) to over 27 days at the lowest lesion densities (<1 lesion/cm²).

Model for estimating spore production per lesion. The model's parameters were estimated with data from experiment 1 and 10 leaves in experiment 2 were used for validation. Cumulative spore production per lesion and lesion density were log-transformed (Equation 1). After examination of the data distribution (Fig. 4), the data were split into two groups, one with lesion densities below 2 lesions/cm² (0.7 on the Ln scale) and the other with lesion densities above 2 lesions/cm². Each data subset was analysed separately.

For lesion densities below 2 lesion/cm², parameter b(t) in Equation 1was not significantly different from zero, suggesting that spore production per lesion was independent of lesion density.

Before day 6 of sporulation, average spore production per lesion was lower in this data subset than for densities above 2 lesion/cm², and this reversed after day 8 (Fig. 4). This suggests that spore production per lesion started slowly at very low lesion densities, but quite rapidly became higher than for greater densities.

For lesion densities greater than 2 lesions/cm², cumulative spore production per lesion was correlated with lesion density according to Equation 1. The relationship between lesion density and cumulative spore production per lesion on day 3 (t = 3) was poorly explained by the model but the model quality increased at day 6 and R² remained above 0.92 between day 9 and day 27 (Table 1).

Parameters a(t) and b(t) of Equation 1were then related to lesion age using Equations 2.1 and 2.2 :

a(t) = 0,754 -exp (-0,174 x t + 1.981) (R² = 1.00, P = 3E-5)
b(t) = -0,749 + exp (-0,144 x t + 0,153) (R² = 0.99, P = 2E-4)

Cumulative spore production per lesion was then related to lesion density and lesion age by Equation 4. The cumulative spore production per lesion for each collection date (t = 3, 6, 9, 12, 15, 19, 23 and 27) estimated with Equation 4was plotted against the corresponding measured values from experiment 1 (Fig. 5). The model fitted the data correctly (R² = 0.98, P = 2E-177) and the residue distribution showed no bias. Equation 4 was validated with data from experiment 2 (Fig. 5), which showed a good fit for the model (R² = 0.98, P = 9E-67).

Lesion size assessment. A subset of 22 leaves in experiment 2 were used to measure individual lesion size. Lesions were larger for lower lesion densities from day 4 after onset of sporulation, and this difference increased with time (Fig. 6). Lesions stopped growing around day 11 at the highest densities and around day 18 at the lowest densities. The maximal lesion size on day 18 was 4.7 mm² for a density of 0.3 lesion/cm² and the minimal lesion size was 0.7 mm² for densities >20 lesions/cm². Visually, the relationship between lesion size and lesion density was similar to that between cumulative spore production per lesion and lesion density.

Lesion size model. Data for lesion densities above and below 2 lesion/cm² were analyzed separately. For lesion densities greater than 2 lesions/cm², the sporulating area was related to lesion density using Equation 6. The relationship between lesion density and sporulating area was correctly described by the model (Table 1 Uredospore production per mm² of sporulating area. Daily spore production per mm² of sporulating area was modeled by dividing the estimated spore production per lesion per day (Equation 5, Fig. 7B) by the estimated lesion size (Equation 8, Fig. 7A). The strong effect of density on spore production disappeared when spore production was related to the sporulating area (Fig. 7C). The overall spore production for the whole sporulation period ranged from 0.43 mg per mm² of sporulating area at 3 lesions/cm² to 0.35 mg per mm² of sporulating area at 30 lesions/cm². Daily spore production per mm² of sporulating area increased to a maximum and then decreased towards zero (Fig. 7C). The model predicted a maximum value around 0.034 mg/mm² for spore production, which can be considered independent of lesion density. This maximum was reached 4 days after onset of sporulation for all lesion densities. Over 80% of the spores were produced before day 15 of sporulation, regardless of lesion density. However, there was a slight density effect after the maximum was reached, with lower spore production and a shorter sporulation period for greater lesion densities (Fig. 7C).

N and C contents of spores and primary leaves. The carbon and nitrogen levels (%) of the spores were constant and did not depend on lesion density (Fig. 8). Spore contained 49.9% (σ = 0.5), 50.0% (σ = 1.3) and 48.9% (σ = 0.9) carbon in experiments 1, 2 and 3 respectively . The carbon levels of primary leaves before inoculation were similar in experiments 1 and 3. Primary leaves contained 40.2% (σ = 0.9) and 39.8% (σ = 0.6) carbon in experiments 1 and 3 respectively .

The nitrogen levels (%) of primary leaves before inoculation differed greatly in experiments 1 and 3: leaves contained 6.4% (σ = 0.3) N in experiment 1 and only 2.8% (σ = 0.4) N in experiment 3.

In contrast, spores contained 4% (σ = 0.05) N in experiments 1 and 2, and 3.4% (σ = 0.1) N in experiment 3. The N/C ratios of the leaves differed in experiments 1 (N/C = 16) and 3 (N/C = 7.1), but those for the spores were almost similar (N/C = 7.9 in experiments 1 and 2, N/C = 7 in experiment 3).

DISCUSSION

Our study confirmed that uredospore production results in a significant loss of biomass in diseased crops. Thus, uredospore production has to be introduced as a damage function in diseased plant models. The weight of the spores (20 mg) produced by a primary leaf exceeded the mean dry weight of the leaf (13 mg), which is consistent with the findings of Mehta and Zadoks [START_REF] Mehta | Uredospore production and sporulation period of Puccinia recondita f. sp. triticina on primary leaves of wheat[END_REF]. For the dynamics of uredospores production, the pattern of leaf rust sporulation, expressed in mg of spores per lesion per day was similar to those obtained by other authors, expressed as the number of spores per lesion per day [START_REF] Mehta | Uredospore production and sporulation period of Puccinia recondita f. sp. triticina on primary leaves of wheat[END_REF][START_REF] Sache | Effect of density and age of lesions on sporulation capacity and infection efficiency in wheat leaf rust[END_REF]. The weight of spores produced by a single lesion was greatly reduced at high lesion density. A similar density effect on spores production has been reported for several rust fungi [START_REF] Baart | Effects of infection density on the size of barley and wheat leaf rust colonies before and on the size of uredia after the start of sporulation[END_REF][START_REF] Leonard | Factors affecting rates of stem rust increase in mixed plantings of susceptible and resistant oat varieties[END_REF][START_REF] Sache | Comparison of the wheat brown and yellow rusts for monocyclic sporulation and infection processes, and their polyciclic consequences[END_REF][START_REF] Yarwood | Uredospore production by Uromyces phaseoli[END_REF]. Our data suggest that the density effect on spore production shows a threshold at around 2 lesions per cm² below which sporulation is independent of density. Below this threshold, spore production per lesion started slowly, but increased rapidly after 7 days of sporulation. This could be due to the rings of secondary lesions that appeared one week after onset of sporulation at the lowest lesion densities and suggests that the secondary lesions increased the host tissue area exploited by the fungus.

Our results showed that increasing lesion density also greatly reduced lesion size. But spore production became independent of lesion density when expressed as spores per mm² of sporulating area instead of spores per lesion. Other authors assumed that the density effect on spore production should result from competition between adjacent uredia, reducing the sporulation capacity of individual uredia [START_REF] Baart | Effects of infection density on the size of barley and wheat leaf rust colonies before and on the size of uredia after the start of sporulation[END_REF][START_REF] Leonard | Factors affecting rates of stem rust increase in mixed plantings of susceptible and resistant oat varieties[END_REF][START_REF] Sache | Comparison of the wheat brown and yellow rusts for monocyclic sporulation and infection processes, and their polyciclic consequences[END_REF][START_REF] Yarwood | Uredospore production by Uromyces phaseoli[END_REF]. According to our model, most of the effect of density on spore production per lesion could be accounted for by lesion size. A residual density effect remained, however, since the total weight of spores produced per mm² of sporulating area was slightly lower for high lesion densities than for low lesion densities. This residual effect resulted from a slightly faster decrease in spore production and a shorter sporulation period at high lesion densities. The fast decrease in spores production per mm² of sporulating area for all lesion densities suggest that there is a limitation in leaf nutrients available for the fungus.

The large difference in spore production in experiments 1 and 3 suggests that the nitrogen content of the leaves significantly affects spore production. Although the experimental design was the same for all experiments, seed germination and first leaf development was not optimal in experiment 3. This resulted in the leaves having a low nitrogen content and in spore production being more than 70% lower at all lesion densities than in experiments 1 and 2. Several authors have shown that the nitrogen content of the leaves has a great influence on disease development [START_REF] Leitch | Influence of nitrogen on the development of Septoria epidemics in winter wheat[END_REF][START_REF] Snoeijers | The effect of nitrogen on disease development and gene expression in bacterial and fungal plant pathogens[END_REF].

In the experiments, the carbon and nitrogen levels of the spores remained constant at all lesion densities. The spores contained about 50% carbon and between 3.4% and 4% nitrogen. Since the C and N contents of the spores were independent of lesion density, losses of carbon and nitrogen from infected leaves can be directly estimated from the dry weight of spores. In experiment 1 and 2, for lesion densities greater than 15, the spore production per cm 2 of leaf was maximal (Fig. 1) and the total amount of carbon and nitrogen exported reached respectively 2.2 mg and 0.18 mg per cm² of leaf.

In experiment 3, exported carbon and nitrogen were reduced by more than 70%, corresponding to 0.5 mg C and 0.04 mg N per cm² of leaf. As well, carbon and nitrogen exported per mm² of sporulating area can be estimated from spore production per mm² of sporulating area. According to our model, exported carbon per mm² of sporulating area ranged from 0.21 to 0.18 mg and exported nitrogen per mm² of sporulating area ranged from 0.017 to 0.014 mg when lesion density was increased from 3 to 30 lesions per cm 2 .

The weight of spores produced in experiment 3 was strongly reduced relative to experiment 1 but the nitrogen content of the spores remained almost constant, suggesting that the nitrogen content is strongly regulated in the spores. The difference in the nitrogen levels of the spores (4% and 3.4%) in experiments 1 and 3 was much smaller than that of the leaves (6.4% and 2.8%). In experiment 1, the nitrogen level of the leaves was probably not limiting for spore production. In experiment 3, the spores contained more nitrogen (3.4%) than the leaves (2.8%), which suggests a priority for nitrogen allocation to the spores rather than to the plant. In field epidemics, leaf rust should then affect yield quality by diverting nitrogen fluxes from infected leaves to spikes during the grain filling period.

These experiments were conducted under environmental conditions that were optimal for brown rust and spore production was measured on primary leaves. As in other studies [START_REF] Mehta | Uredospore production and sporulation period of Puccinia recondita f. sp. triticina on primary leaves of wheat[END_REF], secondary leaves were cut for facilitating measurements. It is difficult to say whether this affected spore production or not : more seed assimilates were available for the primary leaf but the global photosynthesis of the plant was reduced. Such measurements are technically impossible under field conditions. However, we assume that the pattern of spore production per unit of sporulating area on adult plants is also constant over time and independent of lesion density when nitrogen is not limiting.

The nitrogen content of the host leaves may vary under field conditions, which should influence the amount of spores produced by the pathogen.

The sporulating area should be a suitable variable for simulating the effect of brown rust on wheat because brown rust spore production per sporulating area unit is independent of lesion density and because the main effect of brown rust on photosynthesis is directly related to the diseased surface area [START_REF] Shtienberg | Effects of foliar diseases on gas exchange processes : A comparative study[END_REF]. Lastly the nitrogen and carbon content of spores can be used to estimate amounts of nitrogen and carbon exported by the pathogen and its consequences for yield quality.

Table 1. Statistics for the cumulative spore production model (Equation 1) and lesion size model (Equation 6) for leaf rust (P. triticina) on wheat. (lesion/cm² of leaf). For densities lower than 2 lesions/cm², the amount of spores produced was too small and data are not available. In experiment 3, the nitrogen content of the leaves was limiting.

Fig. 3 .

 3 Fig. 3. Daily spore production per lesion for leaf rust (P.triticina) on wheat (cv. Soissons), for different lesion densities (solid circles: 0.9 lesion/cm², open diamonds: 4.6 lesion/cm² and solid triangles: 22.8 lesion/cm²) as a function of time, in days after onset of sporulation. Spores were harvested every 2 or 3 days. Data are from experiment 1.

Fig. 4 .

 4 Fig. 4. Cumulative spore production per lesion for leaf rust (P.triticina) on wheat as a function of lesion density, after a double Log-transformation, for different dates. Spore weight is in mg, lesion density is in lesions per cm² of leaf and time is in days after onset of sporulation. Two different models were used to fit the data below (solid symbols) and above (open symbols) 2 lesions/cm² (0.7 on the Ln scale). Data are from experiment 1.

Fig. 5 .

 5 Fig. 5. Comparison of simulated and measured data for cumulative spore production per lesion of leaf rust (P.triticina) on wheat (mg/lesion) for each leaf and collection date. Doted line : y=x. Open circles : data from experiment 1, Solid circles : data from experiment 2, for validation.

Fig. 6 .

 6 Fig. 6. Leaf rust (P.triticina) lesion size (mm²) as a function of lesion density (lesions per cm² of leaf)for different days after onset of sporulation. Data set is from experiment 2. Lesion size is defined as the sporulating area.

Fig. 7 .

 7 Fig.7. Simulated data for lesion growth in mm 2 (A, Equation8), daily spore production in mg per lesion (B, Equation5) and spore production in mg per sporulating area unit (C) for leaf rust (P.triticina) on wheat. Data are given for 5 different lesion densities : 3 (open triangles), 5 (solid triangles), 10 (open circles), 20 (solid circles) and 30 (open squares) lesions/cm².

Fig. 8 .

 8 Fig. 8. Carbon (A) and nitrogen (B) levels (%) of leaf rust (P. triticina) spores collected in experiments 1 (open circles), 2 (solid circles), and 3 (open triangles) in relation to lesion density

Fig. 1

 1 Fig.1,

  Cumulative spore production per lesion for leaf rust (P.triticina) on wheat (cv. Soisson), as a function of time and lesion density. Spore weight is in mg, lesion density is in lesion per cm² of leaf and time is in days after onset of sporulation. Three sets of data are represented : experiment 1 (open circle), 2 (solid circle) and 3 (open triangles). In experiment 3, nitrogen content of the leaves was limiting. Ordinate scale is changed after day 12.

	CAPTIONS FOR FIGURES
	Fig. 1. Amount of spores produced by leaf rust (Puccinia triticina) on a primary leaf of susceptible
	wheat (cv. Soissons) against lesion density (lesions/cm²). Spore production is expressed in mg of
	spores (fresh weight) per cm² of leaf. Three sets of data are presented : experiment 1 (open circles), 2 Equation 1 b t a 3 6 9 1 2 1 5 1 9 2 3 2 7 a(t) -3.39 -1.87 -0.83 -0.14 0.23 0.57 0.69 0.75 (solid circles) and 3 (open triangles). In experiment 3, the nitrogen content of the leaves was limiting.
	b(t) R 2	0.04 0.01	0.23 0.52	0.41 0.92	0.55 0.94	0.62 0.96	0.70 0.96	0.73 0.96	0.75 0.96
	p-value Fig. 2.	0.68	4E-5	3E-14	6E-16 Equation 6 c	5E-17	2E-17	5E-17	1E-17
	t a	1	4	6	8	1 1	1 4	1 8
	u(t)	-0.89	0.12	0.71	1.40	1.89	2.03	2.22
	v(t) R 2	0.33 0.42	0.35 0.45	0.44 0.81	0.59 0.78	0.70 0.86	0.73 0.97	0.78 0.96
	p-value	4E-2	2E-3	5E-6	6E-6	7E-7	7E-9	4E-7

a Time, in days after onset of sporulation. b Equation

1

: ln [Y t (d)] = a t -b t x ln(d) c Equation

6

: ln [S t (d)] = u t -v t x ln (d)


Léo Perrin

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