Théophile Gaudin

Patricia Rotureau

Isabelle Pezron

Guillaume Fayet
email: guillaume.fayet@ineris.fr

Estimating the adsorption efficiency of sugar-based surfactants from QSPR models

Keywords: QSPR, sugar-based surfactants, efficiency, bio-based compounds

Adsorption efficiency, measured as the surfactant concentration at which the surface tension of the aqueous solution decreases by 20 mN/m, characterizes their affinity for surfaces and interfaces, which is crucial for a cost-effective use of surfactants. In this article, the first Quantitative Structure Property Relationship models to predict this efficiency were proposed based on a dataset of 82 diverse sugarbased surfactants and using different types of molecular descriptors. Finally, an easy-to-use model was evidenced with good predictivity assessed on an external validation set. Moreover, it is based on a series of fragment descriptors accounting for the different structural trends affecting the efficiency of sugarbased surfactants. Due to its predictive capabilities and to the structure-property trends it involves, this model open perspectives to help the design of new sugar-based surfactants, notably to substitute petroleum-based ones.

Introduction

In the context of development and use of more environmentally-friendly products, efforts are nowadays in progress towards the design of new surfactants issued from renewable resources to substitute petroleum-based surfactants that constitute most part of the market. Among them, sugar-based surfactants are an important subfamily of bio-based surfactants (contributing up to 40% of biosurfactant consumption [START_REF] Anbu | Procuring sugar surfactants is a cleaner and greener alternative to synthetic surfactants[END_REF]), characterized by their polar head constituted by carbohydrates such as glucose, maltose or sucrose, and their derivatives. For this reason, sugar-based surfactants can be obtained from renewable resources such as starch [START_REF] Kjellin | Surfactants from Renewable Resources[END_REF], and are often biocompatible and easily biodegradable [START_REF] Matsumura | Surface activities, biodegradability and antimicrobial properties of n-alkyl glucosides, mannosides and galactosides[END_REF]. So, they are commonly considered among the most promising alternatives to conventional petroleum-based non-ionic surfactants (Hill & LeHen-Ferrenbach, 2009), particularly regarding soft detergents or personal care products, cosmetics and pharmaceutical formulations [START_REF] Rojas | Interfacial Properties of Sugar-Based Surfactants[END_REF].

The main performance characteristics of surfactants that are used to select surfactants in application formulations relate to their effectiveness and their efficiency. The target effectiveness, i.e. the maximum performance to reach, is the maximum lowering of the surface tension in aqueous solution. This maximum is almost reached when the concentration in surfactants favors their aggregation to form micelles in the solution. This concentration is the critical micelle concentration (CMC). So, effectiveness is characterized by measuring the surface tension at CMC, denoted γCMC [START_REF] Rosen | The relationship of structure to properties in surfactants. IV. Effectiveness in surface or interfacial tension reduction[END_REF].

Efficiency [START_REF] Rosen | Relationship of Structure to Properties in Surfactants: II. Efficiency in Surface or Interfacial Tension Reduction[END_REF] represents the amount of surfactant needed to reach a given performance. In surfactant solutions, two phenomena are in competition, adsorption at solvent/air interface and aggregation into micelles. While CMC is more related to aggregation, adsorption efficiency (commonly simply named efficiency) is measured as the quantity of surfactants needed to decrease the surface tension of the aqueous solution by 20 mN/m, in negative logarithmic unit by convention (-log C20, denoted pC20) [START_REF] Rosen | Surfactants and Interfacial Phenomena[END_REF]. This level of reduction of surface tension is expected as generally sufficient to characterize a quasi-saturated interface [START_REF] Rosen | Relationship of Structure to Properties in Surfactants: II. Efficiency in Surface or Interfacial Tension Reduction[END_REF].

In applications, a high efficiency enables to reduce the amount of surfactant used [START_REF] Fisher | Surface activity of sucrose palmitates[END_REF][START_REF] Rosen | Relationship of Structure to Properties in Surfactants: II. Efficiency in Surface or Interfacial Tension Reduction[END_REF]. Indeed, using efficient surfactants allows to limit the amount of surfactant in formulation for a same target application and, as a consequence, limit their cost and, in some cases, avoid their possible toxicity.

If CMC and pC20 characterize different aspects of the behavior of surfactants in solution (aggregation for CMC and adsorption for pC20), they are both related to the hydrophobic effect [START_REF] Tanford | Interfacial free energy and the hydrophobic effect[END_REF], that minimizes the contact between solvent and alkyl chains. So, it is not surprising to note that some common trends have been highlighted between the molecular structure and both the CMC (in log) and the pC20 in the experimental literature [START_REF] Rosen | Surfactants and Interfacial Phenomena[END_REF]. For instance, in general, pC20 was observed to increase with alkyl chain length [START_REF] Rosen | Relationship of Structure to Properties in Surfactants: II. Efficiency in Surface or Interfacial Tension Reduction[END_REF][START_REF] Zhu | Surface Properties of N-Alkanoyl-Nmethyl Glucamines and Related Materials[END_REF], and to slightly decrease with polar head size, which also affects CMC [START_REF] Crook | Molecular Weight Distribution of Nonionic Surfactants. I. Surface and Interfacial Tension of Normal Distribution and Homogeneous p,t-Octylphenoxyethoxyethanols (OPE's)[END_REF][START_REF] Myers | Surfactant Science and Technology[END_REF][START_REF] Rosen | Surfactants and Interfacial Phenomena[END_REF].

The relationship between the molecular structure and pC20 was quantified by Rosen [START_REF] Rosen | Relationship of Structure to Properties in Surfactants: II. Efficiency in Surface or Interfacial Tension Reduction[END_REF] under a thermodynamic formalism by considering the free energies of transfer of the CHn groups and of the polar head from solvent to interface. This approach has notably been used to evidence structural trends of a series of conventional surfactants (without any sugar-based surfactants), like the increase of efficiency with the length of the alkyl chain or its decrease in the case of ionic surfactants.

To reach predictive models, in particular for sugar-based surfactants, the quantitative structure-property relationship approach represents a relevant predictive method that already demonstrated success for physico-chemical properties [START_REF] Katritzky | Quantitative Correlation of Physical and Chemical Properties with Chemical Structure: Utility for Prediction[END_REF][START_REF] Nieto-Draghi | A General Guidebook for the Theoretical Prediction of Physico-Chemical Properties of Chemicals for Regulatory Purposes[END_REF] notably for surfactants [START_REF] Creton | Prediction of Surfactants' Properties using Multiscale Molecular Modeling Tools: A Review[END_REF][START_REF] Hu | A Review on Progress in QSPR Studies for Surfactants[END_REF] and, in particular, sugar-based ones [START_REF] Gaudin | New QSPR Models to Predict the Critical Micelle Concentration of Sugar-Based Surfactants[END_REF][START_REF] Gaudin | Investigating the impact of sugar based surfactants structure on surface tension at critical micelle concentration with structure-property relationships[END_REF]. But up to now, none were dedicated to the adsorption efficiency of surfactants.

In this study, a series of six QSPR models dedicated to the adsorption efficiency of sugar-based surfactants were developed based on different types of descriptors to access models accounting at best for the different structural trends influencing the adsorption efficiency of sugar-based surfactants. The inclusion of quantum chemical descriptors allows to access to physically meaningful description of molecular structures, notably in terms of charge distributions, whereas topological and, furthermore, constitutional descriptors are easier to calculate. Moreover, fragment-based descriptors were introduced.

Computed separately for the polar head and the alkyl chain of the surfactant, they are particularly representative of the specificity of the surfactants' structure and may allow to distinguish the specific impacts of polar heads and alkyl chains on their efficiency.

Computational details

Experimental dataset

The experimental data used in this paper were issued from a large data collection on the properties of sugar-based surfactants [START_REF] Gaudin | Développement de modèles QSPR pour la prédiction et la compréhension des propriétés amphiphiles des tensioactifs dérivés de sucre[END_REF]. More than 2500 data on different amphiphilic properties (including, notably, critical micelle concentrations (CMC), surface tensions at CMC, Krafft points and adsorption efficiencies) for more than 600 different sugar-based surfactants were extracted from literature constituting the largest database for this family of compounds, to the best of our knowledge.

The 172 gathered data related to efficiency were analyzed in detail to select only the most reliable ones for the development and the validation of QSPR models. Indeed, the performances of QSPR models are critically dependent on the number and quality of the data employed for its development. For this reason, all the selected data were obtained using a robust protocol and under homogeneous conditions. Indeed, all selected pC20 were measured near room temperature (i.e. between 20°C and 25°C), since temperature can strongly influence pC20 values [START_REF] Mahmood | Effect of Temperature Changes on Critical Micelle Concentration for Tween Series Surfactant[END_REF].

Moreover, all pC20 values collected in the dataset (cf. Table 1) were measured by tensiometry (using, e. g. the Wilhelmy plate method [START_REF] Le Neindre | Tensions superficielles et interfaciales[END_REF] or the Du Noüy Ring (du Noüy, 1925)) upon the following general principle. Aqueous solutions at various concentrations of surfactants are prepared and the measured surface tension are plotted versus concentration in logarithmic scale. This curve enables the determination of various properties, like the CMC, γCMC, and the efficiency. In the case of efficiency, the concentration at which the surface tension has decreased by 20 mN/m with respect to the pure water is extracted, and then, adsorption efficiency is calculated as the negative logarithmic transformation of this concentration, pC20.

<< TABLE 1 >>

The collected data allow to figure out the experimental variance that can be found in literature for a single molecule by different experimentalists. As shown in Table 2, this variance reaches 0.7 (log unit) for 1-O-octanoyl-D,L-xylitol.

<< TABLE 2 >>

After data curation, the final dataset (in Table 1) was constituted of 82 pC20 of diverse sugar-based surfactants, with various polar heads (cyclic, acyclic and even mixed), with linear, branched and/or unsaturated alkyl chains, and with various linkages (ether, thioether, ester, amide and methylamide).

The distribution of pC20 data is relatively homogeneous, with data ranging between 1.70 and 6.46 (log unit) and a maximum between 3 and 3.5 (log unit), as shown in Figure 1.

<< FIGURE 1 >>

To perform an external validation of the model (to evaluate its predictive power), the dataset was divided into two parts taking care of their similarity in terms of property distribution and chemical diversity to ensure that the validation set is at best representative of the applicability domain of the model. The partition was performed by a property-ranged approach to ensure similar distributions of efficiency values in both sets. Surfactants were classified by increasing order of pC20 and the molecules of the validation set were selected regularly to obtain a 1:3 partition, i.e. by selecting the 2 nd , 5 th , 8 th molecule, etc. Hence, a training set of 55 surfactants was selected for the development of the model and a validation set of 27 surfactants was kept apart for the external validation. The similarity between both sets in terms of chemical diversity was also checked based on a principal component analysis on all computed descriptors. The molecules of both sets revealed well-distributed in the global chemical space of the investigated surfactants, as shown in Figure 2.

<< FIGURE 2 >>

Molecular descriptors

The molecular structures of the 82 studied sugar-based surfactants of the dataset were optimized from Density Functional Theory (DFT) at B3LYP/6-31+G(d,p) level with Gaussian 09 software [START_REF] Frisch | Gaussian 09[END_REF] after preliminary conformation analyses to identify the most stable conformation [START_REF] Gaudin | Conformations of n-alkyl-α/β-dglucopyranoside surfactants: Impact on molecular properties[END_REF]. Frequency calculations were performed at same level to ensure they correspond to local minima, i.e. presenting no imaginary frequency. This level of calculation was found relevant for sugar-based surfactants to evidence the influence of their molecular structure on a series of molecular descriptors in a previous work [START_REF] Gaudin | Conformations of n-alkyl-α/β-dglucopyranoside surfactants: Impact on molecular properties[END_REF] and it was successfully used in previous QSPR study targeting their CMC [START_REF] Gaudin | New QSPR Models to Predict the Critical Micelle Concentration of Sugar-Based Surfactants[END_REF].

The structures of the 35 polar heads and 18 alkyl chains constituting these 82 surfactants were also computed on the same scheme. These fragments were separated before the first heteroatom and each fragment was saturated in hydrogen, as illustrated in Figure 3.

<< FIGURE 3 >>

Based on these structures, i.e. for each surfactant and each fragment, more than 300 descriptors were computed using CODESSA software . These descriptors are of four types:

-constitutional descriptors, related to the number of specific types of atoms, functional groups and bonds (e.g. number of carbon, of single bonds); -topological descriptors, that characterize the atomic connectivity including information about the size, composition and degree of branching (e.g. Wiener index); -geometrical descriptors, related to the 3D molecular structure (e.g. molecular volume, bond length); -quantum-chemical descriptors, including atomic charges, electronic and binding information.

Within the investigated dataset, some sugar-based surfactants were enantiomeric mixtures [START_REF] Savelli | Effects of tail alkyl chain length (n), head group structure and junction (Z) on amphiphilic properties of 1-Z-R-d,l-xylitol compounds (R=CnH2n+1)[END_REF], for D and L sugar alcohol polar heads, or anomeric [START_REF] Boyère | Monocatenary, branched, double-headed, and bolaform surface active carbohydrate esters via photochemical thiol-ene/yne reactions[END_REF] mixtures, for polar heads containing a free anomeric alcohol. In these cases, the different isomers were considered as conformations of the same compound. The different isomers (possibly in different conformations) were calculated at B3LYP/6-31+G(d,p) level and the most stable one was used to compute the molecular descriptors.

Model development and validation

In this study, multi-linear regressions (MLR) were developed to favor a simpler use for predictions and an easier interpretation of the role of each descriptor in the predictive power of the model from both a statistical and phenomenological point of view.

The selection of the descriptors included into the models was performed using the Best MultiLinear Regression (BMLR) approach, as implemented in CODESSA software . This algorithm was described in details and successfully used in previous works (Fayet, Rotureau, Joubert, & Adamo, 2010a[START_REF] Fayet | QSPR modeling of thermal stability of nitroaromatic compounds: DFT vs. AM1 calculated descriptors[END_REF] notably for the CMC of sugar-based surfactants [START_REF] Gaudin | New QSPR Models to Predict the Critical Micelle Concentration of Sugar-Based Surfactants[END_REF]. The final model was chosen as the best compromise between correlation and number of descriptors to avoid against any overparameterization.

To evaluate the performances of models, a series of internal and external validations were performed.

The goodness of fit was measured by the determination coefficient (R²), the mean absolute error (MAE) and the root mean square error (RMSE) between predicted and experimental values for the training set.

Moreover, Student's t-test at a confidence level of 95% was performed to check the relevance of each descriptor into the regression.

The robustness of the models was assessed by using leave-one-out (LOO) and leave-many-out (LMO) cross-validations. The Q²LOO, Q²3CV, Q²7CV and Q²10CV coefficients issued at various levels of partition (for LOO, 3-fold, 7-fold and 10-fold cross-validations, respectively) were checked to be as close as possible to R² and close one to each other.

A Y-scrambling test was performed to ensure that models were not issued from chance correlations. 500 random permutations of experimental property values were performed. New models were refitted [START_REF] Lindgren | Model validation by permutation tests: Applications to variable selection[END_REF] at each permutation and the average (R²YS) and standard deviation (SDYS) in the R² of the new models were calculated. In absence of chance correlation, low values of R²YS are expected. As proposed by Rücker [START_REF] Rücker | Y-Randomization and Its Variants in QSPR/QSAR[END_REF], R²YS should be superior to 2.3 SDYS for a model to be considered as not issued from chance correlations with a 99% confidence level.

Then, an external validation was performed to evaluate the predictive power of the selected models, on the validation set of 27 surfactants. The coefficient of determination R²EXT, the mean absolute error MAEEXT and the root mean square error RMSEEXT were calculated between experimental and predicted values of efficiency. A series of additional external validation metrics were also used (listed in Table 3), for which thresholds values have been proposed by Chirico et al. [START_REF] Chirico | Real External Predictivity of QSAR models. Part 2. New Intercomparable Thresholds for Different Validation Criteria and the Need for Scatter Plot Inspection[END_REF] to estimate that a QSPR model is reliable.

<< TABLE 3 >>

In a last step, the applicability domain (AD), i.e. the domain in which a prediction is valid, was defined.

As a QSPR model is issued from a similarity principle, it is expected to be reliable for molecules similar to those used to fit it. So, the AD of each model has been defined by the range of values of the calculated

Results

As explained in the previous sections, different types of descriptors were developed from the simplest constitutional counts to quantum chemical descriptors. Moreover, some descriptors were calculated from the whole molecular structure whereas others were related to the alkyl chain or to the polar head.

Finally, six new QSPR models were developed in this study for pC20. Three models were based on integral descriptors: 1) based on all types of descriptors, 2) limited to topological and constitutional descriptors or 3) focused on constitutional descriptors to favor simpler models. On the same scheme, three other models were based on fragment descriptors.

Models based on integral descriptors

Model with all descriptors

From the 326 integral descriptors calculated for the whole surfactant molecule, a four-parameter model (Eq. 1) was found as the best compromise between correlation and number of descriptors among the 17 equations sorted out by the BMLR method: The model is characterized by good fitting (R² = 0.90, RMSE = 0.37 (log unit)) and robustness (Q²CV = 0.88, Q²10CV = 0.87, Q²7CV = 0.87, Q²3CV = 0.86). From the Y-scrambling test, the criterion of Rücker [START_REF] Rücker | Y-Randomization and Its Variants in QSPR/QSAR[END_REF] revealed satisfied: R² -R²YS = 0.81 > 2.3 SDYS = 0.12. So, the model was not issued from a chance correlation.

When tested in external validation, a good predictivity was obtained for this model with RMSEIN = 0.49. It has to be noticed that the only surfactant of the validation set that felt out of AD was octyl glycol (Figure 4) due to a slightly too low HACA2,TMSA (with 9.36•10 -3 whereas the AD ranged between 1.19•10 -2 and 4.04•10 -2 ) and a too high value of NO,avg (2.97•10 -2 vs. an AD between 1.95•10 -6 and 2.01•10 -2 ). This molecule has the smallest polar head of the dataset (notably when compared to octyl--D-glucoside in Figure 3), with only two hydroxyl groups, which explains its finding at the limits of the AD. However, the predicted pC20 for this surfactant presented a deviation of 0.39 (log unit), i.e. within the standard error in the validation set (RMSEIN = 0.49 (log unit)).

<< FIGURE 4 >>

Models with topological and constitutional descriptors

The use of the only 74 constitutional and topological descriptors led to a five-parameter model (Eq. 2) as best compromise between correlation and number of descriptors among the 7 equations sorted out by the BMLR method: pC20 = 1.093 nS -9.64 2 ASIC + 0.224 R² -R²YS = 0.83 > 2.3 SDYS = 0.12.

The predictivity of Eq. 2 was found better to that of Eq. 1 with RMSEIN = 0.41 (log unit) and better Two molecules were found out of AD. For the S-butyl-1-thio-D,L-xylitol, the relative number of carbon nC,rel was 0.265 for an AD between 0.270 and 0.313 and the error in prediction for this molecule (0.52) was only slightly higher than RMSEIN. For the octyl glycol, 1 χ v = 5.1 is lower than the range of the AD at [5.3;20.0] and predicted pC20 was overestimated by 0.80, as shown in Table 1.

Model with constitutional descriptors

The five-parameter model in Eq. 3 was chosen among the six equations sorted out by the BMLR method when focusing on the only 36 constitutional descriptors of the entire surfactants: Once again, the predictive power revealed slightly lower than Eqs. for these 3 surfactants for an AD between 0.7283 and 0.9929. So, it is not surprising to note that these molecules were correctly predicted with deviations between experimental and predicted efficiencies of 0.05 to 0.42 (i.e. within the RMSEIN level at 0.44 (log unit)). Once again, octyl glycol is found out of AD due to its value of 0 SICh (4.1 vs. AD = [5.2;22.8]) but the predicted value of pC20 was finally close to experiment (3.51 (log unit) vs. pC20,exp = 3.19 [START_REF] Shinoda | Surface Chemical Properties in Aqueous Solutions of Non-ionic Surfactants Octyl Glycol Ether, α-Octyl Glyceryl Ether and Octyl Glucoside[END_REF]). At last, a pC20 of 1.67 (log unit) was predicted for 1-O-hexanoyl-D,L-xylitol, i.e. minor to the AD (1.70 to 6.46 (log unit)) but this prediction remains within the expected uncertainty of the model, when compared to the experimental value of pC20 at 2.10 (log unit) [START_REF] Savelli | Effects of tail alkyl chain length (n), head group structure and junction (Z) on amphiphilic properties of 1-Z-R-d,l-xylitol compounds (R=CnH2n+1)[END_REF].

pC20 =

Topological/Constitutional descriptors

To avoid quantum chemical calculations, the second fragment-based model focused on the 150 constitutional and topological descriptors. A four-parameter model (Eq. 5) was finally selected.

Model based on constitutional descriptors

At last, focusing on the 72 fragment-based constitutional descriptors, a four-parameter model (Eq. 6)

was found to be the best compromise between correlation and number of descriptors among the 8 equations proposed by the BMLR method.

<< FIGURE 5 >>

Only one surfactant of the validation set revealed out of AD, N-octadecyl-N-methyl lactobionamide, which presents the longest saturated alkyl chain of the dataset (containing 18 C atoms) leading to a predicted pC20 slightly higher than the AD range (6.74 (log unit) > 6.46 (log unit)). However, the observed error in prediction for this molecule, 0.40 (log unit), is in line with the error obtained in the validation set (RMSEIN = 0.43 (log unit)).

Finally, this model is particularly appealing since it is based on very simple descriptors and remains reliable enough for good-quality and fast estimation of pC20 of sugar-based surfactants in the perspective of molecular screening or discovery, notably for formulation specialists.

Discussion

A summary of the six models developed in this study is proposed in table 4 and the applicability domain associated to the descriptors they used are provided in Table 5. These models present good predictive powers with standard errors between 0.41 and 0.49 for the validation set, which is reasonable compared to experimental deviations observed in literature for a single surfactant in Table 2 and, as indicated in previous section, they fulfilled all the Chirico criteria.

<< TABLE 4 >> << TABLE 5 >>

When looking at the descriptors included in the models, they are related to four main structural trends:

the alkyl chain length, the size of the polar head, the presence of a S linkage and the presence of a double bond in the alkyl chain.

In addition to the molecular weight of the alkyl chain (in Eq. 6), two other descriptors are related to the length of the alkyl chain. The Randic index of order 2 in the alkyl chain 2 χc and the information content descriptors 2 CICc are two topological descriptors of the alkyl chain that relate to the size of this fragment.

Besides, 2 χc has been also evidenced to increase with alkyl chain length in previous work [START_REF] Gaudin | New QSPR Models to Predict the Critical Micelle Concentration of Sugar-Based Surfactants[END_REF]. In all cases, these descriptors present the largest t-test. This is in agreement with experimental knowledge, since increasing alkyl chain length has been identified as the main factor increasing the efficiency of surfactants due to a higher affinity of the surfactant for interfaces [START_REF] Rosen | Relationship of Structure to Properties in Surfactants: II. Efficiency in Surface or Interfacial Tension Reduction[END_REF].

Four of the QSPR models include descriptors related to the polar head size. The first ones are the number of rings in the whole molecule, and in the polar head only, nrings,h. In practice, they correspond to a unique descriptor since none of the alkyl chains of the investigated surfactants contain rings. This descriptor is directly linked to the number of sugar residues. Indeed, in the data set, surfactants with more than one sugar residue contain at least one ring in its polar head and all acyclic molecules only contain one sugar residue. So, no ring in the polar heads is equivalent to a small polar head, and the more the number of rings is large, the more the polar head is large. The second ones are Structural Information Content descriptors, 0 SICh and 1 SICh, two topological indices that also increase with polar head size, as shown in Figure 6. In all cases, the regression coefficients associated to these descriptors are negative, which means that the size of the polar head decreases the efficiency of sugar-based surfactants. This trend is again meaningful: because of the low affinity of alkyl chains with water (the so-called hydrophobic effect [START_REF] Tanford | Interfacial free energy and the hydrophobic effect[END_REF]), the more alkyl chains are adsorbed at the surface, the more the surface tension decreases [START_REF] Rosen | Relationship of Structure to Properties in Surfactants: II. Efficiency in Surface or Interfacial Tension Reduction[END_REF]. At equal chain length and with bigger polar heads, a lower amount of alkyl chains per unit surface is adsorbed at saturation due to sterical hindrance between polar heads. Thus, at a given concentration, the associated decrease of energy per unit surface (or surface tension decrease) is expected to be lower with bigger polar heads, which implies a higher concentration of molecules to achieve a 20 mN/m surface tension decrease, i. e. a lower pC20.

<< FIGURE 6 >>

Then, the (relative) number of S atoms (nS and nS,rel) appears in four of the developed models. In the dataset, the sulfur is always found in the linkage. Sulfur linkage has been evidenced to add an hydrophobic contribution in surfactants (in addition to the carbon chain) either based on theoretical calculations, in a previous study [START_REF] Gaudin | New QSPR Models to Predict the Critical Micelle Concentration of Sugar-Based Surfactants[END_REF], and by experimentalists [START_REF] Marchant | Polysaccharide Surfactants: Structure, Synthesis, and Surface-Active Properties[END_REF]. In these new models, these descriptors always positively contribute to the calculated efficiency, which is in line with the mentioned experiments and calculations suggesting that sulfur linkage increases efficiency.

The last trend encountered into the models is the presence of a double-bond in the alkyl chain. Indeed, ,, single rel c n is lower (minor to 1) for surfactants containing one double-bond in the alkyl chain. In Eq. 6, ,, single rel c n has a positive regression coefficient which is relevant. Indeed, as experimentally observed [START_REF] Myers | Surfactant Science and Technology[END_REF][START_REF] Rosen | Surfactants and Interfacial Phenomena[END_REF], surfactants with unsaturated alkyl chains have lower efficiency due to the increased hydrophilicity of their chain.

Finally, it can be noticed that the physical factors reflected by the descriptors included in these models for pC20 are the same than those found for CMC in previous work [START_REF] Gaudin | New QSPR Models to Predict the Critical Micelle Concentration of Sugar-Based Surfactants[END_REF]. This is in accordance with the observed correlation between pC20 and CMC as evidenced for the 24 common molecules of both validation sets of the present article and the previous one for CMC (in Figure 7).

When comparing the different models developed in this study, it is interesting to note that simple models only based on constitutional descriptors performed as well as, and even slightly better, than models including quantum-chemical descriptors, with RMSEIN of 0.45 vs. 0.49 (log unit) for integral descriptors and 0.43 vs. 0.44 (log unit) for fragment descriptors. Thus, the only 2D structure of sugar-based surfactants seems sufficient to access reliable pC20 estimates.

Finally, among the developed models, the simple fragment-based constitutional model (Eq. 6) represents the most relevant one. Its performances are very high with an error of only 0.43 (log unit) (in terms of RMSEIN), which is reasonable compared to the observed deviations in experimental data found in literature (see Table 2). Moreover, it is very easy to use, requiring knowledge of the only 2D-structure of surfactants. This model even demonstrated similar deviations in prediction than the relationship between pC20 and CMC, as expressed in Eq. 7, according to the Abbott criterion (Abbott, 2016) considering that the concentration to decrease the surface tension of water by 20mN/m (i.e. C20) is ten times lower than CMC, which was highlighted as relevant for sugar-based surfactants as illustrated in Figure 7. pC20 =log CMC + 1 (7)

<< FIGURE 7 >>

Indeed, when applied to the validation set molecules, Eq. 7 did not offer more reliable estimates than the new QSPR model with a mean absolute error of 0.30 (log unit) vs. 0.31 (log unit) for the QSPR model (as shown in Table 6). This enforces the interest of the new QSPR model that can be applied without knowledge of any experimental data and can then be applied in an in-silico design or screening strategy before any experimental characterization and even synthesis.

<< TABLE 6 >> 5 Conclusion

This paper presents the first QSPR models dedicated to the prediction of efficiency of surfactants. Based on different types of descriptors, a series of new QSPR models were developed and validated focusing on sugar-based surfactants in the perspective of their use to guide the design of bio-based formulations in substitution to petroleum-based surfactants. The final proposed model is very simple, based only on constitutional descriptors of the polar head and of the alkyl chain of surfactants. A good predictive power was highlighted with a RMSEIN of 0.43 (log unit) evaluated on an external validation set. This simple fragment-based approach is very easy to use based on the only knowledge of the 2D structure of the surfactants and represents a powerful alternative to the systematic experimental campaigns in particular at early R&D stages as they allow to access to efficiency estimates of new surfactants even before synthesis. This enables to identify the surfactants that could be used for target applications at the lowest concentrations. To the end, the structure-efficiency trends encountered in these models can guide the design of new high potential surfactant's structures. Indeed, a similarity was found between the molecular factors that affect the critical micelle concentration and efficiency. Specifically, our analyses suggest that, beyond bearing long alkyl chains, sugar-based surfactants with small polar heads, saturated alkyl chains and sulfur linkages are expected to have higher efficiencies. N-Oleyl-N-Methyl Gluconamide 5.37 20 [START_REF] Burczyk | Synthesis and Surface Properties of N-Alkyl-N-methylgluconamides and N-Alkyl-N-methyllactobionamides[END_REF] 5.79 5.82 6.30 5.48 5.56 5.78 S-Decyl 5-Thio-D-Arabinonolactone 5.40 20 [START_REF] Lalot | Synthesis and amphiphilic properties of S-alkylthiopentonolactones and their pentitol derivatives[END_REF] 5.35 5.46 5.08 5.28 5.16 5.18 S-Decyl 1-Thio-L-Xylitol 5.40 20 [START_REF] Lalot | Synthesis and amphiphilic properties of S-alkylthiopentonolactones and their pentitol derivatives[END_REF] 5.23 5.41 5.17 5.17 5.19 5.11

N-Tetradecyl-N-Methyl Lactobionamide 5.40 20 [START_REF] Burczyk | Synthesis and Surface Properties of N-Alkyl-N-methylgluconamides and N-Alkyl-N-methyllactobionamides[END_REF] 5.02 4.97 5.01 5.16 5.11 5.23

Oleyl-β-D-Maltoside 5.40 25 [START_REF] Milkereit | Structures of micelles formed by synthetic alkyl glycosides with unsaturated alkyl chains[END_REF] a 5.37 5.17 5.48 5.32 5.45 5.38 N-Hexadecanoyl-N-Methyl Lactitolamine 5.46 25 [START_REF] Wilk | Preparation and properties of new lactose-based surfactants[END_REF] 5.41 5.43 5.51 5.52 5.43 5.61 N-Tetradecyl-N-Methyl Gluconamide 5.55 20 (Burczyk et al., 2001) 5.55 5.33 5.15 5.45 5.39 5.43 Oleyl-β-D-Maltotrioside 5.70 25 [START_REF] Milkereit | Structures of micelles formed by synthetic alkyl glycosides with unsaturated alkyl chains[END_REF]

List of captions

  descriptors and of the experimental property in the training set. To evidence the actual performances of the model inside its AD, all external validation criteria were calculated again considering only the molecules of the validation set belonging to the applicability domain (R²IN, MAEIN, RMSEIN, Q²F1,IN, Q²F2,IN, Q²F3,IN, CCCIN, IN m r² , , Δr²m,IN).

pC20 = 1 .

 1 952 2 ACIC -77.58 HACA2,TMSA -113.8 NO,avg -0.302 nrings + 2.80 (1) with 2 ACIC the Average Complementary Information Content of order 2, 2,TMSA HACA the area-weighted surface charge of hydrogen bonding acceptor atoms divided by the total molecular surface area, based on the Mulliken partial charges (Mulliken, 1955), , O avg N the average nucleophilic reactivity index for a O atom and rings n the number of rings in the molecule.

  In particular, it fulfilled all the criteria of Chirico et al. (R²IN = 0.83 > 0.70, Q²F1,IN = 0.82 > 0.70, Q²F2,IN = 0.82 > 0.70, Q²F3,IN = 0.83 > 0.70, CCCIN = 0.89 > 0.85, m r² IN = 0.70 > 0.65, Δrm²IN = 0.16 < 0.20) in its applicability domain.

  external validation metrics (R²IN = 0.88, Q²F1,IN = Q²F2,IN = 0.86, Q²F3,IN = 0.88, CCCIN = 0.93, m r² IN = 0.83 Δrm²IN = 0.01), for the 25 molecules of the validation set that belong to AD.

  1 and 2 with RMSEIN = 0.45 (log unit). However, all criteria of Chirico are satisfied (R²IN = 0.84, Q²F1 = 0.83, Q²F2 = 0.83, Q²F3 = 0.86, CCC = 0.91, m r² = 0.77 and Δrm² = 0.12). The two molecules out of AD are the same than for Eq. 2, based on the same criteria (S-butyl-1-thio-D,L-xylitol due to nC,rel = 0.265 vs. [0.270;0.313] and octyl glycol due toMw,rel = 5.13 vs. [5.38;6.96]) and for both of them pC20 the overestimation was of 0.69 and 0.55, respectively.3.2 Models focused on fragment-based descriptors3.2.1 Model with all types of descriptorsA total of 627 fragment-based descriptors were calculated when considering the descriptors of the polar head and of the alkyl chain. From this large set of descriptors, the five-parameter model in Eq. 4 was selected as giving the best compromise between correlation and number of descriptors among the 13 equations sorted out by the BMLR method: pC20 = -4.97 FHBSAh + 1.60•10 -2 2 CICc -1.616 RNCSc -6.31•10 -2 0 SICh (4) -2.95 2 ASICc + 10.22with FHBSAh the fractional H-bonding surface area of the polar head based on Mulliken partial charges[START_REF] Mulliken | Electronic Population Analysis on LCAO-MO Molecular Wave Functions[END_REF], 2 c CIC the complementary information content of order 2 of the alkyl chain, RNCSc is the relative negatively charged surface area of the alkyl chain based on Zefirov partial charges (Zefirov, Palyulin, Oliferenko, Ivanova, & Ivanov, 2001), 0 h SIC is the structural information content of order 0 of the polar head and 2 c ASIC is the average structural information content of order 2 for the alkyl chain. An excellent fitting was obtained with the training set data with a coefficient of determination R² = 0.95 and a low error (RMSE = 0.28 (log unit)), the model appears to be robust (Q²CV = Q²10CV = Q²3CV = 0.93, Q²7CV = 0.92) and not issued from chance correlation as evidenced by Y-scrambling (R²YS = 0.09 and SDYS = 0.05). This model demonstrated a good predictive power satisfying all criteria of Chirico with R²IN = 0.87, Q²F1,IN = 0.85, Q²F2,IN = 0.85, Q²F3,IN = 0.86, CCCIN = 0.91, m r² IN = 0.70, Δrm²IN = 0.15 and an error of RMSEIN = 0.44 (log unit) in its applicability domain. 5 surfactants of the validation set were found out of AD of this model. S-Hexyl, S-Octyl and S-Decyl 5-Thio-D-Xylonolactone were found nearly on the threshold of the AD in terms of FHBSAh with 0.7276

pC20 = 1 .

 1 129 nS,h + 0.753 2 χc -6.78 2 ABICc -3.26•10 -2 1 SICh + 4.74(5) with nS,h the number of S atoms in the polar head, 2 χc the Randic index of order 2 in the alkyl chain, 2 ABICc the Average Bonding Information Content of order 2 in the alkyl chain and 1 SICh the Structural Information Content of order 1 in the polar head.Based on four parameters, this model presents good correlation (R² = 0.92, RMSE = 0.33 (log unit)) and robustness (Q²CV = 0.91, Q²10CV = Q²7CV = 0.90, Q²3CV = 0.89). The Y-scrambling test ensures that the model was not obtained by chance correlation considering the Rücker criterion at a 99% confidence level: R² -R²YS = 0.85 > 2.3SDYS = 0.11. This model exhibits also good predictivity in its applicability domain, in which all the molecules of the validation set were included. Once again it fulfilled Chirico criteria (R²IN = 0.87, RMSEIN = 0.42 (log unit), Q²F1 = Q²F2 = Q²F3 =0.87, CCC = 0.92, m r² = 0.75, Δrm² = 0.13). The predictivity of this model including constitutional and topological descriptors is similar to the quantum chemical model (RMSEIN = 0.42 vs. 0.44 (log unit) for Eq. 4).

  pC20 = 26.36 nS,rel,h + 2.70•10 -2 Mw,c -0.199 nrings,h + 58.77 nsingle,rel,c -58.71 (6) with nrel,S,h the relative number of S atoms in the polar head, Mw,c the molecular weight of the alkyl chain, nrings,h the number of rings in the polar head and nrel,single,c the relative number of single bonds in the alkyl chain.This model is well-fitted with the training set surfactants (R² = 0.91, RMSE = 0.35 (log unit)), and presents a good robustness (Q²CV = 0.89, Q²10CV = Q²7CV = Q²3CV = 0.88). The Y-scrambling confirms that it is not issued from chance correlation: R²YS = 0.08 and SDYS = 0.11.At last, Eq. 6 demonstrates similar predictive performances than more complex fragment-based models. As shown in Figure5, the errors observed for the molecules of the validation set within AD are low (RMSEIN = 0.43 (log unit)) and all validation metrics fit with the Chirico criteria (R²IN = 0

Figure 1 .

 1 Figure 1. Distribution of pC20 values in the dataset.

Figure 2 .Figure 3 .

 23 Figure 2. Repartition of molecules of training and validation sets in the chemical space of the

Figure 4 .Figure 5 .

 45 Figure 4. Octyl glycol.

Figure 6 .

 6 Figure 6. Correlation of Structural Information Content (orders 0 and 1) of the polar head with

Figure 7 .

 7 Figure 7. Correlation between log CMC and pC20 compared with Abbott criterion.

Figure 1 .

 1 Figure 1. Distribution of pC20 values in the dataset.

Figure 2 .Figure 3 .

 23 Figure 2. Repartition of molecules of training and validation sets in the chemical space of the whole

Figure 6 .

 6 Figure 6. Correlation of Structural Information Content (orders 0 and 1) of the polar head with

Figure 7 .

 7 Figure 7. Correlation between log CMC and pC20 compared with Abbott criterion.

  1 χ v -0.796 nrings + 42.89 nC,rel -4.10 (2) with nS the number of S atoms, 2 ASIC the Average Structural Information Content of order 2, 1 χ

v the Kier & Hall index of order 1, nrings the number of rings, and nC,rel the relative number of C atoms. The model was well fitted for the training molecules (R² = 0.92, RMSE = 0.34 (log unit)) and revealed robust from LOO and LMO cross-validations with Q²nCV = 0.87-0.90. The Y-scrambling demonstrated that it did not originate from a chance correlation according to Rücker's criteria:

  46.45 nC,rel -0.802 Mw,rel + 8.52•10 -2 nH + 1.452 nS -0.534 nrings -7.68 (3) with nrel,C the relative number of C atoms, Mw,rel the relative molecular weight, nH the number of H atoms, nS the number of S atoms and nrings the number of rings.

	If the goodness of fit of this model is slightly lower than Eqs. 1 and 2 with R² = 0.88 and RMSE = 0.41
	(log unit), it proved robust (Q²nCV = 0.83-0.85). The Y-scrambling ensured that the model was not issued
	from a chance correlation with low values of R² for the models obtained after randomization (R²YS =
	0.09; SDYS = 0.05).

Table 1 . Experimental and calculated pC20 values from and the new QSPR models

 1

	surfactant	exp pC20 T (°C)	ref	predicted pC20 eq. 1 eq. 2 eq. 3 eq. 4 eq. 5 eq. 6
				TRAINING SET
	Octanoyl-β-D-Glucosylamine	1.70	25	(Plusquellec et al., 1994) a	2.35 2.20 2.55 2.18 2.63 2.57
	1-O-Pentanoyl-D,L-Xylitol	1.90	25	(Savelli et al., 1999)	1.81 1.57 1.68 1.53 1.63 1.63
	Octanoyl-β-D-Galactosylamine	2.00	25	(Plusquellec et al., 1994) a	2.32 2.20 2.55 2.58 2.63 2.57
	S-Pentyl-1-Thio-D,L-Xylitol	2.20	25	(Savelli et al., 1999)	2.95 2.79 3.02 2.24 3.03 3.21
	Hexyl-D-Maltonamide	2.50	25	(Zhang & Marchant, 1996)	1.86 2.18 2.19 2.10 1.94 2.19
	1-O-Octanoyl-D,L-Xylitol	2.80	25	(Savelli et al., 1999)	2.98 3.01 3.00 2.93 2.73 2.77
	1-O-Heptyl-D,L-Xylitol	2.90	25	(Savelli et al., 1999)	2.96 2.87 2.83 2.42 2.86 2.77
	3,7-Dimethyloctyl-β-D-Maltotrioside	3.00	25	(Minamikawa & Hato, 2005) a	3.39 3.36 3.32 3.20 3.13 3.31
	N-Decanoyl-N-Methyl Lactitolamine	3.06	25	(Wilk et al., 2001)	3.85 3.67 3.86 3.11 3.21 3.33
	Octyl-D-Maltonamide	3.10	25	(Zhang & Marchant, 1996)	2.58 2.90 2.91 2.72 2.84 2.95
	Octyl-D,L-Glycerol	3.19	25	(Shinoda et al., 1959) a	3.10 3.52 3.52 3.37 3.39 3.15
	1-O-Heptanoyl-D,L-Xylitol	3.20	25	(Savelli et al., 1999)	2.54 2.52 2.61 3.12 2.27 2.39
	S-Hexyl 1-Thio-D-Lyxitol	3.25	20	(Lalot, Stasik, Demailly, Beaupère, & Godé, 2004)	3.45 3.37 3.53 3.65 3.46 3.59
	N-Decyl-N-Hydroxymethyl Gluconamide	3.30	22	(Piasecki & Pilakowska-Pietras, 2007) a	4.18 3.95 3.93 3.81 3.86 3.91
	Nonyl-β-D-Glucoside	3.30	20	(Ericsson, Söderman, Garamus, Bergström, & Ulvenlund, 2004) a	3.44 3.23 3.18 3.35 3.65 3.33
	3,7-Dimethyloctyl-β-D-Maltoside	3.31	25	(Minamikawa & Hato, 2005) a	3.17 3.06 3.28 3.45 3.35 3.51
	S-Hexyl 5-Thio-D-Arabinonolactone	3.36	20	(Lalot et al., 2004)	3.50 3.46 3.62 3.79 3.44 3.66
	6-O-Dodecanoylstachyose	3.40	20	(Söderberg, Drummond, Neil Furlong, Godkin, & Matthews, 1995) a 3.88 4.01 3.76 3.47 3.60 3.49
	N-Octyl Glucoheptonamide	3.40	25	(Syper, Wilk, Sokołowski, & Burczyk, 1998) a	2.80 2.91 2.86 2.95 3.08 3.15
	1-O-Nonyl-D,L-Xylitol	3.40	25	(Savelli et al., 1999)	3.81 3.85 3.61 3.42 3.72 3.53
	N-Decyl-N-Hydroxymethyl Glucoheptonamide	3.50	22	(Piasecki & Pilakowska-Pietras, 2007) a	3.69 4.05 3.93 3.73 3.82 3.91
	Decyl-D-Maltonamide	3.60	25	(Zhang & Marchant, 1996)	3.21 3.59 3.55 3.59 3.66 3.71
	6-O-Dodecanoylraffinose	3.60	20	(Söderberg et al., 1995) a	3.78 3.85 3.66 3.69 3.75 3.69
	S-Hexyl 1-Thio-L-Xylitol	3.66	20	(Lalot et al., 2004)	3.44 3.37 3.53 3.69 3.46 3.59
	Decyl-β-D-Maltoside	3.70	25	(Aveyard, Binks, Chen, Esquena, & Fletcher, 1998) a	3.89 3.60 3.28 3.78 3.81 3.51
	Decyl-D-Lactobionamide	3.73	25	(Syper et al., 1998) a	3.70 3.59 3.55 3.57 3.66 3.71

Table 2 . Experimental values of pC20 gathered from different sources.

 2

	a	5.11 5.12 5.37 5.07 5.22 5.18

*data extracted from graphs

Table 4 . Summary of the performances of the new QSPR models.

 4

	model	ndesc				descriptors					R²	RMSE	R²IN RMSEIN nout
	integral/all types (Eq. 1)	4	2	2, TMSA ACIC HACA ,	,	N	, O avg	,	rings n	0.90	0.37	0.83	0.49	1
	integral/constitutional and topological (Eq. 2)	5		nS, 2 ASIC, 1 χ v , nrings, nC,rel	0.92	0.34	0.88	0.41	2
	integral/constitutional (Eq. 3)	5		,, , C rel w rel n M	, , , H S n n n rings	0.88	0.41	0.84	0.45	2
	fragments/all types (Eq. 4)	5	2 FHBSA CIC RNCS SIC ASIC 0.95 0 2 , , , ,	0.28	0.87	0.44	5
				h		c	c				h		c
	fragments/constitutional and topological (Eq. 5)	4		nS,h, 2 χc, 2 BSICc, 1	SICh			0.92	0.33	0.87	0.42	0
	fragments/constitutional (Eq. 6)	4		, , S rel h n	,	, w,c rings h , M n	,	gle rel c sin , , n	0.91	0.35	0.85	0.43	1

ndesc: number of descriptors; nout: number of molecules out of AD of the model;

Table 6 . Estimation of pC20 based on Abbott's assumption (Eq. 7)

 6

		log CMC		pC20
	Surfactant	exp	QSPR (Eq. 6)	Abbott (Eq. 7)	exp	ref
	S-Butyl-1-Thio-D,L-Xylitol	-0.74	2.83	1.74	1.80	(Savelli et al., 1999)
	1-O-Hexanoyl-D,L-Xylitol	-1.24	2.01	2.24	2.10	(Savelli et al., 1999)
	2-Amino-2-Deoxy-Octyl-β-D-Glucoside	-1.64	2.95	2.64	2.70	(Boullanger & Chevalier, 1996) a
	Octyl-β-D-Glucoside	-1.70	2.95	2.70	3.00	(Matsumura et al., 1990) a
	Octyl Glycol	-2.31	3.15	3.31	3.20	(Shinoda et al., 1959) a
	N-Decyl-N-Methyl Lactobionamide	-2.64	3.71	3.64	3.30	(Burczyk et al., 2001)
	1-O-Nonanoyl-D,L-Xylitol	-2.36	3.15	3.36	3.30	(Savelli et al., 1999)
	S-Hexyl 5-Thio-D-Xylonolactone	-2.30	3.66	3.30	3.40	(Lalot et al., 2004)
	1-O-Octyl-D,L-Xylitol	-2.17	3.15	3.17	3.40	(Savelli et al., 1999)
	3,7-Dimethyloctyl-β-D-Glucoside	-2.40	3.71	3.40	3.50	(Minamikawa & Hato, 2005) a
	N-Decyl-N-Methyl Gluconamide	-2.89	3.91	3.89	3.60	(Burczyk et al., 2001)
	S-Hexyl 1-Thio-L-Ribitol	-1.99	3.59	2.99	3.70	(Lalot et al., 2004)
	Decyl-β-D-Glucoside	-2.68	3.71	3.68	3.90	(Aveyard et al., 1998) a
	N-Dodecanoyl-N-Methyl Lactitolamine	-3.35	4.09	4.35	4.00	(Wilk et al., 2001)
	S-Octyl 5-Thio-D-Xylonolactone	-3.28	4.42	4.28	4.20	(Lalot et al., 2004)
	Dodecyl-D-Maltonamide	-3.50	4.47	4.50	4.40	(Zhang & Marchant, 1996)
	S-Octyl 5-Thio-D-Arabinonolactone	-3.32	4.42	4.32	4.70	(Lalot et al., 2004)
	N-Tetradecanoyl-N-Methyl Lactitolamine	-4.17	4.85	5.17	4.90	(Wilk et al., 2001)
	N-Oleyl-N-Methyl Lactobionamide	-4.27	5.58	5.27	5.10	(Burczyk et al., 2001)
	S-Decyl 5-Thio-D-Xylonolactone	-4.64	5.18	5.64	5.40	(Lalot et al., 2004)
	S-Octyl 1-Thio-L-Xylitol	-2.92	4.35	3.92	5.40	(Lalot et al., 2004)
	S-Octyl 1-Thio-L-Ribitol	-3.42	4.35	4.42	5.60	(Lalot et al., 2004)
	N-Hexadecyl-N-Methyl Lactobionamide	-5.03	5.99	6.03	6.00	(Burczyk et al., 2001)
	N-Octadecyl-N-Methyl Lactobionamide	-5.48	6.74	6.48	6.30	(Burczyk et al., 2001)
		MAE b	0.31	0.30
	a data extracted from graphs; b MAE = Mean Absolute Error

Acknowledgements

This work was performed, in partnership with the SAS PIVERT, within the frame of the French Institute for the Energy Transition (Institut pour la Transition Energétique (ITE) P.I.V.E.R.T. (www.institutpivert.com) selected as an Investment for the Future ("Investissements d'Avenir"). This work was supported, as part of the Investments for the Future, by the French Government under the reference ANR-001-01. Calculations were performed using HPC resources from GENCI-CCRT (Grant 2013-t2013086639).

) [START_REF] Tropsha | The Importance of Being Earnest: Validation is the Absolute Essential for Successful Application and Interpretation of QSPR Models[END_REF])

) [START_REF] Schüürmann | External Validation and Prediction Employing the Predictive Squared Correlation Coefficient -Test Set Activity Mean vs Training Set Activity Mean[END_REF] Q²F2 > 0.70 y is the average of experimental properties; y ˆ is the average of calculated properties; 2 0 R is the coefficient of determination forcing the origin for the axis; 2 m r and 2 ' m r are calculated by using experimental data on the ordinate axis and on the abscissa axis, respectively.


Rodrigo De Toledo

Bin Wang

Bruno Lévy

Geometry Textures and Applications †

Geometry textures are a novel geometric representation for surfaces based on height maps. The visualization is done through a GPU ray casting algorithm applied to the whole object. At rendering time, the fine-scale details (mesostructures) are reconstructed preserving original quality. Visualizing surfaces with geometry textures allows a natural LOD behavior. There are numerous applications that can benefit from the use of geometry textures. In this paper, besides a mesostructure visualization survey, we present geometry textures with three possible applications: rendering of solid models, geological surfaces visualization and surface smoothing.

Introduction

A set of geometry textures can be used to represent complex surfaces as an alternative for triangle mesh representation [dTWL07]. The goal is to interactively display finely tessellated geometric models. Geometry texture visualization is implemented on the GPU and the main rendering effort is on the pixel pipeline. Transferring the workload to the pixel pipeline brings the benefit of a natural LOD, since rendering time is proportional to the number of rendered pixels. Therefore, when a complex object is far away from the viewer, less computation must be done.

The main idea in our technique is to reconstruct the finescale geometric details over a simple proxy of the original model. Fine-scale geometric details are known in the literature as mesostructures. Mesostructures are commonly simulated as a pattern in visualization systems. In a different situation, we want to reconstruct the complete object. In our case, the mesostructure changes along the object surface, and memory becomes an issue. For this reason we are interested in height maps, which are a compact way to represent surfaces. The input of our rendering technique is a set of height maps generated from the original model to represent † This paper is based on Geometry Textures, by Rodrigo de Toledo, Bin Wang, and Bruno Lévy, which appeared in th eProceedings of SIBGRAPI 2007 -XX Brazilian Symposium on Computer Graphics and Image Processing.

the real geometry information. We directly use the maps during rendering, executing a ray-casting algorithm on the GPU. In our approach, the geometry is passed to the GPU as a texture (the geometry texture), which is used by the fragment shader. Geometric details are reconstructed with the correct shading, self-occlusion and silhouette.

There are many applications that can benefit from visualization with intrinsic LOD behavior. We have investigated two of them: complex solid objects rendering and geological surfaces visualization. In the first one, models are partitioned to produce several geometry textures. On the second application, we are able to visualize a set of geological horizons, each one rendered as a geometry texture. Finally, we also research a third application that make use of image operations on geometry textures to smooth detailed surfaces.

The remaining of this paper is organized as follows:

• Section 2 classifies major work on mesostructure visualization according to how the techniques represent mesostructures. We conclude by demonstrating the reason for choosing height maps as our mesostructure representation. • In Section 3, we discuss previous work that also use a set of height maps to represent and render solid models. • The basic rendering algorithm for geometry textures is explained in Section 4. • Section 5 presents how to convert complex solid models into our geometry-texture representation.

• Then, Section 6 describes three possible applications for our technique. • Finally, in Section 7, we draw some conclusions and discuss different future work.

Mesostructure

The appearance of an object is a result of the microstructure in its surface. A complete model that takes into account the microstructure and all the physical events occurring in this scale would be impractical and far from interactive rates. For this reason a lot of effort was dedicated in representing the appearance in a mesoscale level, between the micro and macro scales. The surface structure we need to represent in this level is named mesostructure.

When touching an object, one can feel its texture. Actually, textures give both tactile and visual information in such a way that we can imagine the sensation of touch by just looking at an object. The texture is a result of the object mesostructure and it can be simulated in several ways in visualization systems. Its simplest representation is the color map, which is a simple 2D image applied over a virtual object [Cat74,BN76]. Blinn extended texture mapping with the bump mapping technique [Bli78], which gives visual effects of bumps and depressions by disturbing surface normals according to a bump map.

More complex mesostructure representations store mesoinformation in other formats: height maps, volume data, shading functions and polygonal meshes (see Figure 1). We describe some major work in the following sections.

Height-map mesostructure

The first technique that used a height map to represent and render mesostructures was displacement mapping [START_REF] Cook R | Shade trees[END_REF]. It subdivides the macro geometry into a large number of small polygons whose vertices are displaced in the normal direction according to the associated displacement map. Its drawback is the excessive number of generated polygons.

To avoid extra polygons, the VDM method (Viewdependent Displacement Mapping [WWT * 03]) adopts the idea of previously sampling mesostructure appearance under various lighting and view directions. In preprocessing, a height map is taken as input and synthesizes a set of VDM images for different view directions and different curvatures, recording the VDM data as a 5D function. Although resulting in good quality, this method transforms a 16KB map into a 64MB volume of data, which prevents its application for our purpose, since we use hundreds of maps.

Policarpo et al.

[POC05] use height maps to interactive mesostructure visualization. Their method is the most similar to ours, using GPU ray casting. However, they do not devote attention to complex and highly-tessellated surfaces.

Hirche et al. work [HEGD04

] is also centered in a heightmap ray-casting per-pixel implementation. However, they extrude some prisms from the base mesh (their macrogeometry), and the ray casting is done by using the prism walls, quadrupling the number of faces, which is unnecessary in our method.

Volumetric mesostructure

The use of volumetric mesostructure brings a more flexible way to represent details. The mesostructures found in real world are better approximated by volumes, which have no overlapping restriction. Another advantage is the possibility to add opacity per-voxel information, which can be used for simulating more complex lighting phenomena as refraction and subsurface scattering.

Most of the rendering applications use prism extrusion from the base mesh to create a 3D parameterized space. This can lead to self-intersections near concave features, which is solved by some methods [PKZ04, PBFJ05].

In Lensch et al. work [LDS02] volumetric textures are applied over arbitrary macro-geometry mesh. They use back to front slicing technique, drawing the polygons inside each prism formed by the intersection of the orthogonal viewing slices. To produce a fast intersection computation, they have created a structure based on normal edge classification. So, the slices are rapidly computed. They have created 3 different implementations: in software, in hardware, and a hybrid one, which is the fastest. * 04] is a generalization of VDM that accepts volume mesostructure definition. Their main structure, named GDM (Generalized Displacement Maps), is also a five-dimensional function accessed in real-time, but as opposed to VDM they use prisms, computing the ray exit point and evaluating the intersection in each prism domain.

Wang et al. work [WTL

In Shell Texture Function (STF [CTW * 04]), Chen et al. compute subsurface scattering, using photon tracing in preprocessing and ray tracing for final visualization, but the rendering are not interactive (images are generated in about 100 seconds). In their subsequent work, SRTF (Shell Radiance Texture Functions) [SCT * 05], the visualization achieves interactive frame rates. However, it is not possible to reproduce detailed silhouettes since SRTF is rendered without ray tracing. Indeed, these techniques are a hybrid approach between ray casting and shading functions (described in Subsection 2.4). * 06] have developed interactive good results by rasterizing polygonal mesh mesostructure. They carefuly stitch together geometry elements finding corresponding elements in adjacent mesostructures patches. They succesfully align elements through local deformation although presenting some distortion on regions with very high curvature.

Polygonal mesostructure

Zhou et al. [ZHW

In contrast to Zhou et al. work, Shell Maps [PBFJ05] do not render the final geometry. They create prisms arround the surface (similar to some previously mentioned techniques) and map them on their counterpart prisms in texture space. They create a general procedure to ray cast mesostructures in texture space. Although the possibility to apply their procedure for different mesostructure representations, in their examples they use polygonal and procedural ones, rendered in non-interactive rates.

Representations of the Shading Function

Shading models are a very complex and important subject in visualization domain. Its complexity comes from the fact that these models try to represent and simulate the lighting phenomena happening in microscale level to obtain results in macro and mesoscale levels. Generally, in interactive applications, the shading model in use is the diffuse/specular one, which is only an approximation of real lighting effects. For more accurate shading, a bidirectional reflection distribution function (BRDF) is more appropriate to categorize how light is reflected from a point on the surface.

BRDF = f (θ in , φ in , θout , φout ) (1)
So, based on the BRDF model, to compute the outgoing light Lo in a given direction ωo we need to integrate the incoming light in all directions:

Lo( ωo) = Ω BRDF( ω i , ωo)L i ( ω i )cos(θ i )d ω i (2)
where Ω represents the hemisphere vectorial space since the light beams coming from the back are not considered. ω describes the spherical coordinate angles (θ, φ) of a given direction vector.

This subsection covers the methods based on BRDF principles. Most of them use a set of images as the input information about the mesostructure shading function. This image collection is the result of capturing real pictures of the desired materials from different points of view, uniformly distributed in a hemisphere. In some methods [START_REF] Liu | Synthesizing bidirectional texture functions for real-world surfaces[END_REF]VT04] this is done synthetically. Dana et al. [START_REF] Dana | Reflectance and texture of real-world surfaces[END_REF] introduced BTF (bidirectional texture function) in an experimental work based on the BRDF principles. They captured a set of images, using different material samples and varying the light and viewing directions. The images are stored in a publicly available database. So, the BTF is a 6D function, which has the same 4 BRDF variables (Equation 1) plus 2 for position on the image:

BT F = f (θ in , φ in , θout , φout , u, v) (3)
Some methods have extended the BTF approach. Liu et al. [START_REF] Liu | Synthesizing bidirectional texture functions for real-world surfaces[END_REF] address two related problems: the synthesization of a continuous BTF and the specification of new synthesized BTF's without the captured images. In [TZL * 02], Tong et al. succeeded in rendering arbitrary surfaces with BRDF material although in low speed (about 1 frame per second).

Applying BTF on surfaces for real-time rendering is not easy; the lookup value at (u, v) is a hard task since BTF original information consists of a large amount of images. Polynomial Texture Maps [MGW01] is a BTF simplification since it removes the viewing direction from the equation, reducing the number of images taken for each surface.

PT M r,g,b (θ in , φ in , u, v) (4)
Instead of recording and accessing all the pictures, which would still be very costly, they do a convenient approximation to store per-texel information: split chromaticity pertexel luminance approximated as a biquadratic expression.

TensorTexture [VT04] technique uses synthetic mesostructure. It also differs from BTF in storing information. It applies a dimensionality reduction after a multilinear analysis of the image data tensors. However, this approach had proved to be too slow for interactive rendering. The CPU implementation has an average time of 1.6 seconds per image. We summarize the visual effects and characteristics of some of the main mesostructure visualization techniques in Table 1. Some of the presented numbers are approximations, but they are meaningful to compare different techniques. For example, in Extra Memory, we have evaluated how much per-texel information is needed to represent the mesostructure (computing the uncompressed data). Preprocessing marks those that present expressive pre-computations. In Rendering Speed column, up arrows mean fast rendering while down arrows mean slow rendering (no arrows are used for average speed).

Comparing mesostructure techniques

[Cat74] 3 1 ↑↑ Normal Bump Mapping [Bli78] X 3 1 ↑ Height Map Displacement Map [Coo84] X X X 1 10 3 - VDM [WWT * 03] X X X X 2 12 1 X ↑ Ray-casting [POC05] X X X X 1 1 - Prism Ray-casting [HEGD04] X X X 1 4 - Volumetric Mesostructure Semi-Transparent [LDS02] X X X 2 7 3 - GDM [WTL * 04] X X X X 2 14 4 X - Shell Texture F. [CTW * 04] X X X X X 10 NA X ↓↓ Polygonal mesh Shell Maps [PBFJ05] X X X X NA 4 X ↓↓ Mesh Quilting [ZHW * 06] X X X NA 10 3 X ↓ Shading Function BTF [DvGNK99] X X X X 2 9 1 X ↓ PTM [MGW01] X X X 5 1 X ↑ TensorTexture [VT04] X X X X 2 11 1 X ↓↓
Based on Table 1 we decide to use height-map ray-casting algorithm for mesostructure rendering. It is the best option in number of visual effects that do not contain significant negative points (those in red boxes).

Related Work on Multiple Height Maps

The use of multiple height maps for representation of a complex model is an interesting idea due to the compactness. For rendering, the use of GPU height-map primitives is promising. The drastic reduction of vertices in the scene (the usual bottleneck for complex object visualization) may result in better performance. On the other hand, the GPU heightmap primitives visualization converges the effort in the pixel shader. For this reason they have a natural LOD behavior. In this section, we describe some work previously done that are related to this approach. RTM (Relief Texture Mapping [START_REF] Oliveira | Relief texture mapping[END_REF]) is the first work on object visualization based on multiple height maps. RTM starts by capturing the depth of an object from six orthogonal points of view. In rendering time, a warping-based technique is applied on the faces of the bounding box to reconstruct the original model. [START_REF] Baboud | Rendering geometry with relief textures[END_REF] use multiple height maps to represent and render complex objects. They store the height maps of the six orthogonal points of view using an adapted perspective frustum, which increases the detail information about features that are almost perpendicular to the view point. During rendering, they use a clipped frustum to reduce the number of discarded fragments.

Badoud and Décoret

Porquet et al. [START_REF] Porquet | Real-time high-quality view-dependent texture mapping using per-pixel visibility[END_REF] have developed a technique to render complex surfaces by using a rough geometric approximation on which colors and normals are applied according to previously captured information, including height maps. In preprocessing, they capture the maps from several points of view. In rendering time, the three closest views to the camera projection are used by the fragment shader. Based on the height maps, the fragment shader reconstructs the equivalent information of the three points of view to finally find which one is the nearest sample to the current fragment, defining the color and the normal to be used. The main disadvantage of this method is the lack of silhouette details.

The multi-chart geometry images method [SWG * 03] is similar to our geometry textures method, although it does not use height maps. It extends Geometry images [START_REF] Gu | Geometry images[END_REF] by splitting a model into multi charts before parameterizing each one onto a square. It is especially different from our method during rendering. To render the geometry images, it is necessary to reconstruct many triangles from the images, overcharging the vertex pipeline, which is exactly what we want to avoid. Since geometry image patches are defined on the surface itself rather than height maps, they cannot apply any optimization at the GPU fragment shader.

Geometry Texture Rendering

Height-map GPU ray casting

A height map (a.k.a. height field, relief map or depth texture) is a 2D regular table where a height is specified for each entry. One way of representing a height map is a grayscale image with brightness representing height (see Figure 1(c)). Terrains are an example of a surface well suited to be represented by height maps.

A height map can be represented by the function:

hm(x, y) : [0, 1] 2 → [0, 1].
We can define a Boolean function z(P), for (P.x, P.y, P.z) ∈ [0, 1] 3 , as: z(P) ⇐ ( P.z ≤ hm(P.x, P.y) ) ? true : false, In our implementation, we use the faces of a parallelepiped to generate the fragments that will ray cast the height map. The coordinates of its vertices are between 0 and 1 (as in a canonical cube), defining the local coordinate space. The height map data is passed to the GPU through a grayscale texture. The x and y axes of the parallelepiped are exactly coincident with the u and v coordinates of the texture, while z coincides with the height direction.

In GPU, the vertex shader sends to the fragment shader the viewing direction already in local space. This way, each fragment knows the path of the ray inside the canonical cube, and can easily compute its exit point. The fragment-shader's main task is to answer the two ray-casting questions: is there an intersection between the ray and the height map? Where is the closest hit point?

As explained in [POC05], one strategy is to uniformly sample this ray path in N points P i and use function z(P) to find the ray-surface intersection. This could be implemented by a conditional loop varying P i , from the entering to the exit point, and stopping when z(P i ) is false. In each step P i is translated by ∆, which is the ray path divided by N.

When a ray does not intersect any height sample, the fragment should be discarded without any contribution to the frame buffer or to the z-buffer. For fragments not discarded, the actual color can be retrieved from a color texture, and the normal can be either computed directly from the height map or given as a third texture to be used for shading. The coordinates used in the texture lookup are (P.x, P.y) right after the iterations. Finally, the depth is computed to register the correct z-value for the z-buffer. Two-step searching. Even if ∆ is short enough not to miss a sample, the point where the ray touches the object is not exactly computed. As proposed in [POC05], we use a second searching step between points P i-1 and P i . This second step is a binary search that divides the searching domain by two in each step. Therefore, with M steps in the binary search we multiply the precision by a 2 M factor. Notice that this second searching procedure is done in the same fragment code, it is not a multi-pass algorithm.

We propose the following improvements for the heightmap GPU visualization:

Non-rectangular height map domain. A height sample with value equal to 0 can be considered either as the surface base or as a representation of empty space. We have chosen the last case, since representing empty space is essential for geometry textures. Figure 4(d) has an example.

Balancing between steps. Binary search is much more efficient than linear search. However, it cannot be exclusively executed because, along the ray path, there could be several intersections. In other words, the z(P) function may change different times along the parametric ray for each fragment. On the other hand, when the ray is totally perpendicular to the height map (in a perfect top view), z(P) will change sign at most once. In this case, the binary search can perfectly and efficiently compute the intersection without the previous linear search. Based on this observation, we have implemented a balance between the linear and the binary steps according to the viewing slant. When in a vertical view, we reduce the linear search iterations while increasing the binary search iterations (N↓, M↑). For an almost horizontal viewing direction we do the opposite, prioritizing the linear search (N↑, M↓). This way, we achieve up to two times faster rendering for some cases (see Figure 2) than without balancing. Some other methods were proposed to accelerate GPU ray casting of height maps [START_REF] Donnelly | GPU Gems 2 -Per-Pixel Displacement Mapping with Distance Functions[END_REF]PO07]. However, they need extra information about distance functions.

Geometry Textures

The problem approached in our work is different from standard mesostructure rendering. We are interested in reconstructing the details of an entire object, which is a nonrepetitive pattern, since it changes over the model domain. In our case, the input is a set of height maps converted from the original model to represent the complete geometrical information.

Geometry texture is a geometric representation for surfaces. Its domain is a parallel hexahedron and in its interior a height-map represents the surface (see Figure 3). The domain is not restricted to a perfect rectangle, so empty samples are expected in each geometry texture. The construction of geometry texture patches from a complex model has no global folding restriction and the patches are well fitted around the surface contour. In rendering time, geometry textures use our ray-casting algorithm implemented in a fragment shader. As a result, the geometry is correctly reconstructed and rendered.

Conversion

The input for our conversion procedure is a polygon mesh and the output is a set of geometry textures. There are three main steps when converting triangle meshes into geometry textures (see Figure 4(a)): mesh partitioning, bounding-box determination and maps extraction.

Among the three conversion steps, mesh partitioning is the most difficult and critical for the success of the algorithm. Some conditions are expected when partitioning the input model mesh into charts:

• The chart must not contain a folding situation in its z.

• The chart domain should be as square as possible to reduce empty spaces. • For visualization purposes, neighboring charts must share their boundary, resulting in some overlapping between them. Right after partitioning we add an extra triangle ring around each chart (see Figure 4b). As explained in [dTWL07], the overlap area is necessary to avoid cracks, while in rendering time, z-buffer information is correctly generated to achieve a seamless reconstruction.

Bounding-box determination includes finding a good orthonormal coordinate system and domain for each one of the charts. Direction z is taken as the median normal of all triangles in one chart, with this we minimize the occurence of self-folding partitions. Then, x and y must be chosen based on the minimization of empty spaces in the domain. We have used the 2D-PCA (Principal Component Analysis) algorithm to determine directions x and y projected on the plane defined by z. Finally, we project the coordinates of all triangles to obtain the exact domain in each direction. ( x, y, z) forms the local coordinate system of the chart.

In the map extraction phase, there are three maps we want to obtain: height, color and normal maps. We have developed a GPU solution to extract the three maps from one chart. Based on the local coordinate system of each chart, we render its triangles in an orthogonal view projection perpendicularly to direction z, filling out the viewport with a user-defined resolution (for example, 256 × 256). We repeat the procedure three times capturing the image buffer:

• To obtain the color map we render the colors without illumination. • To generate the normal map we assign (x, y, z) global coordinates of normal n i of each vertex v i to (r, g, b). During rasterization, within each triangle, the normals are interpolated and then normalized again. • To obtain the height map we start by finding in CPU the maximum and minimum coordinates (hmax and h min ) in direction z. We assign a value h i ∈ [0, 1], computed as h i = vi.z-hmin hmax-hmin , to each vertex v i . This value is interpolated inside the triangles and outputted as luminance.

Note that for space-reduction purposes, the image buffer is captured and stored using 8-bit precision.

The rendering algorithm loads the maps as textures. However, before that, we apply a texel-dilation procedure to prevent sampling problems when using mip-mapping. These problems may occur when the graphics hardware access border information for non-rectangular domain, using a null value (from out of the domain) to interpolate the returning value. Our dilation algorithm fills empty texels around the map domain with interpolated values. The procedure is done for normal and color maps. See Figure 4.

In the following subsections we discuss two different strategies we have implemented for mesh partitioning: PGP (Periodical Global Parameterization [RLL * 06]) and VSA (Variational Shape Approximation [START_REF] Cohen-Steiner D | Variational shape approximation[END_REF]).

Partitioning with PGP

Periodic Global Parameterization is a globally smooth parameterization method for surfaces. PGP can be applied to meshes with arbitrary topology, which is a restriction in other parameterization methods [RLL * 06]. Moreover, we can extract a quadrilateral chart layout from this parameterization, which is based on two orthogonal piecewise linear vector fields defined over the input mesh. This orthogonality is especially interesting to guide our partitioning procedure in order to reduce empty spaces in the domain. These vector fields are obtained by computing the principal curvature directions, resulting in a parameterization that follows the natural shape of the surface (left picture on Figure 5).

PGP avoids excessive empty spaces. However, often charts obtained with PGP partitioning present some folding problems. For this reason we have decided to investigate VSA as an alternative partitioning method.

Partitioning with VSA

Variational Shape Approximation is a clustering algorithm for polygonal meshes that can be used for geometry simplification [START_REF] Cohen-Steiner D | Variational shape approximation[END_REF]. Our partitioning algorithm based on VSA uses each cluster as an initial chart, which is further increased to overlap neighboring domains (see Figure 4).

The main advantage of using VSA compared to PGP is that it reduces the number of folding cases. VSA uses L 1,2 metric, which is based on the L 2 measure of the normal field. This means that it takes into account the normal direction of the vertices to cluster them. A folding situation occurs only if vertices in a chart have a sharp difference in their normal direction (over 90 degrees). With VSA, each chart only contains vertices with similar normal directions.

In the next section we describe some experiments to compare the PGP and the VSA methods.

Comparing PGP and VSA

In our tests we have partitioned with both PGP and VSA two different models: bunny and buffle (Figure 5).

Our first test compares how good PGP and VSA are at avoiding empty spaces. We have counted for each chart the number of used and empty texels. This was done in the four test cases (PGP bunny, VSA bunny, PGP buffle and VSA buffle). For simplicity, the charts were extracted without overlapping and the maps without dilation. The maximum map size was 128 × 128. A map could be smaller than that (adapted to the chart size), as long as each dimension remained a power of two (respecting OpenGL's restriction). In our second test we have counted how many charts have folding problems, again in the four test scenarios. A way to check if a chart has a folding situation is to verify if there is any inverted triangle when rasterizing all triangles in local direction z. Given that the triangles in the mesh respect a counterclockwise rotation, if at least one of them is rasterized clockwise then the mesh folds itself in direction z. Figure 5 shows a partitioned buffle model with inverted triangles marked in green. In Table 3 we present the number of folding charts and the total number of inverted triangles found. Clearly, VSA is more appropriate to avoid folding problems.

In conclusion, both methods have their own advantages. Table 3: VSA significantly reduces the number of folding charts (identified by inverted triangles).

While PGP is better for reducing empty spaces, VSA is much more efficient in avoiding self-folding charts. We consider that the folding problem is the most important issue. Self-folding charts cannot be represented by our geometry texture, which is a strong restriction. On the other hand, empty-space reduction is merely an optimization.

Based on these facts, we have chosen VSA as our partitioning method. The results presented in the next section have been achieved using VSA.

Applications

Once we have obtained a set of geometry textures, we can render them using the height-map ray-casting algorithm on GPU. In the following sections we discuss some applications using geometry textures visualization.

Rendering of solid models

We have used the dragon model with 871,414 triangles for tests (see Figure 6). We have generated three sets of 453 geometry textures, varying their maximum resolution (128 2 , 256 2 and 512 2 ). The tests were done with a GeForce 8800 GTX graphics card.

Since geometry textures have their performance bottlenecked per pixel, we have varied the zoom level in our tests. We have plotted the results in Figure 7. situations: with Compiled Vertex Array (CVA) and with Vertex Buffer Object (VBO). Both methods are advanced features in OpenGL and without them the speed would be less than 1 FPS for such model size. For the dragon model with 871K triangles, we have obtained 27 FPS with CVA and 135 FPS with VBO (plotted as dashed lines in Figure 7).

The horizontal axis in Figure 7 represents the number of pixels of the dragon's ray-casting area, which is formed by the rasterization of the geometry-textures boundingbox faces (without recounting overlapping fragments). The higher the number of pixels, the larger the model is on the screen. For a practical idea, we have highlighted some window sizes (640 × 480, 800 × 600, 1024 × 768, 1280 × 1024) which the model would fit.

The ray-casting algorithm is the same independently of resolution. However, we have verified that the lowest maximum resolution (128 2 ) was the fastest one. This is a result of less graphics card memory in use (4.21MB), optimizing caching and fetching. As we will see, for quality purposes, 256 2 would be a good choice for maximum resolution for the dragon's 453 geometry textures used in test. Its performance is better than the original mesh rendered with VBO for resolutions smaller than 800 × 600.

Horizons visualization

Horizons are geological surfaces representing interfaces between two contiguous sediment layers. In general, horizons are extracted from volumetric seismic data [OK04]. For each seismic trace (a vertical column in the volume), at most one sample belongs to the same horizon. Thus, height maps are a natural choice to represent this surfaces, given that there is a unique representation on vertical. In oil industry, a set of horizons for one geographical region is very important to determine possible bottom and top limits of a reservoir. resented by height-maps. To visualize them we use our geometry texture technique. As a result, we obtain high performance when this set is far from the camera. Figure 9 presents rendering speed in three different situations: in top, profile and oblique views.

Originally, these horizons are height maps with 1588 × 1188 resolution, they could be reconstructed with about 16 million triangles. As already mentioned, the use of geometry textures brings a LOD behavior that can be once more observed on horizons-rendering speed (see Figure 9). Horizontal (or profile) view is not so efficient as top view because of the linear-binary balance (see Section 4.1). Note that the speed values are clearly slower than in dragon example because different graphics cards were used for tests.

Smooth surfaces

Once we have a new geometry representation based on images (height maps), image operations can be applied on these maps to transform geometry . For example, one could use a low-band filter to smooth the geometry (or a high-band filter to highlight small geometric features).

We have experimented a Gaussian blur on both height and normal maps to smooth the surface. The result is a surface without most of its high frequency details as shown in Figure 10. However, it is clear that it is not possible to deliberately apply the smooth operation because each geometry-texture patch does not have complete information about the neighbors. As a consequence, some undesirable features start to appear on regions with strong curvature. This is not the only issue. When applying Gaussian filter with significant radius, as in Figure 10(d), each patch becomes more flat and it is possible to see the seams.

Conclusion and Future Work

We have proposed a new representation for highlytessellated models using a set of geometry textures, which are rendered using a GPU implementation of height-map ray casting. Our results have shown that this new representation is suitable for natural models (i.e. non-CAD models). The final rendered images have similar quality compared to traditional polygonal rasterization methods.

The following items are some of the positive aspects of our technique:

• The rendering speed naturally follows a LOD behavior.

This means that when the model is small on the screen (i.e., far from the camera), rendering is faster. This is a result of relieving the vertex-stage burden, transferring computation bottleneck to pixel stage. • Depending on the chosen resolution, our representation requires less memory than polygonal representa- Our potential drawbacks are:

• Recent graphics cards that implement VBO extension have very high performance for polygonal rendering. As a consequence, when compared to VBO performance, our technique is faster only if the model is not so big on the screen. However, if the scene contains multiple models, polygonal rendering would proportionally lose performance, while our solution would keep a stable frame rate. • Another adverse point is the sophisticated conversion algorithm. Our procedure is composed by multiple passes, including the partitioning step, which is considerably elaborate. This may be improved by using multilayer height maps as shown in [START_REF] Santos | Solid height-map sets: modeling and visualization[END_REF].

We suggest the following ideas for future work:

Self shadows

Self shadows for height-map ray casting is a straightforward algorithm. However, in the case of geometry textures, multiple maps are in use simultaneously and they do not have information about neighbor maps. Alternatively, it is always possible to use shadow mapping [START_REF] Williams | Casting curved shadows on curved surfaces[END_REF] to produce shadc The Eurographics Association and Blackwell Publishing 2008.

ows since geometry textures keep z-buffer up to date. However, shadow mapping suffers from aliasing problems and needs sophisticated algorithms to overcome this problem (see [START_REF] Barroso | Improved real-time shadow mapping for cad models[END_REF] for a survey about this subject). For this reason, ray cast for shadows, including neighboring information, is a valuable topic for future research.

Other image operations

We have implemented a simple blur operation to smooth surfaces represented by geometry textures (Section 6.3). It is possible to use other operations that may result in different geometry modifications. For example, a high-band filter can be used to highlight small geometric features. Other possible applications are operations to insert geometric patterns and simulations of cellular network [START_REF] Gobron | Generating surface textures based on cellular networks[END_REF].

Combined use of explicit and implicit surfaces for GPU ray casting

This work implements a GPU ray casting for height maps that uses the faces of a bounding-box to trigger the algorithm. This idea has also been explored for implicit surfaces rendering [TL04,dTLP07,LB06]. We suggest, as a future research, the combination of implicit and explicit algorithms. For example, it is possible to produce some holes in natural models by subtracting some implicit spheres from the geometry texture set. Another example could be rendering implicit surface objects (e.g., a torus) applying height-map mesostructure information.

Acknowledgement

To Tecgraf design group. To the anonymous reviewers for their excellent contributions.

Figure 1 :

 1 Figure 1: Different representations for mesostructures: (a) color maps, (b) normal maps, (c) height maps, (d) volumetric mesostructure [CTW * 04], (e) polygonal mesh [ZHW * 06], and (f) shading function (picture from Columbia University website).

Figure 2 :

 2 Figure 2: Comparing not-balanced and balanced searches (right column). With balanced search, we obtain better performance (85 fps × 66 fps on a GeForce 6800) when seen from above and better quality when seen in a profile view.

Figure 3 :

 3 Figure 3: Geometry texture sample. A height map defined inside a hexahedron's domain.

Figure 4 :

 4 Figure 4: (a) Complete procedure for converting polygon meshes into a set of geometry textures. (b) Mesh partitioning. (c) Bounding-box determination. (d) Height-map extraction (e) Height and normal maps of a patch. (f) Map dilation (twopixel dilation).

c

  The Eurographics Association and Blackwell Publishing 2008.

Figure 5 :

 5 Figure 5: Buffle partitioning with PGP and with VSA. Triangles with folding problem have their vertices marked in green.

FPSFigure 7 :

 7 Figure 7: We have compared three different geometry texture resolutions (128 2 , 256 2 , 512 2 ) and the original mesh rendered with VBO (Vertex Buffer Object) and CVA (Compiled Vertex Array) of the dragon model. Frame rates registered on a GeForce 8800.

  To compare performance between geometry textures and common triangle-mesh visualization, we have measured the rendering speed of the original dragon model in two special c The Eurographics Association and Blackwell Publishing 2008.

Figure 6 :

 6 Figure 6: The dragon model partitioned into 453 geometry textures and rendered with maximum resolution 256 × 256 for each patch.

Figure 8 Figure 8 :

 88 Figure 8 shows an example of horizons set that are rep-

c

  The Eurographics Association and Blackwell Publishing 2008.

Figure 9 :

 9 Figure 9: Rendering speed of the scene in Figure 8 varying screen resolution in three different point of views: viewing from the top, in profile and obliquely. NVidia GeForce 7900 GTX.

Figure 10 :

 10 Figure 10: Smoothing Buffle model. (a) Original (b) 2-texels Gaussian blur (c) 4-texels Gaussian blur (d) 8-texels Gaussian blur.

Table 1 :

 1 Comparison

between different interactive mesostructure visualization. Negative points are marked with a red background.

Table 2 :

 2 Table 2 shows the results, indicating that PGP is better at avoiding empty spaces. Considering empty spaces, PGP is better than VSA.

	Model Method Total texels Empty texels	%
	Bunny PGP	1008128	240357	23.84%
	Bunny VSA	461312	179194	38.84%
	Buffle PGP	2498048	563652	22.56%
	Buffle VSA	979456	453069	46.25%

c The Eurographics Association and Blackwell Publishing 2008.