Katarzyna Brymora

Florent Calvayrac
email: florent.calvayrac@univ-lemans.fr

Surface anisotropy of iron oxide nanoparticles and slabs from first principles : influence of coatings and ligands as a test of the Heisenberg model

Keywords: surface anisotropy, magnetic nanoparticles, Heisenberg model, ab initio, functionalization

Surface anisotropy of iron oxide nanoparticles and slabs from first principles: Influence of coatings and ligands as a test of the Heisenberg model

Katarzyna Brymora, Florent Calvayrac

Introduction

Numerous experimental and theoretical studies have been performed in the last decade to provide an understanding of the properties of magnetic nanoparticles, such as super-paramagnetism or surface-enhanced anisotropy[1] [START_REF] Lu | Magnetic nanoparticles: Synthesis, protection, functionalization, and application[END_REF].

It has been proven possible to create nanoparticles built with alternating layers of various materials exhibiting different magnetic behaviors, which will influence the magnetic moment structure at the surface or interface [START_REF] Tronc | Surface effects in noninteracting and interacting gamma-fe2o3 nanoparticles[END_REF]. An induced magnetic anisotropy (exchange bias) can appear in such systems, corresponding to a shifted hysteresis loop and a ferrimagnetic alignment of the moments near the center of the nanoparticle as well as a pinning of the magnetic moments on the surface.

It has been demonstrated that surface modification by organic ligands have a strong influence on the magnetic structure of the nanoparticles (as described in the pioneering work of [START_REF] Berkowitz | Spin Pinning at Ferrite-Organic Interfaces[END_REF]) and also cause spin pinning near the surface [START_REF] Kseolu | Effect of surfactant coating on magnetic properties of fe3o4 nanoparticles: {ESR} study[END_REF].

Surface effects alone can change the magnetic structure, as it was demonstrated for instance in the case of cobalt [START_REF] Luis | Enhancement of the magnetic anisotropy of nanometer-sized Co clusters: Influence of the surface and of interparticle interactions[END_REF], cobalt oxide [START_REF] Hajra | Enhancement of magnetic anisotropy in mechanically attrited Cr2o3 nanoparticles[END_REF], magnetite powders [START_REF] Kihal | High field surface magnetic study of Fe3o4 nanoparticles[END_REF], and maghemite nanoparticles [START_REF] Nadeem | Spin-glass freezing of maghemite nanoparticles prepared by microwave plasma synthesis[END_REF]. Those effects have been reviewed in the case of iron oxide in [START_REF] Tronc | Surface-related properties of -Fe 2 O 3 nanoparticles[END_REF].

From the theoretical point of view, those effects and the related phenomena have been considered mainly from the phenomenological side. Typically, Monte-Carlo calculations on the classical Heisenberg model were performed [START_REF] Van Leeuwen | Quenching of magnetic moments by ligand-metal interactions in nanosized magnetic metal clusters[END_REF], and results were compared to experimental data obtained on CO-functionalized NiPt clusters, as well as to DFT calculations where magnetic moments were computed in the collinear local density functional approach.

The Monte-Carlo-Metropolis approach was also used in the case of magnetite nanoparticles in [START_REF] Mazo-Zuluaga | Surface anisotropy, hysteretic, and magnetic properties of magnetite nanoparticles: A simulation study[END_REF]. In this work, the authors have demonstrated that the surface anisotropy constant can heavily influence the exchange-bias behavior. But this value remains a parameter and cannot even be precisely inferred from experimental data; only a range of possible values is estimated from the resulting magnetic behavior and the comparison to experiment.

The problem of the ab initio computing of the magnetic anisotropy at the surface of a nanoparticle will be the focus of this paper. In literature, this parameter is a phenomenological input in large scale classical calculations based on modified Heisenberg models. On the example of a small cluster, namely Fe 13 O 8 we link various magnetically constrained calculations to a Heisenberg model in order to estimate magnetic properties of the nanoparticle from first principles. We study the change in magnetic properties due the presence of a ligand (dopamine) or of a nearby gold cluster. We also consider a magnetite surface, with an eventual cobalt oxide layer. We discuss the resulting pertinence of the description by a Heisenberg Hamiltonian.

Theory/calculation

Structure of the Chosen Systems

The first objective was to obtain an optimized structure of Fe 13 O 8 , which according to mass spectrometry shows a higher abundance than other iron oxide clusters with different compositions [START_REF] Sun | Theoretical study on gold-coated iron oxide nanostructure: Magnetism and bioselectivity for amino acids[END_REF]. We chose this system because it is small enough to be easily modelled by repetitive first-principle calculations and yet provide interesting surface effects, and was already structurally studied [START_REF] Wang | Geometry and electronic structure of magic iron oxide clusters[END_REF]. We used an ab initio ultrasoft pseudopotential scheme with a plane-wave basis as implemented in the Quantum Espresso (QE) suite ( [START_REF] Giannozzi | Quantum espresso: a modular and opensource software project for quantum simulations of materials[END_REF]). The plane wave basis set was defined by an energy cutoff of 30 Ry (408 eV), confirmed to be sufficient by the test of convergence. A mixing factor of 0.17 was employed. Integration in the first Brillouin zone was performed using 1x1x1 points sampling, since the system is an isolated cluster in a large computational cubic box of the size of 30 Å. The GGA density functional from PBE [START_REF] Perdew | Generalized gradient approximation made simple[END_REF] was used with the corresponding pseudopotentials computed by A.dal Corso with the "rrkj3" code [START_REF] Corso | Spin-orbit coupling with ultrasoft pseudopotentials: Application to au and pt[END_REF] and taken from the Quantum ESPRESSO pseudopotential data base. The optimization procedure was conducted without any symmetry. The structural results are close to the one previously reported in the literature [START_REF] Wang | Geometry and electronic structure of magic iron oxide clusters[END_REF].

We also considered two cases of infinite surfaces: first, a 001 oriented slab of magnetite with atomic positions taken from [START_REF] Grazulis | Crystallography Open Database (COD): an open-access collection of crystal structures and platform for world-wide collaboration[END_REF]. This system provides an interesting approach to magnetic nanoparticles of medical interest. The small surface to volume ratio of those nanoparticles of a diameter close to 10 nm makes possible the study of the grafting of ligands by considering a locally flat surface in order to reduce computational cost [? ]. Another interest of such surfaces lies in their catalytic properties [? ].

In a second step, we coated this surface in a close to epitaxial condition with 5 crystallographic units of a 012 surface of cobalt oxide, using periodic boundary conditions, and a vacuum of 12 atomic units on top, generated thanks to the ASE package [START_REF] Bahn | An object-oriented scripting interface to a legacy electronic structure code[END_REF], with the same computational conditions as previously described for the isolated cluster. The atomic positions, represented on figure 3, changed by less than 2 % in comparison to the input values after optimization, and the cell parameters changed by less than 1 %. A potential application of such a system might be found in the pinning of magnetic moments, and hence a change in hysteresis curves, from the coupling to antiferromagnetic materials [? ? ] From discussions with experimental teams [? ] developing ferrites made of hybrid oxide nanoparticles we then tried a combined approach, generating a simple molecular dynamics simulation of a maghemite cluster coated with cobalt oxide, at 1000K, based on simple, Buckhingam-like ionic potentials [START_REF] Fongang | Coupled structural and magnetic properties of ferric fluoride nanostructures: Part II, a Monte Carlo-Heisenberg study[END_REF] and extracting several representative clusters of increasing size around a random atom. The atomic positions in these clusters were then optimized using the BFGS procedure within the Quantum Espresso suite with the same conditions discussed above, using periodic boundary conditions with a cubic cell of 32 atomic units. With the largest clusters discussed the systems are close to the bulk. We think that these clusters, even though the crystallography or the stoechiometry deviate from pure allotropic forms, reflect what can be found in a typical molecular dynamics run. One should note however that before optimization of the atomic positions, the magnetic moments around each iron or cobalt atom were found close to zero. A typical cluster is given on figure ??.

Ab initio Magnetic Computation

The change in local magnetic moments has been computed using the Quantum Espresso (PWSCF) code. We would like to note that such magnetic moments are sometimes referred to in the literature as "spins", when they are actually expectations on the values of spin components integrated over a sphere of reasonable but arbitrary radius centered on each atom. For the purpose of this calculations, we used non-collinear density functional theory [START_REF] Barth | A local exchange-correlation potential for the spin polarized case : I[END_REF].

We first considered four cases, Fe 13 O 8 alone, with one dopamine molecule added, with six dopamine molecules (Fig. 1), and with a structurally optimized 20 atoms gold cluster in the framework of non-collinear magnetism calculations. We then considered a magnetite surface coated or not by cobalt oxide. In the case of magnetic computations we found that a reduction of the cutoff energy by half ensured consistency of the results (a convergence test showed that a cutoff energy of 30 Ry is a sufficient value to compute structural properties), and a 0.17 mixing factor for self-consistency was employed. A smearing factor of 0.02 Ry was used. We performed calculations assuming the systems to be isolated (in the case of the cluster) with a Martyna-Tuckerman correction [START_REF] Martyna | A reciprocal space based method for treating long range interactions in ab initio and force-field-based calculations in clusters[END_REF]. We checked the results by performing calculations with relativistic pseudopotentials and spin-orbit coupling which we generated from the values suggested in the QE distribution for cutoff (1.4 and 1.6 a.u for Oxygen with 6 active electrons and a projector on empty 3d states), and the ultrasoft pseudopotentials without spin-orbit as described previously.

We enforced a set of arbitrary magnetization states different from the ground-state result but close to it within 2 µ b in each direction, and extracted the corresponding magnetic field from the converged values, using the constrained magnetic calculation option from the latest version of the Quantum Espresso code, which gives the magnetic field as a function of the distribution of magnetic moments. We used a penalty factor of 0.001 for this purpose in order to speed up convergence under this external constraint.

In the case of the isolated clusters, we found a magnetic structure rather hard to describe in simple terms. In the case of the magnetite surface, we found, as known from the literature, a slightly ferrimagnetic state with a resulting magnetization of zero along the x and y axes and of 0.17 µ B in the z direction, orthogonal to the surface, with an absolute magnetization of 2.48 µ B . In the case of the magnetite surface coated by cobalt oxide, we found an enhancement of total magnetization, to 9.89 µ B in the z direction, as expected experimentally.

Magnetic Model

The previous procedure gives a distribution of local magnetic moments ( S i ), magnetic fields ( H), and total energies (H). We have collected these values from various QE runs under different constraints, varying the total magnetization in either direction by a few µ B , and fitted the parameters from the Heisenberg Hamiltonian [START_REF] Mazo-Zuluaga | Surface anisotropy, hysteretic, and magnetic properties of magnetite nanoparticles: A simulation study[END_REF], given as:

H = -2  J  S i • S j (1) -K V i S 2 x,i S 2 y,i + S 2 y,i S 2 z,i + S 2 x,i S 2 z,i -K S k S k • e k 2 -gµ B H • i S i
It is to be noted here that the Zeeman energy (last term) is absent from QE results.

The first sum involves nearest neighbors interactions between iron atoms. In reference [START_REF] Mazo-Zuluaga | Surface anisotropy, hysteretic, and magnetic properties of magnetite nanoparticles: A simulation study[END_REF] these were computed as from coordination numbers. In the bulk three different coordination numbers appear: z AA = 4, z BB = z BA = 6, and z AB = 12. These numbers apply for the core of the nanoparticle. In our case, these numbers were computed from the coordinates of the iron atoms, enforcing a cutoff radius such that no atom had more than 12 neighbors.

The second term in the Hamiltonian is the core cubic magneto-crystalline anisotropy and reference [START_REF] Mazo-Zuluaga | Surface anisotropy, hysteretic, and magnetic properties of magnetite nanoparticles: A simulation study[END_REF] chose a value of K V = 0.002 meV / spin

The third term accounts for the single-ion site surface anisotropy where the unitary vector reads

e k = j P k -P j | j P k -P j | (2)
with P i the position vector of each iron atom on the surface and the sum runs over iron neighbors of j .

In reference [START_REF] Mazo-Zuluaga | Surface anisotropy, hysteretic, and magnetic properties of magnetite nanoparticles: A simulation study[END_REF] the exchange parameters were set at a value of J AA = -0.11 meV, J BB = +0.63 meV, and J AB = -2.92 meV corresponding to a mix of ferromagnetic and anti-ferromagnetic interactions. Those values were taken from [START_REF] Uhl | A first-principles study of exchange integrals in magnetite[END_REF] where they were fitted on ab initio results using a method similar in principle to the one with presently discussed : bulk spin waves were fitted to non-collinear spin calculations.

Fitting Procedure

The results of the ab initio calculations (energies) were fitted using the Monte-Carlo Metropolis result method, based on random configurations of the parameters.

We used a basic, proven quasi-random recurrence generator [START_REF] Press | Numerical Recipes in Fortran 77: The Art of Scientific Computing[END_REF] and a Metropolis algorithm where we accept configuration changes (in our case, a change of the Heisenberg parameters) (q i = q i + ∆q i ) corresponding to a ∆E = E (q i ) -E(q i ) change in the virtual energy of the system. The virtual energy E is here a penalty function representing the distance in between the set of energies found with ab initio calculations and these found with the Heisenberg model applied to the distribution of moments. The change is accepted if a random number y uniformly drawn in between 0 and 1 is lower than P (∆E, T ).

To fit the ab initio results using the Metropolis simulated annealing method we choose the following parameters: a reference energy H 0 (fit parameter), the set of J  (enforcing J  = J <j,i> ), K V , K S and g. We first used a small set of J  corresponding to a small cutoff in neighborhood search, then increased the number of neighbors to increase the quality of the fit. In order for the results to keep a physical meaning we kept the total number of fitted parameters (maximum of 40) well under the number N of samples used to define the virtual energy or penalty function from the energies in Rydbergs:

E = 1 N N i |H ab initio -H Heisenberg | .
In the case of the isolated cluster, since the set of J  is not too large, we allowed those to take any value, but in the case of the surfaces, the large number of J  gave an excellent fit anytime, whatever the conditions, and the distribution of values obtained was hard to read. Therefore, in this latter case, we used the same model already used in [START_REF] Fongang | Coupled structural and magnetic properties of ferric fluoride nanostructures: Part II, a Monte Carlo-Heisenberg study[END_REF], namely two parameters for each kind of super-exchange(iron-iron, iron-cobalt or cobalt-cobalt) related to the super-exchange angle θ in between atoms separated by an oxygen atom, using the relation

J θ = J 180 • cos 2 θ + J 90 • sin 2 θ ( 3
)
which can be written in the form

J θ = J 90 • + (J 180 • -J 90 • ) cos 2 θ (4)
where J 90 • and J 180 • are the coupling constant corresponding to the superexchange angle of 90 • and 180 • , respectively. Those two parameters per couple of atoms (therefore six in total) are the only ones fitted in the model with the volume and surface anisotropies.

In the case of the isolated cluster we also modified the Heisenberg Hamiltonian in the model to allow for a local change in surface anisotropy ending up with:

H = H 0 -2  J  S i • S j (5)
-

K V i S 2 x,i S 2 y,i + S 2 y,i S 2 z,i + S 2 x,i S 2 z,i - k K Sk S k • e k 2 -gµ B H • i S i
We also had to chose several parameters of the simulated annealing procedure such as the fictitious temperature, the annealing law, and the dependence of random changes to the temperature.

All the software used is archived on http://perso.univ-lemans.fr/~fcalvay/surf_anis_code

Results and Discussion

Iron Oxide Clusters

We generated sets of N = 101 configurations randomly drawn from ab initio results corresponding to a total enforced magnetization running from l to 5 µ B along the x axis, 1 to 5 along the y axis, and 37 to 41 µ B along the z axis.

The best result (penalty function as defined above less than 10 -4 Ry) was, as could be expected from the increased number of degrees of freedom in the model, found using the largest number of fitting parameters (full set of J  and set of surface anisotropy constants K Sk ). A typical fit of ab initio values versus Heisenberg model is given in Fig. 4; in this case it can be seen that the Heisenberg model seems to model correctly the ab initio results. A further confirmation is found in the fact that the g factor is obtained with a value lower than 10 -2 confirming that the absence of Zeeman energy in the QE code results is found by the fitting procedure.

A histogram of the obtained parameters for the J  is illustrated in Fig. 6. It can be seen that those values are close to the ones of reference [START_REF] Uhl | A first-principles study of exchange integrals in magnetite[END_REF], with an alternation of ferromagnetic and anti-ferromagnetic couplings.

The volume anisotropy was found to have a value of -2 × 10 -05 a.u, coherent with the one used in [START_REF] Mazo-Zuluaga | Surface anisotropy, hysteretic, and magnetic properties of magnetite nanoparticles: A simulation study[END_REF]. The values for the local surface anisotropy constants are given in Table 1. It can be seen that the best fit corresponds to an alternation of positive and negative values for those constants. This fact raises the question of the validity of the Hamiltonian used in [START_REF] Mazo-Zuluaga | Surface anisotropy, hysteretic, and magnetic properties of magnetite nanoparticles: A simulation study[END_REF] in which a constant surface anisotropy was used.

Indeed, we found that the model does not adjust as well when we use a constant surface anisotropy. The best penalty was found at a value of 1.13 × 10 -3 Ry. This corresponds to a value of the surface anisotropy of 1.753 a.u and a volume anisotropy of 6.31 × 10 -05 a.u. The latter value is now positive, but the authors of [START_REF] Mazo-Zuluaga | Surface anisotropy, hysteretic, and magnetic properties of magnetite nanoparticles: A simulation study[END_REF] found that the K S /K V ratio is the more important parameter to predict the magnetic structure of a nanoparticle of intermediate size. Here, the value of 27780 we find for this ratio hints at a hedgehog type magnetic structure (Fig. 2). It might be that the large surface contribution of such a small system as the one we study is responsible for this fact, but at least the value we find is not a free, almost unknown parameter as in the literature. Physically, such a result would correspond to jumps during magnetization reversal, and exchange bias properties. Such results have been experimentally observed and are also reviewed in [START_REF] Mazo-Zuluaga | Surface anisotropy, hysteretic, and magnetic properties of magnetite nanoparticles: A simulation study[END_REF]. When we tried the same method with full spin-orbit coupling and relativistic pseudopotentials, the increased disorder in the magnetic moments resulted in a penalty function of 2 × 10 -3 Ry, with essentially similar results for the fitting parameters.

Iron Oxide Clusters and Dopamine

Linking to previous work [? ] on the coating of magnetic nanoparticles by various ligands in order to make them more biocompatible or to link to antibodies in order to better target tumors, we then added a dopamine molecule next to the cluster from the previous section and, after optimizing the atomic positions, tried the same fitting procedure with a Heisenberg model. It turned out that we could not achieve a fit with a penalty function better than 5 × 10 -3 Ry, which more or less corresponds to the distribution of energies in the ab initio results, and unrealistic coupling constants as well as a strongly unstable distribution of surface anisotropy constants. We concluded that a Heisenberg model might be too simple to describe such a system, where electrons donated by the dopamine molecule can lead to some itinerant magnetism, or at least to some symmetry breaking. The accuracy of ab initio results could also maybe be improved to the expense of computational time, but since the fitting procedure works in isolated cases the origin of the lack of fit seems to come more from the model than from the quality of the results on which it is fitted.

In order to address the question of the symmetry breaking, we added 6 dopamine molecules (Fig. 1) symmetrically distributed around the cluster. The simulation time was found to be too large to compute as many constrained points as in the previous section, but, by plotting the magnetization as a function of the magnetic field, as can be seen in Fig. 7, we could check that the response of the system seems to be unchanged from this functionalization.

Iron Oxide Clusters and Gold Cluster

We also tried to introduce in the system a small gold cluster (which could act as a nano-antenna in plasmonics) which was found after structural optimization to adhere to the iron oxide cluster. In this case, the Heisenberg model did not fit very well either the computed ab initio values of energies. However, as can be seen in Fig. 7, although the absolute values of external magnetic field to achieve the same total magnetization along the x axis are strongly different from the previous cases (with or without dopamine), the slope of the curve is similar.

Magnetite surface and magnetite coated with cobalt oxide

In the case of the magnetite surface, generating 124 samples for 6 parameters, we found the best fit, with a correlation coefficient of 0.87 in between the ab initio results and the fit for a surface anisotropy of 784.00 meV, a volume anisotropy of 6.96x10 -3 meV, a Land g factor close to zero (which we see as a proof of correctness of our model), and coupling constants used to generate the J  of 1.102 meV and 2.076 meV respectively. The results of the fit are shown on figure 5 and, in this case also, we conclude that the Heisenberg model corrected with a surface anisotropy term gives a good description of the magnetism of the system. The values we find for volume anisotropy is in good agreement with the literature, and the ratio of surface to volume anisotropy is in the range expected by experimentalists.

In the case of the magnetite surface coated by iron oxide, in which we could only generate 55 samples due to the much higher numerical cost, we found that the model was much harder to fit (with a correlation coefficient of only .54 in between the ab initio values and the fitted values). The best fit was obtained with a surface anisotropy of 546.77 meV, volume anisotropy of 2.63x10 -3 meV, and coupling constants of 0.98 meV and 2.11 meV in the case of iron-iron, 0.77 meV and 2.09 meV in the case of iron-cobalt, and 0.97 meV and 2.37 meV in the case of cobalt-cobalt pairs. In this case, either a better ab initio calculation would have been needed (using for instance the LDA+U method in order to better describe the insulating behavior of cobalt oxide), or the magnetic model has to be improved, including for instance

Mixed iron oxide and cobalt oxide clusters

We then addressed the case of clusters of iron, cobalt and oxygen atoms of increasing size, which we submitted to a global magnetic constraint, namely a global angle of respectively 5,30,45 and 75 degrees with respect to the z axis, cases to which we will refer as numbers 0 to 3 later. This constraint allows to get a value of the magnetic field from the ab initio calculations.

In order to estimate the change in the anisotropy term from the cobalt/iron mixing, after various functional tries, we found that the best fit was obtained when we allowed a change in the volume and surface anisotropy term depending on the number of iron and cobalt atoms present in the cluster. The additional term was multiplicated by adjustable parameters a i (where i represents iron and cobalt atoms), so if there are N different species in the system with n i atoms of each kind per cell for the volume term or per surface site in the case of surfaces, the volume and surface anisotropies read

K S,V,i = K 0 S,V a n i i (6)
so that the final Hamiltonian we used reads

H = -2  J  S i • S j (7)
- i K V,i S 2 x,i S 2 y,i + S 2 y,i S 2 z,i + S 2 x,i S 2 z,i - k K S,k S k • e k 2 -gµ B H • i S i
The advantage of the modification of the volume and surface anisotropies we used in equation 6 is that such a term allows simply to take into account the changes in anisotropy due to a varying ratio of cobalt to iron atoms.

We tried to include a factor in the Hamiltonian in order to modelize the effect of the various oxygen stoechiometries of the clusters on the energy but for the functional form we tried (multiplying each anisotropy term by the ratio of magnetic to oxygen atoms) we found that the corresponding parameter was converging to zero.

The values of the parameters converged to values presented in table 2 As can be seen on figure 8 the fitting procedure of the Hamiltonian converged to a satisfying level, with a correlation coefficient close to 1 in between ab initio values of the magnetic energy and energies given by the model. We then dropped from the fitting procedure several configurations chosen at random, in order to check the predictive capabilities of the model (points labeled with an impulse), which can be seen as satisfying, despite obviously less good than the ones given by the model when reusing the input configurations used for fitting as in the previous case. The model can nevertheless be seen as adequate, since it uses only 12 fit parameters for 30 couples of input values, and has some predictive power.

From the physical results one can see that the values of the exchange constants kept the same order of magnitude as in the previous case of pure surfaces, although the values changed, with an interesting information for Fe/Co pairs. The same goes for the surface and volume anisotropies, with an enhancement in the case of iron oxide in comparison to cobalt oxide, the mixed situations being intermediate. One should note than in the total energies, the main contribution comes from the surface term, followed respectively by the volume and exchange contributions. It is then no wonder that the latter term is particularly difficult to adjust.

A problem remaining to be solved is that for the time being, we do not have at our disposal a clear physical way to predict the magnetic moments emerging from a given configuration of atoms. A machine learning approach based on physically relevant data such as relative angles and distances can be envisaged ; nevertheless the present study gives surface anisotropy values from an adjustment over ab initio results, and if values of magnetic moments are set to physically reasonable values as found in the bulk, can give a phenomenological description of magnetic nanoparticles of large size if the geometric configurations of atoms do not diverge too much from the bulk.

Conclusions

In this work, we adjusted a classical Heisenberg model of magnetism using the Metropolis simulated annealing method including surface anisotropy effects on magnetically constrained, non collinear ab initio results obtained a small iron oxide cluster functionalized or not by one or several dopamine molecules or a nearby gold cluster. We conclude that the Heisenberg model seems to apply well to the simpler systems (namely, a free iron oxide cluster or a magnetite surface), allowing us to give some absolute values of the surface anisotropy constant, although a locally varying surface anisotropy alternating positive and negative values seem to provide a better description.

This could allow us to describe the magnetic behavior of a nanoparticle of size 1 to 10 nm, which ab initio calculations can hardly tackle for the time being because of computing power limitations, hoping that the large surface proportion of iron atoms in the small cluster we have studied does not influence the results.

In the case of functionalized cluster by one or several molecules of dopamine, or by a nearby gold cluster, the Heisenberg picture does not apply as well as for the simpler system, but we could nevertheless observe that the linear relation in between magnetic field and magnetization was unchanged in all those cases even if absolute values changed.

In the case of clusters of varying sizes and compositions, with periodic boundaries conditions, we were able to give a model describing hybrid ferromagnetic and antiferromagnetic systems, describing the enhancement in surface anisotropy due to the juxtaposition and mixing of the systems and giving estimations of the corresponding values.

Assuming fixed magnitudes for the magnetic moments of the atoms, the hamiltonian we present in the present publication can be used to describe the magnetic order in similar systems including transition metal atoms , as we checked on examples non included in the fitting procedure.

Figure 1 :

 1 Figure 1: Fe 13 O 8 cluster with six dopamine molecules.

Figure 2 :

 2 Figure 2: Result of a typical non-collinear constrained calculation of the iron oxide cluster ; here we imposed magnetic moments of 5,1 and 45 µ B on each axis.

Figure 3 :

 3 Figure 3: Atomic positions in a magnetite slab coated with cobalt oxide

Figure 4 :

 4 Figure 4: Results of the ab initio calculations fitted using the Monte-Carlo Metropolis (without spin-orbit)in the case of the isolated Fe 13 O 8 cluster.

Figure 5 :

 5 Figure 5: Results of the ab initio calculations fitted using the Monte-Carlo Metropolis in the case of the magnetite surface

Figure 6 :

 6 Figure 6: Histogram of exchange constants found by the fitting procedure on the iron oxide cluster.

Figure 7 :

 7 Figure 7: Magnetic moments vs. magnetic field extracted from the constrained calculation for the studied clusters

Figure 8 :

 8 Figure 8: Results of fitting the Heisenberg model on mixed iron oxide and cobalt clusters under varying magnetic constraints

Table 1 :

 1 Local surface anisotropy constants

	Fe atom Surface anisotropy in a.u.
	1	0.5592594×10 -01
	2	-0.5450376×10 -01
	3	0.4755301×10 -01
	4	-0.2952414×10 -01
	5	0.4511738×10 -01
	6	0.5525300×10 -01
	7	-0.4852317×10 -01
	8	-0.2241407×10 -01
	9	0.8430323×10 -01
	10	-0.7067873×10 -02
	11	-0.7141519×10 -01
	12	-0.7437612×10 -01
	13	-0.8089262×10 -01

Table 2 :

 2 Parameters obtained for mixed iron oxide / cobalt oxide clusters

Acknowledgments

We thank S.Ammar and I.Labaye for fruitful discussions. K.B. was supported by a grant from the French Education and Research Minister. We thank GENCI/IDRIS and CRIHAN for computational time (projects x2014096171 and 007 respectively)

[1] R. H. Kodama, Magnetic nanoparticles, Journal of Magnetism and Magnetic Materials 200 (13) (1999) 359 -372.


Florent Calvayrac
email: florent.calvayrac@univ-lemans.fr

Kullback-Leibler divergence as an estimate of reproducibility of numerical results

Keywords: Kullback-Leibler, numerical, reproducibility, software forge

In large software projects using numerical solutions of equations, small changes in compiler options or parallelization methods can induce slight variations in the last digits of floating point numerical results, due for instance to the noncommutativity of operations. Unfortunate changes in the code (bugs) can induce even larger deviations in the results. We propose to use the Kullback-Leibler divergence estimated from the compression ratio of the results compared to reference results in order to automatically quantify those changes and automatize regression tests of numerical codes in software forges. We use the TDDFT PW-TELEMAN project as an example.

I. INTRODUCTION

It is well known that a priori semantically identical changes in a program computing a numerical solution to a mathematical problem, for a given numerical precision, can nevertheless influence numerical results, not speaking of programming errors. One thinks for instance of the compiling options used, the compiler suite chosen, the number of processors used in a parallel implementation, or underlying hardware or operating system, Indeed, numerical operations in a computer are not associative, therefore the final result might depend on the order of execution of these operations, which is unpredictible on a parallel computer or a GPU. Besides, IEEE standards for the floating point representation of real numbers and operations are not complely respected when high-performance compiler options are chosen, such as when the floating point division is replaced by an inverse multiplications, or power functions or square roots are approximated. Some processors such as the Intel family do not respect the IEEE standard by default (they actually exceed it) unless explicitely specified in the compiler options, with a performance penalty.

Therefore, a comparison of output files of numerical programs on various installations will always lead to some discrepancies, especially in the last digits of the results, even if the code and the inputs are unchanged. Regression tests after a code change for performance reasons, or an addition of features, based on a straightforward comparison of results will always detect bugs, when a manual inspection will show that bugs have not been introduced in the sources.

Except for manual inspection and plotting of results, it is hard to quantify if the results of a numerical code have changed in a significant way when some change has been applied in the process leading to the production of those results. This is especially true for codes using iterations in imaginary time to find stationary solutions of partial differential equations. In this case, the number of iterations to achieve a given numerical precision can vary when some parameter is changed, but nevertheless the end result is essentially the same if no mistake has been made in the coding process, up to a few digits in the last part of the numerical representation. Only a specific extraction of the results of the last iteration, along with a problem-specific distance criterion in between the reference result and the instance being tested, can give a satisfying automatic measure of the quality of the numerical result. Nevertheless, such an extraction is not automatic in the sense that it is problem-dependent.

Besides, in some application domains of numerical computations such as the aeronautical world, safety rules ask that the results of a specific code, for instance in finite elements calculations, can be reproduced bit for bit. This can be tricky considering that even in a space of a few years many unsuspected parameters can change in a computer system ; some authors have even proposed that virtual machines should be provided with a full environment and run on emulators in order to ensure full reproductibility of numerical experiments. Some vendors such as Intel now provide libraries that guarantee numerical reproducibility of results, in exchange for a performance penalty.

In this paper we discuss a method to automatically estimate the changes in the output of a numerical code, without having to define problem-dependent metrics, or having to dig too deep in the hardware or software used, for instance by changing the rules of floating point operations in order to guarantee a nonchanging result, as some authors propose, or by doing statistical analysis of results, which can be problem-dependent and time consuming.

II. METHOD

We implemented a simple and universal method which, given two sets P and Q of results of a numerical code, estimates the socalled Kullback-Leibler [START_REF] Kullback | On Information and Sufficiency[END_REF] divergence D from cross-entropies D = H(P, Q) -H(P ). This formula is not symmetrical under exchange of P and Q, but could be easily symmetrized, without bringing much more information. If the two sets of results are identical, D will be zero.

present approximate implementation for this estimation consists in the computation of

D e = S(C(P + P )) -S(C(P + Q)) S(C(P + P ))
where S is the file size obtained by using the C compression program. Here + is the file concatenation operator andis the traditional substraction. For several wellknown compression programs C such as bzip2, gzip,compress, or zip, the information entropy in the compressed file is indeed strongly increased, and the reduction in file size after compression roughly reflects the information entropy of the original file. We think that using a compression program helps to create an algorithmic expression of the contents of the output of the numerical code, even if the output, as a text file, contains varying information when for instance the compiler options are changed, the compiler changes, or the number of iterations changes, since the results are essentially similar from one iteration to the next and compress very efficiently with modern file compression programs. Many programs used in physics or chemistry indeed only output one text file from which various numerical results have to be extracted after the run, depending on the need, and those results are mixed with various informations such as iteration number, convergence criteria, parallelization choices, etc, which are fairly repetitive and thus are well suited to nondestructive file compression programs. The same goes for numerical results, in which we found a quite high potential for compression, since they do not change significally from one small step iteration to the next, and exhibit many similarities in a given file even if the number of iterations to achieve convergence differ.

III. RESULTS AND DISCUSSION

First, we will discuss some of the numerical optimizations which we noticed induced slight changes in numerical results of a typical large-scale numerical project, namely the socalled PW-TELEMAN effort on which we have been working for a number of years in an international collaboration [START_REF] Calvayrac | Nonlinear electron dynamics in metal clusters[END_REF], [START_REF] Wang | Collision dynamics of proton with formaldehyde: Fragmentation and ionization[END_REF]. This code has been recently open-sourced in a software forge on http://www.pw-teleman.org where the present implementation is also avalaible in the form of scripts. This code solves the time-dependent functional theory equations in real time on a regular numerical grid, within the local approximation in time and space, which generates a set of non-linearily coupled one-particle Schrödinger equations, to be first solved to a stationary solution, which is then submitted to a time-dependent perturbation, allowing to compute a wealth of observables such as electronic emission currents.

A. Examples of optimizations changing numerical results

The first step consisted in optimizing the FFT routines which are used to compute the kinetic energy of the electrons by going to reciprocal space and also to solve Poisson's equation. Historically the NETLIB library was used [START_REF] Swarztrauber | Vectorizing the FFTs[END_REF]. In 2012, a move to the FFTW3 library [5], optionally coupled to the Intel MKL library, has been successfully implemented, and has allowed a typical speedup of 50 %. A second effort, done in parallel to the previous step, was devoted to a further parallelization of the code. A MPI parallelization already existed since more than 15 years, with a distribution of propagation of the wave functions over the processors and a reduction of the density in order to compute the coulombic potential as well as the exchange-correlation potential by sharing the grid across the processors. An implementation in OpenMP in the same spirit has been also achieved in 2012 by P.-G.Reinhard. However, the migration from FFTW3 to FFTW3-OMP does not bring a significant speedup of the calculations. Another implementation, using MPI instead, has been done on the grid points, on top of that done on the wave functions. Indeed, the size of the computation box is one of the major bottlenecks of our code. One can occasionally uses about 250 000 grid points, but the standard is more close to a million, and sometimes even more than a million and a half. Meanwhile, the CPU time almost linearly scales with the number of points. This is why in 2013, a MPI parallelization on the grid points has been implemented. The speedup compared with a serial calculation, using FFTW3, ranges from 2 (for 24 wave functions, box size of 963, and 16 processors) to 15 (for 240 wave functions, box size of 963 and 48 processors). There is of course a compromise between the number of processors, the box size and the number of wave functions to be found. Indeed, increasing the number of processors does not systematically enhance the speedup because one then increases the number of communications between the processors.

During the past two years, the use of GPU cores has also been tested, first by using readymade versions of FFTW and linear algebra routines (CULA). With one GPU core per CPU core, the typical gain is about 1.5 compared to the FFTW3 version. We then started to reprogram in the CUDA language the calculation of the exchange-correlation potential and of the multipoles used in the FALR method to compute the coulombic potential, as well as various field transform operations. On a mixture of cases, the typical speedup went to a factor of 3, using either a combination of cheap CPUs and GPUs (GTX280) or top of the range ones such as K20. A further speedup to a factor of 4-5 was achieved using asynchronous memory transfers in order to increase the utilization of GPUs, the constant memory transfers bringing a huge performance penalty, and finding the optimal register size and block sizes for each GPU (typically, 72 registers on a K20, block sizes of (64,1,1) for 3D grid operations and (512,1,1) for FFTs).

When using a MPI parallelization on the wave functions on top of the use of GPUs (one GPU per core), a speedup by a factor 3 vs All possible compilations options and available compilers (here, the latest versions of GFORTRAN and Intel Fortran) are combined with a shell script, generating a set of executable files, which are then run on the specified examples, with a varying number of processors and

B. Discussion of the results

We present on figure 1 the results of a typical analysis of output files with the presently discussed method, for the same input file, namely a carbon dimer represented on a 128 3 cubic grid of points separated by 0.4 a.u, inducing first the computation of the ground state, then 1000 time steps. The corresponding input file is given as C2-128 in the samples of the distribution of PWTELEMAN. All the combinations of options and a ranging on the number of processors used gave rise to more than 1000 results ; here we give the results for the lowest values of the divergence, close to average values (5 percent), and the higher values obtained. We used the latest version of bzip2 as a compression program.

The results indeed show that changes in compiler options or in the number of processors will result in a value of D e of a few percent, reflecting the slight changes in the numerical results, due to the noncommutativity of numerical operations, which hopefully are limited to the last digits. Various messages in the output of the programs also change, explaining the small change in the results. If the compiler is changed, going from the latest GFORTRAN implementation to the commercial Intel suite of compilers, or the parallelization methods (MPI/OpenMP) or the way the FFT is computed (NETLIB/FFTW3 library / CUDA) those results change to about ten percent. Reassuringly, the use of GPU leads to a value of D e of less than one percent on monoprocessor systems. Programming bugs that either lead to no executable file, to a program failing to reach completion, or to a notable change in results will be immediately detected by a value of D far exceeding this ten percent value. This allows for an automatic quantification of regression in the code when a change is made in the sources, discarding small numerical artefacts related to performance increases or variations in systems architecture, be it hardware or software relard. A threshold has to be defined first by manual inspection of the results, but we are confident that a fifteen percent rule is reasonable, which we could try by voluntarily or not introducing bugs in the code base, which immediatly result in a divergence of fifty percent or more of the results.

IV. CONCLUSION

In the present paper, we have demonstrated in the case of a typical numerical code that an approximate method to compute the Kullback-Leibler divergence of results after changes are made in the production process of those results can help automatically estimate the reproducibility of numerical results, even if the compiler, the hardware, the parallelization method is changed. This way, bugs can be detected without having to define problem-dependent, specific metrics for the changes in numerical results, or to spend too much effort in arcane hardware, numerical, or software problems in order to ensure bit for bit reproducibility.

Fig. 1 .

 1 Fig. 1. Typical results on the PWTELEMAN library for various options

ACKNOWLEDGMENT

The author would like to thank GENCI and CRIHAN for computational time (projects projects x2014096171 and 007 respectively), and all the indirect contributors to the present paper, especially K.Brymora, D.Brusson, P.-G.Reinhard, E.Suraud, M.Dinh, as well as many other contributors to the PWTELEMAN library.


Sandra Devin

Camille Vrignaud

Kathleen Belhassein

Aurélie Clodic

Ophélie Carreras

Rachid Alami

Evaluating the Pertinence of Robot Decisions in a Human-Robot Joint Action Context: The PeRDITA Questionnaire

The domain of human-robot Joint Action is a growing field where roboticists, psychologists and philosophers start to collaborate in order to devise robot abilities that are as efficient and convenient for the human partner as possible. Besides studying Joint Action and developing algorithms and schemes to control the robot and manage the interaction, one of the current challenges is to come up with a method to properly evaluate the progresses made by the community. Several questionnaires have already been proposed to the community that deal with the evaluation of humanrobot interaction. However, these studies mainly concern either specific basic behaviors during Joint Action or human-robot interactions without effective physical Joint Action. When it comes to high level decisions during physical human-robot Joint Action, there are fewer contributions to the topic, and also, the methods to evaluate them are even rarer. The aim of this paper is to propose a reusable questionnaire PeRDITA (Pertinence of Robot Decisions In joinT Action) allowing us to evaluate the pertinence of high level decision abilities of a robot during physical Joint Action with a human.

I. INTRODUCTION

In industrial as well as in human assistance contexts, the robot of the future needs to be able to perform Joint Action with humans. Joint Action is defined as "any form of social interaction whereby two or more individuals coordinate their actions in space and time to bring about a change in the environment" [START_REF] Sebanz | Joint action: bodies and minds moving together[END_REF]. Our study concerns the more precise framework of two "partners" face to face, a robot and a human, sharing the common goal of a collaborative task achievement.

This domain of human-robot Joint Action is a growing field where roboticists, psychologists and philosophers start to collaborate in order to devise robot abilities that are as efficient and convenient for the human partner as possible. Besides developing algorithms and schemes to control the robot and manage its interactions, one of the current challenges is to come up with a method to properly evaluate the progresses made by the community. Since measurement of physiological data on the user could be hard to implement, a usual way to do it is through user studies where naive users answer a questionnaire about their experience of the interaction. Several questionnaires have already been proposed to the community [START_REF] Bartneck | Measurement instruments for the anthropomorphism, animacy, likeability, perceived intelligence, and perceived safety of robots[END_REF], [START_REF] Heerink | Measuring acceptance of an assistive social robot: a suggested toolkit[END_REF] and other studies used "homemade" questionnaires [START_REF] Heerink | Relating conversational expressiveness to social presence and acceptance of an assistive social robot[END_REF], [START_REF] Fischer | Between legibility and contact: The role of gaze in robot approach[END_REF]. However, our research interest is based on physical interactions that are involved in humanrobot collaborative task achievement, where the human and the robot share the task and the space. The questionnaires mentioned above mainly concern rather specific and basic behaviors or tasks without concrete interaction. When it comes to the evaluation of high level decision making processes concerning a concrete physical human-robot interaction, the methods to evaluate these contributions are even rarer.

Concerning this decisional level, the ideal teammate/assistant robot should not only make, whenever necessary, the pertinent decisions with minimum intrusion and maximum latitude offered to its human partner but should also constantly exhibit that it has a good estimate of its human partner mental state with respect to the shared task.

We are contributing to the field, and we develop incrementally architectures [START_REF] Lemaignan | Artificial cognition for social human-robot interaction: An implementation[END_REF] and schemes [START_REF] Devin | An implemented theory of mind to improve human-robot shared plans execution[END_REF], [START_REF] Devin | About decisions during humanrobot shared plan achievement: Who should act and how?[END_REF] for humanrobot collaborative task achievement, but we did not find a questionnaire to evaluate the robot high level decision making processes in this context.

Our goal here is to propose and incrementally refine a questionnaire PeRDITA (Pertinence of Robot Decisions In joinT Action) that is not specific to a robot or a task. A pilot study has been done using a preliminary version of this questionnaire, where subjects had to interact with a PR2 robot to achieve a joint task.

In the sequel, we will first present previous work concerning the user experience and other questionnaires used to evaluate human-robot interaction in Sec. II. Then we will present the questionnaire, how we constructed it in Sec. III and how we evaluated it through a user study presented in Sec. IV. The results are presented in Sec. V. Finally, in Sec. VI, we will discuss the re-usability of the questionnaire and the difficulties to evaluate human-robot Joint Action.

II. BACKGROUND A. Existing questionnaires

Several questionnaires have already been developed to evaluate human-robot interaction. [START_REF] Hoffman | Evaluating fluency in human-robot collaboration[END_REF] allows us to evaluate several aspects such as the "trust" in the robot or the "fluency" of the interaction. It has been used in several studies such as [START_REF] Gombolay | Decision-making authority, team efficiency and human worker satisfaction in mixed human-robot teams[END_REF] or [START_REF] Dragan | Effects of robot motion on human-robot collaboration[END_REF]. The Godspeed questionnaire series [START_REF] Bartneck | Measurement instruments for the anthropomorphism, animacy, likeability, perceived intelligence, and perceived safety of robots[END_REF] enable us to measure the perception of the robot by the human with questions relative to anthropomorphism or perceived intelligence. However, these questionnaires are focused on the evaluation of the perception of the robot and does not deal with the evaluation of the interaction and the usability of the system. The SUS (System Usability Scale) questionnaire [START_REF] Brooke | System usability scale (sus): a quick-and-dirty method of system evaluation user information[END_REF] measures the interaction of a user with an electronic system with 10 claims that subjects need to evaluate using Lickert scale. On the contrary of the Godspeed series, the SUS questionnaire evaluates the usability of the system but lacks of measure concerning the perception of the robot or the interaction. [START_REF] Heerink | Measuring acceptance of an assistive social robot: a suggested toolkit[END_REF] presents a toolkit to measure acceptance for assistive social robots. This toolkit is based on the UTAUT (Unified Theory of Acceptance and Use of Technology) questionnaire [START_REF] Venkatesh | User acceptance of information technology: Toward a unified view[END_REF]. It has been well conceived in order to evaluate the perception and usability of the robot and more particularly for social robots. However, the questionnaire is more oriented toward the perception of the robot than the interaction and the collaboration. The Robotic Social Attributes Scale [START_REF] Carpinella | The robotic social attributes scale (RoSAS)[END_REF], which was inspired by the Godspeed questionnaire and the literature from social psychology, measures the judgment and the perception of people in relation to the social attributes of the robot, like warmth or competence. Finally, several studies as [START_REF] Heerink | Relating conversational expressiveness to social presence and acceptance of an assistive social robot[END_REF], [START_REF] Fischer | Between legibility and contact: The role of gaze in robot approach[END_REF] use "homemade" questionnaire conceived for their experiment and not always easily reusable.

B. Need for a new questionnaire

We saw in the previous subsection that several questionnaires already exist to evaluate human-robot interaction. However, they mainly focus either on some specific basic behaviors or on evaluating human-robot interaction without concrete physical interaction. Even if these questionnaires are interesting in their respective fields of application, when it comes to evaluate high level decisions of a robot, few works have been done on the subject.

The decisions that we consider here correspond to physical (e.g. pick&place) and verbal actions that are involved when a human and a robot have to satisfy a joint goal: what action to perform, who will do and when. Indeed, when the task is a little complex, when various ways exist to achieve a same goal or when the spatial resource itself is shared by the human and the robot, it is important for the human to have a robot partner:

• that tracks and is permanently aware of the current state of the task • that can comply with the human decisions • that makes explicit its internal state • that does all this with minimal intrusive behavior The objective of researchers who contribute to the development of the robot high level decision abilities for Humanrobot joint action is to come up with a robot that is able make the right decisions at the right time.

Our aim here is to provide a questionnaire partially inspired by UX models, which allows us to evaluate the progresses made by the community concerning the pertinence of high level decision abilities of the robot during human-robot Joint Action. [START_REF] Mahlke | User experience of interaction with technical systems[END_REF]. The user experience is based on the humantechnology interaction and is composed of three parts: the perception of instrumental qualities, the perception of non-instrumental qualities and the emotional user reaction.

C. The UX Model

The acceptability of complex technological systems as computer or robots is studied by researchers in social sciences. To do so they define what they call the User eXperience [START_REF] Hassenzahl | User experience-a research agenda[END_REF] as:

"a consequence of a user's internal state (predispositions, expectations, needs, motivation, mood, etc.), the characteristics of the designed system (e.g. complexity, purpose, usability, functionality, etc.) and the context (or the environment) within which the interaction occurs (e.g. organisational/social setting, meaningfulness of the activity, voluntariness of use, etc.)." This definition has been designed as a model by [START_REF] Mahlke | User experience of interaction with technical systems[END_REF] as presented in Fig. 1. This model is based on the humantechnology interaction and is composed of three parts: the perception of instrumental qualities, the emotional user reaction and the perception of non-instrumental qualities.

a) Perception of instrumental qualities: The instrumental qualities of a technology, or also called pragmatic attributes, are strongly linked to the acceptability of the technology defined in [START_REF] Dillon | User acceptance of information technology[END_REF] as:

"the demonstrable willingness within a user group to employ information technology for the task it is designed to support." b) Perception of non-instrumental qualities: In the definition of [START_REF] Hassenzahl | The thing and i: understanding the relationship between user and product[END_REF], the non-instrumental qualities of a technology, or also called hedonic attributes, depend of the user and refer to the pleasure obtained by the use of the technology. It includes several notions such as the stimulation procured during the interaction, identification mechanisms and representations. This aspect is evaluated through the perception of the technology by the user. In the human-robot interaction context, the criteria to take into account are the aesthetics of the robot, symbolic aspects and motivational aspects during the interaction. c) Emotional user reaction: The emotions of the user after the interaction with the technology will impact the final use of the technology. A positive emotion will support a future use while a negative emotion can lead to a reject of the technology.

III. CREATION OF THE QUESTIONNAIRE

A. The dimensions

In order to evaluate the user experience concerning the robot and the interaction, we have chosen to build a questionnaire where subjects have to place themselves in a selfassessment scale. This kind of questionnaire is often used in HRI because it provides quantitative measures on which it is possible to make statistical analysis. We organized the questionnaire on several dimensions, each one measuring a specific aspect of the interaction.

• Evaluation of perception of the instrumental qualities:

The Interaction dimension, based on the French version of the AttrakDiff questionnaire [START_REF] Lallemand | Création et validation dune version franc ¸aise du questionnaire attrakdiff pour lévaluation de lexpérience utilisateur des systèmes interactifs[END_REF] allows us to evaluate the behavioral intention of use. • Evaluation of the perception of non-instrumental qualities: we based this part on the Godspeed questionnaire series [START_REF] Bartneck | Measurement instruments for the anthropomorphism, animacy, likeability, perceived intelligence, and perceived safety of robots[END_REF]. It allows us to evaluate how the human perceived the robot in general. The associated dimension of the questionnaire will be called the Robot perception dimension afterward. In addition to these dimensions, we have added three other dimensions that are more specific to the context of high level decision and physical human-robot Joint Action:

• Verbal: this dimension allows us to evaluate how the human perceived the verbal interaction with the robot (did the robot verbalized the good information at the right time). • Acting: this dimension enables us to evaluate how the human perceived the decisions of the robot concerning its actions (did the robot choose to perform the right actions at the right time). • Collaboration: this dimension measures how the subject perceived the collaboration with the robot in terms of acceptability, usability, or security. As our research interests are more about the pertinence of high level decision abilities of the robot in a context of Joint Action, we focused on the interaction parameters and not on the emotional user reaction. However, in order to better fit with the UX models, it might be interesting to add in this questionnaire this specific dimension, which is possible by using existing scales already validated such as the Self-Assessment Manikin [START_REF] Bradley | Measuring emotion: The selfassessment manikin and the semantic differential[END_REF] or the AffectButton [START_REF] Broekens | Affectbutton: A method for reliable and valid affective self-report[END_REF].

B. The questions

Several antonym items are used by dimension (between 3 and 8). Subjects have to answer a question by placing Fig. 2. A joint activity between a PR2 robot and a subject. They have to achieve together an "Inventory" task where they have to scan several colored cubes and store them into a box of the same color. Only the robot can perform the scanning. In the situation depicted by the photo, the robot is scanning a cube before storing it.

themselves in a scale of 100 between these antonym items (semantic differential). For a better understanding, the English version of the questions and items can be found in Tab. I.

IV. THE PILOT STUDY

The questionnaire presented in this paper has been tested in the context of a pilot study where subjects had to perform a joint task with a PR2 robot.

A. Aim of the study

The aim of the study used to test the questionnaire was to compare two versions of a system allowing to execute Shared Plans in the context of human-robot collaboration. The first version was the initial system presented in [START_REF] Fiore | On planning and task achievement modalities for human-robot collaboration[END_REF] and the second version was the same system with two major improvements concerning Shared Plan elaboration and execution. The first improvement consists in taking into account an estimation (by the robot) of the mental state of its human partner during Shared Plan execution in order to better communicate concerning potential divergent beliefs [START_REF] Devin | An implemented theory of mind to improve human-robot shared plans execution[END_REF]. The second improvement was to consider more flexible Shared Plans where the robot does not decide everything in advance but defers some decisions concerning actions allocation and instantiation to the last moment in order to provide more latitude to the human [START_REF] Devin | About decisions during humanrobot shared plan achievement: Who should act and how?[END_REF]. The pilot study also had the purpose to compare two different modes of last moment decision implemented in the new system. The modes and the two systems will be presented in more detail below.

B. The task

During the study, the subject and the robot have achieved together an "Inventory" task where they had to scan several colored cubes and stored them into a box of the same color. At the beginning of the interaction, both agents had a stack of colored cubes they could access (and only them could access). The cubes were blue, green or red. The stack of the human was located in another room, in a way that, to get a new cube, the human had to leave the sight of view of the robot (see Fig. 4). For a cube to be scanned, the agents needed to put it on one out of the two possible areas on the table in front of the robot (see Fig. 4). Once a cube was placed on a scanning area, the robot could scan it by orienting its head and turning on a red light in the direction of the object (see Fig. 2). If the robot scanned an object while the human was not looking at the scene (e.g. when he was in another room to pick a new cube), the human was not aware that the object had been scanned unless the robot told him. Once the cube was scanned, it could be stored in a box of the same color (e.g. the blue cubes in a blue box). The robot had access to a blue box, the human to a green box, and both had access to a red box. Consequently, only the robot could store the blue cubes, only the human could store the green cubes and both could store the red cubes. The two boxes of the human were located in another room (see Fig. 4).

This task required the execution of a Shared Plan between the human and the robot and contained three main interesting points concerning the execution of this Shared Plan. First, since the cubes and the boxes of the human were located in another room, the action of the robot to scan a cube could be missed by the human and it could result a lack of knowledge that would need to be solved. Second, the presence of the red cubes that could be stored in a box by both agents required a decision: who would store a given red cube? Finally, the presence of two scan areas which were identical and could be used in the same way required the agent to take decision concerning which area to use. However, after few pretests, we noticed several possible problems that we fixed by introducing small adaptations to the task:

• Introduction of a red video tape box: In certain cases, the configuration coupled to the decision of the subjects led to not having any decision in the task concerning the red cubes. Indeed, there were cases where, each time there was a red cube to store, one of the two agents were busy (either the human was in another room to pick or store an object or the robot was performing another action). To ensure that, at each interaction, there was at least one decision to take between the human and the robot, we added to the objects to scan and store a red video tape box. The human and the robot both had a red video tape box in the same placement as their stacks of cubes. At the end of the task, when all the cubes were scanned and stored (and so both agents were available), only one of the two video tape boxes (the one of the human or the one of the robot) needed to be put on a scan area. Then, similarly to what happens with the cubes, the robot scanned the video tape box. Finally, since the video tape box was red, it needed to be stored in a red box either by the human or the robot. • Distraction task: We noticed that some subjects tried not to miss any action of the robot (they stayed in front of the robot each time there was a cube to scan and they hurried in the places where they could not see the robot). Consequently, there was not missing knowledge during the task for these subjects. To ensure that all subjects missed some actions of the robot, at one predefined point of the task, the experimenter asked the subject to leave the task for a while to perform another task consisting in building with Lego bricks a construction shown in a picture. Once the construction was achieved, the subject was free to go back to the main task.

C. The Conditions

As said before, the aim of this study was to compare two versions of a system allowing us to execute Shared Plans during human-robot Joint Action:

• The Reference System (RS): In this system, the robot computes a Shared Plan where all decisions are taken in advance (who should do which action with which specific object). It means that for the task previously described, the robot will decide at the beginning of the interaction who will store which red objects and which scan area to use for each cube. However, if, at one point, the human does not act as planned, the robot will be able to compute a new plan and adapt. Moreover, in this system the robot does not compute the mental state of its partner concerning the Shared Plan. Consequently, in this task, the robot will not be able to identify whenever the human is not aware of a scan action and that this knowledge is important for the rest of the task. For this system, we choose to use two different modes corresponding to the different solution that this system allows concerning communication:

-SILENT: In this mode the robot does not inform the human either about the choices it made or the information the human misses. The only possible verbalizations are whenever the situation is blocked by an action the human should perform and is not performing. -VERBOSE: In this mode the robot informs the human about all decisions it made (it informs before performing each action and each time the human has an action to perform). Moreover, the robot informs the human about all action he misses.

• The New System (NS): In this system, the robot estimates the mental states of the human concerning the Shared Plan. It is able to identify whenever the human misses an information concerning an object which has been scanned and to communicate about this information. Moreover, the robot postpones some of the decisions concerning the Shared Plan to execution. For the task of the study, the robot postpones the decisions concerning who should store the red objects and which scan area to use. Then, there are two possibilities to take the decisions on who should store a red object:

-NEGOTIATION: the robot can ask to the human if he wants to store an object and then it acts according to the answer. -ADAPTATION: the robot can wait a few time to see if the human executes the action, and, if not, it executes it. An example of Shared Plans computed by both systems can be found in Fig. 3. As the Shared Plan for the task of the study is quite big, we present here a minimal example where there is only one red cube to deal with (which is initially located in the human stack).

D. Methodology

21 subjects took part in the study (8 women and 13 men). The participants ranged in age from 21 to 50 (mean = 27.38 y, sd = 7.07) and had normal or corrected-to-normal vision. They were all fluent in french. Each subject of the study had to interact with the robot to achieve the task previously described, and in the four conditions described previously. The order in which they were confronted to the different conditions was randomized. There were four different configurations for the stacks of the human and the robot. The attribution of each configuration to a condition was also randomized for each participant. At their arrival, the participants were introduced to the robot and the environment of the study by the experimenter. Then, participants were asked to read instructions explaining the task and its constraints. The experimenter checked the good understanding of the instructions and showed the placements of the different objects of the task. The participants were then asked to perform a quick familiarization task. In this task, the human and the robot had only one cube in their stacks (a blue for the human and a green for the robot). They had to put them in the scan areas, scan them and then store them in the appropriate boxes. There was no video tape box in the familiarization task. After each interaction with the robot (for each condition), the participants were asked to fill the questionnaire presented in this paper in order to evaluate their feeling concerning the robot and the interaction. In addition to this questionnaire, after each interaction with the robot (including the familiarization task), we asked participants to answer a small yes/no questionnaire. This questionnaire contains general questions about what happened during the interaction (e.g. "Do you think all the cubes have been scanned?"). The aim of this questionnaire was to remind the key points of the interaction to the subjects.

We formulate two hypotheses for this study:

• Hypothesis 1: The new system will be preferred to the old one by the users. • Hypothesis 2: For the new system, the negotiation mode will be preferred by the user to the adaptation one. Indeed, even if in simulation better results were found for the adaptation mode in [START_REF] Devin | About decisions during humanrobot shared plan achievement: Who should act and how?[END_REF], we believe that naive users will be more comfortable with a robot asking whenever there is a choice. The entire experiment lasted between 45 minutes and one hour per participant.

V. RESULTS

A. Questionnaire

We calculated Cronbach's alpha [START_REF] Cronbach | Coefficient alpha and the internal structure of tests[END_REF] for each dimension of the questionnaire. These values can be found in Tab. II. To consider that the internal consistency reliability of a dimension is correct, alpha should be of 0.7 or higher. All the different dimensions have a cronbach's alpha greater than 0.7 except the Verbal dimension. It appears that these items do not all measure the same attribute. Therefore, this dimension will have to be reconsidered on a more suitable task to (a) Shared Plan for the Reference System (RS), all actions are already allocated to an agent and all objects are chosen when the plan is built.

(b) Shared Plan for the New System (NS), some decisions are deferred to last moment in order to provide more latitude to the human: the allocation of some actions and the choice of some objects or places Fig. 3. Two different Shared Plans for the two systems compared in the study. The full shared plan can not be drawn here since it contains dozens of actions. We illustrate here a "minimal" example of a shared plans which concerns only one red cube situated in the stack close to the human. Note: An alpha of 0.7 and higher means the dimension has an acceptable internal consistency reliability. build a robust and validated questionnaire. However, some participants found that the fact that the robot was speaking a lot was comforting because they did not have to take decisions or to interpret robot actions. They also point out the fact that, even if they found it comforting the first time, they would quickly find it "annoying" if they had to interact with the robot several times in this mode. This illustrates the difficulty of correctly evaluating the pertinence of verbal interventions of a robot by naive subjects on a short-term study. We will discuss this point in Sec. VI.

B. Results of the study

All the scores obtained for the different conditions can be found in Tab. III. a) Comparison of the two systems: We first compared the Reference System (RS) to the New System (NS) in order to test the first hypothesis. We compared the two systems looking at the total score of the questionnaire and each dimension individually. We applied student T-tests when the data were normally distributed and Wilcoxon tests when the data were not normally distributed. The obtained results can be found in the first column of Tab. IV. We can see that for the total of the questionnaire and for all dimensions except the Verbal one, the new system has been evaluated significantly better than the reference system (p < 0.05). Consequently, we can consider the first hypothesis as validated. The difference was particularly visible for the Acting dimension of the questionnaire (p 0.003). It shows that the algorithms developed for the robot to be able to take the appropriate decisions at the right time during Shared Plan achievement have been appreciated by the subjects.

b) Comparison of the negotiation and adaptation modes: We then compared the negotiation and the adaptation modes of the new system in order to validate the second hypothesis. We compared the two modes looking at the total score of the questionnaire and each dimension individually. We applied student T-tests when the data were normally distributed and Wilcoxon tests when the data were not normally distributed. The obtained results can be found in the second column of Tab. IV. Here we can see that, even if the means of the negotiation mode are higher than the ones of the adaptation mode, no significant difference was found except for the Verbal dimension. Given that the internal consistency reliability of this dimension of the questionnaire is not acceptable, we cannot conclude about these particular results. Therefore, the second hypothesis is not validated: no difference has been found between the negotiation mode and the adaptation mode. Note: The RS and the NS columns are the means respectively for the Reference System and the New System. The presented results are the means for all subjects and their associated standard deviation. . The scores of the dimensions correspond to the addition of the scores of each question of the dimension (100 scale). The Total line is calculated by adding the scores of all dimensions previously harmonized in a 100 scale.

VI. DISCUSSION AND CONCLUSIONS

We have presented in this paper the PeRDITA questionnaire, specifically designed to evaluate high level decision abilities of a robot during physical Joint Action with a human. Our aim was to propose this questionnaire as a preliminary work in order to build a strong and validated scale to measure the pertinence of robotic decisions on a Joint Action context. Following the results of this first study and correlation analyzes carried out, the questionnaire was modified with, especially the Perception of the Robot dimension which was replaced by a dimension on the Perceived Competence of the Robot. Indeed, it seemed more relevant to focus on this aspect in order to more accurately evaluate the pertinence of robot decisions, considering that the perception can be measured by the Godspeed questionnaire series [START_REF] Bartneck | Measurement instruments for the anthropomorphism, animacy, likeability, perceived intelligence, and perceived safety of robots[END_REF]. This new version of the PeRDITA questionnaire will be used in future studies and it remains to keep working on and test it on other tasks and contexts.

The pilot study presented here has also given to us the opportunity to get more insights on experiments with naive subjects. Indeed, there is a real difficulty to correctly evaluate technical contributions to decision for human-robot interaction due to several reasons:

• one of the first challenge is to find a task for the human and the robot to perform together. Indeed, this task should be sufficiently interesting to permit the evaluation of the system but not too complex if we do not want the subject to focus too much on the task rather than on the robot behavior. Moreover, the task should be adapted to the current perception and manipulation abilities of robots. • another difficulty when the robot interacts with a human is to isolate the decisional aspect from the other robot abilities. Indeed, we figured out that, because they are not used to interact with robots, it is quite hard for the subjects to distinguish two different behaviors of the robot (even if the difference seems huge to roboticists) because they have too much things to observe in the robot behavior. • as the implementation of the robot decisional abilities usually relies on the other components of the robotics architecture (e.g. perception, manipulation), it is complicated and time consuming to obtain a global system sufficiently robust for a user study. • finally, as the subjects during the pilot studies are naive, we can question the obtained results. Indeed, we have experimented the fact that it is difficult, in a one session, while trying to avoid introducing heavy biases, to ask naive users to distinguish between a oneshot use of a robot and its potential daily use [START_REF] Kuhnert | The gap between human's attitude towards robots in general and human's expectation of an ideal everyday life robot[END_REF]. Then "annoying", "repetitive", "intrusive", "delayed" , "lacking of fluidity", "superfluous" behaviors will be certainly more severely evaluated. For the time being, the users are basically happy to "play" with the robot. This was perhaps enforced by the fact that in all the versions that we have proposed to the users, since the robot observes correctly the state and produces valid plans, the tasks is always finally achieved. Some UX models take into account the temporal aspect of the interaction [START_REF] Kim | How user experience changes over time: A case study of social network services[END_REF], however, it usually implies to make long term user studies which are not easily feasible in our research context. One challenge would be to find a way to evaluate this aspect without a need of a long term study, maybe by performing user studies with subjects who are not naive but more used to robotics systems.

Fig. 1 .

 1 Fig.1. UX model by[START_REF] Mahlke | User experience of interaction with technical systems[END_REF]. The user experience is based on the humantechnology interaction and is composed of three parts: the perception of instrumental qualities, the perception of non-instrumental qualities and the emotional user reaction.

Fig. 4 .

 4 Fig. 4. Set up of the task used during the user study. The human and the robot had to bring the colored cubes from their respective stacks to the scan areas. Then the robot had to scan the cubes and finally the cubes had to be stored in the box corresponding to their colors. The areas in red in the figure can be reach only by the human, the ones in green only by the robot and the one in blue by both agents.

  The first column corresponds to the comparison of the Reference System (RS) and the New System (NS). The second column corresponds to the comparison of the negotiation and the adaptation modes of the new system.

TABLE I QUESTIONS

 I OF EACH DIMENSION OF THE QUESTIONNAIRE. SUBJECTS HAVE TO PLACE THEMSELVES BETWEEN THE TWO ANTONYM ITEMS IN A SCALE OF 100.

	Dimension	Question	Item
	Collaboration	In your opinion, the collaboration with the robot to perform the task was:	Restrictive/Adaptive
			Useless/Useful
			Unsettling/Satisfactory
			Annoying/Acceptable
			Insecure/Secure
	Interaction	In your opinion, generally, the interaction was:	Negative/Positive
			Complicated/Simple
			Not practical/Practical
			Unpredictable/Predictable
			Ambiguous/Clear
	Robot perception	In your opinion, the robot is rather:	Machinelike/Humanlike
			Artificial/Living
			Inert/Animated
			Apathetic/Responsive
			Unpleasant/Pleasant
			Disagreeable/Agreeable
			Stupid/Intelligent
			Incompetent/Competent
	Verbal	In your opinion, the robot verbal interventions were:	Incomprehensible/Clear
			Insufficient/Sufficient
			Superfluous/Pertinent
	Acting	In your opinion, the robot actions were:	Inappropriate/Appropriate
			Useless/Useful
			Unpredictable/Predictable

TABLE II

 II

	CRONBACH'S ALPHA FOR THE DIFFERENT
	DIMENSIONS OF THE QUESTIONNAIRE.
	Dimension	Cronbach's alpha
	Collaboration	0.86
	Interaction	0.92
	Robot perception	0.88
	Verbal	0.6
	Acting	0.87

  Robot Perception (/800) 382.05 ± 121.71 437.38 ± 126.41 409.71 ± 125.72 472.57 ± 143.79 457.67 ± 143.05 465.12 ± 141.86

				TABLE III
			RESULTS FOR EACH DIMENSION OF THE QUESTIONNAIRE
	Dimension	SILENT	VERBOSE	RS	NEGOTIATION	ADAPTATION	NS
	Collaboration (/500)	301.29 ± 99.91	316.95 ± 101.54	309.12± 99.8	366.01 ± 98.43	357.43 ± 91.49	361.76 ± 93.96
	Interaction (/500)	344.29 ± 98.23	355.95 ± 110.7	350.12 ± 103.53	383.52 ± 87.17	378.33 ± 99.95	380.93 ± 92.67
	Verbal (/300)	218.48 ± 53.83	227.29 ± 51.72	222.88± 52.33	254.43 ± 40.64	209.05 ± 63.93	231.74 ± 57.68
	Acting (/300)	205.48 ± 61.76	200.86 ± 62.92	203.17 ± 61.62	238.33 ± 60.86	236.81 ± 41.43	237.57 ± 51.43
	Total (/500)	318.19± 77.64	331.97 ± 81.2	325.08 ± 61.9	373.25 ± 74.66	352.98 ± 73.47	363.11± 59.09

TABLE IV P

 IV -VALUES FROM STUDENT T-TESTS AND WILCOXON TESTS (W).

ACKNOWLEDGEMENTS

This research was supported by the French National Research Agency (ANR) grant reference ANR-16-CE33-0017 (JointAction4HRI).

*This work was supported by ANR project JointAction4HRI (ANR-16-CE33-0017)


Vikram Pakrashi
email: pakrashv@tcd.ie

Franck Schoefs
email: franck.schoefs@univ-nantes.fr

Jean Bernard Memet
email: jbmemet@yahoo.fr

Alan O' Connor
email: alan.oconnor@tcd.ie

Improved Image Analysis Based Corrosion Assessment using Preprocessing Techniques

Keywords: Image Processing, Probability of Detection, Probability of False Alarm, Non-Destructive Techniques

The importance of non-destructive techniques (NDT) in structural health monitoring programmes is being critically felt in the recent times. The quality of the measured data, often affected by various environmental conditions can be a guiding factor in terms usefulness and prediction efficiencies of the various detection and monitoring methods used in this regard. Often, a preprocessing of the acquired data in relation to the affecting environmental parameters can improve the information quality and lead towards a significantly more efficient and correct prediction process. The improvement can be directly related to the final decision making policy about a structure or a network of structures and is compatible with general probabilistic frameworks of such assessment and decision making programmes. This paper considers a preprocessing technique employed for an image analysis based structural health monitoring methodology to identify sub-marine pitting corrosion in the presence of variable luminosity, contrast and noise affecting the quality of images. A preprocessing of the gray-level threshold of the various images is observed to bring about a significant improvement in terms of damage detection as compared to an automatically computed gray-level threshold. The case dependent adjustments of the threshold enable to obtain the best possible information from an existing image. The corresponding improvements are observed in a qualitative manner in the present study.

Introduction

Non-Destructive-Testing (NDT) and evaluation for structures and infrastructure networks have become extremely important and topical in the recent times [START_REF] Yong | The Real Time Health Monitoring System of a Large Structure Based on Non-Destructive Testing[END_REF]. NDT based assessments guarantee no significant mechanical interference with the structural response. Information obtained from NDT can be used on a wide range of applications in the field of engineering involving (but not limited to) structural health monitoring, decision making, optimisation of repair work and 'do-nothing' options for an individual infrastructure system or a network of systems. Recent European Union (EU) projects (SAMARIS 2005, Rudlin and[START_REF] Rudlin | The ICON database. Assisting underwater inspections[END_REF] have emphasized the importance of maintenance and management of infrastructure networks on a large scale from a national and international point of view. Since NDT based assessments directly affect the decision making process of infrastructure owners, managers and engineers and significantly affect national and international trade, evaluation and improvement of the techniques are considered to be critical. A typical and important example of such assessment can be related to offshore structures, specifically a metallic pile wharf. Being in a marine environment, the piles undergo corrosion and can lead to serious structural degradation from the safety and serviceability viewpoints over the course of time. Regular monitoring and evaluation of these structures related to corrosion is extremely important. Manual inspection techniques using divers for this purpose suffer from subjectivity and lack of expertise. There are associated time and financial constraints as well. As a result, subjective techniques (potentially unmanned) which provide data for experts off site for further processing are deemed useful. Since corrosion on a steel pile is usually associated with a significant change of colour [START_REF] Tsushima | Corrosion Inspection of Steel Tube Inner Wall[END_REF] of the affected location, image processing based techniques can be successfully used [START_REF] Tsushima | Corrosion Inspection of Steel Tube Inner Wall[END_REF][START_REF] Pakrashi | An Image Analysis Based Damage Classification Methodology[END_REF] for the detection and quantification of the corroded region. Such image processing technique based information also forms a basis of probabilistically identifying the efficiency of the technique itself through the use of Receiver Operating Characteristics (ROC) [START_REF] Schoefs | Spatial dependence of Receiver Operating Characteristic curves for Risk Based Inspection of corroded structures: application to on-pile wharf[END_REF]) which plots the Probability of Detection (PoD) versus the Probability of False Alarm (PFA). The POD is defined as the probability of correcting identifying a damage when it exists, while the PFA is defined as the probability of identifying a damage when there is not any.

A special problem arising with the case of image processing based corrosion detection process is that the data (which are essentially videotapes or photographs obtained by a diver) are of varying qualities and tend to deteriorate with time. Due to the variability of natural conditions the chief variables affecting the image quality are luminosity, contrast and noise. The quality of assessment can significantly vary based on the conditions of these variables. Although, it is not possible to improve the quality of data, it is often possible to modify the images as pre-processing to qualitatively obtain the best results possible. This paper considers the detection of corrosion in an aluminium plate through image processing and synthetically introduces varying image qualities. A possible pre-processing technique from a qualitative point of view is proposed in this regard to improve the probabilities of detection. The pre-processing allows improving certain cases of deteriorated image conditions without having to resort to improvement in the methodology or the measurement, thus improving the quality of the assessment technique.

Image Processing Based Detection

Problem Description

An aluminium plate affected by pitting corrosion due to its residence in brackish sub-marine conditions over a period of twenty years is considered for the detection. The photograph of the corroded plate is available and is used as a basis for detection of corrosion. The detection is based on the fact that the corroded regions are in contrast with the general background of the plate. So the objective of an image processing based detection technique depends of its efficiency to separate the foreground consisting of corroded regions, from the background. Since the corroded regions are spatially distributed over the plate, a successful separation of the background from the foreground is considered non-trivial.

Detection Scheme

The image processing based detection scheme usually starts with obtaining a photograph or a video of the damaged region and its subsequent conversion to computer readable digital version. The photograph or the video needs to be benchmarked in terms of a known measured length in horizontal and vertical direction for future reference with pixel numbers. The digitised image is essentially a rectangular grid of pixels where each pixel is characterised by its hue-saturation value. The digitised coloured image is converted into a binary image and the edges of the black and white images are identified. Based on the definition of the minimum size of damage and on the number of successfully identified closed geometries, a prediction of damage regions within the image can be obtained. For a successful detection, the statistical properties of the closed regions in terms of pixels can be related to actual value by referencing the length benchmarking. Figure 1 provides a schematic for such detection process.

Possibility of Improvement

It is observed from section 2.2 that the separation of the background from the corroded foreground of the image forms the basis of the detection. The background, in this context can be described by the gray-level threshold of the image available for analysis. The gray-level threshold is obtained by minimizing the interclass variances of the black and white pixels using Otsu's method employing MATLAB 7.0 image processing toolbox [START_REF] Matlab | Image Processing Toolbox[END_REF]. Pixels below the threshold assume a value zero and turn black, while the pixels above the threshold turn white. As a result, the coloured original digital image is converted into a simpler black and white binary image. Since the feature of interest, the corroded regions are in contrast with their surroundings in the image this conversion is helpful for identifying the corroded regions.

However, the automatically computed gray-level threshold does not guarantee an optimized solution for appropriate detection of corroded region in the image. This is because of the fact that the identification of corroded regions or the regions of interest is essentially a local and image specific phenomenon on the two-dimensional photographic plane and thus the simultaneous maximisations of the interclass variances in the sub-regions of interest are not always possible. To improve the detection process, image specific adjustments of the threshold are proposed to be carried out to obtain the best possible information from the post-processing of the existing image. Incorporation of a correction to the automatically computed gray-level threshold through heuristic search to find the best value of the threshold maximising the visual information of the picture in terms of corrosion identification is thus proposed qualitatively.

Results

A photograph of a corroded aluminium plate used previously for image processing based detection of corrosion [START_REF] Pakrashi | An Image Analysis Based Damage Classification Methodology[END_REF]) has been used in the present study. Figure 2 shows a digital photograph of the corroded aluminium plate. The luminosity, the contrast and the noise of the photograph have been synthetically changed to obtain different image qualities. These variations are similar to what can be expected under field conditions. Three different levels (high, normal and low) of each of such affecting factors have been chosen. The noise is characterised as additive Gaussian noise and has been described by the parameters of its normal distribution. The luminosity is changed by changing the amount of 'whites' in each pixel while the contrast change is characterized by tuning the differences between the adjacent colours in each pixel. These changes can be performed easily through commercial photo editors, where the luminosity and contrasts are usually marked from a scale of zero to hundred.

For each of the combination of luminosity, contrast and noise, the image processing based detection of corrosion is carried out for automatically computed gray-level threshold and for heuristically fit gray-level threshold. Figure 3 illustrates the adjustments of the gray-level threshold required for various images corresponding to various environmental conditions. The non-adjusted threshold values refer to that obtained from Otsu's method, while the adjusted values correspond to the values obtained from heuristic thresholding. It is observed, that for certain combinations of luminosity, contrast and noise the heuristically set thresholding is significantly different from that obtained by automatic thresholding. The qualitative improvement of the various conditions is observed for each of the image. The qualitative level of improvement due to heuristic adjustment of gray-level threshold has been presented in Table 1. The improvement has been in terms of the probability of detection of the actual corroded area. The level of detection is described in a qualitative sense for each of the combinations of luminosity, contrast and noise for automatically obtained and heuristically fitted gray-level threshold. The conditions where significant improvement is obtained by the proposed preprocessing are marked in gray. About 64% of the cases are observed to improve in terms of corrosion detection by following the proposed pre-processing of image data. Thus, preprocessing techniques can be found to be useful in improving NDT based identification and assessment even when no new data is available and improvement in the measurement technique is not possible.

Conclusions

The importance of pre-processing techniques for the purpose of improvement in NDT based assessment schemes is discussed in this paper. A typical application problem related to corrosion identification using image processing based methods is taken up as an illustrative example in this regard. Various environmental factors, like luminosity, contrast and noise can alter and deteriorate the quality of the optical data for an image processing based information. It is observed that the separation of the corroded regions from the background of an image is non-trivial for a wide range of qualities of the image. The automatically computed gray level threshold for image processing is found qualitatively to be inferior to a heuristically fitted gray level threshold for the purpose of separating the corroded regions from the image background.

Figure 1 .

 1 Figure 1. Schematic of Image Based Detection Scheme

Figure 3 .

 3 Figure 3. Image Adjustment by heuristically adjusting the Gray-level Threshold.

Figure 2 .

 2 Figure 2. Photograph of the Corroded Plate