Nesrine Chehata

Li Guo

Clément Mallet

AIRBORNE LIDAR FEATURE SELECTION FOR URBAN CLASSIFICATION USING RANDOM FORESTS

Keywords: Lidar, Full-waveform, Classification, Feature selection, Random Forests, Urban scenes

Various multi-echo and Full-waveform (FW) lidar features can be processed. In this paper, multiple classifers are applied to lidar feature selection for urban scene classification. Random forests are used since they provide an accurate classification and run efficiently on large datasets. Moreover, they return measures of variable importance for each class. The feature selection is obtained by backward elimination of features depending on their importance. This is crucial to analyze the relevance of each lidar feature for the classification of urban scenes. The Random Forests classification using selected variables provide an overall accuracy of 94.35%.

INTRODUCTION

Airborne lidar systems have become an alternative source for the acquisition of altimeter data providing unstructured 3D point clouds that describe the Earth's topography. The altimeter accuracy of a topographic lidar measurement is high (<0.1 m). Depending on the geometry of illuminated surfaces, several backscattered echoes can be recorded for a single pulse emission. Many authors have shown the potential of multi-echo lidar data for urban area analysis and building extraction [START_REF] Sithole | Experimental comparison of filter algorithms for bare-earth extraction from airborne laser scanning point clouds[END_REF]. 3D point cloud classification can be based on geometric and textural attributes [START_REF] Matikainen | Classification tree based building detection from laser scanner and aerial image datas[END_REF]. Other works include the lidar intensity [START_REF] Charaniya | Supervised parametric classification of aerial lidar data[END_REF] or combine lidar and multispectral data [START_REF] Secord | Tree detection in urban regions using aerial lidar and image data[END_REF]Zakhor, 2007, Rottensteiner et al., 2005). Since 2004, full-waveform (FW) lidar systems have emerged with the ability to record the complete waveform of the backscattered 1D-signal laser pulse. In [START_REF] Gross | Segmentation of tree regions using data of a full-waveform laser[END_REF][START_REF] Wagner | 3D vegetation mapping using small-footprint full-waveform airborne laser scanners[END_REF], FW lidar features were used to detect vegetated areas. In urban scenes, the potential of such data has been barely investigated. For the analysis of laser scanner data, various classification techniques have been applied such as unsupervised classification by ISODATA [START_REF] Haala | Extraction of buildings and trees in urban environments[END_REF] and supervised classification as Bayesian networks (Stassopoulou et al., 2000), Dempster shafer fusion theory [START_REF] Rottensteiner | Using the dempster-shafer method for the fusion of lidar data and multi-spectral images for building detection[END_REF], Support Vector Machines [START_REF] Secord | Tree detection in urban regions using aerial lidar and image data[END_REF][START_REF] Charaniya | Supervised parametric classification of aerial lidar data[END_REF][START_REF] Mallet | Analysis of fullwaveform lidar data for classification of urban areas[END_REF] or classification trees [START_REF] Ducic | 3D Vegetation mapping and classification using fullwaveform laser scanning[END_REF][START_REF] Matikainen | Classification tree based building detection from laser scanner and aerial image datas[END_REF].

In this work, we study different lidar features, multiecho and full-waveform to classify urban scenes into four classes: Buildings, vegetation, natural ground and artificial ground. Artificial ground gathers all kinds of streets and street items such as cars, traffic lights whereas the natural ground includes grass, sand, and bare-earth regions. No filtering is applied before the classification. The objective is to select the most relevant features for classifying urban scenes and to provide an accurate classification with a small number of features. We propose to achieve both objectives using Random Forests. It is an ensemble classifier based on decision trees. It returns good classfication results and provides also feature selection. The paper is organized as follows. The lidar features will be detailed in Section 2. In Section 3, multiple classifiers are presented and especially Random Forests. The feature selection process is detailed in Section 3.3. Experimental results are then discussed in Section 4 and finally, conclusions are drawn in Section 5.

AIRBORNE LIDAR FEATURES

Multi-echo and full-waveform lidar data are available. The feature vector is composed of 21 components: 17 multi-echo and 4 full-waveform lidar features. The multi-echo lidar features are separated into height-based, echo-based, eigenvalue-based and local 3D-plane based features. The resulting feature vector f v for each site is given by:

f v = [Δz Δz f l σ 2 z C; N N e ; λ 1 λ 2 λ 3 A λ P λ S λ L λ ; N z σ 2 Nz R z D Π ; A w σ α] T
All these features are computed using a volumetric approach within a local neighborhood V P at each lidar point P . The local neighborhood includes all the lidar points within a cylinder, with a fixed radius, centered at the point. Lidar points are then projected into a 2D image geometry (0.5 m resolution). Feature images are obtained by computing, for each pixel, the mean corresponding value of the lidar points included in a 3×3 raster kernel (cf.

Figure . 1). Table 1 summarizes input lidar features which are separated into five groups. Lidar features are detailed hereby.

Height-based lidar features

The first group is based on 3D point heights:

• Δz: Height difference between the lidar point and the lowest point found in a large cylindrical volume whose radius has been experimentally set to 15 m. This feature will help discriminating ground and off-ground objects (cf. Figure.

1(b)).

• Δz f l : Height difference between First and Last pulses of the waveform of the current lidar point. It helps discriminating building roofs and ground.

• σ 2 z : The height variance. This feature has high values for vegetation (cf. Figure . 1(c)).

• C: The local curvature is a discrete version of the Laplace operator. It is the maximum value of the gradient differences on heights, which are computed in four main directions based on a raster grid [START_REF] Steinle | Automated extraction and reconstruction of buildings in laserscanning data for disaster management[END_REF]. A 3×3 raster kernel has been chosen.

Echo-based features

• N : Total number of echoes within the waveform of the current lidar point. This feature is high for vegetation and building facades.

• N e : Normalized number of echoes obtained by dividing the echo number by the total number of echoes within the waveform of the current lidar point. This feature highlights the vegetation since multiple reflections can occur on it (cf. Figure. 1(d)).

Eigenvalue-based lidar features

The variance-covariance matrix is computed within the local neighborhood (V ) P . The Eigenvalues λ 1 > λ 2 > λ 3 are used as features. Besides, they provide additional features and help discriminating planes, edges, corners, lines, and volumes [START_REF] Gross | Extraction of lines from laser point clouds[END_REF]. These features describe the spatial local distribution of the 3D points.

Anisotropy = A λ = λ1 -λ3 λ1 (1)
Planarity = P λ = λ 2 -λ 3 λ1 (2)
Sphericity = S λ = λ 3 λ1 (3)
Linearity = L λ = λ 1 -λ 2 λ 1 (4)
λ 3 has low values for planar objects and higher values for nonplanar objects (cf. Figure . 1(e)). The planarity feature shows high values especially for planar objects. Conversely, the sphericity feature gives high values for isotropic distributed 3D neighborhood.

Local plane-based lidar features

The planarity of the local neighboorhood will help discriminating buildings from vegetation. The local plane ΠP within VP is estimated using a robust M-estimator with norm L1.2 [START_REF] Xu | Epipolar Geometry in stereo, motion and object recognition[END_REF].

• N z : Deviation angle of Π P normal vector from the vertical direction. This feature highlights the ground (cf. Figure .1(f)).

• σ 2 z : Variance of deviation angles within V P . It discriminates planar surfaces such as roads and building roofs from vegetation.

• Rπ: Residuals of the local plane estimated in a small vertical cylinder (0.5 m radius). Residuals Rπ are calculated w.r.t the estimated plane as follows:

Rπ = � i∈ν P (d i ) l l ( 5
)
where di is the distance between the lidar point i ∈ νP and the plane. Here l=1.2. Residuals should be high for vegetation.

• D π : Distance from the current point P to the local estimated plane Π P .

Full-waveform lidar features

The remaining features are more specific to FW lidar data and are obtained by modelling the lidar waveforms. The amplitude and echo width are described in [START_REF] Wagner | Gaussian decomposition and calibration of a novel smallfootprint full-waveform digitising airborne laser scanner[END_REF].

• A: echo amplitude. High amplitude values can be found on building roofs, on gravel, and on cars. Asphalt and tar streets have low values. The lowest values correspond to vegetation due to a higher target heterogeneity and attenuation (cf. Figure. 1(g)).

• w: echo width. Higher values correspond to vegetation since it spreads the lidar pulse. A low width is likely to correspond to ground and buildings. However, it may increase with roof slope (cf. Figure. 1(h)).

• α: echo shape describing how locally spread the peak is. It is obtained from a waveform decomposition based on a generalized Gaussian modeling function [START_REF] Chauve | Processing full-waveform lidar data: modelling raw signals[END_REF].

The authors also show that very low and high shape values correspond mainly to building roofs and vegetation.

• σ: echo cross-section corresponds more or less to the peak energy. σ = A × w. It is the basic quantity to describe the scattering of a wave by an object. The cross-section values are high for buildings, medium for vegetation and low for artificial ground.

MULTIPLE CLASSIFIERS

Over the last two decades, many multiple classifiers have been proposed. Several classifiers are trained and their results combined through a voting process. In this paper, we focus on multiple classifiers that are built at data level. The modified data sets are applied to train each classifier in the ensemble, the base classifier should be unstable, that is, small changes in the training set will lead to large changes in the classifier output. Neural [START_REF] Briem | Multiple classifiers applied to multisource remote sensing data[END_REF].

Random Forests

Random Forests are a variant of bagging proposed by [START_REF] Breiman | Random forests[END_REF]. It is a decision tree based ensemble classifier that has excellent performance in classification tasks comparable to boosting [START_REF] Breiman | Random forests[END_REF], even Support Vector Machines (SVMs) [START_REF] Pal | Random Forest classifier for remote sensing classification[END_REF]. It can be used with multi-class problems. It is non-parametric and does not require assuptions on the distribution of the data. This is an interesting property when different types or scales of input attributes are used. Moreover, Random Forests run efficiently on large data sets and can handle thousands of input variables without variable deletion. They do not overfit. They have a good predictive performance even when most predictive variables are noisy. Therefore, there is no need for variable preselection [START_REF] Strobl | Unbiased split selection for classification trees based on the Gini index[END_REF]. In addition, Random Forests is a classification algorithm that directly provide measures of variable importance (related to the relevance of each variable in the classfication process). These outstanding features make it suitable for the classification of remote sensing data such as multispectral data [START_REF] Pal | Random Forest classifier for remote sensing classification[END_REF] or multisource data [START_REF] Gislason | Random forests for land cover classification[END_REF].

Random Forests are a combination of tree predictors such that each tree depends on the values of a random vector sampled independently and with the same distribution for all trees in the forest [START_REF] Breiman | Random forests[END_REF]. As in Breiman's method, in training, the algorithm creates multiple bootstrapped samples of the original training data, then builds a number of no pruning Classification and Regression Trees (CART) from each bootstrapped samples set and only a randomly selected subset of the input variables is used to split each node of CART. For classification, each tree in the Random Forests gives a unit vote for the most popular class at each input instance. The label of input instance is determined by a majority vote of the trees. The number of variables M , randomly chosen at each split, is considered as the single user-defined adjustable parameter. This parameter is not critical and is often set to the square root of the number of inputs [START_REF] Gislason | Random forests for land cover classification[END_REF].

In this work, Random Forests are applied to classify airborne lidar data on urban scenes and to select the most important features for this task. This avoids the user to manually select relevant attributes. Random Forests have not yet been used with airborne lidar data.

Variable importance

Aside from classification, Random Forests provide measures of variable importance based on the permutation importance measure which was shown to be a reliable indicator [START_REF] Strobl | Unbiased split selection for classification trees based on the Gini index[END_REF].When the training set for a particular tree is drawn by sampling with replacement, about one-third of the cases are left out of the sample set. These out-of-bag (OOB) data can be used to estimate the test accuracy and the permutation importance measure.

The importance of variable m can be estimated by randomly permuting all the values of the m th variable in the OOB samples for each tree. A measure for variable importance [START_REF] Breiman | Random forests[END_REF]) can be the difference in prediction accuracy (i.e. the number of observations classified correctly) before and after permuting variable m, averaged over all trees. A high decrease of prediction accuracy indicates the importance of that variable.

Feature selection

The objective of feature selection is to identify small sets of lidar features that can still achieve a good predictive performance and so that correlated features should not be selected. In this paper, we use a backward elimination of features using OOB errors. This feature selection process was proposed in (Díaz-Uriarte and de Andrés, 2006), for biological application, to select genes of microarray data. Using data simulations, the authors showed its robustness to noise or redundant features.

To select the most relevant features, we iteratively fit Random Forests. At each iteration, a fraction of the features (the least important ones) is eliminated and a new forest is built. By default, the fraction is fixed to 0.2. It allows a relatively fast operation, and increases the resolution as the number of considered features becomes smaller. After fitting all forests, the selected set of features is the one whose OOB error rate is within u = 1 standard error of the minimum error rate of all forests. This is similar to the "s.e. rule" commonly used in classification trees literature [START_REF] Breiman | Classification and regression trees[END_REF]. This strategy can lead to solutions with fewer features while achieving an error rate that is not different, within sampling error, from the best solution.

EXPERIMENTAL RESULTS

Data set

The data acquisition was carried out with the RIEGL LMS-Q560 system over the city of Biberach (Germany). The main technical characteristics of this sensor are presented in [START_REF] Wagner | Gaussian decomposition and calibration of a novel smallfootprint full-waveform digitising airborne laser scanner[END_REF] and summarized in Table 2.

The city of Biberach includes artificial grounds, natural grounds, vegetation and buildings. This lidar data set has been used for classifying urban scenes using Support Vector Machines (Mallet

Variable importance results

The variable importance estimate for the training data is depicted on Figure 2 for each group feature. Considering all classes, it is obtained by the mean decrease permutation accuracy. The most important features are height-based: the height difference and the height variance. Echo-based features are not important for urban scene classification when using other attributes that describe more accurately the local distribution of 3D points such as eigenvalue-based or 3D plane-based features. Moreover, the First-Last height difference is not important to classify urban objects since it is used with the height variance σ 2 z . In fact, both variables can be correlated and the latter has more values which allows to distinguish rooftops building and ground for instance.

As for the eigenvalue-based features, λ 3 is the most important eigenvalue, as expected. In fact, it returns the lowest values for planar objects. The sphericity S λ shows a high importance whereas the correlated anisotropy feature A λ shows a lower one. This illustrates the advantage of permutation accuracy measure since redundant features should be less important. Among 3D-plane based features, the distance to plane D Π seems to be the most important one. Finally, for full-waveform features, echo amplitude, and width are the most important for all classes. The FW cross-section σ is less important as it is correlated to the former features.

Feature selection results

To fix the Random Forests parameters, (Díaz-Uriarte and de [START_REF] Díaz-Uriarte | Gene selection and classification of microarray data using random forest[END_REF] showed that the relation of OOB rate with the number of split variables M is largely independent of the number of trees T (for T between 1000-40000). In addition, the default setting of M is a good choice of OOB rate. Therefore, the feature selection was run with M = 4 and T = 1000 trees. Figure 3 is obtained by the backward iterative elimination of features using OOB errors. The graph is shown in a forward way to illustrate the more relevant features. On the x-axis, eigenvalue and FW Lidar features appear respectively in blue and red colors. The selection process returns a feature vector of 17 attributes, where Δ z f l , N , N e and σ are eliminated. Another strategy may consist in keeping the set which first makes the variable importance decrease. In our case the final set may corresponds [Δ z ,σ z ,w,D Π ,A,S λ ]. The corresponding total error is 6%. One can observe that four feature groups are represented: the height based group is the most important one, then two FW features are selected which confirms the contribution of full-waveform lidar data for urban scene classification.

Classification results

Based on the 17 selected features, the Random Forests classification was run and variable importance computed for each class. Underlying parameters have been fixed to M = 4 which means that four variables are considered at each split and the number of trees was set experimentally to 60. The study area is visible on figure 4. We observe that errors occur mainly on building edges. The corresponding lidar points might be ambiguous since they correspond to transition points between building and artificial ground classes. Besides, these confusion errors are amplified due to the interpolation process of lidar points in 2D geometry.

The confusion matrix for test data is given in Table 4. The raster neighborhood size is 3×3. Artificial ground and buildings are well classified with lower error rate. However, the algorithm has more difficulties in classifying natural ground and vegetation. The former class suffers from smaller training set than the other classes. As for vegetation, confusions essentially occur with artificial ground due to the lidar data interpolation. In fact, in non-dense vegetated areas, the lidar beam is likely to reach the ground underneath and the resulting waveform has mixed properties. Therefore, some lidar feature pixels may be a combination of both classes. 4.4.1 2D Geometry impact Since lidar features are processed in a 2D geometry, we studied the effect of the neighborhood size on classification and variable importance results. A neighborhood size of 5×5 was tested. The corresponding confusion matrix is illustrated in Table 5 Figure 4 shows the corresponding classification result.

When increasing the neighborhood size, classification results are only enhanced for the building class, but are worse for all the other classes. In fact, buildings can be represented by large homogeneous segments with regard to lidar features. Therefore when dealing with larger neighborhoods, the training pixels have more representative mean values. It is similar for the artificial ground class however due to some small roads between buildings, the large neighborhood may include building points which increases the number of artificial ground pixels that are misclassified to building class (cf. Table 5). The total error rate seems to be smaller with a large neighborhood, however this is due to the fact that building class has the higher number of pixels.

Variable Importance of selected features

The variable importance was reprocessed for the 17 selected features in order to study more precisely the relevant features for each class. Two data sets were used with different window sizes: 3×3 (cf. Figure 5) and 5×5 (cf. Figure 6).

Firstly, one can observe that the variable importance values are higher and better distinguished when using a small window size since training pixels are more homogeneous. Secondly, when using a large window size, pixels may be mixed in the neighborhood, many variables give the same importance (cf. Figure . 6) for all classes which reveals the classification ambiguity. Consequently, the variable importances seem to be more reliable with a small neighborhood size. It will be discussed hereby.

We confirm that the height difference is the most important feature for all classes, which is the only topographic variable. For artificial ground, considering the different group features, the most important attributes are: P λ for planarity, D Π since it is a flat surface. FW selected attributes seem to have the same importance for this class. As for building class, the most important features in different groups are: S λ ,N z and A. For plane-based features, the distance to the plane D Π is more sensitive to the scan trajectory (the plane is better estimated along the scan trajectory since there are more points) and to the roof area (a large roof returns a higher distance to plane). For this reason it is less important than the deviation angle. For natural grounds, many variables seem to be important. This result should be interpreted with caution due to the few number of corresponding training samples. Finally, for vegetation pixels, the most relevant features of each group are: S λ as it returns isotropic local distribution, R z since no robust plane can be fitted to vegetation and finally the echo width w as already stated in [START_REF] Wagner | 3D vegetation mapping using small-footprint full-waveform airborne laser scanners[END_REF]. For the latter classes, variable importance seems to be more dispersed between different attributes. This dispersion is correlated to the higher error rates on both classes (cf. Table 4).

Computing time

The computing time for feature selection process (21 features) to 17 selected lidar features is 367 minutes. This value is not critical since the feature selection is done only one time provided a set of input attributes. The classification process for 17 lidar selected features needs 11.36 minutes. Random Forests do not need an intensive computing time, however a considerable amount of memory is needed to store a N by T matrix in memory.

CONCLUSION

In this work, Random Forests were successfully applied to the lidar feature selection to classify urban scenes. 21 lidar features were proposed and separated into five groups: height-based, echobased, eigenvalue-based, plane-based and FW features. The method is a decision tree based ensemble classifier. It provides accurate classification and a variable importance estimate based on the permutation accuracy criteria. A feature selection was processed by iterative backward feature elimination. A minimal feature vector with 6 features provides a low OOB error rate. The most relevant feature for all classes is the height difference. Echo-based attributes seem to be non-relevant.Two FW features A and W appear in the final set, which confirms the contribution of fullwaveform lidar for urban scene classification. The 2D window size impact on variable importance has been studied. Small sizes should be used to enhance the feature discrimination and to improve classification accuracy. Some non-relevant lidar features in our context, should be more useful for a finer classification such

Figure 1 :

 1 Figure 1: Orthoimage and some representative Lidar features.

Figure 2 :

 2 Figure 2: Variable importance of Lidar data by mean decrease permutation accuracy.

Figure 3 :

 3 Figure 3: Iterative elimination Lidar feature selection.

  (a) Classification results -Lidar Features : 5×5 window size (b) Difference with the ground truth

Figure 4 :

 4 Figure 4: Classification result (T =60 trees and M =4.)

Figure 5 :

 5 Figure 5: Variable importance per class -3×3 window size

Table 1 :

 1

	Type	Symbol Feature
		Δz	Height diff. to the ground
	Height	Δz f l	Height diff. first-last echoes
	features	σ 2 z	Height variance
		C	Local curvature
	Echo	N	Number of echoes
	features	N e	Normalized number of echoes
		λ1	Highest eigenvalue
		λ 2	Medium eigenvalue
	Eigenvalue	λ 3	Lowest eigenvalue
	features	A λ	Anisotropy
		P λ	Planarity
		S λ	Sphericity
		L λ	Linearity
		Nz	Deviation angle
	Local plane	σ 2 n z	Variance of deviation angles
	features	R z	Residuals to the local plane
		DΠ	Distance to the local plane
		A	Echo amplitude
	FW Lidar	w	Echo width
	features	σ	Cross-section
		α	Echo shape

Synthesis of Lidar features for classification Networks and Decision Trees are two examples of unstable classifiers. The most widely used methods are bagging and boosting. Boosting iteratively reproduces the training set for each classifier by increasing the weight to the incorrectly classified samples in previous classifier. Bagging is the acronym of "bootstrap aggregating". It is made of the ensemble of bootstrap-inspired classifiers produced by sampling with replacement from training instances and uses these classifiers to get an aggregated classifier. It aims at reducing the variance of a classifier

Table 2 :

 2 Specifications of Biberach data set.

	Flight height Footprint size	PRF	Pulse density
	500 m	0.25 m	100 KHz	2.5 (pt/m 2 )
	et al., 2008) with fewer features. The number of available refer-
	ence samples is 797364 and they are split almost evenly between
	training and test samples.
	Class	Training samples Test samples
	Buildings	187673	188015
	Vegetation	15982	15723
	Nat. Ground	2174	2149
	Art. Ground	192704	192944
	Total samples 398533	398831

Table 3 :

 3 Data Set.

  .

	Class	Art.Grnd Build.	Nat.Grnd Veget. Error %
	Art.Grnd 188562	3325	5	1052	2.3
	Build.	13946	173545 5	519	7.7
	Nat.Grnd 500	20	1622	7	24.5
	Veget.	2604	566	0	12553 20.2
	Table 4: Confusion matrix for test data using 60 trees and 4 split
	variables. Total error rate=5.65%, 3×3 window size
	Class	Art.Grnd Build.	Nat.Grnd Veget. Error %
	Art.Grnd 186047	5483	33	1381	3.6
	Build.	7364	180154 2	495	4.2
	Nat.Grnd 1403	33	703	10	67.3
	Veget.	3020	712	2	11989 31.1
	Table 5: Confusion matrix for test data using 60 trees and 4 split
	variables. Total error rate=4.99%, 5×5 window size

ACKNOWLEDGEMENTS

The Random Forests implementation software by L. Breiman and A. Cutler was used in experiments. For this work, we used the R interface of the software which is distributed freely on .


Jérome Demantké

Clément Mallet

Nicolas David

Bruno Vallet

DIMENSIONALITY BASED SCALE SELECTION IN 3D LIDAR POINT CLOUDS

Keywords: point cloud, adaptive neighborhood, scale selection, multi-scale analysis, feature, PCA, eigenvalues, dimensionality

This papers presents a multi-scale method that computes robust geometric features on lidar point clouds in order to retrieve the optimal neighborhood size for each point. Three dimensionality features are calculated on spherical neighborhoods at various radius sizes. Based on combinations of the eigenvalues of the local structure tensor, they describe the shape of the neighborhood, indicating whether the local geometry is more linear (1D), planar (2D) or volumetric (3D). Two radius-selection criteria have been tested and compared for finding automatically the optimal neighborhood radius for each point. Besides, such procedure allows a dimensionality labelling, giving significant hints for classification and segmentation purposes. The method is successfully applied to 3D point clouds from airborne, terrestrial, and mobile mapping systems since no a priori knowledge on the distribution of the 3D points is required. Extracted dimensionality features and labellings are then favorably compared to those computed from constant size neighborhoods.

INTRODUCTION

Point cloud data from airborne and terrestrial devices provide a direct geometrical description of the 3D space. Such information is reliable, of high accuracy but irregular and not dense. However, the underlying structures and objects may be detected among sets of close 3D points. The local geometry is estimated by the distribution of points in the neighborhood. Finding the best neighborhood for each point is a main issue for a large variety of common processes: data downsampling, template fitting, feature detection and computation, interpolation, registration, segmentation, or modelling purposes. The notion of neighborhood and its fundamental properties are fully described in [START_REF] Filin | Neighborhood systems for aiborne laser data[END_REF]. The neighbors of a lidar point are traditionally retrieved by finding the k nearest neighbors or all the points included in a small restricted environment (sphere or cylinder) centered on the point of interest. The main problem stems from the fact that the k and environment radius values are (1) usually heuristically chosen, and (2) assumed to be constant for the whole point cloud, instead of being guided by the data. This does not ensure that all theses neighbors belong to the same object as the current point. Therefore, its local description may be biased when including several distinct structures, and provides erroneous feature descriptors. Moreover, the relative variation in the spatial extent of geometrical structures is ignored. For aerial datasets, problems will occur at the borders between objects and for objects which size is inferior or close to the neighborhood size. For terrestrial datasets, in addition, the point density may significantly fluctuate due to foreground object occlusion, dependence on distance and relative orientation of the objects [START_REF] Soudarissanane | Incidence angle influence on the quality of terrestrial laser scanning points[END_REF], leading to data sparseness or irregular sampling. This paper aims at proposing a methodology to find the optimal neighborhood radius for each 3D point on a lidar point cloud. An "optimal" neighborhood is defined as the largest set of spatially close points that belong to the same object as the point of interest. The inclusion of points lying on different surfaces is prohibited. The context of the study is rather general: in order to be applicable both on terrestrial (TLS) and airborne (ALS) datasets, the method is simply based on the point location, without requiring knowledge on intensity, echo number or full waveforms. The topology resulting from the sequential acquisition of the data is also considered to be lost ("unorganized" point cloud), preventing the adoption of specific scan line grouping methods [START_REF] Hadjiliadis | Sequential classification in point clouds of urban scenes[END_REF]. Furthermore, the process is designed out of the scope of any application, even if the final goal is indeed to be beneficial to any application requiring a correct local description around each 3D point. The problem of scale selection has been mainly tackled for surface reconstruction and feature extraction of scanned opaque objects. Several approaches have therefore been developed for noisy point clouds and irregular sampling issues. Most of them are surface-based i.e., they try to fit a curve or a surface of some form to the 3D point cloud [START_REF] Pauly | Point-based multiscale surface representation[END_REF]. Finding the optimal group that well represents the local geometrical properties is performed using indicators such as the normal and/or the curvature [START_REF] Hoppe | Surface reconstruction for unorganized point clouds[END_REF][START_REF] Zwickler | Pointshop 3d: An interactive system for point-based surface editing[END_REF][START_REF] Dey | An adaptive MLS surface for reconstruction with guarantees[END_REF][START_REF] Belton | Classification and segmentation of terrestrial laser scanner point clouds using local variance information[END_REF]. For instance, it is retrieved by minimizing the upper bound on angular error between the true normal and the estimated one. Starting for the minimal possible subset around the point of interest, the neighborhood is iteratively increased until the angular variance reaches a predefined threshold [START_REF] Mitra | Estimating surface normals in noisy point cloud data[END_REF]. Such works have been theoretically improved by [START_REF] Lalonde | Scale selection for classification of point-sampled 3-D surfaces[END_REF], no more requiring knowledge on the data distribution, and applied to mobile mapping datasets. An alternative work, based of the expression of the positional uncertainty, is introduced in [START_REF] Bae | A closed-form expression of the positional uncertainty for 3D point clouds[END_REF] for processing TLS datasets. However, these methods are effective for smoothly varying surfaces and may not be adapted to anthropic surfaces acquired with various kinds of lidar systems. Furthermore, the model-based assumption does not hold when dealing with objects without predefined shapes (e.g., vegetated areas) or with noise stemming for relief high frequencies (e.g., chimneys and facades for ALS data or pedestrians and points inside buildings for TLS data). Consequently, in our context, a more suitable solution is to directly compute shape features [START_REF] Gumhold | Feature extraction from point clouds[END_REF][START_REF] Belton | Classification and segmentation of terrestrial laser scanner point clouds using local variance information[END_REF], i.e., low-level primitives that may capture the variability of natural environments. The proposed methodology is developed in Section 2. The shape features are described in Section 3. The computation of the optimal neighborhood radius embedded in a multi-scale framework is proposed in Section 4. Results on various laser scanning datasets are presented in Section 5, and conclusions are drawn in Section 6.

PROPOSED METHOD

Description

The methodology aims at finding the optimal neighborhood radius for each lidar point, working directly and exclusively in the 3D domain, without relying on surface descriptors (such as normals) or structures (such as triangulations or polygonal meshes). It is composed on two main steps:

1. Computation of three dimensionality features for each point, between predefined minimal and maximal neighborhood scale. These features describe the distribution of the points in the 3D space, and more exactly, the matching between the local point cloud and each of the three dimensionalities (linear, planar or volumetric).

2. Scale selection: retrieval of the neighborhood radius for which one dimensionality is most dominant over the two others.

The three dimensionality features (a1D-a2D-a3D) are computed exhaustively, at each point and for each acceptable neighborhood scale, from the local covariance matrix. An isotropic spherical neighborhood, centered on the point of interest, is adopted for this purpose. These low-level features, as well as the automatic set up of the radius lower and upper bounds are described in Section 3. Then, the optimal radius is retrieved by comparing the behaviours of these three features between the minimal and maximal acceptable radius. Two radius-selection criteria are tested to evaluate each scale and find the most relevant value. This multi-scale analysis is performed in order to capture variation in shape when aggregating points for an object distinct from the object of interest (edge effect or outliers). It is also useful in case of significant density variation and lack of support data for gathering points over a large volume while ensuring the conservation of the favorite dimensionality. Finally, our method presents three interesting characteristics:

• Definition of a confidence index of the saliency of one dimensionality over the two other ones.

• Multi-scale analysis and automatic set up of the bounding scales. • Labelling of each point according to its privileged dimensionality, providing an interesting basis for segmentation and classification algorithms.

Datasets

In order to assess the relevance of the proposed approach for various point densities, point distributions and points of view, three kinds of lidar datasets are tested: airborne, terrestrial static, and acquired with a mobile mapping system (named ALS, TLS, and MMS, respectively).

ALS : Three datasets are used. The first one (ALSG) has been acquired over Biberach (Germany), covering both residential and industrial areas as well as a city center with small buildings (point density of 5 pts/m 2 ). The second dataset (ALSR) concerns a residential area in Russia, with 5 pts/m 2 [START_REF] Shapovalov | Nonassociative markov networks for 3D point cloud classification[END_REF].

Finally, the third one covers the dense city center of Marseille (France), with high buildings, and thus sparse points on the building facades (ALSF ). Three parallel strips are present: the point density therefore varies between 2 and 4 pts/m 2 (for one strip and for the overlapping areas, respectively).

TLS : The terrestrial scans acquired over the Agia Sanmarina church (Greece) have been processed [START_REF] Bae | A closed-form expression of the positional uncertainty for 3D point clouds[END_REF]. The dataset first offers a large variety of structures of various sizes as well as sparse vegetation on the ground. Furthermore, the point density significantly varies with the orientation of the surfaces with respect to the scanner position.

MMS : Datasets over two urban areas (France and United States, respectively MMSF and MMSU ) from distinct mobile mapping systems have been processed [START_REF] Munoz | Contextual classification with functional max-margin markov networks[END_REF]. Such datasets also include man-made objects of various sizes and shapes, with varying point densities. The two specificities of MMS datasets are (1) gaps in the point cloud due to the occlusion of foreground objects, and (2) vertical privileged directions in the point clouds due to the sequential acquisition by lines.

SHAPE FEATURES

In this work, we focus on shape features computed on the neighborhood of a lidar point. The neighborhood V r P of a point P at scale r is defined as the set of points P k verifying :

P k ∈ V r P ⇔ P -P k ≤ r. (1)
Thus, the neighborhood environments are 3D spherical volumes. They ensure isotropy and rotation invariance, such that the computed shape descriptors are not biased by the shape of the neighborhood. Furthermore, r is the single parameter to be optimized.

A classical approach consists in performing a Principal Component Analysis (PCA) of the 3D coordinates of V r P [START_REF] Belton | Classification and segmentation of terrestrial laser scanner point clouds using local variance information[END_REF][START_REF] Gross | Segmentation of tree regions using data of a full-waveform laser[END_REF]. This statistical analysis uses the first and second moments of V r P , and results in three orthogonal vectors centered on the centroid of the neighborhood. The PCA synthesizes the distribution of points along the three dimensions [START_REF] Tang | First order augmentations to tensor voting for boundary inference and multiscale analysis in 3-D[END_REF], and thus models the principal directions and magnitudes of variation of the point distribution around the center of gravity. These magnitudes are combined to provide shape descriptors for each of the three dimensions. More advanced features based on harmonics or spin images [START_REF] Frome | Recognizing objects in range data using regional point descriptors[END_REF][START_REF] Golovinskiy | Shape-based recognition of 3D point clouds in urban environments[END_REF] are not necessary since the segmentation task, which indeed requires contextual knowledge, is not tackled in this paper.

Principal Component Analysis

Let xi = x i y i z i T and x = 1 n i=1,n xi the center of gravity of the n lidar points of V r P . Given M = x1 -x ... xn -x T , the 3D structure tensor is defined by C = 1 n M T M.
Since C is a symmetric positive definite matrix, an eigenvalue decomposition exists and can be expressed as C = RΛR T , where R is a rotation matrix, and Λ a diagonal, positive definite matrix, known as eigenvector and eigenvalue matrices, respectively. The eigenvalues are positive and ordered so that λ1 ≥ λ2 ≥ λ3 > 0. ∀j ∈ [1, 3], σj = λj, denotes the standard deviation along the corresponding eigenvector -→ vj . Thus, the PCA allows to retrieve the three principal directions of V r P , and the eigenvalues provide their magnitude. The average distance, all around the center of gravity, can also be modeled by a surface. The shape of V r P is then represented by an oriented ellipsoid. The orientation and the size informations are divided between R and Λ : R turns the canonical basis into the orthonormal basis ( -→ v1, -→ v2, -→ v3) and

√

Λ transforms the unit sphere to an ellipsoid (σ1, σ2 and σ3 being the lengths of the semi-axes). As enhanced in Figure 1 and for instance in [START_REF] Gross | Segmentation of tree regions using data of a full-waveform laser[END_REF], such an ellipsoid reveals the linear, planar or volumetric behaviour of the neighborhood i.e., whether the point set is spread in one, two or three dimensions (blue, gray, and green ellipsoids in Figure 1, respectively).

Figure 1: Three examples of ellipsoids computed over three areas of interest of distinct dimensionalities for the TLS dataset (a tripod over a low and sparse vegetation -height colored).

Dimensionality features and labelling

Various geometrical features can be derived from the eigenvalues. Several indicators have already been proposed [START_REF] West | Context-driven automated target detection in 3D data[END_REF][START_REF] Toshev | Detecting and parsing architecture at city scale[END_REF], and the following ones have been selected (Figure 2) to describe the linear (a1D), planar (a2D), and scatter (a3D) behaviors within V r P :

a1D = σ1 -σ2 µ , a2D = σ2 -σ3 µ , a3D = σ3 µ ,
where µ is the normalization coefficient. Both choices µ = σ1 and µ = d=1,3 σ d are conceivable and imply a1D, a2D, a3D ∈ [0, 1]. The dimensionality labelling (1D, 2D or 3D) of V r P is defined by: d

* (V r P ) = arg max d∈[1,3] [a dD ]. (2)
If σ1 σ2, σ3 0, a1D will be greater than the the two others so that the dimensionality labelling d * (V r P ) results to 1. Contrariwise, if σ1 , σ2 σ3 0, a2D i.e., the planar behavior will prevail. At last, σ1 σ2 σ3 implies d * (V r P ) = 3.

We chose µ = σ1 because then : a1D + a2D + a3D = 1, so that the three features can be considered as the probabilities of each point to be labelled as 1D, 2D, or 3D. It will help us to select the most appropriate neighborhood size by finding which radius favors the most one dimensionality (see Equation 3).

Figure 2: Behaviours of the three shape features (TLS dataset).

SCALE SELECTION

Optimal neighborhood radius

As presented in Section 2, the dimensionality features are computed for increasing radius between a lower bound and an upper bound (rmin and rmax, respectively). Their set up is presented in Section 4.2. They represent the minimal and maximal acceptable neighborhood radius according to the area of interest and the sensor. The [rmin, rmax] space has been sampled in 16 values, which is a suitable trade-off between tuning accuracy and computing time. Since the radius of interest is usually closer to rmin than to rmax, the r values are not linearly increased but with a square factor. This allows to have more samples near the radius of interest and less when reaching the maximal values.

Two radius-selection criteria have been developed, namely the entropy feature (E f ) and the similarity index (Si). First, a measure of unpredictability is given by the Shannon entropy of the discrete probability distribution {a1D, a2D, a3D}:

E f (V r P ) = -a 1D ln(a 1D ) -a 2D ln(a 2D ) -a 3D ln(a 3D
).

(3)

The lower E f (V r P ) is, the more one dimensionality prevails over the two other ones. The relevance of scale selection is demonstrated in Figure 3. This criterion allows to define an optimal radius r * E f that minimizes E f (V r P ) in the [rmin, rmax] space :

r * E f = arg min r∈[r min , rmax] E f (V r P ). (4)
A dimensionality labelling is then provided by : d * (V

r * E f P
). Second, Si(V r P ) is defined as the ratio of neighbors P k which dimensionality labelling is the same as P at scale r:

Si(V r P ) = 1 n P k ∈V r P 1 {d * (V r P )=d * (V r P k )} . (5)
where 1 {.} is the Indicator function and n is the number of points within V r P . Si evaluates the homogeneity of the labelling within V r P by measuring the labelling similarity between neighboring points. Finally, we have:

r * Si = arg max r∈[r min , rmax] Si(V r P ). (6)
Si aims at reducing the noise of the results since r * Si is the scale for which the labelling of the current point is the most similar to the labellings of its neighbors at the same scale. Both criteria have been tested for our datasets. Results and conclusions are presented in Section 5.1.

Figure 3: Illustration of the relevance of the entropy feature for scale selection (building roof in ALSG). Left: 3D point cloud (height colored). Right: 3D point cloud colored with E f computed for one point of interest (pink dot). The color of each neighbor P k corresponds to the E f value, computed on the smallest neighborhood containing P k . The entropy first decreases until the neighborhood reaches the edge of the roof (blue circle -optimal size). Then, the entropy increases and becomes maximum when the neighborhood gathers distinct objects (red circle), here the ground and a chimney. Figure 4 provides an example of the behaviour of the shape fea-tures and the radius-selection criterion for two areas of the TLS dataset. Figure 4c corresponds to an area with higher noise than in Figure 4d due to distinct facade orientations. Thus, for Fig- ure 4c (white rectangle of Figure 4a, for low r values, the point set locally looks scattered and a3D prevails. Then, the increase of the size leads to the decrease of the noise effect. The planar feature becomes predominant. For Figure 4d, the facade is orthogonal to the laser beam resulting in lower noise. The neighborhoods remain planar with the increasing scale.

Bounding scales

The optimal neighborhood radius is retrieved between predefined lower and upper bounds. They depend on various characteristics, and are therefore specific to each dataset. The choice of the lower bound is driven by (1) the point cloud noise; (2) the sensor specifications; and (3) computational constraints :

1. Local scattering may appear for linear or planar surfaces.

It may stem from sensor noise or unfavorable geometrical configurations (e.g., laser beam nearly parallel to a surface). Shape features are then biased and the point cloud may locally be erroneously labelled as "volumetric". The knowledge of the intensity of such noise is a prerequisite for solving this issue. Alternatively, the minimum size can be estimated applying the methods mentioned in Section 1 for this purpose.

2. The laser beam deflection system or the kind of mapping device may create point clouds with very irregular landscape sampling (low values in one direction and larger values in the orthogonal one). This is particularly true for MMS since datasets are acquired line by line with forward motion of the device. To cope with this issue, the rmin value is increased until V r P blends points belonging to at least two different scan lines.

3. A minimal number of points is required for the PCA. We consider relevant statistics start with 10 points. If the two first above-mentioned issues are not problematic, the point density directly provides the lower bound.

The selection of the upper bound is not critical and can be tuned with the knowledge of the lower frequencies of the relief i.e., the size of the largest objects in the scene. For TLS and MMS datasets, this corresponds to facades (typically 3 m) whereas for ALS data, ground regions and large buildings are involved. As E f and Si values may remain constant for these large planar areas, a cut-off value around 5 m works well, even if in practice 3-4 m are sufficient.

RESULTS

Comparison between radius-selection criteria

Both criteria E f and Si have been tested, and radius selection results can be significantly different. The entropy feature criterion E f has been preferred to the similarity index Si for two main reasons.

First, as illustrated in Figure 4, E f and Si may provide distinct optimal radius that may correspond to distinct behaviours and privileged dimensionalities. For instance, in Figure 4c, Si erroneously selects a radius for which the volumetric dimensionality is predominant whereas E f allows to detect the correct planar behaviour of the underlying surface. Si favors the most homogeneous scale, but, in practice, several radii may provide the same result (Si = 1). The smallest one is kept which has been revealed to be problematic for the TLS dataset since the optimal radius corresponds to the noise level. For ALS and MMS, such problem of erroneous dimensionality labelling does not occur. However, small radius are favored for large objects since Si may saturate, preventing the selection of larger radius. Consequently, the choice of the lower bound rmin for the multi-scale analysis becomes critical. Second, for each 3D point at each radius of interest, Si depends on the dimensionality labelling of the neighbors at this neighborhood size (i.e., E f ). Two iterations are therefore necessary to retrieve the optimal radius which requires more computational time and memory.

Scale selection

The six datasets have been processed with the proposed approach. Scale selection results are displayed in Figures 4 and5, using the E f criterion. The expected behaviours are reached showing the efficiency of the multi-scale analysis using E f :

• Planar areas exhibit high values for ALS and TLS datasets (see Figures 4b and5a). This is particularly true for the TLS dataset since the multi-scale analysis allows to overtake the noise level.

• Lowest scales correspond to border areas, allowing to retrieve building and vegetation edges for ALS datasets, windows in MMS datasets (Figures 5c andd), or small architectural elements in TLS.

• The algorithm allows to select high values when no predominant dimensionality is found for small and medium radius values. A sufficient number of 3D points is collected to retrieve a coherent dimensionality : 1D for wires and poles (Figures 5c andd), 2D for building facades in ALS datasets (e.g., highest values in Figure 5b which correspond to a few number of points lying on the facades of high buildings) or 3D for vegetated areas (all datasets).

• The method deals well with varying point densities. Such variation may come from the number of overlapping strips for ALS datasets (a lower radius size is necessary when two The effectiveness of the scale selection process can be assessed by retrieving the predominant dimensionality that has been retrieved (1D, 2D or 3D ?) for various objects of interest. Several examples are presented in Figure 5, and comparisons with manually labelled ALS and MMS data have been performed (cf. Figure 6). For both datasets, planar objects are correctly retrieved, with few errors corresponding to building edges or small urban items, especially for ALS data. Besides, objects with one privileged dimension such as poles, wires or trunks are mainly labelled as 1D (M M SU dataset, in Figure 6b). However, since these objects are in fact cylindrical with a small width (e.g., trunks or traffic lights), they may look planar or volumetric for medium-sized neighborhoods. Conversely, volumetric objects such as trees may locally look planar. This happens for large objects low point densities or when no multiple scatterings are found with vegetated areas. Such phenomena is limited for genuine 3D acquisitions (TLS or MMS datasets), whereas it is clearly enhanced for 2.5D data such as for the ALSR data (Figure 6a). High vegetation areas are principally labelled as planar, which is due to the dense canopy cover. Nevertheless, as displayed in Figure 5a, such effect almost disappears when 3D points are acquired within tree canopies.

Comparisons with constant neighborhood size

The adaptive size strategy allows to retrieve a correct number of points to estimate the local dimensionality of the point set. Such strategy may be compared with the results achieved with neighborhoods of fixed size for a whole dataset. Firstly, the three shape features are computed with both strategies for four classes of interest of the ALSR dataset (Figure 7). One can see that the planar behaviour of ground and building points is improved with the adaptive size. Besides, for vegetated areas, the volumetric behaviour is slightly enhanced, however this happens in conjunction with an increase of the planar behaviour (dense canopies).

Figure 7: Improvements in shape feature computation using an optimal neighborhood radius per point. adD and a * dD are the shape feature for dimensionality d, computed with neighborhoods of constant size and adaptive size, respectively. The constant neighborhood size corresponds to the 30 nearest neighbors of each point.

Secondly, in addition to improved shape features, Figure 8 shows that the privileged dimensionality is also better retrieved with the adaptive strategy (MMSF dataset). This is particularly true for trees (1D→3D), small building facades elements (1D→2D), and tree trunks (2D→1D). Furthermore, since the geometrical analysis is less affected by the 3D point distribution, the labelling procedure is far less noisy.

Figure 8: Dimensionality labelling for the MMSF dataset using a constant neighborhood size for the whole point cloud (left), and the optimal value per point selected with E f (right). The constant neighborhood size corresponds to a 1 m radius.

CONCLUSIONS AND FUTURE WORK

The objective of this paper was to provide a simple and general method for finding the optimal neighborhood radius for each 3D point of a lidar point cloud, with the simple knowledge of the point locations. The most appropriate scale of analysis has been automatically selected among a range of potential values using three low-level geometrical shape features describing the "dimensionality" of the local point set (1D-2D-3D). Results on distinct lidar point clouds acquired with airborne, terrestrial and mobile mapping systems under various conditions showed the effectiveness of the proposed method. No a priori knowledge or assumption on the point cloud distribution, density, laser scan pattern, object size or underlying structures was required to achieve such results. Several experiments demonstrated how favorably our method performed compared to constant neighborhood sizes, and how relevant such approach is. Finally, the prevailing dimensionality and the scale of interest for the underlying object are found for each point, providing interesting cues for many potential subsequent segmentation or classification algorithms.

Improvements will focus on two main issues. Firstly, as illustrated in Figure 3, in addition to the global minimum of the entropy feature providing the optimal neighborhood radius, the other minima are likely to be relevant for multi-scale characterization. Secondly, each 3D point is analyzed separately resulting in noisy dimensionality labelled point clouds. Therefore, such results would be improved and fastened by taking into account for close points the scale already computed for optimal neighbors. Further work will consist in detecting and characterizing boundaries between objects by using and quantifying variations in the entropy feature. Finally, segmentation and classification issues will be tackled in order to fully benefit from the computation of non biased shape features and from the retrieval of a correct neighborhood for each point.

Figure 4 :

 4 Figure 4: Optimal neighborhood radius retrieval (TLS dataset). 3D point cloud colored with (a) the intensity and (b) r * . (c-d) Variation of the shape features and radius-selection criteria over the neighborhood radius over 50 points within the white (c) and black (d) rectangles of (a).

Figure 5 :

 5 Figure 5: Radius selection (r) and dimensionality labelling for (a) ALSG, (b) ALSF , (c) MMSU , and (d) MMSF datasets.overlapping strips exist for an area of interest, see Figure5b), from occlusions or from the fluctuating incidence angle of the laser beam on the surfaces for TLS or MMS datasets (Figure5d).

Figure 6 :

 6 Figure 6: dimensionality labelling statistics for two datasets available with ground truth. (a) ALSG dataset: the five main classes have been conserved. (b) MMSU dataset: the 59 classes have been condensed into four classes according to their shape (1D: wires, poles, trunks etc. -2D: wall, door, facade etc. -3D: foliage, grass etc. -Clutter: objects without specific shapes). The number of points for each class is indicated inside brackets.

ACKNOWLEDGEMENTS

The authors would like to thank the ISPRS working group V/3 for providing the "Agia Sanmarina Church" dataset.


Mihail Garajeu

Renaud Masson

Michel Bornert

Christelle Duguay

Arnaud Monnier

Joane Meynard

Porosity dependence of thermal conductivity in UO 2 nuclear fuels

BACKGROUND AND KEY ISSUE

Limits of classic conductivity UO 2 laws o Example of two UO 2 fuels with the same % porosity o Description of UO 2 fuels with only the % total porosity is not adequate to describe the thermal behaviour of these fuels.

ooo««o

  Determination of a new thermal behaviour law which gives the influence of porosity (shape, orientation and network) on thermal conductivity for non-irradiated UO 2 Two porosity types classified by pore morphology embedded into an urania matrix o Two different descriptions of the studied material have been considered. Study of the characteristic dimensions of both porosity types  Spatial scale separation between both porosity families o Opportunity to study separately the influence of each family Study of the influence of the spherical porosity κ(c s ) Study of the influence of the filamentlike porosity κ(c l ) Study of the influence of porosity 𝜿(𝒄 𝒔 , 𝒄 𝒍 ) = 𝜿 𝒄 𝒔 𝜿(𝒄 𝒍 ) C. MAXWELL, A treatise on electricity and magnetism, 1873 ** H. L. DUAN et al., Phys. Rev. B 73(17): 174203(13), May 2006 𝜿(𝒄 𝒔 , 𝒄 𝒍 ) = 𝜿 𝒄 𝒔 𝜿(𝒄 𝒍 ) SES 2018 | J. MEYNARD | PAGE Study of the influence of the lenticular porosity κ(c l ) Penny-shaped inclusions Disk-shaped cracks Study of the influence of the spherical porosity κ(c s ) « Diluted medium » model for spherical pores equiprobably distributed in the matrix (first-order approximation of Maxwell model*) Diluted medium » model** for penny-shaped pores equiprobably distributed in the matrix (first-order approximation in 𝑐 𝑙 𝜔 ) Diluted medium » model*** for disk-shaped cracks equiprobably distributed in the matrix 𝜿 𝝆 𝒇 = 1 -8 3 𝜌 𝑓 𝑰 *** B. SHAFIRO et M. KACHANOV, J. Appl. Phys., Vol. 87, No. 12, 2000 penny-shaped inclusion model Penny-shaped inclusion model more representative than the classic conductivity UO 2 law Remaining model/experience bias which does not allow a distinction between UO 2 fuels based on their thermal behaviour Evaluation of the representativeness of the "penny-shaped" model SES 2018 | J. MEYNARD | Development of the penny-shaped inclusion model which is more representative than classic UO 2 laws  The modelling of porosity dependency of thermal conductivity in UO 2 nuclear fuels is improved. o The penny-shaped inclusion model is not sufficient to discriminate different UO 2 nuclear fuels.  Necessity to improve modelling Work in progress o Finalization of the disk-shaped crack model  Determination of the crack density 𝜌 𝑓  Evaluation of the representativeness of filament-like pores by disks o Evaluation of filament-like pore interconnectivity with additional porosity