come

Impaired regeneration as a component of muscle wasting in immobilization


Cécile Labadie

Christian Ginies

Frederic F Carlin

Catherine M G C Renard

Frederic F Carlin

Hydrosols Cécile Labadie

Céline Cerutti

Christian Giniès

Frédéric Carlin

Characterisation of spoilage microflora in Rose flower and Orange Blossom hydrosols

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Rose flower and Orange Blossom hydrosols are steam distillation products mainly used as food flavoring agents or ingredients of cosmetic products. These are subjected to microbial proliferation that can alter organoleptic properties and prevent cosmetic use due to non-compliance to professional microbiological standards. The aim of this study was to ascertain whether there were specific microorganisms in Rose water and Orange Blossom water, their growth parameters, and their impact on the aromatic compounds. Then, we determined the ability of some selected spoilage and pathogenic bacteria to grow in hydrosols.

Rose flower and Orange Blossom hydrosol samples from the Mediterranean Basin studied in this work contained a maximal amount of 0.06 % of essential oil giving the flavoring properties and quality; this concentration was not high enough to inhibit bacterial growth. These hydrosols also contained floral compounds carried over by foaming and priming during the distillation process. These non volatile compounds (sugars, fatty acids, and carboxylic acids), can be used as nutrients by microorganisms. Microbiological concentrations are generally comprised between 10 5 and 10 7 total counts cfu/mL and bacteria were the only micro-organisms detected in both products. Approximately 40 strains were isolated in both hydrosols from different locations. 16S rRNA gene sequencing identified the major species as Pseudomonas sp., Burkholderia sp., and two undefined species belonging to Acetobacteraceae and Rhodospirillaceae. Changes in populations were observed over 3 and 6 months storage period at ambient temperature and at 5°C. Changes in aromatic profiles during storage were also followed. Growth capacities of pathogenic species and spoilage bacteria had been followed in hydrosols. These results provide a better understanding of hydrosols microflora and may open the door for new ways to improve quality of Rose flower and Orange blossom hydrosols.


Anne--Chris3ne Lalmanach

Research project avian defensins as alterna3ve to an3bio3cs

Defensins

• Molecular arsenal of innate immunity.

• Ca3onic pep3des, β--sheet structure, 6 cysteins highly conserved (3 S--S bonds). • Three families: α (only in mammals), β --> throughout evolu3on, θ (only in primates).

•

  Avian β--defensins (AvBDs), 14 members (for review: van Dijk et al, Male chickens, 3 weeks old (broilers), from Lynn et al, Immunogene3cs, 2004 A--C Lalmanach _ Rencontres Recherche -Industriels SIMV BioFIT Lille, 2--3 Décembre 2014 heterophils granulocytes (Harwig et al, FEBS Led, 1994; Evans et al, Overexpression in intes3nal EC from chicken resistant to Salmonella (Derache et al, Dev Comp Immunol, 2009) • Structural features AvBD2 natural purified or synthe3c pep3de, 3D structure solved by NMR 6 Prepara3on of AvBDs (1)• Recombinant avian β--defensins (AvBD)Various system used: E coli (Origami strains), Pichia pastoris, Insect cells… Unsuccessfully

with trypsin, chymotrypsin, elastase, cathepsin D and H.

--Par3al suscep3bility of AvBD2 to cathepsin B and S.

--Total sucep3bility of AvBD2 to cathepsin L and K.

--Total resistance of AvBD7 to 9 proteases tested (conserva3on of an3microbial ac3vity) 9

Resistance of AvBDs to proteolysis


domain: sdv info sde

text:
Contribute to reference method and reference system

Laboratory conditions not real conditions (complex gas mixture)

Comparison and calibration

Reference source

Reduce uncertainty

  questions (gas, system, trend) • Diversity of animal farms: size, species, etc.• Emitting intensity: activity, duration,

  Increase in aerobic activity: CO 2 , H 2 O losses (N 2 ?) Increase in anaerobic activity: CH 4 increase => collected? Decrease in biological activity or size of the system increase the climate influence (température, precipitations) on the emissions e.g. stocking density (or daily gain) increase heat production and water input per m² => litter more anaerobic e.g. compost stabilization induce emission decrease A G R I C U L T U have usually a small impact on overall GHG emissions During long term processes special attention must be given to GHG emissions Compost turning itself has a small impact on emission, induced processes should be considered Long term compost stabilization can induce emission of N 2 O or CH 4 when C and H 2 O (moisture) are not adequate Litter accumulation for several poultry batches increase emissions (idem for cattle manure) Excretion removal from animal house reduce emissions A G R I C U L T U

Thresholds for Life Cycle Assessment/multicriteria comparisons? fertilizers Introduction 3. Which trend per gas and per system? « ecological footprint » Society Animals Crops Natural resources, mineral and organic, flora & fauna, waters, soils, air, seas

	Population increase
	& needs for non food products
	unchanged production systems
	Baseline
	(business as usual)
	Increase recycling,
	decrease emissions
	with ecological
	intensification

Emitting intensity

Dilution of nutrients and increase in biological interactions/diversity reduce emissions

Emitting processes are not governed in the same way for extensive/intensive animal groups (e.g. house/grazing) Increasing biological interactions in concentrated substrates can help to decrease emissions but increases the risk


Data (2013) available

Region Site nb Visit nb Quadrats Boundary vegetation

	Armorique	3	1	20 x 1sqm	Not sampled
	Camargue	3	1	20 x 1sqm	Not sampled
	Coteaux	3	1	5 x 4sqm	Not sampled
			1 in 2012	5 x 10sqm (2012)
	EastAnglia	4	2 in 2013	5 x 5sqm (2013)	Not sampled
				20 x 1sqm (v1)	Sampled in suppl. transect (B-
	Goettingen	3	2	5 x 4sqm (v2)	boundary; only during v1)
	Lleida	3	1	20 x 1sqm	Not sampled
	Ontario	4	2	5 x 10sqm	included in Boundary transect
	PVDS	3			Formating in progress

Variability in the number of visit, sampling area, sampling design 2 Plant « abundance » = percent cover (from 0 to 100 %)

Total species richness Species richness and taxonomy 4 Total species richness Plant identified at the family, genus, species and subspecies levels and… unknown species ! How to deal with… We* propose to…

	Cumulated richness curves (accumplot)
	Differences in sampling area Differences in the number of visits Goettingen Galium aparine Viola arvensis Poa trivialis Papaver rhoeas Sisymbrium officinale 0 100 Convolvulus arvensis region species per Alopecurus myosuroides dominant Cirsium arvense The 10 most Bromus sterilis Elytrigia repens Armorique Lolium perenne Agrostis stolonifera Dactylis glomerata Holcus mollis Poa annua Poa trivialis Holcus lanatus Medicago sativa Trifolium repens Polygonum aviculare Camargue Phragmites australis Derbesia sp. Bolboschoenus maritimus Lemna minuta Echinochloa phyllopogon Cyperus difformis Typha domingensis Leersia oryzoides Lemna minor Polypogon monspeliensis species richness	Standardize percent cover per m² Average plant cover over both visits Exclude plant identified at the family level Rule out subspecies Rapahanistrum => Raphanes raphanistrum Manage plant identified at the genus level using the « Genus » R script developed by Yann C. & Clélia Glycine max sites Bromus madritensis Ambrosia artemisiifolia Bromus rubens Elymus repens 200 300 Diplotaxis erucoides Convolvulus arvensis Fumaria officinalis Taraxacum officinale Pastinaca sativa Equisetum arvense Anacyclus clavatus Vicia cracca Bromus diandrus Medicago sativa Avena sterilis Phleum pratense Papaver rhoeas Bromus inermis Lolium rigidum Ontario (2012) Lleida e.g. Raphanes raphanistrum subsp. East Anglia (2013) Coteaux Alopecurus myosuroides Bromus sp. Anisantha sterilis Convolvulus arvensis Hordeum vulgare Avena fatua Galium aparine Equisetum telmateia Veronica persica Rubus sp. Lolium perenne Sinapis arvensis Potentilla reptans inodorum Alopecurus myosuroides Geranium pusillum Veronica hederifolia Cirsium arvense Pisum sativum Geranium dissectum Tripleurospermum

Differences in plant identification

Exclude unknwown, « other » plant (i.e. moss, algae, …)

* Based on email exchanges with Clélia, the Solsona, Lleida and Goettingen teams

Preliminary results

6 803 species identified over all regions (without including PVDS)


domain: math info sdv

text:
Etienne E Klein

Aurore Bontemps

Sylvie Muratorio

Comparison of

published or not. The documents may come

It seems implicitly accepted in the literature that estimates based on seedlings or seeds provide effective or basic dispersal kernels respectively. Our results showed that it is not true for estimates obtained with CSM or SEMM because they rely on the composition of seed/seedling pools and not seed/seedling counts such as IM or GSM. This has important consequences for life-stage studies where the discordance of dispersal kernels estimated from different cohorts is used to investigate post-dispersal density-dependent mortality.

  that estimate seed dispersal kernels from genotypes of established seedlings Etienne KLEIN a , Aurore BONTEMPS b , and Sylvie ODDOU-MURATORIO b a UR546 Biostatistiques et Processus Spatiaux (BioSP), F-84914 Avignon, France b UR629 Ecologie des Forêts Méditerranéennes (URFM), F-84914 Avignon, France Keywords: spatial ecology, movement ecology, molecular ecology, seed dispersal Abstract: The seed dispersal kernel is a major determinant of spatial population dynamics and spatial distribution of genetic diversity. Among the main methods to estimate it, Inverse modelling (IM) and Gene Shadow Model (GSM) rely on seed or seedling counts in traps whereas Competing Source Model (CSM) and Spatially Explicit Mating Models (SEMM) rely on compositions of seed or seedling pools. GSM, CSM and SEMM exploit genetic information from molecular markers whereas IM only exploit seed/seedling counts ignoring their origins. In presence of post-dispersal density-dependent mortality (DDM) the effective dispersal kernel, describing the spatial distribution of established seedlings relatively to the seed source, is notoriously different from the basic dispersal kernel, describing the spatial distribution of seed deposition sites relatively to the source. Using simulated datasets, we investigated whether IM, GSM, CSM and SEMM applied to established seedlings estimate the basic or the effective dispersal kernel. We demonstrated that CSM and SEMM estimate the basic seed dispersal kernel, independently on the presence of post-dispersal mortality. By contrast, GSM estimates the effective dispersal kernel. Besides, for all methods, seed migration was inflated in presence of DDM, due to lower mortality among randomlydistributed immigrants relatively to local seedlings.


Alexandre Ponomarenko

Thierry Ameglio

Thierry Améglio

Freezing of water in capillary tubes

> Perspectives

FREEZING: TWO DIFFERENT TIME SCALES SUCROSE CONCENTRATION EFFECT

> A variation

> During the rst 100 ms, an ice sheet propagates quickly at the interface between the liquid and the tube.

Quick propagation of an ice sheet

Trees are able to undergo several months of cold temperatures every year. This is very surprising since the temperature may go down to -30°C, far below the freezing temperature of water. A direct consequence is the tree line, the limit in altitude where trees can survive.

How do trees survive the negative temperatures, until which critical temperature ? What sets the critical temperature ? What are the freezing dynamics of water in a tree ?

Since trees are not transparent, recording acoustic events is a promising technique to resolve those questions.

> Here we present some aspects of freezing of water in capillary tubes, monitored both with optics and acoustics. > Here we quantify the in uence of sucrose concentration on the quantity of acoustic events.

> Context -Motivation

The experiment shows a critical sucrose concentration above which there is no more acoustic emissions during freezing.

In a tree, acoustic emission would only re ect freezing in the cells below this concentration.

Explore di erent kind of tubes : open and closed geometries , rigid and exible tubes, hydrophil hydrophobe surfaces.

As in a tree undergoing freezing, we want to observe the freezing dynamics of a system where water with sugar and pure water are connected by a porous medium.

Work with porous media to understand propagation of freeze between cells and tissues in a tree.

  a time scale of the order of 1 minute, the entire volume of the drop freezes. The freezing front rst closes the drop at both ends (1-10s) and then propagates to the center (10-40s). Acoustic events are emitted only in the second step.


Emilio José Velez

Lucie Marandel

Stéphane Panserat

Karine Dias

Yujie Gao

Elisabeth Plagnes-Juan

Inge Geurden

Sandrine Skiba

Iban Seiliez

E J Vélez

S Skiba- Cassy

Eating for two: Consequences of parental nutrition on offspring autophagy

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Pregestational nutrition events may result in permanent changes in the growth potential, health and metabolic status of the next generation. Possible biological mechanisms for 'imprinting' the nutritional event in the next generation comprise epigenetic phenomenon, such as DNA and histone methylation. In the present study, we demonstrate that feeding broodstock fish with a diet deficient in methionine (one of the main methyl-group donor needed for all biological methylation reactions including DNA and histones methylation) strongly perturbs the nutritional control of the expression of several autophagy-related genes in the next generation. Our current research aim at defining the epigenetic mechanisms involved in this effect.

Three keywords*

Methionine Epigenetic Autophagy

Poster : Yes


Ana Paula Do

Espírito Santo

Frederic F Carlin

Caroline Garcia

Catherine M G C Renard

Development of functional apple juice naturally enriched with folates through fermentation by probiotic Lactobacilli

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Development of Functional Apple Juice Naturally Enriched with Folates through Fermentation by Probiotic Lactobacilli

Espirito-Santo, Ana Paula ; Carlin, Frédéric ; Garcia, Caroline ; Renard, Catherine M. G. C. INRA, UMR408 "Sécurité et Qualité des Produits d'Origine Végétale". F-84000 Avignon, France. Avignon Université, UMR408 "Sécurité et Qualité des Produits d'Origine Végétale". F-84000 Avignon, France.

The fermentation of a plant matrix by probiotic microorganisms is a challenge due to initial acidic environment and frequent presence of natural antimicrobials. However, undesirable compounds can be consumed and beneficial elements can be produced through fermentation of a plant food matrix by lactic acid bacteria. We evaluate here the possibility to ferment microfiltrated apple juice (MAJ) by commercial probiotic Lactoballi strains and explore the modifications of sugars and organic acid profiles and color of the MAJ, as well as the production of folates.

MAJ (var Golden) was fermented by single cultures of Lactobacillus acidophilus L10, L. casei L26, L. paracasei L33, L. plantarum 299v and L. rhamnosus LGG at 40°C for 48h. Bacteria were inoculated at 7 log CFU/ml into the fruit juice and their growth during fermentation (4, 8, 24, 32,48h) and viability during shelf-life at 5°C for 28 days were evaluated through the pour plate method on MRS agar medium. CIELab* parameters lightness (L*), redness (a*), and yellowness (b*) were determined using a CR-400 Minolta chromameter and color difference (ΔE = ((L*-L* 0 ) 2 +(a*-a* 0 ) 2 +(b*-b* 0 ) 2 ) 0.5 ) was calculated. Sugars and organic acids were quantified by colorimetric enzymatic methods adapted to microplate. Folate polyglutamates were deconjugated using chicken pancreatic enzymes prior to HPLC with fluorescence detection (excitation 295 nm/ emission 356 nm). Mono-and diglutamyl folates were quantified against standard curves of 5CH 3 -tetrahydrofolate, 10 formyl-tetrahydrofolate and 5 formyl-tetrahydrofolate. Measurements were made in triplicate.

The pH of MAJ decreased from 3.75 to 3.45 on average during fermentation. The strains reached 8.5 -10 log CFU/ml in 48h of fermentation. Strains L10, L26 and LGG maintained counts over 8 log CFU/ml and L299v and L33, over 6 log CFU/ ml during 28 days of shelf-life. The consumption/ production of glucose followed different patterns amongst the strains. LGG reduced significantly glucose concentration (from 23 to 20 g/l). All the strains reduced significantly concentrations of saccharose (from 8 to 2 g/l) and fructose (from 213 to 100 g/l). Production of L-lactic acid was highest in juice fermented by LGG. The concentration of formic acid was constant in L33 and L299v. L-malic acid was reduced from 183 to 3 mg/l by all strains. L10, L26 and L299v produced succinic acid. The juice lost color as parameters a* and b*, but not L*, were significantly reduced during fermentation, the highest ΔE being obtained in the juice fermented by LGG. Both L299v and LGG were able to produce folates (10 and 13 µg/l of folic acid equivalents, respectively).

The probiotic strains of Lactobacilli tested can ferment microfiltrated apple juice and maintain cell counts above the minimum limit required for a probiotic product during 4 weeks of shelf-life. Fermentation reduced calories from sugars by almost 50%, notably through fructose reduction, which can be interesting for fructose-intolerant individuals. Production of folates by L. plantarum L299v and L. rhamnosus LGG in apple juice is reported for the first time, opening new possibilities for the development of naturally folate-enriched fruit juices.


Samir Boris Jung

The Janus faces of bicarbonate therapy in the ICU


大纲 Jaap Van Milgen

Energy utilization and energy systems

中 国 • 猪 营 养 国 际 论 坛 技 术 资 料 .022
Energy utilization and energy systems Jaap van Milgen

饲料原料的优先顺序取决于所用的能量体系

The ranking of feed ingredients depends on the system of expression

中 国 • 猪 营 养 国 际 论 坛 技 术 资 料

LeLe纤维对能量消化率的影响

  .628蛋白质+2.34脂肪-1.84灰分 GE = 17.33 + 0.628 CP + 2.34 fat -1.84 ash (NRC, 1998) 总能=22.6粗蛋白质+38.8脂肪+17.5淀粉+16.7糖+18.6残留物 残留物=有机物质-(粗蛋白质+脂肪+淀粉+糖) GE = 22.6 CP + 38.8 fat + 17.5 starch + 16.7 sugars + 18.6 residue residue = OM -(CP + fat + starch + sugars) (Noblet et al., 1994. J. Anim. Sci. 72:344-354) Goff and Noblet (2001) J. Anim. Sci. 79: 2418-2427 中 国 • 猪 营 养 国 际 论 坛 技 术 资 料 Goff and Noblet (2001) J. Anim. Sci. 79: 2418-2427 ME/DE, % 粗蛋白质 ,占干物质% Crude protein, % in DM Sauvant et al. (2004) Tables of composition and nutritional value of feed materials 中 国 • 猪 营 养 国 际 论 坛 技 术 资 料 Lys Met Cys Thr Trp Ile Leu Val Phe Tyr His Arg Ser Gly Ala Glu Gln Pro Asp Asn能量值

从可消化养分评估能量值 Estimating energy values from digestible nutrients

		消化能DE	代谢能ME	净能NE	ME:DE	NE:ME
		(kJ/g)	(kJ/g)	(kJ/g)	(%)	(%)
	蛋白质Protein	23.3	20.4	12.1	88	59
	油脂Lipid	38.7	39.3	35.0	101	89
	淀粉Starch	17.5	17.5	14.3	100	82
	糖Sugars	16.8	16.5	11.9	98	73
	纤维Fiber	16.7	15.5	8.6	93	56

Noblet et al. (1994) J. Anim. Sci. 72:344-354

Acknowledgements:

大纲 Outline


text:
Kiên Kiêu
email: kien.kieu@jouy.inra.fr

Katarzyna Adamczyk-Chauvat

Keywords: Stochastic geometry, tessellation, point process, pseudolikelihood, Campbell measure, Kullback-Leibler divergence

Recently a new class of planar tessellations, named T-tessellations, was introduced. Splits, merges and a third local modication named ip where shown to be sucient for exploring the space of T-tessellations. Based on these local transformations and by analogy with point process theory, tools Campbell measures and a general simulation algorithm of Metropolis-Hastings-Green type were translated for random T-tessellations.

The current report is concerned with parametric inference for Gibbs models of T-tessellations. The estimation criterion referred to as the pseudolikelihood is derived from Campbell measures of random T-tessellations and the Kullback-Leibler divergence. A detailed algorithm for approximating the pseudolikelihood maximum is provided. A simulation study seems to show that bias and variability of the pseudolikelihood maximum decrease when the tessellated domain grows in size.

In the last part of the report, it is shown that an analogous approach based on the Campbell measure and the KL divergence when applied to point processes leads to the well-known pseudo-likelihood introduced by Besag. More surprisingly, the binomial regression method recently proposed by Baddeley and his co-authors for computing the pseudolikelihood maximum can be derived using the same approach starting from a slight modication of the Campbell measure.

Introduction

Recently, a new class of planar tessellations was introduced [START_REF] Kiêu | A completely random T-tessellation model and Gibbsian extensions[END_REF]: Gibbsian Ttessellations. Briey, a T-tessellation is a tessellation with only vertices at the intersection of three edges, two of them being aligned. In [START_REF] Kiêu | A completely random T-tessellation model and Gibbsian extensions[END_REF], it was shown that the space of T-tessellations can be explored using three simple and local operators: splits, merges and ips. A model was proposed to be considered as a completely random T-tessellation model (CRTT). And Gibbsian variants were discussed. Also tools and results such as Campbell measures, Papangelou kernels, Georgii-Nguyen-Zessin formulae were translated from point process theory to T-tessellations. A Metropolis-Hastings-Green algorithm based on splits, merges and ips was derived for the simulation of Gibbsian T-tessellations. Such an algorithm denes a Split, Merge and Flip (SMF) Markov chain converging to the Gibbs distribution to be simulated. As a companion tool, a C++ library named LiTe [START_REF] Adamczyk-Chauvat | LiTe[END_REF] was developped. Using LiTe, it is possible to simulate a rather large class of T-tessellation Gibbs models.

The main purpose of this report is to introduce a method for estimating the parameters of a T-tessellation Gibbs model. This inference method is based on a so-called pseudolikelihood. The approach developped here involves Campbell measures and the Kullback-Leibler divergence. It is shown that the well-known pseudolikelihood introduced by Besag [START_REF] Besag | Statistical analysis of non-lattice data[END_REF] for point process can also be built from the Campbell measure and the Kullback-Leibler divergence.

The proposal for the pseudolikelihood for Gibbsian T-tessellations is provided in Section 2.2. Section 2.3 focuses on the special case of a canonical exponential family. Practical computation of the pseudolikelihood maximum is discussed in Section 2.4. The behaviour of the maximum pseudolikelihood estimator is investigated based on some numerical simulations in Section 2.5. Section 2.6 is devoted to an unexpected drawback of the pseudolikelihood.

In Section 3.2, it is shown that a similar approach based on the Campbell measure and the Kullback-Leibler divergence provides a new way to obtain the widely used pseudolikelihood for point processes. Even the recently logistic regression method for computing the pseudolikelihood estimator can be obtained in such a way as shown in Section 3.3.

Appendix A is a technical prerequisite where the Kullback-Leibler divergence is extended to non-probability measures. Appendix B provides a result about the mean of a functional summed over all subpatterns of a nite Poisson point process. Such a result is used for proving that the distribution of a Gibbsian T-tessellation is fully characterized by its Papangelou conditional intensities (detailed proof provided in Appendix C).

Pseudolikelihood for T-tessellations 2.1 Gibbsian T-tesselllations

A general theory of planar T-tessellations was sketched in [START_REF] Kiêu | A completely random T-tessellation model and Gibbsian extensions[END_REF]. At the time being, the theory is limited to bounded domains. Let D ⊂ R 2 be a convex bounded polygon.

Below, D is referred to as the domain. The space of T-tessellations of D is denoted by T . A T-tessellation has only convex cells and can also be seen as a collection of segments. A segment is a maximal union of aligned and connected edges. There are two classes of segments: non-blocking (single edge) and blocking (multiple edges) segments. Three local operators allow to explore T : Split a cell is divided by a new segment.

Merge a non-blocking segment is removed.

Flip an edge at the end of a blocking segment is removed while the incident blocked segment is extended.

Note that the inverse of a split is a merge (and vice versa), while the inverse of a ip is a ip. Below S T , M T and F T refer to the spaces of splits, merges and ips that can be applied to tessellation T . Note that S T is continuous while M T and F T are nite. In [START_REF] Kiêu | A completely random T-tessellation model and Gibbsian extensions[END_REF], uniform measures with innitesimal elements denoted by ds, dm and df are dened. The last two measures are just counting measures. The measure ds has a total mass equal to the sum of the perimeters of T cells, denoted by u(T ), divided by π. The probability measure πds/u(T ) on S T is characterized by the two following facts: Cells of T are split with probabilities proportional to their perimeters.

Given a cell C of T is split, the splitting line is a uniform and isotropic random line hitting C.

The so-called CRTT model introduced in [START_REF] Kiêu | A completely random T-tessellation model and Gibbsian extensions[END_REF] plays the role of the Poisson point process in point process theory. Its distribution is denoted by µ.

Denition 1. For a given non-negative real number ν, the CRTT on D with parameter ν is dened by the following distribution on T :

µ ν (dT ) ∝ I{T ∈ T (L)} ν n n! dl 1 . . . dl n , ∀L = {l 1 , . . . , l n }, (1)
where T (L) is the subset of T-tessellations with segments supported by the line pattern L.

For sake of brevity, we write µ the distribution of the CRTT with parameter ν = 1.

CRTT stands for Completely Random T-Tessellation.

All other models considered below have a density with respect to µ. Their distributions can be written as

P (dT ) = h(T ) µ(dT ). (2)
In general, h is known analytically only up to a multiplicative normalizing constant. Even for the CRTT where h is constant, the normalizing constant is not known at the moment. This is why inference by maximum likelihood is not straightforward. The density h is said to be hereditary in the sense dened in [START_REF] Kiêu | A completely random T-tessellation model and Gibbsian extensions[END_REF] if:

For every pair (T, s), T ∈ T , s ∈ S T , h(T ) = 0 ⇒ h(sT ) = 0.

For every pair (T, f ),

T ∈ T , f ∈ F T , h(T ) = 0 ⇒ h(f T ) = 0.
The condition was erroneoulsy stated in [START_REF] Kiêu | A completely random T-tessellation model and Gibbsian extensions[END_REF] with inversed implications.

In [START_REF] Kiêu | A completely random T-tessellation model and Gibbsian extensions[END_REF], two reduced Campbell measures for random T-tessellations were introduced. The split Campbell measure of a random T-tessellation T maps any function φ :

C s = {(s, T ) : T ∈ T , s ∈ S T } → R to C ! s (φ) = E m∈M T φ m -1 , mT . (3)
The split Campbell measure is very similar to the Campbell measure of a point process. Indeed splitting and merging are analog to insertion and removal of points.

The ip Campbell measure is more specic to T-tessellations but its denition is formally very close to the denition of the split Campbell measure. The ip Campbell measure of T maps any function φ :

C f = {(f, T ) : T ∈ T , f ∈ F T } → R to C ! f (φ) = E f ∈F T φ f -1 , f T . (4)
It was shown in [START_REF] Kiêu | A completely random T-tessellation model and Gibbsian extensions[END_REF] that if T ∼ hµ where h is hereditary then

C ! s (ds, dT ) = λ s (s, T ) dsP (dT ), (5)
C ! f (f, dT ) = λ f (f, T ) df P (dT ), (6)
where

λ s (s, T ) = h(sT ) h(T ) , (7)
λ f (f, T ) = h(f T ) h(T ) , (8)
for P -almost all T ∈ T , almost all s ∈ S T and all f ∈ F T . The densities λ s and λ f of the reduced Campbell measures are called the split and ip Papangelou conditional intensities of T. Here we will need a third one: the merge Papangelou conditional intensity dened by

λ m (m, T ) = h(mT ) h(T ) . (9)
Note that

λ m (m, T ) = λ s (m -1 , mT ) -1 . ( 10
)
The split and ip Papangelou conditional intensities uniquely determine the density h when it is hereditary. A proof of the result below is given in Appendix C.

Proposition 1. Let T ∼ P = hµ where h is supposed to be hereditary. The density h is uniquely determined up to a µ-null set by any pair (λ s , λ f ) of split and ip Papangelou conditional intensities of T.

As this report focuses on parametric inference, most often a parametric family of distributions P θ on T will be considered. The issue adressed in this report is to estimate θ given a realization of P θ .

Since a general simulation algorithm of Metropolis-Hastings-Green is provided in [START_REF] Kiêu | A completely random T-tessellation model and Gibbsian extensions[END_REF], inference by Monte-Carlo Maximum Likelihood is possible, see e.g. [START_REF] Geyer | Likelihood inference for spatial point processes[END_REF]. But its practical implementation can be quite demanding. So easier and more automatized inference methods are of interest especially in contexts where estimation has to be repeatedly applied with dierent models or on large data sets.

The pseudolikelihood approach developped below is based on Campbell measures and Kullback-Leibler divergence. Using discretization as in [START_REF] Besag | Statistical analysis of non-lattice data[END_REF] for point processes is hardly concevable for tessellations.

Pseudolikelihood denition and properties

Consider a parametric family of distributions P θ on T , θ ∈ Θ. Every P θ is supposed to be absolutely continuous with respect to µ and the densities of the P θ 's are denoted by h θ . The model is supposed to be identiable:

θ = θ * ⇔ h θ (T ) = h θ * (T ) for µ-almost all T ∈ T . ( 11
)
The densities h θ are supposed to be hereditary. The split, merge and ip Papangelou intensities are denoted λ s,θ , λ m,θ and λ f,θ .

We are now ready for the pseudolikelihood denition.

Denition 2. Given an observed T-tessellation T , the log-pseudolikelihood is dened as

LPL(θ|T ) = - m∈M T log λ m,θ (m, T ) - S T λ s,θ (s, T ) ds- f ∈F T log λ f,θ (f, T ) - f ∈F T λ f,θ (f, T ) . ( 12
)
The maximum pseudolikelihood estimator is dened as

θ = arg max θ∈Θ LPL(θ|T ). (13)
Similarly to the log-likelihood, the log-pseudolikelihood can be derived as an empirical estimator of a contrast function. As a starting point, instead of P θ , consider the split and ip Campbell measures:

C ! s,θ (ds, dT ) = λ s,θ (s, T ) dsP θ (dT ), (14)
C ! f,θ (f, dT ) = λ f,θ (f, T ) df P θ (dT ), (15)
Compared to P θ , the Campbell measures have densities that do not involve unknown normalizing constants. It should be noticed however that the dominating measure above depends on θ. This is why we introduce further measures closely related to the Campbell measures. For every pair (θ, θ * ), consider the measures α s,θ,θ * on C s and α f,θ,θ * on C f dened by

α s,θ,θ * (ds, dT ) = λ s,θ (s, T ) dsP θ * (dT ), (16)
α f,θ,θ * (df, dT ) = λ f,θ (f, T ) df P θ * (dT ). (17)
The measures α s,θ,θ * and α f,θ,θ * can be viewed as candidate Campbell measures. When θ = θ * , they are indeed Campbell measures.

For the time being, let us focus on the case where α s,θ,θ * α s,θ * ,θ * and α f,θ,θ * α f,θ * ,θ * , then

dα s,θ * ,θ * dα s,θ,θ * (s, T ) = λ s,θ * (s, T ) λ s,θ (s, T ) , dα f,θ * ,θ * dα f,θ,θ * (f, T ) = λ f,θ * (f, T ) λ f,θ (f, T ) .
It is standard in statistics to quantify the dissemblance between distributions using the Kullback-Leibler divergence. Since the measures α s,θ,θ * and α f,θ,θ * are not probability measures, using the Kullback-Leibler divergence requires some adaptation. Appendix A provides such an extension to non-negative non-probability measures. The extended Kullback-Leibler divergence of α s,θ,θ * from α s,θ * ,θ * is dened as

KL (α s,θ * ,θ * , α s,θ,θ * ) = Cs λ s,θ * (s, T ) λ s,θ (s, T ) log λ s,θ * (s, T ) λ s,θ (s, T ) + 1 - λ s,θ * (s, T ) λ s,θ (s, T ) α s,θ,θ * (ds, dT )
Note that for every θ ∈ Θ KL (α s,θ * ,θ * , α s,θ,θ * ) ≥ 0 and that the extended Kullback-Leibler divergence cancels for θ = θ * . Replacing in the right-hand side α s,θ,θ * by its denition (16), one gets

KL (α s,θ * ,θ * , α s,θ,θ * ) = Cs λ s,θ * (s, T ) log λ s,θ * (s, T ) λ s,θ (s, T ) + λ s,θ (s, T ) -λ s,θ * (s, T ) dsP θ * (dT ).
The rst part of the integral can be rewritten using the expression of the split Campbell measure (14):

KL (α s,θ * ,θ * , α s,θ,θ * ) = Cs log λ s,θ * (s, T ) λ s,θ (s, T ) C ! s,θ * (ds, dT ) + Cs (λ s,θ (s, T ) -λ s,θ * (s, T )) dsP θ * (dT ). Dene M s (θ, θ * ) = Cs log λ s,θ (s, T ) C ! s,θ * (ds, dT ) - Cs λ s,θ (s, T ) dsP θ * (dT ).
The extended Kullback-Leibler divergence of α s,θ,θ * from α s,θ * ,θ * can be written as

KL (α s,θ * ,θ * , α s,θ,θ * ) = M s (θ * , θ * ) -M s (θ, θ * ).
Only the term M s (θ, θ * ) is informative about the divergence of α s,θ,θ * from α s,θ * ,θ * . Note that M s does not have a constant sign. Instead, for any given θ * , M s (., θ * ) has a global maximum at θ * . Now suppose that θ * is unknown and that a realization T of P θ * is available instead. In view of denition (3), the rst integral of M s can be estimated by

m∈M T log λ s,θ (m -1 , mT ).
From the equivalence [START_REF] Geyer | Markov chain Monte Carlo maximum likelihood[END_REF], one can rewrite the estimator as

- m∈M T log λ m,θ (m, T ).
Deriving an estimator of the second integral of M s is straightforward. The mean under P θ * of the integral S T λ s,θ (s, T ) ds is equal to the second integral involved in the expression of M s (θ, θ * ) above.

Similarly considering the extended KL divergence of α f,θ,θ * from α f,θ * ,θ * leads to the function:

M f (θ, θ * ) = T F T log λ f,θ (f, T ) C ! f,θ * (df, dT ) - T F T λ f,θ (f, T ) df P θ * (dT ).
Furthermore, M f (θ, θ * ) can be estimated without bias by

- f ∈F T log λ f,θ (f, T ) - f ∈F T λ f,θ (f, T ).
We can now state a rst general result about the mean log-pseudolikelihood Proposition 2. If T ∼ P θ * , then for every

θ ∈ Θ E θ * LPL(θ|T) = M s (θ, θ * ) + M f (θ, θ * ), (18)
= M s (θ * , θ * ) + M f (θ * , θ * ) -KL (α s,θ * ,θ * , α s,θ,θ * ) -KL (α f,θ * ,θ * , α f,θ,θ * ) . ( 19
)
Proof. The Proposition has already been proved under the assumption that α s,θ,θ * α s,θ * ,θ * and α f,θ,θ * α f,θ * ,θ * . Let us consider the case where either α s,θ,θ * does not dominate α s,θ * ,θ * or α f,θ,θ * does not dominate α f,θ * ,θ * . For instance, consider the rst case. There exists a non-null subset, with respect to the measure dsP θ * (dT ), of C s where λ s,θ cancels while λ s,θ * does not. In view of expression [START_REF] Besag | Statistical analysis of non-lattice data[END_REF], this is equivalent with the existence of a non-null subset, with respect to the measure C ! s,θ * , where λ s,θ cancels. Hence if α s,θ,θ * does not dominate α s,θ * ,θ * , then M s (θ, θ * ) is equal to -∞. Since, by denition, KL (α s,θ * ,θ * , α s,θ,θ * ) = ∞, the equality between (18) and (19) holds. A similar rationale applies for ips instead of splits.

A straigthforward consequence of Proposition 2 is the following. Corollary 1. If T ∼ P θ * , then for every θ ∈ Θ,

E θ * LPL(θ|T) ≤ E θ * LPL(θ * |T)
Proof. In view of equations (1819), we have for every

θ ∈ Θ E θ * LPL(θ|T) = E θ * LPL(θ * |T) -KL (α s,θ * ,θ * , α s,θ,θ * ) -KL (α f,θ * ,θ * , α f,θ,θ * ) . (20)
Since the extended Kullback-Leibler divergences are non-negative, the inequality follows.

The following result shows that -(M s (., θ * ) + M f (., θ * )) is a contrast function in the sense that its global minimum at θ * is strict. Proposition 3. If T ∼ P θ * , then θ * is a strict global maximum of the mean logpseudolikelihood:

∀θ ∈ Θ \ {θ * }, E θ * LPL(θ|T) < E θ * LPL(θ * |T).

Proof. It remains to show that

E θ * LPL(θ|T) = E θ * LPL(θ * |T) ⇒ θ = θ * .
Consider a θ such that the mean log-pseudolikelihoods are equal. First, observe that in view of Equation (20), both Kullback-Leibler divergences must cancel. This implies that α s,θ,θ * (respectively α f,θ,θ * ) must dominate α s,θ * ,θ * (respectively α f,θ * ,θ * ), otherwise one of the KL-divergences is innite. Hence the mean log-pseudolikelihoods can be equal only if α s,θ,θ * α s,θ * ,θ * and α f,θ,θ * α f,θ * ,θ * . From now on, we focus on that case. The Kullback-Leibler divergence of α s,θ,θ * from α s,θ * ,θ * cancels if and only if α s,θ,θ * = α s,θ * ,θ * except on a α s,θ,θ * -null subset E ⊂ C s . In view of denition ( 16), E is α s,θ,θ * -null if and only there exists a pair

(E 1 , E 2 ) of subsets of C s such that E = E 1 ∪ E 2 , E 1 is
null with respect to the measure dsP θ * (dT ) and λ s,θ cancels on E 2 . The domination α s,θ,θ * α s,θ * ,θ * implies that λ s,θ * must cancel too on E 2 . Therefore α s,θ,θ * and α s,θ * ,θ * coincide on E 2 . More generally, both measures equal except on E 1 which is null with respect to dsP θ * (dT ). In turn, this implies that λ s,θ and λ s,θ * equal for P θ * -almost all T ∈ T and almost all splits s ∈ S T . Hence, λ s,θ * is a split Papangelou conditional intensity for P θ * .

Similarly, it can be shown that λ f,θ is a ip Papangelou conditional intensity for P θ * . From Proposition 1, it follows that h θ is a density of P θ * and that it equals h θ * µ-almost everywhere. Since the model is supposed to be identiable, θ must equal θ * .

The canonical exponential family case

Consider the case of a canonical exponential family:

P θ (dT ) ∝ exp θ T t(T ) µ(dT ), (21)
wher t(T ) ∈ R d is a vector of statistics of tessellation T . Then we have

λ s,θ (s, T ) = exp θ T t(s, T ) , t(s, T ) = t(sT ) -t(T ), λ m,θ (m, T ) = exp θ T t(m, T ) , t(m, T ) = t(mT ) -t(T ), λ f,θ (f, T ) = exp θ T t(f, T ) , t(f, T ) = t(f T ) -t(T ).
The log-pseudolikelihood writes

LPL(θ|T ) = -θ T m∈M T t (m, T ) - S T exp θ T t (s, T ) ds- θ T f ∈F T t (f, T ) - f ∈F T exp θ T t (f, T ) . ( 22
)
The gradient of the log-pseudolikelihood has the following form

∇ LPL(θ|T ) = - m∈M T t (m, T ) - S T t(s, T ) exp θ T t (s, T ) ds- f ∈F T t (f, T ) - f ∈F T t(f, T ) exp θ T t (f, T ) . ( 23
)
The Hessian matrix of LPL has elements of the form

H ij (LPL)(θ|T ) = - S T t i (s, T )t j (s, T ) exp(θ T t(s, T )) ds- f ∈F T t i (f, T )t j (f, T ) exp(θ T t(f, T )).
That is

H(LPL)(θ|T ) = - S T t(s, T )t(s, T ) T exp(θ T t(s, T )) ds- f ∈F T t(f, T )t(f, T ) T exp(θ T t(f, T ))
As a consequence the log-pseudolikelihood is concave. Strict concavity holds if for every

y ∈ R d y T H(LPL)(θ|T )y = - S T y T t(s, T ) 2 exp(θ T t(s, T )) ds- f ∈F T y T t(f, T ) 2 exp(θ T t(f, T )) < 0. Dene S * T = {s ∈ S T : t(s, T ) > -∞} , F * T = {f ∈ F T : t(f, T ) > -∞} . (24)
In the quadratic form above, the integral can be restricted to S * T and the sum to F * T . The log-pseudolikelihood is strictly concave if at least one of the two following conditions is fullled.

Condition 1. The subset t(F * T , T ) of R d is not included in a vector hyperplane.

Condition 2. For any subset

B of S * T almost equal to S * T , t(B, T ) is not included in a vector hyperplane of R d . Above B is almost equal to S *
T if and only if

S T I S * T \B (s) ds = 0.
Condition 1 ensures that for every y ∈ R d , there exists at least a ip f ∈ F * T such t(f, T ) is not orthogonal to y. Similarly Condition 2 ensures that there exists a non-null subset of S * T where t(s, T ) is not orthogonal to y.

Example 1. Consider the distribution dened by the density

h θ (T ) ∝ exp • n s (T )θ (25) where
• n s (T ) refers to the number of internal segments of T . Above the density h θ is dened up to a unknown normalizing constant depending on θ. This distribution denes a CRTT with intensity controlled by parameter θ. In [START_REF] Kiêu | A completely random T-tessellation model and Gibbsian extensions[END_REF], the CRTT model is introduced together with another parametrization based on τ = exp(θ). The parameter τ can be considered as a scaling parameter: increasing τ is equivalent to increasing the tessellated domain D. Let us come back to parametrization (25) which denes a canonical exponential family with

t(T ) = • n s (T ),
and thus

t(s, T ) = 1, t(m, T ) = -1, t(f, T ) = 0.
The log-pseudolikelihood writes

LPL(θ|T ) = |M T | θ - u(T ) π exp(θ) -|F T | .
The log-pseudolikelihood can be rewritten in terms of the number of internal segments of T :

LPL(θ|T ) = • n s,nb (T ) θ - u(T ) π exp(θ) -2 • n s,b (T ) , where • n s,nb (T )
is the number of non-blocking internal segments of T and

• n s,b (T ) the number of blocking internal segments. The function above is strictly concave (negative second derivative). It has therefore a unique maximum. In order to identify the maximum, one solves

LPL ( θ|T ) = • n s,nb (T ) - u(T ) π exp( θ) = 0.
This yields

θ = log • n s,nb (T ) π u(T ) .
That is

τ = • n s,nb (T ) π u(T ) .
Note that for the CRTT model, claimed to be the analog of the Poisson point process, the pseudolikelihood maximum estimator is not the same as the likelihood estimator. Indeed for the CRTT model the likelihood has the form

τ • ns(T ) c(τ ) ,
where c(τ ) is a normalization constant (unknown so far). It follows that the likelihood maximum estimator of τ is a function (still unknown) of the number

• n s (τ ) of internal segments of the tessellation T .

The estimators derived from the likelihood and the pseudolikelihood do not coincide for completely random T-tessellations contrary to what happens for Poisson point processes.

Example 2. Consider the distribution dened (up to an unknown nomalizing constant) by the density

h θ (T ) ∝ exp • n s (T )θ 1 -a 2 (T )θ 2 , (26)
where a 2 (T ) is the sum of T cell squared areas. Given the number of cells the statistics a 2 (T ) is large when the cells have heterogeneous areas. Large values of θ 2 favour tessellations with homogeneous cells (with respect to their areas). Again we have a canonical exponential family with

t(T ) = • n s (T ) -a 2 (T ) .
Expressions for t(s, T ), t(m, T ) and t(f, T ) can be worked out. But this does not lead to a simple analytical expression of the pseudolikelihood.

Algorithm for approximating the pseudolikelihood maximum

A simple approach for approximating the log-pseudolikelihood consists in discretizing the integral on S T involved in the pseudolikelihood. Below we consider the framework of a canonical exponential family. The log-pseudolikelihood and its gradient have the forms given by Equations ( 22) and ( 23).

Let S be a nite set of so-called dummy splits of T . The number of elements of S is denoted |S|. The log-pseudolikelihood ( 22) is replaced by

LPL d (θ|T ; S) = -θ T m∈M T t (m, T ) - u(T ) π|S| s∈S exp θ T t (s, T ) - θ T f ∈F T t (f, T ) - f ∈F T exp θ T t (f, T ) . ( 27
)
The approximation concerns only the integral over S T involved in (22) which is replaced by a weighted discrete sum. All other terms involving merges and ips are exact. Notice that if S is a sample of splits drawn uniformly from S T , then LPL d (θ|T ; S) is an unbiased estimator of LPL(θ|T ).

Remark 1. In the special case of the CRTT model, the approximation is exact. Since t(s, T ) = 1 for any split s, the sum over S in (27) writes

s∈S exp θ T t (s, T ) = |S| exp (θ) .
Thus the term involving dummy splits becomes

u(T ) π exp(θ).
Compare with Example 1.

A straightforward approximation of the log-pseudolikelihood gradient is

∇ LPL d (θ|T ; S) = - m∈M T t (m, T ) - u(T ) π|S| s∈S t(s, T ) exp θ T t (s, T ) - f ∈F T t (f, T ) - f ∈F T t(f, T ) exp θ T t (f, T ) . (28)
Similarly, the log-pseudolikelihood Hessian can be approximated by

H(LPL d )(θ|T ; S) = - u(T ) π|S| S t(s, T )t(s, T ) T exp(θ T t(s, T ))- f ∈F T t(f, T )t(f, T ) T exp(θ T t(f, T )) (29)
Algorithm 1 Algorithm NOIS (Newton optimization and increasing splitting) for approximating the pseudolikelihood maximum.

Require: a T-tessellation T , the stepsize to be used in θ's update, the tolerance parameter δ 1: m ← number of non-blocking segments of T 2: θ ← null vector G ← ∇ LPL d (θ|T ; S) as given by (28) 10:

H ← H(LPL d )(θ|T ; S) as given by (29) 11:

θ ← θ + H -1 G 12:
Add m independent and uniform splits of T to S 13:

Update the list of t(s, T )'s where s ∈ S 14:

L ← LPL d (θ|T ; S) as given by ( 27) 15:

∆ ← variation of L 16: until |∆| ≤ δ (|L| + δ) .
The integral on splits in the log-pseudolikelihood (22) can be written as an expectation with respect to the uniform distribution on the space S T of splits. The estimation criterion consists of deterministic terms and of an expectation with respect to a random variable that can be simulated. Stochastic approximation provides a number of algorithms for optimizing such type of criterion. However the present framework is rather simple here. The expectation is taken under a xed distribution not depending on the searched parameter.

From stochastic approximation we keep the alternation between the update of the parameter and the estimation of the expectation. An iteration of the algorithm consists of the two following stages: Update of θ according to Newton's optimization method based on the logpseudolikelihood Hessian approximation.

Increment of the sample S of dummy splits.

The full procedure is detailed in Algorithm 1. It is referred to as NOIS (Newton optimization and increasing splitting) algorithm below. Some remarks:

The memory space used by the algorithm for storing the t(s, T )'s grows along iterations.

The time for computing s∈S exp θ T t (s, T ) grows along iterations too.

One could add one dummy split to S at each iteration, but the convergence of the sum on dummy splits is slow and the trajectories of the θ n 's are then rather irregular.

The stopping criterion is inspired by the R function optim.

When the parametric family includes the CRTT distributions, one may start with the maximum pseudolikelihood estimate under the CRTT model. Without loss of generality, one may assume that the rst component of t(T ) is equal to

• n s (T ).
The initial value of θ writes:

log • n s,nb (T ) π /u(T ), 0, . . . , 0 T .

Example 3. Consider the model with the following non-normalized density

h θ (T ) ∝ exp   θ 1 • n s (T ) -θ 2 v∈V (T ) π 2 -φ(v)   , (30)
where 0 ≤ φ(v) ≤ π/2 denotes the acute angle between the two segments meeting at vertex v. When θ 2 > 0, T-tessellations with almost rectangular cells are favoured by the model. A realization of such a model with θ 1 = 2.49 and θ 2 = 2.5 is shown in Figure 1 (top). Figure 1 (bottom) shows an example of parameter trajectory obtained with the NOIS algorithm. Some details:

The observed tessellation was generated by a SMF Markov chain starting from the empty tessellation and after a burnin stage of 1,000 iterations.

A sample of 1,000 dummy splits was used for computing the discrete approximation of the pseudolikelihood and of its gradient.

As a starting point for the optimization, the pseudolikelihood estimator assuming the CRTT model was used.

Ten iterations have been performed before the stopping criterion was met with δ = 0.01. Since the T-tessellation used for inference had 31 non-blocking segments, the NOIS algorithm only used 10 × 31 = 310 dummy splits. q q q q q q q q qq The three models are referred to as CRTT, angle and area models below. Parameter values are given in Table 1. Each model has been simulated on three squared domains with varying sizes. The domain sizes were chosen in order to emphasize their eect on the estimator precision. Domain side lengths are provided in Table 1. The maximum pseudolikelihood estimator has been computed on samples of 500 Ttessellations for each model and domain size. In order to generated the T-tessellation, SMF Markov chains has been used as described in [START_REF] Kiêu | A completely random T-tessellation model and Gibbsian extensions[END_REF]. As for any Metropolis-Hastings algorithm, burnin and sampling period (number of Markov chain iterations between consecutive sampled items) have to be determined.

Simulation study

The burnin duration was tuned empirically by monitoring the evolution of the model energy. It was chosen as the number of iterations required for energy stabilization (visual assessment).

Concerning the sampling period, an automatic procedure was used. Along an SMF chain curse, segments are born (splits), change their length (ips) or die (merge). Thus a segment has a survival time expressed in terms of number of iterations. As a criterion, the renewal rate of segments was considered. The sampling period was chosen long enough so that 75 % of segments living at the beginning of the period had died at the end of the period. Sampling periods were approximated using the RLiTe function getSMFSamplingPeriod.

Note that burnin duration and sampling periods were tuned for each model on the largest domain to be considered. Simulations on smaller domains were performed using the SMF Markov chains with parameters chosen for the largest domains.

First sample times determined from the burnin duration and the sampling period are shown for the three models simulated in their largest domains in Figure 2.

In order to compute estimates using the LiTe class PLInferenceNOIS, one needs to specify the tolerance and the maximal number of iterations. Values are given in Table 2. Estimation for the CRTT model is apart since the maximum pseudolikelihood has with a null maximum number of iterations. The maximal number of iterations for the angle model was chosen large enough so that the stopping criterion could be met in most cases. Concerning the area model, it turned out that the stopping criterion was met within very few iterations (sometimes only one for the smallest model). This resulted in rather biased estimates for the parameter θ 1 . This poor performance may be due to at pseudolikelihood functions as illustrated below. In order to bypass this problem, the tolerance parameter δ is chosen negative so that the stopping criterion cannot be met and the maximum number of NOIS iterations is chosen large. Estimate distributions for the CRTT model are shown in Figure 3 (top row) as boxplots. Dispersion is moderate and decreases when the domain size increases. Small bias: it is not visible as soon as the total number of cells is larger than a hundred. For fty cells (smallest domain size), a small negative bias cannot be excluded.

Estimate distributions for the angle model are shown in Figure 3 (middle row). As for the CRTT model, the estimate dispersion and bias seem to decrease when the domain size increases. The estimate distributions show some asymmetry.

Let us consider the area model. As mentionned above, the stopping criterion of the NOIS algorithm does not seem to work for the area model. This problem is illustrated based on a simulated tessellation. The pseudolikelihood is approximated using 5,000 dummy splits, see Figure 4. From now on this approximation is considered as exact. In the region around the maximum, the log-pseudolikelihood is rather at: relative variations of the log-pseudolikelihood are small. As a consequence, the NOIS algorithm stops too early and nd an estimate a little bit too far from the true maximum. A possible enhancement of the stopping criterion: stop when the pseudolikelihood and the estimates do not vary enough any more. An alternative solution could be to reparametrize the density (26) in order to get two statistics of T with more comparable values.

Estimation for the area model behavess like the two previous models, see Figure 3 (bottom row). Both dispersion and bias decrease when the domain size increases. Thus the estimate of θ is equal to θ = 0.24.

Fitting a wrong model

Simulations of T-tessellations under the CRTT model with parameter θ = 0.24 are shown in Figure 5. The most striking feature of simulated tessellations is their heterogeneity compared to the observed T-tessellation. This is expected since the CRTT model yields completely random T-tessellation while the observed tessellation shows some kind of regularity. But there is another important dierence between the observed and the simulated tessellations: there are more cells in the latter. This dierence is unexpected. Such a behaviour does not occur with point processes. In the point process framework, the situation is as follows. Consider a given point pattern of say n points. Suppose that the homogeneous Poisson model is chosen for tting. The intensity parameter is estimated by n divided by the area of the considered domain. This estimate is obtained both as the maximum pseudolikelihood and as the maximum likelihood estimators. Therefore simulating the tted Poisson model would yield point patterns of size n in expectation. Somehow pseudolikelihood tting leads to the best Poisson point process representing the observed point pattern. In the Ttessellation framework, the tted model seems to be biased for reasons that remain to be investigated.

Pseudolikelihood for point processes

Pseudolikelihood was rst introduced by Besag for spatial random elds with discrete support [START_REF] Besag | Statistical analysis of non-lattice data[END_REF], later for a Strauss point process [START_REF] Besag | Some methods of statistical analysis for spatial data[END_REF]. The extension to a rather general class of models for point processes was provided by Jensen and Møller [START_REF] Jensen | Pseudolikelihood for exponential family models of spatial point processes[END_REF]. General convergence results (consistancy, central limit theorem) were obtained by Billiot et al. [START_REF] Billiot | Maximum pseudolikelihood estimator for exponential family models of marked Gibbs point processes[END_REF]. In this report, we focus on point processes within a bounded domain denoted by D. The way the pseudolikelihood is introduced here is original. Previously the point process was discretized by countings on a mesh (a square lattice for example). This yields a random eld with discrete support. By letting the tile size decrease to zero, one obtains the point process pseudolikelihood as a limit of the random eld pseudolikelihood (the limit does not depend on the considered mesh). Similarly to random T-tessellations, one can derive the pseudolikelihood starting from the Kullback-Leibler divergence and candidate Campbell measures. Such an approach is developed below. It should be noticed that inference based on pseudolikelihood can also be considered as a particular instance of large class of methods so-called Takacs-Fiksel methods [START_REF] Coeurjolly | Takacs-Fiksel method for stationary marked Gibbs poind processes[END_REF].

Below only the main steps of the pseudolikelihood derivations are sketched. Full details can be recovered by analogy with the pseudolikelihood derivation for random T-tessellations.

cells 326 cells 322 cells 297 cells 211 cells 298 cells 324 cells 342 cells 326 cells

Figure 5: Center: a T-tessellation supposed to be a realization of the CRTT model with unknown parameter. Around: realizations of the CRTT model tted using the maximum pseudolikelihood estimator.

Finite point processes

Let D denote the space of nite point patterns in a planar compact domain D. Considering only bounded domains simplies the presentation and makes the analogy with random T-tessellations more straightforward. Any point process X denes a reduced Campbell measure C ! on D × D which maps any measurable function φ :

D × D → R to C ! (φ) = E x∈X φ(x, X \ {x}). (31)
The Campbell measure above is very similar to the split Campbell measure dened for random T-tessellations. A split of a T-tessellation is the insertion of a non-blocking segment. Its analog for point patterns is the insertion of a new point. A merge of a T-tessellation is the removal of a non-blocking segment. Its analog for point patterns is the removal of an (arbitrary) point of the pattern. Let µ denote the distribution of a Poisson point process with intensity 1 on D. Consider a distribution P µ on D with density h :

P (dX) = h(X) µ(dX).
Let us suppose that h is hereditary :

∀x ∈ D, ∀X ∈ D, h(X) = 0 ⇒ h(X ∪ {x}) = 0. (32)
Then the reduced Campbell measure is absolutely continuous with respect to the product of the Lebesgue measure on R 2 with P and its density denoted λ with respect to the product measure is dened as the Papangelou conditional intensity given by

C ! (dx, dX) = λ(x; X) dxP (dX)
The Papangelou conditional intensity has the following expression

λ(x; X) = h(X ∪ {x}) h(X) . (33)
Note that one can nd in the literature expressions of the Papangelou conditional intensity slightly dierent from (33). For instance, for x ∈ X, Baddeley and Turner [START_REF] Baddeley | Practical maximum likelihood for spatial point patterns[END_REF] dene λ(x; X) as h(X)/h(X \ x) instead of 1. The dierent versions of the Papangelou conditional intensity are consistent with (33) as long as they coincide with (33) almost-everywhere with respect to the product of the Lebesgue measure on R 2 with P .

Contrast function and pseudolikelihood

Consider a parametric family of distributions P θ on D. It is supposed that every P θ µ and has a hereditary density h θ with respect to µ. Also the model is supposed to be identiable. The associated Papangelou conditional intensities are denoted λ θ . For every pair (θ, θ * ), consider the measure α θ,θ * on D × D dened by

α θ,θ * (dx, dX) = λ θ (x, X) dxP θ * (dX). (34)
The Kullback-Leibler divergence of α θ,θ * from α θ * ,θ * can be expressed as

KL (α θ * ,θ * , α θ,θ * ) = M (θ * , θ * ) -M (θ, θ * ),
where

M (θ, θ * ) = D×D log λ θ (x, X) C ! θ * (dx, dX) - D×D λ θ (x, X) dxP θ * (dX) (35)
Note that M (., θ * ) has a global maximum at θ = θ * . Similarly to random T-tessellations, it can be shown that the maximum is strict. Let X be a realization of X ∼ P θ * , then

LPL(θ|X) = x∈X log λ θ (x, X \ {x}) - D λ θ (x, X) dx (36)
is an unbiased estimator of M (θ, θ * ). Equation (36) coincides with the log-pseudolikelihood introduced by Besag [START_REF] Besag | Statistical analysis of non-lattice data[END_REF][START_REF] Besag | Some methods of statistical analysis for spatial data[END_REF][START_REF] Jensen | Pseudolikelihood for exponential family models of spatial point processes[END_REF] for point processes. The maximum pseudolikelihood estimator is θ = arg max θ LPL(θ|X).

Generally no analytical expression of the integral involved in the log-pseudolikelihood (36) is available. Thus computing θ is not straightforward. The most commly used method for computing θ is the Poisson regression method. This method was rst proposed by Berman and Turner [START_REF] Berman | Approximating point process likelihoods with GLIM[END_REF] for the inference of heterogeneous Poisson point processes. The extension to a larger class of Gibbs processes was developped by Baddeley and Turner [START_REF] Baddeley | Practical maximum likelihood for spatial point patterns[END_REF].

A weaker but more tractable contrast function

Below we introduce a family of contrast functions. They can be considered as approximations of the contrast function -M associated with the pseudolikelihood. Their main feature is that their minimum is easily computed using standard tools of statistics.

Let ρ be an integrable function on D. In practice, ρ may be taken proportional to the indicator function of D. Dene

α ρ θ,θ * (dx, dX) = (λ θ (x, X) + ρ(x)) dxP θ * (dX). (37)
When ρ ≡ 0, the measure dened in (37) is just α θ,θ * as dened in Equation (34). Consider the function

KL (α θ * ,θ * , α θ,θ * ) -KL α ρ θ * ,θ * , α ρ θ,θ * . (38)
This function cancels at θ = θ * and it is non-negative according to Lemma 1. The following inequalities hold:

0 ≤ KL (α θ * ,θ * , α θ,θ * ) -KL α ρ θ * ,θ * , α ρ θ,θ * ≤ KL (α θ * ,θ * , α θ,θ * ) .
Heuristically, the function (38) tends to the upper bound when ρ tends to innity while it tends to the lower bound when ρ tends to zero. Hence the function (38) of θ

Approximates KL (α θ * ,θ * , α θ,θ * ) as ρ tends to innity.

Is less discriminative than KL (α θ * ,θ * , α θ,θ * ) for a nite ρ.

Simple computations show that up to some additive terms not depending on θ the KL-divergence can be written as

KL α ρ θ * ,θ * , α ρ θ,θ * = - D×D log (λ θ (u, X) + ρ(u)) C ! θ * (du, dX) - D D ρ(u) log (λ θ (u, X) + ρ(u)) P θ * (dX)du + D D λ θ (u, X) P θ * (dX)du + . . .
Thus the dierence (38) can be written as

- D×D log λ θ (u, X) λ θ (u, X) + ρ(u) C ! θ * (du, dX)+ D D ρ(u) log (λ θ (u, X) + ρ(u)) P θ * (dX)du + . . .
As a consequence the M function introduced in (35) can be redened as follows

M ρ (θ, θ * ) = D×D log λ θ (u, X) (λ θ (u, X) + ρ(u)) C ! θ * (du, dX) - D D ρ(u) log (λ θ (u, X) + ρ(u)) P θ * (dX)du. ( 39
)
The function M ρ has a global maximum θ = θ * . Below it is rewritten as an expectation in order to derive an unbiased estimator. The rst term can be rewritten as

D×D log λ θ (u, X) (λ θ (u, X) + ρ(u)) C ! θ * (du, dX) = E θ * x∈X log λ θ (x, X \ {x}) λ θ (x, X \ {x}) + ρ(x)
,

where E θ * is the mean when X ∼ P θ * . In order to rewrite the second term, consider a point process Y on D with intensity ρ. Then we have

D D ρ(u) log (λ θ (u, X) + ρ(u)) P θ * (dX)du = E E θ * y∈Y log (λ θ (y, X) + ρ(y)) , (40)
where the rst mean is taken with respect to Y. Therefore an unbiased estimator of

M ρ is x∈X log λ θ (x, X \ {x}) λ θ (x, X \ {x}) + ρ(x) - y∈Y log (λ θ (y, X) + ρ(y)) . (41)
The unbiasedness of (41) can be checked as follows:

The mean with respect to X ∼ P θ * of the rst sum is equal to the rst integral in (39) by denition of the reduced Campbell measure (31).

The mean with respect to both Y and X ∼ P θ * of the second sum is equal to the second integral in (39) in view of identity (40).

Let θ be the estimator of θ * that maximizes the above criterion with respect to θ. It maximizes also

LPL ρ (θ|X) = x∈X log λ θ (x, X \ {x}) λ θ (x, X \ {x}) + ρ(x) + y∈Y log ρ(y) λ θ (y, X) + ρ(y) (42)
since both criteria dier only by a term not depending on θ. In order to obtain an even more homogeneous expression, assume that the conditional probability given X that X ∩ Y = ∅ is zero. Then almost surely X = X \ {y} and the estimating criterion can be rewritten as

LPL ρ (θ|X) = x∈X log λ θ (x, X \ {x}) λ θ (x, X \ {x}) + ρ(x) + y∈Y log ρ(y) λ θ (y, X \ {y}) + ρ(y). (43)
The estimation criterion above was rst proposed by Baddeley et al. [START_REF] Baddeley | Logistic regression for spatial Gibbs point processes[END_REF]. Our derivation here is dierent: based on a contrast function rather than a score function.

Thus we get the log-likelihood of a binomial regression. The sample to be considered is t z , z ∈ X ∪ Y with t z = 1 when z ∈ X. And the success probabilities are of the form λ θ (z, X \ {z}) λ θ (z, X \ {z}) + ρ(z) .

Let us focus on the case of an exponential family. Then we have

Pr [t z = 1] = exp θ T t(z, X \ {z}) exp (θ T t(z, X \ {z})) + ρ(z) , = 1 1 + exp (log ρ(z) -θ T t(z, X \ {z}) . That is logit Pr [t z = 1] = log ρ(z) -θ T t(z, X \ {z}).
Therefore in order to compute θ one can use computing tools already available for the binomial regression with a logistic link function.

In our paper [START_REF] Kiêu | A completely random T-tessellation model and Gibbsian extensions[END_REF], a new class of stochastic models for planar tessellations was introduced. Theoretical concepts and results mainly inspired by point process theory were provided. Also a Metropolis-Hastings-Green type algorithm was proposed for simulating Gibbs models. The current report is a rst step towards parametric inference from observed data.

The key ingredient for introducing the pseudolikelihood for Gibbsian T-tessellations is the Kullback-Leibler divergence and contrast function theory. Applying such an approach to the density h would lead to the classical likelihood which is unpracticable because of the unknown normalizing function involved in h. The trick here is to consider Campbell measures instead. As shown in this report, applying a similar approach to point process theory yields the well-known pseudolikelihood introduced by Besag [START_REF] Besag | Statistical analysis of non-lattice data[END_REF] who discretized point processes into spatial random elds with discrete supports. It was also shown that the approximation of the maximum pseudolikelihood based on a binomial regression [START_REF] Baddeley | Logistic regression for spatial Gibbs point processes[END_REF] can be derived from a contrast function involving the Kullback-Leibler divergence and the Campbell measure.

Whether such an approach can be extended to other types of random spatial structures is a open question. In our opinion, the key point is to identify basic operators that allow to explore the whole space of spatial structures. Based on such operators, one may dene one or several Campbell measures leading in turn to a pseudolikelihood.

Alternative inference methods are also of interest. As a variant, one could consider two contrast functions associated with splits and merges on one side and on ips on the other side. This would lead to two pseudolikelihoods and two estimators. Another possibility is Monte-Carlo Maximum Likelihood (MCML), see e.g. [START_REF] Geyer | Markov chain Monte Carlo maximum likelihood[END_REF]. Implementing MCML is possible using the simulation algorithm proposed in [START_REF] Kiêu | A completely random T-tessellation model and Gibbsian extensions[END_REF]. Bayesian approaches would deserve some investigations too.

In this report, the behaviour of the maximum pseudolikelihood estimator was investigated by means of simulations showing that bias and dispersion decrease as the number of cells in the observed tessellation increases. Theoretical studies would be of interest too, in particular asymptotics. However such studies require the development of a theoretical framework where random T-tessellation would be extended to the whole plane (currently random T-tessellations are bounded to a compact polygon).

The intriguing behaviour of the maximum pseudolikelihood when tting a wrong model to observed data should be investigated either theoretically or empirically.

Applying the maximum pseudolikelihood to real data may require some preprocessing. As an example, consider the maximum pseudolikelihood for the CRTT model. Its computation requires the determination of the number of non-blocking segments and of the total cell perimeter. In practice, a T-tessellation is often provided as a list of polygons (cells). Therefore one has to compute the number of non-blocking segments from the list of polygons. This is not a trivial task both because real tessellations may not be exact T-tessellations and because of unavoidable numerical errors related to number representations in digital computing. The library LiTe provides some methods which A Extension of the Kullback-Leibler divergence to non probability measures Let α and β be two nite non-negative measures on a measurable space E. Let the extended Kullback-Leibler divergence of β from α be dened as

KL(α, β) =        E dα dβ (z) log dα dβ (z) + 1 - dα dβ (z) β(dz), if β α, +∞ otherwise. (44)
In the equation above, 0 log 0 is assumed to be 0. If α and β have the same total mass, Equation (44) simplies into

KL(α, β) = E dα dβ (z) log dα dβ (z) β(dz).
The simplication above applies especially when α and β are probability measures. The classical Kullback-Leibler divergence is a particular case of the extended one. Since the function x → x log x + 1 -x is non-negative, the extended Kullback-Leibler divergence dened in Equation ( 44) is also non-negative. Furthermore the function above cancels only at x = 1. It follows that the extended Kullback-Leibler divergence cancels only when α equals β except on a β-null subset of E.

A small lemma that will be used further in this report.

Lemma 1. Let α, β and γ be non-negative nite measures on a measurable space E.

Then the following inequality holds

KL(α, β) ≥ KL(α + γ, β + γ).
The result is rather intuitive: α + γ and β + γ are more similar than α and β.

Proof. First consider the case where β does not dominate α. Then KL(α, β) = +∞ and the inequality holds. Now assume that β α. Let E be a measurable subset E ⊂ E such that (β + γ)(E) = 0. Both β and γ cancel on E. Since β α, α cancels on E too. It follows that

(α + γ)(E) = 0. Hence (β + γ)(E) = 0 implies that (α + γ)(E) = 0, i.e. β + γ α + γ.
Let µ be a measure dominating α, beta and γ. Let a, b and c be the densities of α, β and γ with respect to µ. We have

KL(α + γ, β + γ) = E a(z) + c(z) b(z) + c(z) log a(z) + c(z) b(z) + c(z) + 1 - a(z) + c(z) b(z) + c(z) (b(z) + c(z)) µ(dz), = E (a(z) + c(z)) log a(z) + c(z) b(z) + c(z) + b(z) -a(z) µ(dz).
The dierence between divergences can be written as

KL(α, β) -KL(α + γ, β + γ) = E a(z) log a(z) b(z) -(a(z) + c(z)) log a(z) + c(z) b(z) + c(z) dµ(z).
We have to check that the integrand is non-negative. The non-negativity results from the following inequality

x log x y ≥ (x + h) log x + h y + h , for all h ≥ 0.
The latter inequality is a consequence of the decreasing of the function

x y → x log x y
when moving away from the diagonal y = x. The decreasing is proved by

∇x log x y , 1 1 = log x y + 1 -x y , 1 1 = 1 - x y + log x y ≤ 0.
Above ., . denotes the scalar product.

B Subcongurations of a Poisson point process Lemma 2. Let X be a Poisson point process with intensity measure ν on a space D.

The total mass ν(D) is supposed to be nite. Then, for any measurable real-valued function φ dened on the space of nite point patterns in D, we have The right-hand side statement holds for T = T 0 . It follows

T 0 / ∈ G ⇒ ∀o ∈ M T 0 ∪ F T 0 , (o -1 , oT 0 ) / ∈ E s ∪ E f .
The right-hand side statement holds for o = o 1 :

T 0 / ∈ G ⇒ (o -1 1 , o 1 T 0 ) / ∈ E s ∪ E f , ⇒ h(T 0 ) = λ t 1 (o 1 , o -1 1 T 0 )h(o -1 1 T 0 ), ⇒ h(T 0 ) = λ t 1 (o 1 , T 1 )h(T 1 ),
where for i = 1, . . . , n

t i = s if o i is a split, f if o i is a ip.
Repeating this computation, one gets

h(T ) = h(D) n i=1 λ t i (o i , T i ). (46)
Now we show that G is a µ-null set. Let and that µ(G s ) = 0. The proof for µ(G f ) = 0 is similar. In Equation (46), the density h is determined only up to the multiplicative constant h(D). It should be noticed that h(D) is necessarily positive (otherwise h would be null on T due to its hereditariness). The constant h(D) is determined from the fact that the integral of h on T is equal to 1.

  3: Compute t(m, T ) for m ∈ M T 4: Compute t(f, T ) for all f ∈ F T 5: Compute the sum of the t(f, T )'s 6: S ← sample of m independent and uniform splits s of T 7: Compute the t(s, T )'s 8: repeat 9:

  At the time being, it is not possible to investigate the asympotic behaviour of the maximum pseudolikelihood estimator. Gibbs models of T-tessellations are dened only in bounded domains. Extensions to the whole plane, existence and unicity of Gibbs measures are still open questions. In order to provide some hints on the behaviour of

Figure 1 :

 1 Figure 1: Top: T-tessellation generated by the model with penalization on acute angles between segments. Bottom: maximum pseudolikelihood estimation. Background: discrete approximation of the log-pseudolikelihood (increasing from red to white). Grey arrows: approximated log-pseudolikelihood gradient. Circles: intermediate estimates found by NOIS algorithm. Cross: true parameters.

Figure 2 :

 2 Figure 2: Simulation of the CRTT (top row), angle (middle row) and area (bottom row) models. Left: rst tessellation sampled at the end of the burnin stage. Right: energy evolution during the rst SMF iterations. Vertical lines: iterations where tessellations are sampled.

2 Figure 3 :Figure 4 :

 234 Figure 3: Maximum pseudolikelihood estimates (500 replicates) for the three models (CRTT, angle and area). Abscissa: mean number of cells for each of the three domain size. For each model and domain size, the estimate distribution is represented as a boxplot. First and last deciles are shown as the whisker ends. True parameter values represented as horizontal lines.

  ) = exp(ν(D)) E φ(X). (45) Proof. Using the general expression of the distribution of a Poisson point process, one getsE Y ∈X φ(Y ) = e -ν(D) n≥0 1 n! D n Y ⊂{x 1 ,...,xn} φ(Y ) ν n (dx 1 . . . dx n ).A subset of a point pattern can be mapped into a subset of indices: ..,n}D n φ({x i : i ∈ I}) ν n (dx 1 . . . dx n ). 1 , . . . , x m }) ν m (dx 1 . . . dx m ).The summand does not depend on I. Therefore the sum over I is just equal to the number of subsets of m elements from a n-size set:(D) n-m m!(n -m)! D m φ({x 1 , . . . , x m }) ν m (dx 1 . . . dx m ), = e -ν(D) m≥0 e ν(D) m! D m φ({x 1 , . . . , x m }) ν m (dx 1 . . . dx m ), = e ν(D) E φ(X).C A random T-tessellation is determined by its Papangelou conditional intensitiesBelow a proof of Proposition 1 is given. From equations (78), it follows thath(sT ) = λ s (s, T )h(T ), h(f T ) = λ f (f, T )h(T ),for µ-almost all T ∈ T and almost all s ∈ S T and all f ∈ F T . That is there exists a null set E s ⊂ C s with respect to the measure dsµ(dT ) such that h(sT ) = λ s (s, T )h(T ) for all (s, T ) / ∈ E s . Similarly there exists a null set E f ⊂ C f with respect to the measure df µ(dT) such that h(f T ) = λ f (f, T )h(T ) for all (f, T ) / ∈ E f . Let G =    T ∈ T : ∃ T ∈ L⊂L(T ) T ( L), ∃o ∈ M T ∪ F T , (o -1 , o T ) ∈ E s ∪ E f    ,where L(T ) is the pattern of lines supporting the segments of T . The unionL⊂L(T ) T ( L)is the subset of T-tessellations that is spanned by applying successive merges and ips to T (remember that a merge remove a line supporting a segment and that a ip does not change the supporting lines).Let T ∈ T . There exists a sequence of splits and ips o n , . . . , o 1 which transform gradually the empty tessellation D into T :D = T n , o n T n = T n-1 , . . . , o 2 T 2 = T 1 , o 1 T 1 = T 0 with T 0 = T . Note that for i = 1, . . . , n, o i ∈ S T i ∪ F T i i.e. o -1 i ∈ M T i-1 ∪ F T i-1 . Suppose that T / ∈ G. T 0 / ∈ G ⇔ ∀ T ∈ L⊂L(T ) T ( L), ∀o ∈ M T ∪ F T , (o -1 , o T ) / ∈ E s ∪ E f .

TT

  ( L), ∃o ∈ M T , (o -1 , o T ) ∈ E s ( L), ∃o ∈ F T , (o -1 , o T ) ∈ E f    . Obviously G = G s ∪ G f . Let F s = T ∈ T : ∃m ∈ M T , (m -1 , m T ) ∈ E s . We have T ∈ G s ⇔ ∃ T ∈ L⊂L(T ) T ( L), T ∈ F s . Therefore µ(G s ) ≤ E T ∈ L⊂L(T) T ( L) I Fs ( T ), T ∼ µUsing Denition 1, one can rewrite the right-hand side as a mean with respect to the unit Poisson line process L:µ(G s ) ≤ cte E T ∈T (L) T ∈ L⊂L(T ) T ( L) I Fs ( T ), ≤ cte E T ∈T (L) T ∈ L⊂L T ( L) I Fs ( T ), ≤ cte E |T (L)| T ∈ L⊂L T ( L)I Fs ( T ).The constant before the mean is the unknown normalizing constant of the distribution µ. Now let us show that the random variableT ∈ L⊂L T ( L) I Fs ( T )is null almost surely. Since it is non-negative, it is sucient to show that its mean is null. Since the union is disjoint, we haveE T ∈ L⊂L T ( L) I Fs ( T ) = E L⊂L T ∈T ( L)I Fs ( T ).Applying Lemma 2 (see Appendix B), we can rewrite the mean of the sum on the line subsets:E T ∈ L⊂L T ( L) I Fs ( T ) = exp(b(D)/π) E T ∈T (L) I Fs ( T ),where b(D) is the perimeter of the domain D. Therefore it follows from Denition 1E T ∈ L⊂L T ( L) I Fs ( T ) = exp(b(D)/π)µ(F s )From the denition of F s we haveµ(F s ) ≤ E m∈M T I Es (m -1 , mT ), T ∼ µ,≤ T S T I Es (s, T ) dsµ(dT ), ≤ 0. Hence E T ∈ L⊂L T ( L) I Fs ( T ) = 0, and T ∈ L⊂L T ( L) I Fs ( T ) is null almost surely. In turn, this implies that E |T (L)| T ∈ L⊂L T ( L) I Fs ( T ) = 0

Table 1 :

 1 Simulation settings. For each model, θ values are given. Tessellated domains are squares with side lengths w 1 , w 2 and w 3 . Numbers of iterations for the burnin stage and between sampled tessellations (sampling period) are also provided.

	model θ 1	θ 2	w 1 w 2 w 3 burnin period
	CRTT 0.64		2.5 1.75 1	12500	3704
	angle 2.49	2.5 2.5 1.75 1.4 30000	9473
	area 0.53 835.2 2.8 1.9 1.5 11000	7223

Table 2 :

 2 Estimation settings used by the NOIS algorithm. For each model, the values of the tolerance parameter δ and the maximal number of NOIS updates.

	model tolerance max. iterations
	CRTT		0
	angle	0.005	150
	area	-0.005	100
	an explicit simple form. It can be computed using method Run of class PLInferenceNOIS


domain: sdv sde

text:
Frédéric Rees

Cyril Germain

Thibault Sterckeman

Jean-Louis Morel

Increase of Cd and Zn uptake by the hyperaccumulator Noccaea caerulescens grown in biochar-amended soils

Increase of Cd and Zn uptake by the hyperaccumulator Noccaea caerulescens grown in biochar-amended soils F.REES 1,2 , C. Germain 1,2 , T. Sterckeman 1,2 and J.L. Morel 1,2 1 Université de Lorraine -Laboratoire Sols et Environnement UMR 1120, Vandoeuvre-lès-Nancy,, France ; 2 INRA -Laboratoire Sols et Environnement UMR 1120, Vandoeuvre-lès-Nancy,, France ABSTRACT: Biochar, i.e. the solid product of biomass pyrolysis, has recently been investigated as a soil amendment in metal contaminated soils, mostly for its ability to decrease the phytoavailability of trace elements and to support plant growth. Several works have shown its efficiency for limiting the uptake of metals by plants, but the combination of biochar with metal-hyperaccumulating plants has never been tested. This work was conducted to examine the effects of biochar amendments on metal uptake by a Cd-and Znhyperaccumulator (Noccea caerulescens), compared to a non-hyperaccumulating plant (Lolium perenne). Plants were grown in controlled conditions on one acidic (A) and one alkaline (B) soil contaminated by Cd, Pb and Zn by smelter activities, both amended by a wood-derived biochar at variable rates up to 10 % (w/w). Biochar amendments in both soils decreased the availability of metals, but also of other major cations such as Ca 2+ , as shown by the analysis of pore water and soil extracts. This effect was linked to the observed increase of soil pH with increasing biochar dose. While shoot metal uptake of L. perenne constantly decreased with biochar addition in both soils, an increase of shoot Cd uptake of N. caerulescens with 5% (w/w) biochar was recorded on both soils, and of Zn uptake on soil B. We explain this increase of metal hyperaccumulation by a decrease of competition with Ca for metal uptake. Biochar therefore affects plant metal uptake by decreasing the availability of both cationic trace elements and major cations.

This study reveals that biochar may be used not only as a sorbent or liming agent for decreasing the mobility of metals in contaminated soils, but also as an enhancer of phytoextraction to increase the removal of metals by hyperaccumulating plants.