Roger Martineau

Daniel Sauvant

Daniel R Ouellet

Cristiano Côrtes

Jean Vernet

Isabelle Ortigues

Hélène Lapierre
email: helene.lapierre@agr.gc.ca

I Ortigues-Marty

Relation of net portal flux of nitrogen compounds with dietary characteristics in ruminants: A meta-analysis approach

Keywords: portal, amino acid, ammonia, urea

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.

INTRODUCTION

Decrease of N intake (NI) with the aim of increasing efficiency of N utilization and decreasing the negative environmental effects is a main goal of modern animal production. It needs, however, to be achieved through balanced rations so that true AA supply and consequent animal productivity are not compromised. A first step toward reaching this objective involves adequately defining the relationships between feed intake and composition, and availability of nitrogenous compounds to the animal. In ruminants, nutrients truly available to the animal greatly differ from nutrients ingested due to extensive rumen fermentation. The N digested is mainly absorbed as free AA, used by the animal for protein synthesis, or as ammonia, which needs to be detoxified by the liver into urea [START_REF] Lapierre | The route of absorbed nitrogen into milk protein[END_REF]. Therefore, the forms in which N is absorbed are critical in determining the proportion of N digested that is available to support protein synthesis. Besides N intake, the recycling of arterial urea into the gut provides an additional source of N for microbial protein synthesis that also needs to be quantified [START_REF] Lapierre | Nitrogen recycling in the ruminant: A review[END_REF].

One way to quantify the flow of nutrients truly available to the animal is to measure the net transfer of metabolites across the portal-drained viscera (PDV; gut, spleen, pancreas, and associated mesenteric fat). Early studies describing procedures for catheter placement and blood flow measurement were reported in lambs [START_REF] Katz | A method for simultaneous cannulation of the major splanchnic blood vessels of the sheep[END_REF], beef cattle (Sniffen and Jacobson, 1975) and dairy cows [START_REF] Baird | Determination of portal and hepatic metabolite production rates in the adult dairy cow[END_REF]. Since then, the technique has been used widely to study stress (e.g., parasitism and thermal stress). Furthermore, data presented on a regression basis with no numerical data were not included in the database. Finally, the database used for this meta-analysis consisted of 215 treatments from 68 publications obtained from 18 different scientific journals, conference proceedings, and theses or dissertations. The list of these publications is available in the Appendix and a description of the database is presented in Table 1.

Selection of Variables

Previous meta-analyses have indicated that NI was a driving force in N metabolism [START_REF] Castillo | A review of efficiency of nitrogen utilisation in lactating dairy cows and its relationship with environmental pollution[END_REF][START_REF] Yan | Splanchnic nitrogen metabolism by growing beef steers fed diets containing sorghum grain flaked at different densities[END_REF]. Although, the correlations between NPA of nitrogenous compounds and NI may not be as good as those obtained with other parameters such as N digested or MP supply [START_REF] Lapierre | The route of absorbed nitrogen into milk protein[END_REF], these parameters were either not measured or would have relied on a single empirical predictive model. Therefore, each NPA was regressed as a function of NI (i.e., the predictor of interest) and other covariates that could be qualitative or quantitative.

Qualitative Covariates. The qualitative covariates selected were species (i.e., sheep, cattle, or dairy cows) and physiological stage of cattle and sheep (i.e., growing or adult). Dairy cows were separated from cattle because of their different physiological status and level of intake relative to maintenance. Physiological stage of cattle and sheep was categorized on the basis of their BW. Cattle (i.e., growing or finishing cattle including both beef and dairy breeds) and sheep had a large range of BW because research has been carried out on growing as well as on adult animals (Table 1). Therefore, these species were divided into 2 subcategories (i.e., sheep below or above 50 kg; cattle below or above 350 kg) to characterize physiological stage, because it could not be properly assessed in each publication. The BW threshold used to subcategorize physiological stages corresponded to a BW area for which few experiments were found.

Quantitative Covariates. The quantitative covariates were related to the chemical composition of the diet and to their intake. They were selected because of their metabolic links with the NPA and the relative ease of measuring or estimating them using the NRC (2001) model. The covariates selected were DMI (g/d per kg of BW), the proportion of concentrate (CONC; % of DM), NFC concentration (% of DM), and NFC intake (NFCi; g/d per kg of BW), NDF concentration (% of DM), and intake (NDFi; g/d per kg of BW), total digestible nutrients (TDN; % of DM) concentration calculated at 1 × maintenance, CP concentration (% of DM), and finally RDP and RUP concentrations (% of CP and % of DM). Parameters reported in g/d were expressed as a function of BW 1.0 to allow comparison between small and large ruminants (Table 1). Indeed, although across-species metabolic processes may be best described as a function of BW 0.75 [START_REF] Brody | Bioenergetics and Growth[END_REF], intake as well as nutrient digestion and absorption have been shown to be similar between sheep and cattle when expressed as a function of BW 1.0 [START_REF] Vernet | Prediction of blood nutrient supply to tissues of economical interest in ruminants: A first step with the prediction of portal blood flow[END_REF][START_REF] Sauvant | Étude comparative du mode d'expression du niveau d'alimentation chez les ruminants[END_REF]. Feed ingredients as similar as possible to those reported in the publications were selected from the NRC (2001) feed library to obtain the most accurate assessment of the feeding value of each ingredient and reconstitute the ration. The chemical composition (CP, NDF, and ash) of each feed ingredient was used when reported. When not reported for individual ingredients, the CP, NDF, and ash composition of the total diet was calculated to be as close as possible to that reported in the paper using values from the NRC (2001) feed library for all ingredients and making adjustment to the composition of the main forage of the ration. The CP of the ration was reported in all of the publications, whereas NDF and ash (or OM) concentrations were not reported in 48 and 40% of the publications, respectively. Where NDF or ash were not reported, values from the NRC (2001) feed library were used.

Corrections for Differences in Methods

Differences in methods of analyzing and calculating NPA of nitrogenous nutrients introduce a systematic bias into the values reported; therefore, corrections of NPA for differences in methods were applied before performing meta-analyses, as described by [START_REF] Martineau | Technical note: Corrections of net portal flux of nitrogen compounds for differences in methods: First-step of a meta-analysis[END_REF]. In the database, 23.8% (n = 45) and 26.6% (n = 46) of NPA reported on a plasma basis were recalculated on a whole-blood basis for ammonia-N and urea-N, respectively. Net portal absorption was converted from an α-amino-N basis to a total AA-N basis (61.3%; n = 114) by multiplying NPA of α-amino-N by 1.3958 [START_REF] Martineau | Technical note: Corrections of net portal flux of nitrogen compounds for differences in methods: First-step of a meta-analysis[END_REF]. When the NPA of both α-amino-N and individual AA were reported in the same experiment, the NPA of individual AA-N was calculated and used. Corrected data were used for the meta-analyses and are presented in the tables.

Meta-Analyses

Coding of Experiments. For publications with 2 or more experiments, the latter were separated to be individually encoded. Each experiment consisted of at least 2 treatments.

Statistical Model. Relationships between the response (i.e., NPA of AA-N, ammonia-N, or urea-N) and the explanatory variables were studied with a variancecovariance model:

Y = α + βX + species + α i (species) + species × X + e,
where α is the general intercept, β is the slope of the relationship between X and Y common to all experiments, α i is the fixed effect of experiment i on the general intercept α nested within species, and e is the random residual error term [START_REF] Sauvant | Meta-analyses of experimental data: Application in animal nutrition[END_REF]. Each NPA was regressed against NI, the covariate of interest, and outliers were identified and discarded iteratively on the basis of studentized residual errors (less than -3.0 or greater than 3.0; 9, 7, and 11 outliers for NPA of AA-N, ammonia-N, and urea-N, respectively). All models were used without weighting the observations because NPA and intake parameters were divided by the BW and the number of animals per treatment was small and comparable across the studies. Linear and quadratic models were fitted to the data and compared. Normality and homoscedasticity of residuals were assessed graphically (normal probability plot, studentized residuals vs. fitted values). Thorough graphical assessments were carried out at each step of the meta-analyses.

Influence of Covariates. The influence of the quantitative (i.e., dietary factors) and qualitative covariates (i.e., species and physiological stage) was assessed graphically and statistically. Once all outliers were discarded, a set of residual errors (i.e., the difference between an observed value and its corresponding fitted value) was produced with only NI in the model. These residual errors were regressed against each covariate to generate data in Tables 2,3, and 4. Furthermore, the residual errors with NI in the model were tested on smaller data sets extracted from the whole database that excluded experiments with little intra-study variation of the covariate. The standard deviation for the covariate was determined in each experiment of the database and the minimal threshold of intra-study variation was the mean of all of these standard deviations. The use of smaller data sets, termed subdatabases, improves the ability to assess the structural form of the within-experiment relationship between the residuals and each covariate [START_REF] Sauvant | Meta-analyses of experimental data: Application in animal nutrition[END_REF]. The effect of the species and physiological stage was first tested on the whole database and, if significant, then tested on each subdatabase.

The general characteristics of the subdatabases used to test each covariate are presented in Tables 2,3, and 4. The linear or quadratic slope (β) of each covariate and its interaction with species, the root mean square error (RMSE), and P-values less than 0.20 are pre-sented. A quadratic term or an interaction with species was included in the model only if P ≤ 0.10. The influence of a covariate was considered significant at P ≤ 0.05 and a tendency at 0.05 < P ≤ 0.10.

Models of Net Portal Appearance.

The models of net portal appearance were built first with NI as the sole predictor of interest of each NPA. Then, the covariates that showed a significant effect on the residuals were used, one by one, to generate a set of independent models. More complete models were generated using a forward procedure based on a heuristic approach [START_REF] Sauvant | Meta-analyses of experimental data: Application in animal nutrition[END_REF], adding one by one significant and biologically meaningful covariates. The covariates that modified meaningfully the slope of NI when added to the model were retained to control for potential confounders [START_REF] Kleinbaum | Confounding and interaction in regression[END_REF]. Finally, a parsimonious model with the lowest RMSE is proposed for each NPA based on biological relevance and in an attempt to include the same covariates in the predictive models for the 3 NPA.

All statistical analyses were carried out using the GLM procedure (Minitab version 16.1.1.0, 2010, Minitab Inc., State College, PA). Part of the effort of the approach used for this meta-analysis was to try to characterize covariates having an influence on the relationship between the NPA and NI, including the effect of species. Therefore, because the experimental conditions were not randomly distributed and were specific to each study, experiment was considered as a fixed factor in the models. The range of inference for this meta-analysis is, therefore, limited to the domain of the specific experiments in the data set [START_REF] Sauvant | Meta-analyses of experimental data: Application in animal nutrition[END_REF]. In the models of NPA, a quadratic term or an interaction with species was included only if P ≤ 0.10 and an effect was considered significant at P ≤ 0.05 and a tendency at 0.05 < P ≤ 0.10.

The project was submitted to and approved by the Institutional Committee for Animal Care of the Sherbrooke Research Centre (Quebec, Canada), although no animals were required for this study.

RESULTS AND DISCUSSION

Description of Selected Experiments and Data Sets

Rations. A description of animals, dietary characteristics, intake, and NPA is presented by species in Table 1. The BW of sheep and cattle were separated by physiological stages and overall averaged (± SD) 48.2 (± 16.7) and 325 (± 91) kg for sheep and cattle, respectively. Dietary characteristics differed among the species. On average, diets fed to sheep contained less energy (e.g., lower proportion of concentrate and NFC concentration) and more fiber (e.g., higher NDF concentration) than those fed to cattle and dairy cows. Sheep were fed mostly forage diets of average to low 1).

Nitrogen Intake and Net Portal Appearance of Nitrogenous Compounds. Nitrogen intake was similar for sheep and cattle on a g/d per kg of BW basis. However, NI was almost twice as high (P < 0.001) in dairy cows due to higher DMI and CP concentration of diets offered to dairy cows compared with other species (Table 1). Other species differences were noticeable on all covariates and on the NPA values as separated using the Tukey test.

On average, the portal blood flow, expressed as L/h per kg of BW, was lower in cattle than in sheep and dairy cows. The NPA of AA-N was lowest in cattle, but did not differ between sheep and dairy cows; however, the NPA of AA-N accounted for 54% of NI in sheep, whereas it represented only 37% of NI for cattle and dairy cows. The NPA of ammonia-N was similar for cattle and sheep but twice as high in dairy cows; on the other hand, NPA of ammonia-N accounted for 41% of NI with no species differences. The negative NPA of urea-N denotes a net transfer or recycling of urea-N from the blood into the gut. The NPA of urea-N was lowest in dairy cows and highest in cattle; the NPA of urea-N accounted for 42% of NI in sheep and only 28% of NI for cattle and dairy cows. All of these differences observed between species need to be investigated through meta-analyses, taking into account the differences observed in NI and diet composition between species.

Modeling N Intake and Net Portal Appearance of Nitrogenous Nutrients. Figure 1 shows that each NPA is related to NI. This raw relationship does not take into account the variations in the diets fed to the different species, which need to be accounted for with statistical models. The influence of each covariate on the within-experiment relationship between NI and each NPA is presented in Tables 2, 3, and 4. As mentioned previously, the use of smaller subdatabases improves the ability to explore the structural form of the relationship between the covariate and the response, after controlling for NI. However, this approach has some drawbacks, as few studies in dairy cows showed sufficient intra-study variation for CONC, NFC, NDF, TDN, RDP, NFCi, or NDFi. Therefore, extrapolation of results to dairy cows should be done cautiously and more research should be directed toward studying the variation of these covariates in dairy cows.

In the sections below, the influence of significant covariates on each NPA (Tables 2,3, and 4) is discussed. And finally, the complete model, including a set of common covariates for each NPA, is presented and discussed.

Net Portal Appearance of AA-N

Influence of Covariates. The effect of species was significant (P < 0.001) and NPA of AA-N was highest in sheep compared with cattle and dairy cows: 0.240, 0.174, and 0.134 g of N/d per kg of BW for least squares means of AA-N in sheep, cattle, and dairy cows, respectively. These least squares means were calculated at mean NI (0.466 g/d per kg of BW) and are in line with the discrepancy observed between NPA of AA-N and NI among species in Table 1. The effect of physiological stage was not significant (0.191 and 0.195 g of N/d per kg of BW for LSM of AAN-N for growing and adult, respectively; P = 0.64; LSM calculated at a mean NI of 0.422 g/d per kg of BW). Therefore, only the interaction of species with other covariates will be presented in Table 2. The effect of species itself is not reported because it could not be appropriately tested using the residuals that controlled for the study effect.

Most covariates had a significant influence on the effect of NI on NPA of AA-N, except RUP as a % of DM or CP and DMI (Table 2). Energy factors (e.g., proportion of concentrate, NFC, TDN, and NFCi) were associated positively with NPA of AA-N, whereas the opposite was observed for fiber factors (e.g., NDF and NDFi). Other potential covariates related to energy could have been useful; for example, in experiments designed to study the effects of processing method and degree of processing on N metabolism, [START_REF] Alio | Splanchnic nitrogen metabolism by growing beef steers fed diets containing sorghum grain flaked at different densities[END_REF] and [START_REF] Tagari | Portal drained visceral flux, hepatic metabolism, and mammary uptake of free and peptide-bound amino acids and milk amino acid output in dairy cows fed diets containing corn grain steam flaked at 360 or steam rolled at 490 g/L[END_REF] found a positive effect of processing on NPA of AA-N. However, these covariates could not be properly assessed in our data set because of an insufficient number of experiments dealing with that issue.

Regarding the other covariates studied, the CP diet concentration interacted strongly with species, the influence being slightly positive in sheep and strongly negative in cattle. However the species × CP interaction and the CP main effect were no longer significant when other covariates were added in the model, particularly NDF. A deeper analysis of the meta-design of studies with variation in NDF and CP revealed that such variations were obtained differently for sheep and cattle. In cattle experiments, the variation in CP was Interaction Sp × TDN and Sp × TDN 2 were included in the model. Species effect on the slope of TDN: Δ = 0.3386, 0.3888, and -0.7274 and on the slope of TDN 2 : Δ = -0.0023, -0.0027, and 0.0050 for sheep, cattle, and dairy cows, respectively. obtained by diluting the ration with a concentrate (Prior et al., 1981;Reynolds and Huntington, 1988;Huntington, 1989). On the other hand, the variation in sheep experiments was obtained by diluting the ration with a concentrate (Goetsch et al., 1994;Goetsch and Ferrell, 1995;Patil et al., 1995), by feeding a high-versus a low-quality forage (Prewitt et al., 1975;Park et al., 1997), or by adding soybean meal to the forage (Krehbiel et al., 1998). As a result, in cattle, an increase in CP was associated with an increase in NDF, whereas, in sheep, an increase in CP was generally associated with a decrease in NDF (or an increase in TDN), which is positively associated with NPA of AA-N. Not surprisingly, when controlling for both CP and NDF (or TDN), CP is no longer significant, as NDF or TDN are more potent covariates in the relationship between NI and NPA of AA-N.

The quality of protein fed (i.e., RUP as a % of CP) was not found to be associated with NPA of AA-N in this meta-analysis. Similarly, literature findings related to the effect of RUP supplementation on NPA of AA-N have not been consistent. At similar NI across treatments, RUP (% of CP) had a positive effect on NPA of AA-N in sheep (Bohnert et al., 1999;Rémond and Poncet, 2007) and dairy cows (Blouin et al., 2002). On the other hand, a negative effect (Han et al., 2001) or no effect was found on NPA of AA-N when the RUP concentration in the diet was increased (Cleale et al., 1987;Branco et al., 2004;Atkinson et al., 2007). Fukuma et al. (2005) reported a positive response in mature wethers fed increasing levels of RUP through dietary fishmeal supplementation; however, NI increased with the level of fishmeal supplementation, such that no difference could be found among treatments in the residuals (i.e., at equal NI). Overall, RUP per se may not be a major factor associated with NPA of AA-N if availability of degradable N adversely affects microbial protein synthesis.

Model of NPA of AA-N. Based on information gained with the previous analyses, a general NPA model was developed using the whole database. The final model for each NPA was based on the best fit to the data (i.e., lowest RMSE) and on the quest for common covariates linking NPA of AA-N, ammonia-N, and urea-N. Two covariates met these criteria: TDN concentration and NDFi. The model developed for NPA of AA-N (SE in parenthesis and P-value in superscript) is, therefore, NPA of AA-N (g of N/d per kg of BW) = -0.237 0.011 (±0.092) + 0.603 <0.001 (±0.048) × NI + 0.0027 0.024 (±0.0012) × TDN -0.0053 0.053 (±0.0027) × NDFi, with NI and NDFi in grams per day per kilogram of BW and TDN as % of DM; species effect on the intercept (P = 0.010): Δ = 0.102, -0.033, and -0.068 for sheep, cattle, and dairy cows, respectively; and RMSE = 0.0404 g of NPA of AA-N/d per kg of BW; n = 174 (75 experiments). Therefore, in this database, ruminants converted 60% of incremental NI into absorbed AA-N, but with the sheep presenting a higher intercept than cattle and dairy cows. Direct comparisons of NPA of AA-N in different ruminant species fed the same diets are scarce; only one study reported no difference between sheep and steers in NPA of most AA, results being expressed on a metabolic weight basis (Prior et al., 1981). However, NPA of AA in steers would have been 0.57 times the reported values if expressed on a BW 1.0 basis rather than on a BW 0.75 basis and, thus, absorption of most AA would have been lower in steers than in sheep. This is consistent with the finding reported in a review [START_REF] Lapierre | Nitrogen recycling in the ruminant: A review[END_REF]) that sheep absorbed twice as much AA-N relative to digested N than cattle. In that review, AA-N relative to digested N was intermediate in dairy cows. However, the review could not determine whether this trend was an effect of species per se or an indirect effect of diets differing between species. The reasons for the difference observed between species in the current analysis after controlling for dietary characteristics are not clear. A higher NPA of AA-N could result from a higher supply of digestible AA due to increased rumen microbial protein synthesis or smaller losses of AA across the PDV due to either decreased oxidation or smaller losses of endogenous protein in the feces, or both. In addition, salivary urea-N secretion (approximately 60-80% of salivary N) and other N compounds (e.g., enzymes, true proteins from mucus-producing glands, or glycoproteins) are not taken into account in PDV net flux because the salivary glands are not drained by the portal vein. Therefore, it is possible that sheep could benefit of an additional source of N if sheep salivate more than cattle or dairy cows when fed similar diets.

Inclusion of factors positively (TDN) or negatively (NDFi) related to the energy of the diets agrees with the fact that it is widely recognized that NI alone cannot explain protein duodenal flow and consequent NPA of AA-N due to the important contribution of microbial rumen microflora to both dietary protein degradation and microbial protein synthesis [START_REF] Clark | Microbial protein synthesis and flows of nitrogen fractions to the duodenum of dairy cows[END_REF]. Increased availability of energy stimulates microbial protein synthesis [START_REF] Firkins | Ruminal nitrogen metabolism: Perspectives for integration of microbiology and nutrition for dairy[END_REF], which increases the availability and absorption of AA-N. Indeed, in a study conducted to test extreme situations, NPA of AA-N was more influenced by ME intake than by CP concentration in cattle receiving an 11.7 or 16.2% CP diet (Reynolds et al., 1992;[START_REF] Reynolds | Splanchnic amino acid metabolism in ruminants[END_REF]. Although the NRC (2001) system does not provide an estimate of the digestibility of energy specific to the rumen, the latter should be related to TDN diet concentration and inversely to NDFi. Indeed, concomitantly with results from Table 2, TDN had a positive relationship with NPA of AA-N, whereas the opposite was observed for NDFi. Their inclusion in the final model improved the prediction of NPA of AA-N, as the slope (g of NPA of AA-N/g of NI) and RMSE (g of NPA of AA-N/d per kg of BW), with NI as a sole predictor, were 0.522 and 0.0432, respectively, compared with 0.603 and 0.0404 with the inclusion of TDN and NDFi in the final model.

Net Portal Appearance of Ammonia

Influence of Covariates. The effect of species was tested within experiment with NI and the species × NI interaction. The interaction was significant (P = 0.012) and NPA of ammonia-N was lowest in sheep compared with cattle and dairy cows: 0.162, 0.207, and 0.204 g of N/d per kg of BW for least squares means of ammonia-N in sheep, cattle, and dairy cows, respectively. These least squares means were calculated at mean NI (0.478 g/d per kg of BW). The effect of physiological stage was not significant (0.164 and 0.169 g of N/d per kg of BW for LSM of ammonia-N for growing and adult, respectively; P = 0.50; LSM calculated at a mean NI of 0.422 g/d per kg of BW). Therefore, only the interaction of species with other covariates are presented in Table 3.

All covariates had a significant influence on the effect of NI on NPA of ammonia-N but DMI (Table 3). Energy and fiber covariates had the opposite effect to what they had on NPA of AA-N: energy factors (i.e., proportion of concentrate, NFC, TDN, and NFCi) were associated negatively with NPA of ammonia-N, whereas the opposite was observed for fiber factors (i.e., NDF and NDFi). Therefore, at equal NI, provision of energy increases capture of RDP for microbial protein synthesis (see discussion above), thereby decreasing NPA of ammonia-N. Indeed, infusion of starch (1.2 kg) in the rumen or in the abomasum decreased NPA of ammonia-N in dairy cows (Reynolds et al., 1998), suggesting that ruminal or hindgut N use for fermentation was stimulated by the supply of energy.

Concentration of CP and of RDP (% of DM) tended to interact with species: at a similar NI, increasing CP or RDP diet concentrations would very slightly alter the NPA of ammonia-N in the sheep, whereas it would increase NPA of ammonia-N in both cattle and dairy cows. It has been observed that adding dietary urea increased NPA of ammonia-N in wethers (Ferrell et al., 1999) and beef steers (Maltby et al., 2005); however NI was not constant in these studies. In addition, an increase in RDP concentration in the diet resulting from a change in feed ingredients (at a similar NI) increased ammonia absorption in wethers (Bohnert et al., 1999) and tended to increase ammonia absorption in beef heifers (Huntington, 1987); it also numerically increased ammonia-N absorption in dairy cows (Blouin et al., 2002).

The influence of DMI tended to differ among species, being negative in the sheep and positive in cattle and dairy cows. At similar NI, an increment in DMI implies a decreased CP concentration in the diet. As discussed above, sheep experiments with variations in CP and NDF are usually associated with a decrease in CP and an increase in NDF (or decrease in TDN). Therefore, an increase in DMI in sheep as opposed to cattle and dairy cows may be associated with a lack of ruminally available energy to capture RDP for microbial protein synthesis in this subdatabase.

Model of NPA of Ammonia. As observed from the subdatabases of experiments, energy and fiber covariates had the opposite effect on NPA of ammonia-N compared with NPA of AA-N, when analyzed using the whole database. Indeed, accordingly with results presented in Table 3, TDN had a negative relationship with NPA of ammonia-N, whereas the opposite was observed for NDFi. The proposed model (SE in parenthesis and P-value in superscript) is NPA of ammonia-N (g of N/d per kg of BW) = 0.169 0.065 (±0.090) + 0.258 <0.001 (±0.040) × NI -0.0022 0.036 (±0.0010) × TDN + 0.011 0.042 (±0.005) × NDFi -0.00051 0.002 (±0.00016) × NDFi 2 , with NI and NDFi in grams per day per kilogram of BW and TDN as % of DM; species interaction on the slope of NI (P = 0.10): Δ = -0.067, -0.054, and 0.121 g of NPA of ammonia-N/g of NI for sheep, cattle, and dairy cows, respectively; and RMSE = 0.0313 g of NPA of ammonia-N/d per kg of BW; n = 183 (76 experiments). [START_REF] Lapierre | Nitrogen recycling in the ruminant: A review[END_REF] also reported that the slope between NPA of ammonia-N and NI was about twice as high in cattle than in sheep, but it could not be determined if this discrepancy was due to diet differences, and dairy cows were not analyzed separately from growing cattle. Results from the current metaanalysis suggest that, at constant TDN concentration and NDFi, an incremental NI would increase NPA of ammonia-N to a greater extent in dairy cows than in sheep and cattle.

Net Portal-Drained Removal of Urea-N

Influence of Covariates. The effect of species was significant (P = 0.04) as NPA of urea-N was the most negative in dairy cows and the less negative for cattle: -0.154, -0.117, and -0.181 g of N/d per kg of BW for least squares means of urea-N in sheep, cattle, and dairy cows, respectively. These least squares means were calculated at mean NI (0.468 g/d per kg of BW). The effect of physiological stage was not significant (-0.143 and -0.114 g of N/d per kg of BW for LSM of urea-N for growing and adult, respectively; P = 0.09; LSM calculated at a mean NI of 0.403 g/d per kg of BW). Therefore, only the interaction of species with other covariates is presented in Table 4.

Unlike the case for NPA of AA-N or ammonia-N, few covariates had a significant influence on the effect of NI on NPA of urea-N or interacted with species (Table 4). This could be partially explained by the fact that not all of the urea-N returning to the gut is accounted for by the NPA: part of the urea transfer to the gut transits via saliva (see discussion below). An energy factor like NFCi increased recycling of urea-N when controlling for NI. On the other hand, NDFi had a significant quadratic relationship. The effect of DMI mirrored that of NDFi because both covariates are correlated positively in the studies with variation of DMI and NDFi (r = 0.72; P < 0.001). However, it should be pointed out that ruminant studies in the subdatabase did not overlap over the full range of NDFi because only sheep studies (n = 3) had some of their treatments with NDFi levels greater than 15 g/d per kg of BW (none in cattle and dairy cows).

Energy factors (e.g., TDN and NFCi) were associated negatively with NPA of urea-N in line with previous studies showing that provision of rapidly fermentable energy increases the transfer of urea-N across the rumen wall [START_REF] Kennedy | The degradation and utilization of endogenous urea in the gastro-intestinal tract of ruminants: A review[END_REF][START_REF] Norton | Urea synthesis and degradation in sheep given pelleted-grass diets containing flaked barley[END_REF][START_REF] Obara | The influence of energy-rich supplements on nitrogen kinetics in ruminants[END_REF]Theurer et al., 2002). The effect of protein factors was not detected because it was already captured by NI in the residuals.

Model of NPA of Urea-N. It should be first acknowledged that this model relates only to urea-N being transferred or recycled into the gut through the PDV wall from the arterial circulation. Not all of the urea-N recycled into the gastrointestinal tract necessarily transits through the PDV, as a substantial flow of urea-N can be supplied to the rumen via the saliva. Low net PDV removal of urea-N may be counterbalanced by high salivary urea-N secretion. For example, the proportion of urea-N transfer through the saliva or the rumen can differ substantially depending on the type of diet (Huntington, 1989); however, the current data set could not address that aspect. The processing method and the degree of grain processing can increase the fermentation of starch in the rumen [START_REF] Theurer | Grain processing effects on starch utilization by ruminants[END_REF][START_REF] Huntington | Starch utilization by ruminants: From basics to the bunk[END_REF] and urea-N recycling to the gut [START_REF] Alio | Splanchnic nitrogen metabolism by growing beef steers fed diets containing sorghum grain flaked at different densities[END_REF]Delgado-Elorduy et al., 2002;Theu-rer et al., 2002). However, the data set did not contain enough experiments to control for that covariate. The proposed model (SE in parenthesis and P-value in superscript) including the same covariates as for AA-N and ammonia-N is NPA of urea-N (g of N/d per kg of BW) = 0.304 0.001 (±0.089) -0.107 0.004 (±0.036) × NI -0.0044 <0.001 (±0.0010) × TDN -0.020 <0.001 (±0.005) × NDFi + 0.00069 <0.001 (±0.00015) × NDFi 2 , with NI and NDFi in grams per day per kilogram of BW and TDN as % of DM, and RMSE = 0.0295 g of NPA of urea-N/d per kg of BW; n = 163 (70 experiments). Urea returned to the gut provides degradable N, which can be partly used into the rumen to support microbial protein synthesis. Indeed, the recycling of urea-N into the gut is divided between anabolic and catabolic fates. Urea-N transferred to the gut and hydrolyzed into ammonia-N can be 1) used for anabolic purposes (microbial protein synthesis); 2) absorbed and returned to the ornithine cycle to resynthesize urea-N; or 3) excreted in the feces [START_REF] Lapierre | Nitrogen recycling in the ruminant: A review[END_REF]. Increasing the energy supply through increased TDN is related to more negative NPA (or increased recycling) of urea-N at similar NI. This relationship is in line with previous studies showing that provision of rapidly fermentable energy increases the transfer of urea-N across the rumen wall, as previously discussed.

The effect of NDFi to decrease the NPA of urea (or increase the recycling) may be indirectly related to the effect of NDFi on NPA of ammonia-N. In experiments with a sufficient variation of NDFi, NPA of urea-N was strongly associated with NPA of ammonia-N: NPA of urea-N = -0.072 0.001 (±0.023) -0.428 0.002 (±0.128) × NPA of ammonia-N, with NPA of urea-N and ammonia-N in grams of N per day per kilogram of BW and RMSE = 0.0362 g of NPA of urea-N/d per kg of BW; n = 48 (20 experiments). [START_REF] Sunny | Salvage of blood urea nitrogen in sheep is highly dependent on plasma urea concentration and the efficiency of capture within the digestive tract[END_REF] infused increasing doses of urea to increase urea entry rate and plasma concentrations without altering directly the rumen environment. They reported that urea recycled to the gastrointestinal tract was strongly correlated to plasma urea concentration, contrary to previous assumptions. Therefore, experimental conditions increasing absorption of ammonia and urea synthesis may increase recycling of urea into the gastrointestinal tract.

Linking AA-N, Ammonia-N, and Urea-N Together Huntington et al. (1988) suggested that good agreement could be obtained among N available for absorption (i.e., N digested + recycled urea-N) and N available for metabolism (i.e., NPA of AA-N + NPA of ammonia-N). Obviously, such an agreement can only be obtained if unaccounted N available for absorption (e.g., salivary urea-N and glycoproteins, biliary N) and for metabolism (e.g., chylomicron N, peptides, nucleic acid, amino sugars) counterbalance each other [START_REF] Martineau | Technical note: Corrections of net portal flux of nitrogen compounds for differences in methods: First-step of a meta-analysis[END_REF]. If so, digested N would be equivalent to NPA of AA-N plus NPA of ammonia-N minus recycling of urea-N or to the summation of the 3 NPA. Furthermore, the total tract N apparent digestibility (TTDN) could be estimated by dividing the summation of the 3 NPA by NI.

About one-third of the treatments in the whole database did not report the 3 NPA concurrently and were excluded from the analysis. The mean estimated TTND (i.e., summation of the 3 NPA divided by NI) was 0.52 (SD = 0.34; minimum = -1.03; maximum = 2.07; n = 140). Such high variability could be related to measurement errors or to experimental conditions where unaccounted N available for both absorption and metabolism did not counterbalance. When the summation of the 3 NPA was regressed within experiment against NI, 0.62 (±0.06) g of N appeared as NPA of AA-N, ammonia-N, and urea-N (or as digested N) for each gram of incremental NI (P < 0.001; n = 142; 63 experiments). The TTND was reported in only 27 of these 63 experiments and averaged 0.62 (SD = 0.10; n = 82 treatments). The lower estimation of TTDN as the summation of NPA of nitrogenous compounds divided by NI compared with measured TTDN might be related to an underestimation of NPA of ammonia-N in most of the studies where the Berthelot reaction was used to determine ammonia concentrations [START_REF] Martineau | Technical note: Corrections of net portal flux of nitrogen compounds for differences in methods: First-step of a meta-analysis[END_REF].

CONCLUSIONS

The meta-analysis, performed on 90 experiments, indicated that the NPA of AA-N, ammonia-N, and urea-N were linearly related to NI. Also, covariates related to the energy supply of the diet, namely TDN concentration and NDFi, influenced the relationship between NI and the NPA of the 3 nitrogenous compounds. In addition to diet characteristics, species differences were detected. The NPA of AA-N responded similarly between species to incremental NI, TDN, and NDFi (no species by factor interaction), but at similar NI, TDN, and NDFi, the sheep absorbed more AA-N than did cattle and dairy cows. On the other hand, the response of NPA of ammonia-N to incremental NI tended to be twice as high in dairy cows as compared with sheep or cattle. No species difference was detected for the recycling of urea-N as measured by PDV flux, which excludes salivary contribution. These differences could be related to a lower loss of AA across the PDV (undigested endogenous secretions and oxidation) in sheep than in cattle. More research should be directed toward understanding the mechanisms responsible for these species differences, especially because modern diets for ruminants need to be enhanced with the goals of increasing N utilization efficiency, optimizing urea-N recycling, and decreasing environmental pollution associated with N losses.

Figure 1 .

 1 Figure 1. Within-experiment relationships of N intake and net portal absorption (NPA) of AA-N (a), ammonia-N (b), and urea-N (c); Δ = sheep; = cattle; and × = dairy cows.

Table 1 .

 1 Journal of Dairy Science Vol. 94 No. 6, 2011 Description of animals, dietary characteristics, intake, and net portal appearance (NPA) of nitrogenous metabolites in the database 1

				Sheep				Cattle				Dairy cows
	Item	n	Mean	SD	Minimum Maximum	n	Mean	SD	Minimum Maximum	n	Mean	SD	Minimum Maximum
	BW 2 (kg) Growing stage Adult stage Dietary characteristics 3		38 b 66 c	6 14	24 50	49 97	47 278 a 22 427 b	66 37	127 392	348 522	-37 619 a -	-54	-475	-713
	(% of DM) CONC NFC NDF TDN CP RDP RUP RDP (% of CP) RUP (% of CP) Intake 3		24.6 b 24.9 c 56.5 a 60.3 b 12.5 c 8.6 b 4.0 b 67.7 b 32.3 a	30.1 13.2 15.1 6.6 3.0 2.6 1.2 8.5 8.5	0.0 1.0 12.2 47.6 4.1 2.6 1.5 42.5 12.8	100.0 63.7 81.0 79.1 19.8 15.9 7.6 87.2 57.5	69 69 69 69 69 69 69 69 69	55.4 a 48.2 a 29.8 b 71.0 a 14.9 b 10.8 a 4.1 b 71.8 a 28.2 b	33.6 15.8 13.6 8.4 3.3 3.3 1.2 8.8 8.8	0.0 13.9 12.2 53.6 8.9 5.7 1.2 50.0 9.5	92.0 70.1 59.2 81.9 23.7 19.9 6.9 90.5 50.0	33 33 37 33 37 33 33 33 33	46.0 a 38.9 b 34.6 b 69.3 a 16.7 a 10.9 a 5.7 a 65.6 b 34.4 a	14.7 6.8 6.1 3.1 1.7 1.9 1.2 7.5 7.5	11.5 20.2 24.7 63.8 12.0 6.5 3.5 42.0 22.7	64.0 49.8 48.0 74.6 21.9 16.1 9.0 77.3 58.0
	(i; g/d per kg of BW) DMI NI NFCi NDFi Portal blood flow		22.6 b 0.45 b 5.8 c 12.5 a 2.76 a	6.7 0.17 3.9 4.9 0.84	6.3 0.09 0.2 2.8 1.56	41.6 0.90 22.6 27.9 6.00	69 69 69 69 65	17.8 c 0.43 b 8.3 b 5.6 c 2.07 b	4.8 0.16 3.1 3.5 0.50	5.5 0.12 2.9 0.8 1.11	29.4 0.88 14.0 14.0 3.32	37 37 33 37 31	29.1 a 0.77 a 11.7 a 10.1 b 2.90 a	6.6 0.19 2.8 2.9 0.61	12.4 0.36 5.0 4.3 1.84	41.7 1.21 15.0 14.8 4.23
	(L/h per kg of BW) NPA 4 (g of N/d
	per kg of BW) AA-N Ammonia-N Urea-N		0.23 a 0.16 b -0.16 b	0.12 0.10 0.09	-0.05 0.04 -0.47	0.50 0.55 0.04	60 59 57	0.17 b 0.18 b -0.11 a	0.14 0.10 0.05	0.00 0.05 -0.24	1.05 0.50 -0.03	22 29 29	0.29 a 0.32 a -0.22 c	0.10 0.12 0.09	0.07 0.13 -0.44	0.48 0.56 -0.09

Table 2 .

 2 Influence of each covariate on net portal appearance of amino acid-N 1 <0.20 are reported; otherwise NS. Each covariate was tested on the residual errors of the model with N intake as the sole predictor. A quadratic term was included in the model only if P ≤ 0.10.

		Characteristics of each subdatabase 2		Slope P-value		P-value 4
	Item	Exp Trt Mean SD Min Max SDmin	Linear	Quadratic	RMSE 3	Sp × cov	Sp × cov 2
	Diet 5 (% of DM) CONC NFC NDF TDN CP 6 RUP RUP (% of CP)	11 26 19 47 20 50 21 50 30 70 27 70 25 64	30.4 29.7 28.4 17.7 52.0 18.5 12.2 81.0 0.0 90.0 1.6 67.3 65.3 9.8 47.6 81.8 13.0 3.9 4.1 22.3 4.4 1.6 1.2 8.2 34.8 10.0 9.5 54.7	5.13 3.56 3.47 2.05 1.30 0.61 3.77	0.0011 <0.01 0.0022 <0.01 -0.0027 <0.01 0.0039 <0.01 -0.0043 NS 0.0062 0.11 0.0008 0.18	-------	0.043 0.039 0.041 0.038 0.043 0.036 0.037	NS NS NS 0.115 0.010 NS NS	NS NS NS 0.108 NS NS NS
	Intake (i; g/d per kg of BW) DMI NFCi NDFi	29 67 27 63 23 52	21.1 6.9 11.0	7.8 3.9 5.6	5.5 41.6 0.3 17.2 1.5 27.9	2.38 1.26 1.58	-0.0009 NS 0.0063 0.02 -0.0066 0.01	---	0.043 0.042 0.046	NS 0.104 NS	NS NS NS
	1 P-values

2

Characteristics of subdatabases containing only experiments in which a sufficient variation of the covariate under study was observed. Number of experiments (Exp) and treatments (Trt), mean, SD, and minimal (Min) and maximal (Max) values for each covariate are reported, as well as the cut-off SD (SDmin) used to acknowledge a sufficient variation of the covariate.

3 Root mean square error. 4 P-value of the interaction of species (Sp) with covariate (cov) or covariate × covariate (cov 2 ). Each interaction was tested individually and included in the model if P ≤ 0.10. quality. Furthermore, forage was the sole ingredient in 60% of sheep diets, compared with 17 and 11% for cattle and dairy cows, respectively. Finally, most dairy cows were in lactation (n = 31 from 37 treatments) and milk production averaged 28.8 kg/d (SD 9.6; range 15.9-47.7 kg/d). Therefore, diets of dairy cows had a narrower range of NFC, NDF, and TDN concentrations compared with diets of sheep and cattle (Table

Table 3 .

 3 Influence of each covariate on net portal appearance of ammonia-N 1 P-values <0.20 are reported; otherwise NS. Each covariate was tested on the residual errors of the model with N intake as the sole predictor. A quadratic term was included in the model only if P ≤ 0.10. 2 Characteristics of subdatabases containing only experiments in which a sufficient variation of the covariate under study was observed. Number of experiments (Exp) and treatments (Trt), mean, SD, and minimal (Min) and maximal (Max) values for each covariate are reported, as well as the cut-off SD (SDmin) used to acknowledge a sufficient variation of the covariate.

		Characteristics of each subdatabase 2		Slope P-value		P-value 4
	Item	Exp Trt Mean SD	Min	Max SDmin	Linear	Quadratic	RMSE 3	Sp × cov	Sp × cov 2
	Diet 5 (% of DM) CONC NFC NDF TDN CP 6 RDP 7 RDP (% of CP)	14 36 20 54 20 52 21 54 31 76 29 75 22 56	33.8 31.1 49.7 66.1 13.0 9.4 65.4	31.2 19.4 20.4 10.2 4.0 3.0 9.6	0.0 1.6 12.2 47.6 4.1 2.6 47.5	92.0 70.1 81.0 81.9 22.3 18.4 90.5	4.61 3.42 3.38 1.91 1.40 1.20 3.38	-0.0008 <0.01 -0.0015 <0.01 0.0017 <0.01 -0.0030 <0.01 0.0052 0.02 0.0088 0.01 0.0012 0.03	-------	0.034 0.037 0.036 0.034 0.033 0.036 0.028	NS NS NS NS 0.086 0.096 NS	NS NS NS NS NS NS NS
	Intake (i; g/d per kg of BW) DMI 8 NFCi NDFi 9	28 68 28 72 23 60	21.9 7.1 10.6	8.0 4.0 5.6	5.5 0.3 1.5	41.6 17.2 27.9	2.30 1.22 1.30	-0.0007 NS -0.0061 <0.01 0.0037 NS	---	0.032 0.037 0.043	0.060 NS 0.055	NS 0.189 0.183

1 3 Root mean square error.

Table 4 .

 4 Influence of each covariate on net portal appearance of urea-N 1 <0.20 are reported; otherwise NS. Each covariate was tested on the residual errors from the model with N intake as the sole predictor. A quadratic term was included in the model only if P ≤ 0.10.

			Characteristics of each subdatabase 2	Slope P-value		P-value 4
	Item	Exp Trt Mean SD	Min Max SDmin	Linear	Quadratic	RMSE 3	Sp × cov	Sp × cov 2
	Diet 5 (% of DM) CONC NFC NDF TDN 6 CP RDP RDP (% of CP)	11 18 30 19 30 25 22	28 48 40 48 74 64 55	40.9 33.2 33.1 19.5 48.4 22.1 12.2 81.0 3.39 0.0 92.0 5.02 1.6 70.1 3.56 66.9 10.5 47.6 81.9 2.01 13.0 4.1 4.1 22.3 1.41 9.3 3.2 2.6 18.4 1.20 66.0 9.6 47.5 90.5 3.50	-0.00033 0.13 -0.00053 0.16 0.00070 0.13 -0.3748 0.03 0.0026 0.03 ---0.00082 NS -0.00009 NS --0.00020 NS -	0.029 0.029 0.031 0.028 0.032 0.027 0.024	NS NS NS 0.023 NS NS 0.146	NS NS NS 0.024 NS NS NS
	Intake (i; g/d per kg of BW) DMI NFCi NDFi	24 24 21	55 60 51	20.4 7.5 10.3	8.0 4.0 5.8	5.5 41.6 2.21 0.3 17.2 1.22 1.5 27.9 1.35	-0.0117 <0.01 0.00025 <0.01 -0.0038 0.03 --0.0059 0.15 0.00031 0.05	0.038 0.029 0.036	0.182 NS NS	NS NS NS
	1 P-values

2

Characteristics of subdatabases containing only experiments in which a sufficient variation of the covariate under study was observed. Number of experiments (Exp) and treatments (Trt), mean, SD, and minimal (Min) and maximal (Max) values for each covariate are reported, as well as the cut-off SD (SDmin) used to acknowledge a sufficient variation of the covariate. 3 Root mean square error. 4 P-value of the interaction of species (Sp) with covariate (cov) or covariate × covariate (cov 2 ). Each interaction was tested individually and included in the model if P ≤ 0.10. 5 CONC = proportion of concentrate; TDN = total digestible nutrients at 1× maintenance; NFCi = intake of NFC; NDFi = intake of NDF.

6

Journal of Dairy Science Vol. 94 No. 6, 2011

CONC = proportion of concentrate; TDN = total digestible nutrients at 1× maintenance; NFCi = intake of NFC; NDFi = intake of NDF.

Interaction Sp × CP was included in the model. Species effect on the slope of CP: Δ = 0.0084, -0.0106, and 0.0022 for sheep, cattle, and dairy cows, respectively.

P-value of the interaction of species (Sp) with covariate (cov) or covariate × covariate (cov 2 ). Each interaction was tested individually and included in the model if P ≤ 0.10.

CONC = proportion of concentrate; TDN = total digestible nutrients at 1× maintenance; NFCi = intake of NFC; NDFi = intake of NDF.

Interaction Sp × CP was included in the model. Species effect on the slope of CP: Δ = -0.0055, 0.0008, and 0.0046 for sheep, cattle, and dairy cows, respectively.

Interaction Sp × RDP was included in the model. Species effect on the slope of RDP (% of DM): Δ = -0.0082, -0.0013, and 0.0095 for sheep, cattle, and dairy cows, respectively.

Interaction Sp × DMI was included in the model. Species effect on the slope of DMI: Δ = -0.0039, 0.0017, and 0.0022 for sheep, cattle, and dairy cows, respectively.

Interaction Sp × NDFi was included in the model. Species effect on the slope of NDFi: Δ = -0.0079, 0.0040, and 0.0039 for sheep, cattle, and dairy cows, respectively.

ACKNOWLEDGMENTS

This research was undertaken thanks to funding from Dairy Farmers of Canada (Ottawa, ON), the Natural Sciences and Engineering Research Council of Canada (Ottawa, ON), and Agriculture and Agri-Food Canada (Ottawa, ON).


François G Schmitt

Juan Carlos Molinero

Sylvie Zongo Brizard

Nonlinear dynamics and intermittency in a long term copepod time series

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


est destinée au dépôt

  Transcriptional and functional classification of the GOLVEN/ROOT GROWTH FACTOR/CLE-like signaling peptides reveals their role in lateral root and hair formation Ana Fernandez, Andrzej Drozdzecki, Kurt Hoogewijs, , Tom Beeckman, Annemieke Madder, Pierre Hilson To cite this version: Ana Fernandez, Andrzej Drozdzecki, Kurt Hoogewijs, , Tom Beeckman, et al.. Transcriptional and functional classification of the GOLVEN/ROOT GROWTH FACTOR/CLE-like signaling peptides reveals their role in lateral root and hair formation. Plant Physiology, 2013, 161 (2), pp.954-970. 10.1104/pp.112.206029. hal-01190620

Supplemental Data

The following materials are available in the online version of this article. S1. GLV primers for amplification of promoter sequences. S2. Transcript levels in GLV mutant lines. S3. GLV primers for amplification of open reading frame sequences. S4. Synthetic peptides. S5. Odds ratio estimates of root tip angle distribution. Supplemental Movie S1. Expression of GLV4 in the root epidermal cells. The molecular marker was GLV4pro::NLS-2xGFP. The movie pauses every time a nucleus with GFP signal is reached. At the end, a summary of all cells where the GFP signal appeared is shown. Roots were counterstained with PI (red). Supplemental Movie S2. Expression of GLV8 in the root cortical and non-hair epidermal cells. The molecular marker was GLV8pro::NLS-GFP-GUS. The movie pauses every time a nucleus with GFP signal is reached. At the end, a summary of all cells where the GFP signal appeared is shown. Notice that one cortical and 2 non-hairs cells at the right side lack GFP signal because they only appear late and therefore their nucleus is not reached.

Supplemental Table

Supplemental Table

Supplemental Table

Supplemental Table

Supplemental Table

Roots were counterstained with PI (red).


Philippe Evon
email: philippe.evon@ensiacet.fr

Guyonne De Langalerie

Laurent Labonne

Othmane Merah

Thierry Talou

Stéphane Ballas

Thierry Véronèse

De Langalerie

Stems from amaranth (Amaranthus cruentus), an original agricultural by-product for obtaining low-density insulation blocks and hardboards for building applications

Keywords:

Introduction

► Amaranthus cruentus, an annual herb native to temperate and tropical regions (cultivated for the nutritional properties of seeds). ► A promising raw material for the biorefinery of whole plant (all parts potentially usable for food and non-food applications). ► Seeds contain: -A lipid fraction (7-8%), rich in squalene of vegetal origin (5-6%) → FOOD SUPPLEMENT, COSMETICS [1].

-Proteins (15%) → BREAD FLOURS, BIOCIDAL AND ANTIOXYDANT ACTIVITIES [2].

-Starch (up to 55%) → INJECTION-MOULDING APPLICATIONS. ► This study especially aims to investigate the possible uses of stems for material applications.

Tableau 3 .

 3 Density, flexural properties and water resistance of hardboards produced from extrusion-refined bark using hot pressing (200 ºC mold temperature, 19.6 MPa applied pressure, and 5 min molding time). ) 90% purity after two purification steps. (ii) After sieving, four fractions collected: -Fines (< 1 mm): 11%. -SP (1-2 mm): 16%. -MP (2-4 mm): 45%. -BP (> 4 mm): 28%. d, density; σ f , flexural strength ; λ, thermal conductivity. Pith particles d (kg/m 3 ) s f (kPa) λ (mW/m.K) SP 116 ± 1 15.8 ± 1.9 57.5 ± 0.6 MP 100 ± 5 23.0 ± 3.3 54.8 ± 1.5 BP 89 ± 2 84.5 ± 8.2 56.0 ± 1.4 Mix (all fractions) 130 ± 3 59.3 ± 4.5 60.3 ± 1.5 Tableau 1. Characteristics of insulation blocks.


Jean-Denis Durou
email: durou@irit.fr

Frédéric Courteille
email: courteil@irit.fr

Integration of a Normal Field without Boundary Condition

We show how to use two existing methods of integration of a normal field in the absence of boundary condition, which makes them more realistic. Moreover, we show how perspective can be taken into account, in order to render the 3D-reconstruction more accurate. Finally, the joint use of both these methods of integration allows us to obtain very satisfactory results, from the point of view of CPU time as well as that of the accuracy of the reconstructions. As an application, we use this new combined method of integration of a normal field in the framework of photometric stereo, a technique which aims at computing a normal field to the surface of a scene from several images of this scene illuminated from various directions. The performances of the proposed method are illustrated on synthetic, as well as on real images.

Introduction

Photometric stereo is a computer vision technique which uses several images of a scene illuminated from various directions. Usually, this technique consists in two stages: the normal to the surface is first computed in each visible point [START_REF] Woodham | Photometric Method for Determining Surface Orientation from Multiple Images[END_REF]; the normal field is then integrated, using either the calculus of variations [START_REF] Horn | The Variational Approach to Shape From Shading[END_REF], direct integration [START_REF] Wu | A Line-Integration Based Method for Depth Recovery from Surface Normals[END_REF] or frequencydomain methods [START_REF] Frankot | A Method for Enforcing Integrability in Shape from Shading Algorithms[END_REF]. In this paper, we take an interest in the second stage. More precisely, we improve two existing methods of integration in two ways: on the one hand, we show that the knowledge of the height on the boundary, a knowledge which is usually not available, is not necessary; on the other hand, we show how to extend these methods to perspective projection. This theoretical study is propounded in Section 2. In Section 3, a new method of normals integration is designed from this theoretical study, which combines both improved methods. In order to validate this new method, we test it in the framework of photometric stereo, on synthetic, as well as on real images (Section 4). In Section 5, we summarize the main contributions of the paper.

Relation between Normal and Gradient

Due to lack of space, no state-of-the-art on the integration of a normal field is given here (see e.g. [START_REF] Klette | Height data from gradient fields[END_REF][START_REF] Robles-Kelly | A Graph-Spectral Method for Surface Height Recovery from Needle-maps[END_REF][START_REF] Horowitz | Depth from Gradient Fields and Control Points: Bias Correction in Photometric Stereo[END_REF][START_REF] Agrawal | An Algebraic Approach to Surface Reconstruction from Gradient Fields[END_REF]). Suppose that, in each point Q = (x, y) in the image of a surface S, we know the unit outgoing normal -→ n (x, y) to S:

- → n (x, y) =   n X (x, y) n Y (x, y) n Z (x, y)   . (1)
The vectorial function -→ n is called a "normal field". The problem of integrating a normal field consists in searching for a shape S i.e., for three functions X, Y and Z (Z is called the "height"), such that the object point P conjugated with Q has X(x, y), Y (x, y) and Z(x, y) as coordinates. It can be solved only if the model of projection is known. Under the assumption of orthographic projection (cf. points P 1 and Q 1 in Fig. 1), X(x, y) and Y (x, y) are worth:

Orthographic Projection

X(x, y) = x d/f, Y (x, y) = y d/f. (2)
Let us denote g the transverse magnification f /d. The product of both partial derivatives ∂ x -→ P and ∂ y -→ P is normal to the surface (the dependances in (x, y) are omitted, for seek of simplicity):

∂ x - → P ∧ ∂ y - → P =   -Z x /g -Z y /g 1/g 2   , (3)
where Z x and Z y denote the partial derivatives of function Z. The product of this vector by -→ n (x, y) is null, which gives us the three following equations: 4), which is linear in Z x and Z y , is ill-constrained if the rank of its matrix is less than 2 i.e., if n Z 2 = 0 and n X n Z = 0 and n Y n Z = 0, which occurs only if n Z = 0. This particular case must be considered carefully, since -→ n (x, y) is parallel to the image plane Π i in this case. Under the assumption of orthographic projection, the image points which lie on an "occluding contour" satisfy this property. Thus, even if the normal is easy to determine for the image points which lie on an occluding contour (the normal is parallel to Π i and is also normal to the occluding contour), system (4) is ill-constrained and Z cannot be computed for such points. Now let us consider the image points which do not lie on an occluding contour. System (4) is well-constrained and its resolution gives us the gradient of Z:

     g n Z Z x = -n X , g n Z Z y = -n Y , n Y Z x -n X Z y = 0. (4) System (
∇Z = 1/g p q , (5)
where the usual notations p = -n X /n Z and q = -n Y /n Z have been used. Thus, the problem of integrating a normal field has been reformulated as a problem of integrating the gradient of a function of two variables, whose solution is straightforward:

Z(x, y) = Z(x 0 , y 0 ) + (x,y) (x0,y0) p(u, v)du + q(u, v)dv g . (6)
It results from this expression that Z is computed up to an additive constant. Thus, the Root Mean Square Error in Z, denoted |∆Z| 2 , depends on this constant. In the tests, as we will see, no boundary condition is imposed, thus this constant is computed so that |∆Z| 2 be minimal.

Perspective Projection

Under the assumption of perspective projection (cf. points P 2 and Q 2 in Fig. 1), X(x, y) and Y (x, y) are worth:

X(x, y) = x Z(x, y)/f, Y (x, y) = y Z(x, y)/f. (7)
The product of ∂ x -→ P and ∂ y -→ P is a little more complicated than in the orthographic case:

∂ x - → P ∧ ∂ y - → P = Z/f   -f Z x -f Z y Z + x Z x + y Z y   . (8)
Knowing that this vector is parallel to -→ n (x, y), this gives us the three following equations:

     f n Z Z x + n X [Z + x Z x + y Z y ] = 0, f n Z Z y + n Y [Z + x Z x + y Z y ] = 0, n Y Z x -n X Z y = 0. (9)
System ( 9) is not linear in Z x and Z y , but homogeneous in Z. Thus, it is useful to introduce a change in the unknown:

T = ln |Z|. ( 10
)
We obtain a new system which is linear in T x and T y :

     [f n Z + x n X ] T x + y n X T y = -n X , x n Y T x + [f n Z + y n Y ] T y = -n Y , n Y T x -n X T y = 0. (11)
This system is ill-constrained if the rank of its matrix is less than 2 i.e., if:

     f n Z [x n X + y n Y + f n Z ] = 0, -n X [x n X + y n Y + f n Z ] = 0, -n Y [x n X + y n Y + f n Z ] = 0. (12)
As n X , n Y and n Z cannot simultaneously be equal to 0, since -→ n is a unit vector, system (12) holds only if [START_REF] Basri | Photometric Stereo with General, Unknown Lighting[END_REF] holds only if the image point Q lies on an occluding contour, as in the orthographic case. Now let us consider the image points which do not lie on an occluding contour. System [START_REF] Simchony | Direct Analytical Methods for Solving Poisson Equations in Computer Vision Problems[END_REF] is well-constrained and its resolution gives us the gradient of T :

x n X + y n Y +f n Z = 0. Moreover, x n X +y n Y +f n Z = --→ CQ• - → n , then system
∇T = r s , (13)
where the following notations are used:

   r = - n X x n X + y n Y + f n Z , s = - n Y x n X + y n Y + f n Z . ( 14
)
Once again, the problem of integrating a normal field has been reformulated as a problem of integrating the gradient of a function of two variables, whose solution is straightforward:

T (x, y) = T (x 0 , y 0 ) + (x,y) (x0,y0) [r(u, v)du + s(u, v)dv] . (15)
From ( 15) and ( 10), we deduce:

Z(x, y) = Z(x 0 , y 0 ) exp      (x,y) (x0,y0) [r(u, v)du + s(u, v)dv]      . ( 16
) It results from this expression that Z is computed up to a multiplicative constant. It is also noticeable that ( 16) requires the knowledge of the focal distance f , as well as the location of the principal point O (since the coordinates x and y depend on it).

Integrability of a Normal Field

In order to ensure that the normal field is integrable i.e., that Eqs. ( 6) or ( 16) are independent of the integration path, it is necessary and sufficient that p and q (in the orthographic case) or r and s (in the perspective case) satisfy the Schwartz equations ∂p/∂y = ∂q/∂x or ∂r/∂y = ∂s/∂x. In practice, a normal field is never rigorously integrable. There are two answers to this problem. The first one consists in using several integration paths between (x 0 , y 0 ) and (x, y), and to mean the integrals, as Wu and Li do (see [START_REF] Wu | A Line-Integration Based Method for Depth Recovery from Surface Normals[END_REF] and Section 3.1). The second answer considers that Eqs. ( 5) or ( 13) are least square problems, as Horn and Brooks do (see [START_REF] Horn | The Variational Approach to Shape From Shading[END_REF] and Section 3.2). Now let us give the continuous formulation of this second answer, which uses the calculus of variations.

Under the assumption of orthographic projection, the resolution of Eq. ( 5) in the least square sense amounts minimizing the following functional:

F(Z) = (x,y)∈Ω F (x, y, Z, Z x , Z y ) dx dy, (17)
where Ω denotes the "domain of reconstruction", and:

F (x, y, Z, Z x , Z y ) = [Z x -p/g] 2 + [Z y -q/g] 2 , ( 18
)
which measures the departure of the normal field from integrability. The calculus of variations tells us that searching for a function Z that minimizes F(Z) is equivalent to the resolution of the associated Euler equation [START_REF] Horn | The Variational Approach to Shape From Shading[END_REF]:

∂F ∂Z - ∂ ∂x ∂F ∂Z x - ∂ ∂y ∂F ∂Z y = 0. (19)
From ( 18) and (19), we deduce:

∇ 2 Z = 1/g [p x + q y ] . (20)
This is a Poisson equation, which is not particularly difficult to solve, even analytically [START_REF] Simchony | Direct Analytical Methods for Solving Poisson Equations in Computer Vision Problems[END_REF]. More precisely, solving Eq. ( 20) is really equivalent to the search for an extremum of

F(Z) only if Z is constrained on the boundary ∂Ω of Ω.
Otherwise, Eq. (20) has to be complemented with the "natural boundary equation" (see [START_REF] Horn | The Variational Approach to Shape From Shading[END_REF] and Section 3.2.2). Under the assumption of perspective projection, the resolution of Eq. ( 13) in the least square sense amounts minimizing the following functional:

G(T ) = (x,y)∈Ω G(x, y, T, T x , T y ) dx dy, (21)
where:

G(x, y, T, T x , T y ) = [T x -r] 2 + [T y -s] 2 . ( 22
)
The associated Euler equation is:

∇ 2 T = r x + s y , (23)
which is a Poisson equation again. This equation must also be complemented with the natural boundary equation if T is unconstrained on ∂Ω.

A New Method of Normals Integration

In this section, which makes up the algorithmic contribution of the paper, we extend and combine two existing methods of normals integration. On the one hand, we show that the knowledge of the height on ∂Ω is not necessary. Since such a knowledge is usually not available, the need for a boundary knowledge could lead us to use erroneous data. On the other hand, we show how to extend both these methods to perspective projection. Finally, we combine them, in order to design a new method of normals integration which is fast and accurate at the same time.

Improvement of Wu and Li's Method

Propagation methods of integration consist in computing the height from starting points, where the height is known. Such points are usually located on ∂Ω. These methods are fast but work well only if the normal field is integrable. Let us focus on Wu and Li's method [START_REF] Wu | A Line-Integration Based Method for Depth Recovery from Surface Normals[END_REF].

Wu and Li's Method

Wu and Li make a discrete approximation of the integral in Eq. [START_REF] Horn | The Variational Approach to Shape From Shading[END_REF]. They compute the means of the discrete integrals that are computed along the two paths that are displayed in Fig. 2 and they show that the accuracy is optimal if (x 0 , y 0 ) and (x, y) lie on the same diagonal. Suppose for seek of simplicity that the pixels are squared. The image points (x 0 , y 0 ) and (x, y) can thus coincide with pixels (i 0 , j 0 ) and (i, j). From now on, we will use the subscripted notation, e.g. u i,j for u(i, j). Between two neighbouring pixels (i, j) and (i + 1, j + 1), a discrete approximation of ( 6) is: where δ denotes the pixel size (δ = 0.05 in the tests), and: I 1+2 = p i,j + p i+1,j + q i+1,j + q i+1,j+1 , I 3+4 = q i,j + q i,j+1 + p i,j+1 + p i+1,j+1 .

Z i+1,j+1 ≈ Z i,j + δ/(2 g) [I 1+2 + I 3+4 ] , (24)
(25)

Obviously, the knowledge of Z on ∂Ω is not required for the implementation of (24).

First Improvement

Since the shape can be computed only up to an additive constant, we arbitrarily fix the height at a starting point

Q 0 .
Then, in order not to privilege a direction compared to another, we treat the pixels in spiral (cf. Fig. 3-a). The shape computed using the true normals of a synthetic vase is shown in Fig. 3

Second Improvement

Using the theoretical study of Section 2, we know that perspective integration is very similar to orthographic projec-tion. From this statement, we deduce the following approximation:

T i+1,j+1 ≈ T i,j + δ/2 [J 1+2 + J 3+4 ] , (26)
with:

J 1+2 = r i,j + r i+1,j + s i+1,j + s i+1,j+1 , J 3+4 = s i,j + s i,j+1 + r i,j+1 + r i+1,j+1 . (27)
The main difference between ( 24) and ( 26) is that the focal length f and the location of the principal point O must be known to implement (26). In Fig. 4, the graph of the true height Z t of the vase is shown i.e., the surface of equation z = Z t (x, y), which is not the surface of the vase. Contrary to the orthographic case, Z t is not uniform on ∂Ω (cf. Fig. 4-b). The graph of the height Z computed by the improved method of Wu and Li is shown in Fig. 5. It is globally satisfactory, even if |∆Z| 2 = 0.11, which is a little more than in the orthographic case. It is noticeable that the height on ∂Ω is qualitatively right (cf.

Improvement of Horn and Brooks' Method

The global methods of integration [START_REF] Horn | The Variational Approach to Shape From Shading[END_REF][START_REF] Frankot | A Method for Enforcing Integrability in Shape from Shading Algorithms[END_REF] deeply differ from the propagation methods. Apart from their slowness, they have two main advantages: they are more robust to noise; in the case where the Schwartz equation is not satisfied, they provide however an acceptable shape. Let us now focus on Horn and Brooks' method.

Horn and Brooks' Method

Horn and Brooks propose [START_REF] Horn | The Variational Approach to Shape From Shading[END_REF] a resolution of the Poisson equation ( 20) that comes from the following approximation of the expression (17) of F(Z):

E(Z) = (i,j)∈Ω1 Z i+1,j -Z i,j δ - p i+1,j + p i,j 2 g 2 + (i,j)∈Ω2 Z i,j+1 -Z i,j δ - q i,j+1 + q i,j 2 g 2 .
(28) In this expression: Ω 1 denotes the set of pixels (i, j) ∈ Ω such that (i + 1, j) ∈ Ω; Ω 2 denotes the set of pixels (i, j) ∈ Ω such that (i, j + 1) ∈ Ω; Z = (Z i,j ) (i,j)∈Ω ; Ω denotes the set of pixels (i, j) ∈ Ω whose four nearest neighbours are in Ω. The values Z i,j of the pixels lying on ∂Ω are not considered as unknowns, since Horn and Brooks use a boundary condition of the Dirichlet type. For a pixel (i, j) ∈ Ω, we get from (28):

∂E ∂Z i,j = 2 δ - Z i+1,j -Z i,j δ - p i+1,j + p i,j 2 g - Z i,j+1 -Z i,j δ - q i,j+1 + q i,j 2 g + Z i,j -Z i-1,j δ - p i,j + p i-1,j 2 g + Z i,j -Z i,j-1 δ - q i,j + q i,j-1 2 g . ( 29
)
The characterization ∇E = 0 of an extremum of E can thus be written [START_REF] Horn | The Variational Approach to Shape From Shading[END_REF], for (i, j) ∈ Ω:

4 Z i,j -(Z i+1,j + Z i,j+1 + Z i-1,j + Z i,j-1 ) + δ 2 g (p i+1,j -p i-1,j + q i,j+1 -q i,j-1 ) = 0. ( 30
)
This equation is indeed a discrete approximation of the Poisson equation ( 20), if centered finite difference approximations of p x and q y are used and if the Laplacian operator of Z is approximated by:

∇ 2 Z(i, j) ≈ Z i+1,j + Z i,j+1 + Z i-1,j + Z i,j-1 -4 Z i,j δ 2 . (31)

First Improvement

In order to avoid the need for Z on the boundary, it suffices to consider that all the values Z i,j , for (i, j) ∈ Ω, are unknowns. This implies that the equations ∂E/∂Z i,j = 0, for (i, j) ∈ Ω, are not all written under the form (30). For example, if two neighbours (i + 1, j) and (i, j + 1) of (i, j) only are in Ω, then ∂E/∂Z i,j = 0 is written:

2 Z i,j -(Z i+1,j +Z i,j+1 ) =
-δ 2 g (p i+1,j +p i,j +q i,j+1 +q i,j ).

(32) The pixels which require a particular treatment all lie on ∂Ω. In fact, equations such as (32) are nothing else than the discrete version of the "natural boundary equation" that has already been cited in Section 2. The results of Fig. 6 do not use any boundary condition.

Second Improvement

In the perspective case, we saw in Section 2.3 that F(Z) must be replaced by G(T ). The generalization of Horn and Brooks' method for perspective projection is thus straightforward and provide the results of Fig. 7.

|∆Z|2 = 1.11 |∆Z|2 = 0.82 (a) (b)

A New Method of Normals Integration

In order to take advantage of both them, we combine Wu and Li's and Horn and Brooks' methods. Wu and Li's method is fast and provides shapes of good quality, which consequently allows us to reduce the number of iterations of Horn and Brooks' method from 8000 to 500. Thus, the reconstruction is of good quality, even if the normal field is not integrable.

Contrary to orthographic integration, perspective integration does not allow us to directly compute the scene surface, since the graph z = Z(x, y) of the computed height Z is not the scene surface. But, knowing that an image point Q = (x, y) is conjugated with the object point P = (x Z(x, y)/f, y Z(x, y)/f, Z(x, y)), it is easy to compute the scene surface from Z. In Fig. 8, the surface of the vase after resampling of the object points corresponding to the true height Z t is shown. Obviously, the bottom and the top of the vase are contained in two planes parallel to Cyz. The computed shape from the true normals of the vase using our new method of normals integration, after resampling, is shown in Fig. 9. This shape is visually similar to the true shape of the vase.

Application to Photometric Stereo

Among the computer vision techniques for 3Dreconstruction, the photometric methods use the relation between the greylevel and the normal to the surface. Thus, they are appropriate to test the new method of integration of a normal field described in Section 3.

Photometric Methods for 3D-reconstruction

Image Irradiance Equation

The image irradiance equation is the basic equation of the photometric methods for 3D-reconstruction, namely shapefrom-shading and photometric stereo. This equation, which expresses the conservation of light energy, is particularly simple for a Lambertian surface illuminated by a homogeneous parallel beam [START_REF] Woodham | Photometric Method for Determining Surface Orientation from Multiple Images[END_REF]:

-ρ(P ) -→ L • - → n (P ) = I(Q). (33)
In (33), an object point P is characterized by its albedo ρ(P ) and its unit outgoing normal -→ n (P ), whereas its conjugated image point Q is characterized by its greylevel I(Q). Note that, for a colour image, I(Q) as well as ρ(P ) must be considered as vectors in R 3 , and no longer as scalars. The light beam is characterized, in direction and density, by a vector -→ L . Note that both members of Eq. ( 33) are not homogeneous: in fact, rather than an equality, (33) is a relation of proportionality. Finally, it is noticeable that the resolution of (33) is possible only if the relation between P and Q is known i.e., if the model of projection is known.

Shape-from-shading and Photometric Stereo

The knowledge of ρ(P ), -→ L and I(Q) does not allow us to determine -→ n (P ) from Eq. (33). There is indeed an infinity of vectors which form the same angle with vector -→ L . This ambiguity, which is well-known, constitutes the main difficulty of shape-from-shading. We avoid shape-fromshading to illustrate our new method of integration, since it would be difficult to decide on its accuracy, knowing that a normal field computed through using shape-from-shading would not be accurate enough, especially in the case of real images [START_REF] Zhang | Shape from Shading: A Survey[END_REF][START_REF] Durou | Numerical Methods for Shape-from-shading: A New Survey with Benchmarks[END_REF]. We prefer to make the tests in the framework of photometric stereo, a problem which is well-posed in most cases, contrary to shape-from-shading, since several images of the same scene, illuminated under various directions, are available. The photometric stereo technique, which is due to Woodham [START_REF] Woodham | Photometric Method for Determining Surface Orientation from Multiple Images[END_REF], has enjoyed some renewal in the last years [START_REF] Torreão | Geometric-photometric approach to monocular shape estimation[END_REF][START_REF] Noakes | Nonlinearities and Noise Reduction in 3-Source Photometric Stereo[END_REF][START_REF] Horowitz | Depth from Gradient Fields and Control Points: Bias Correction in Photometric Stereo[END_REF]13,[START_REF] Basri | Photometric Stereo with General, Unknown Lighting[END_REF][START_REF] Wu | Dense Photometric Stereo by Expectation Maximization[END_REF][START_REF] Chen | The 4-Source Photometric Stereo Under General Unknown Lighting[END_REF]2].

If several images of a scene are available, then in each image point Q, the equations (33) obtained for each image form a system, whose unknowns are the coordinates of the vector -→ m(P ) = ρ(P ) -→ n (P ). Three images at least, illuminated under directions that are non-coplanar, are thus necessary to determine -→ m(P ). The main difference between shape-from-shading and photometric stereo is that the albedo of the surface must be known, at least partially, so that shape-from-shading may become a sufficiently constrained problem, whereas this is not necessary for photometric stereo. If the albedo is known, then photometric stereo can work with two images only, but if three images are available, then it is not necessary to know the albedo.

Tests

Synthetic Images

We simulate five images of the same vase as that of Section 3, under perspective projection (cf. Fig. 10). Note that a coloured texture is warped on the vase. In each image point, the albedo and the normal are computed by the resolution of a system of five equations and three unknowns. The reconstructed shape using our new method of integration is displayed in Fig. 11. It is visually satisfactory, but one may wonder whether this still holds with real images.

Real Images

We test our method of integration on three photographs of Beethoven's bust (cf. Fig. 12) that are available on the web 1 . The material of the bust seems to be approximately Lambertian. Moreover, estimates of the directions and densities of the light beams are given. The shape computed using our new method of integration is visually satisfactory (cf. Fig. 13).

Conclusion and Perspectives

In this paper, after a rigorous theoretical study, we show how to improve two existing methods of normals integration. On the one hand, we show how to use them without boundary condition, in order to make them more realistic. On the other hand, we extend them to perspective projection and show that this improves the accuracy of the reconstructions. Finally, we show that the joint use of both improved methods provides very satisfactory results.

As an application, we use this new method of integration in the framework of photometric stereo. The performances of the method are illustrated on synthetic, as well as on real images: they are very encouraging. Note that this application of our work amounts taking into account perspective in photometric stereo, a problem which has recently been pioneered in [13].

As a perspective, we now aim at reconstructing scene surfaces from images taken under badly-controlled conditions i.e., for non-Lambertian surfaces illuminated by nonhomogeneous or non-parallel beams, a problem which has already been addressed by Basri et al.

Figure 1 .

 1 Figure 1. Two models of projection: orthographic (points P1 and Q1) and perspective (points P2 and Q2).

Figure 2 .

 2 Figure2. The two paths used by Wu and Li: 1 followed by 2; 3 followed by 4.

Figure 3 .

 3 Figure 3. First improvement of Wu and Li's method: (a) treatment of the pixels in spiral; (b) reconstructed shape of a vase from the true normals, without boundary condition.

Fig. 5 -

 5 b), whereas no boundary condition is used.

Figure 4 .Figure 5 .

 45 Figure 4. Graph of the true height Zt of the vase: (a) perspective view; (b) side view.

Figure 6 .

 6 Figure 6. Improvement of Horn and Brooks' method: reconstructed shapes from the true normals, without boundary condition, at iterations (a) 500 and (b) 8000.

Figure 7 .

 7 Figure 7. Graphs of the heights Z computed from the true normals by the improved method of Horn and Brooks, at iterations (a) 500 and (b) 8000.

Figure 8 .Figure 9 .

 89 Figure 8. True shape of the vase: (a) perspective view; (b) side view.

Figure 10 .

 10 Figure 10. Five images of the vase illuminated under various directions. Angles φ (azimuth) and θ (complementary of the elevation) are in degrees: (a) (φ1; θ1) = (0; 0); (b) (φ2; θ2) = (45; 30); (c) (φ3; θ3) = (135; 30); (d) (φ4; θ4) = (225; 30); (e) (φ5; θ5) = (315; 30).

Figure 11 .

 11 Figure 11. Reconstructed shape from the five images of Fig. 10: side view (left) and perspective view (right).

Figure 12 .

 12 Figure 12. Three photographs of Beethoven's bust available on the web: (a) (φ6; θ6) = (-72.48; 15.12); (b) (φ7; θ7) = (-4.85; 11.46); (c) (φ8; θ8) = (69.31; 16.11).

Figure 13 .

 13 Figure 13. Reconstructed shape from the three photographs of Fig. 12: side view (left) and perspective view (right).