Raphaël Pesci
email: raphael.pesci@metz.ensam.fr

Karim Inal
email: bkarim.inal@ensam.fr

Sophie Berveiller
email: csophie.berveiller@metz.ensam.fr

Etienne Patoor
email: etienne.patoor@metz.ensam.fr

Jean-Sébastien Lecomte
email: ejean-sebastien.lecomte@univ-metz.fr

André Eberhardt
email: feberhardt@lpmm.univ-metz.fr

Inter-and Intragranular Stress Determination with Kossel Microdiffraction in a Scanning Electron Microscope

Keywords: Microdiffraction, intragranular stresses, crystallographic orientation, micronic scale, microstructure evolution

A Kossel microdiffraction experimental set up is under development inside a Scanning Electron Microscope (SEM) in order to determine the crystallographic orientation as well as the inter-and intragranular strains and stresses on the micron scale, using a one cubic micrometer spot. The experimental Kossel line patterns are obtained by way of a CCD camera and are then fully indexed using a home-made simulation program. The so-determined orientation is compared with Electron Back-Scattered Diffraction (EBSD) results, and in-situ tests are performed inside the SEM using a tensile/compressive machine. The aim is to verify a 50MPa stress sensitivity for this technique and to take advantage from this microscope environment to associate microstructure observations (slip lines, particle decohesion, crack initiation) with determined stress analyses.

Introduction

X-Ray Diffraction (XRD) is a non-destructive technique very efficient for stress determination of different orders corresponding to different scales (with various methods and instruments). The first order represents the macroscopic or the pseudo-macroscopic stress, where said stress is determined in the whole material or in different phases. The second order defines the average stress of the grain and the third one is devoted to the stress on the crystalline pattern scale (considering for example precipitates or dislocations). However, between these last two orders, given the growing complexity of materials and their applications, it is often necessary to determine stress on an intermediary scale: hence the development of Kossel microdiffraction on the micronic scale [1]. This is a local analysis tool similar to that used first by [2], an X-ray imaging inside a SEM using a 2D detector, which enables to determine not only the crystallographic orientation, but more importantly the stress and strain states (and this for different crystalline materials). It is very promising because it offers technological progresses when compared to conventional methods such as "classical" XRD or EBSD. Indeed, this diffraction technique inside the SEM enables to determine the crystallographic orientation with a greater precision (higher wavelengths emitted), as well as inter-and intragranular stresses since the spot used is only about one cubic micrometer. It also allows to observe simultaneously the microstructure of the material and its evolution, in particular during in-situ tensile tests, and the analysis can be considered as quasi-instantaneous, since it only takes a few minutes. This means that it becomes possible to realize a stress mapping very quickly (for example along grain boundaries or inside one grain, in different phases or even close to a crack, as seen in Fig. 1) and to associate a stress value to a microstructural event (slip line appearance, particle decohesion or fracture, crack initiation).

Kossel technique

When a focused electron beam excites a material, the latter produces a characteristic X-ray emission: some of this emission is then diffracted on the crystallographic planes according to the Bragg equation (λ=2d.sinθ), forming the so-called Kossel cones. This is very interesting because several reflections occur simultaneously, in all space directions (Fig. 2): in fact, one cone is emitted for each (hkl) diffracting plane (all these planes diffract at the same time). A CCD camera is therefore placed so that its screen intercepts some of these cones emitted from the area affected by the electronic spot (size: about 1 µm 3 ): one obtains thus Kossel lines or complete circles (depending on the orientation of the cones and the distance sample/camera) that are transmitted to a computer for evaluation [3]. precise, and their intersections appear more clearly: such is the case for example for a copper alloy in Fig. 3a. Other materials have also been successfully analyzed such as a bainitic steel coming from pressure water reactors, even though this kind of testing is more complicated (more time consuming) due to their smaller grain size. The obtained Kossel line pattern is then indexed using a home-developed simulation program that particularly takes into account the distance between the center of projection on the material and the center of the screen, the wavelength of the emitted X-rays, as well as the crystalline structure and the lattice parameter of the tested material. Practically, this consists in manually superimposing the simulated Kossel line pattern on the experimental one, until they match; of course, one of the objectives is to automate this system. In addition, the simulation program gives the Miller indices of all the diffracting planes as well as the crystal orientation in space (Fig. 3b).

Comparison to EBSD results and stress sensitivity determination EBSD measurements are here carried out in a grain of a CuAlBe flat specimen concurrently with Kossel microdiffraction, in order to compare the determined crystallographic orientations. The obtained Kossel and Kikuchi line patterns are first indexed (Fig. 4), and the Euler angles then calculated: the two techniques provide similar angles (respectively 92,51°, 11,48°, 85,66° and 92,88°, 11,48°, 85,24°), since the desorientation is less than 0,01°. This result is very important because it enables to validate the indexing program for Kossel line patterns, all the more so as this Kossel technique gives the crystal orientation very rapidly and with greater precision than with EBSD (X-rays have higher wavelengths). Since the Miller indices of the diffracting planes are thus clearly identified, it is possible to introduce a stress state in the developed simulation program in order to detect which planes shift more with the applied stress. This helps to focus our attention on the appropriate planes during in-situ tensile tests (development stage of this technique), in order to calculate from the experimental Kossel line pattern the intragranular stress associated to the spotted microstructure. An in-situ tensile test is therefore performed inside a SEM, using a small tension/compression machine equipped with a temperature regulating system ranging from -160°C to 300°C (Fig. 5a). A CuAlBe SMA is investigated (Fig. 5b): the linear part of the curve with an applied force lower than 160N (between steps 1 and 3) corresponds to the elasticity of austenite, while after step 3, stress induces martensitic transformation during superelastic loading.

Evolution of the Kossel lines pattern during loading

The same tensile test is still investigated in order to show the capacity of this technique to study each phase separately (possibility to pinpoint each lath of martensite, and the austenite between these laths) as well as the stress and strain sensitivity of the associated Kossel curves, since a lattice strain corresponds to Kossel line displacement which can be directly observed with the camera. So first, when considering only the elasticity of austenite (step 1), one obtains the Kossel line pattern presented in Fig. 7a. New Kossel lines is all the more important as the applied strain is high, since when the latter increases again, the resulting displacement is even greater (and so are the diameters of the circles in this case), which is characteristic of a higher lattice strain. For example, when studying the austenite between the martensite laths at different strain levels (here 4 and 5), the associated Kossel line pattern shifts more and more with strain. This is even more obvious when superimposing the obtained patterns with the one of pure austenite (Fig. 9). However, one has to take into account both the relative and cooperative displacements of the Kossel lines under loading, respectively due to the stress (some lines shift more than others) and the reorientation of the grain (all the lines shift in the same way).

Conclusion

Kossel microdiffraction is mainly developed for stress determination. Even if the Kossel line patterns are obtained for many materials and are well indexed, an automatic indexing program remains to be developed. This implies the detection of the curves in the experimental Kossel line pattern, without omitting the fact that the stress state has to be taken into account in the image, meaning that it is not always possible to superimpose exactly the simulated Kossel line pattern on the experimental one. It is what is being done at the moment in order to have a quick and efficient tool in the determination of local stress and strain tensors from the measurement of interreticular distances of diffracting planes [4]. This will be possible by implementing the Ortner method in the program [5], which from the lattice strain and the crystal orientation enables to determine the strain tensor by projection on the coordinate system directions, in order to obtain finally the stress tensor using Hooke's law.

Fig. 1 :

 1 Fig. 1: Stress mapping using Kossel microdiffraction a) Along grain boundaries or inside one grain -b) In different phases of a material (ferrite and cementite in a 16MND5 bainitic steel) -c) Close to a crack

Fig. 2 :

 2 Fig. 2: Principle of Kossel microdiffraction A Kossel cone is emitted for each (hkl) diffracting plane

Fig. 3 :

 3 Fig. 3: a) Kossel line pattern for copper -b) Same pattern fully indexed

Fig. 4 :

 4 Fig. 4: Crystallographic orientation determined in one grain of a CuAlBe specimen a) Using Kossel microdiffraction technique -b) Using EBSD technique

Fig. 5 :

 5 Fig. 5: a) In-situ tensile/compressive machine equipped with a temperature regulating system [-160°C; 300°C] -b) In-situ tensile test performed on a CuAlBe SMA

Fig. 7 Fig. 8 :

 78 Fig. 7: a) Kossel line pattern for pure austenite (step 1) -b) Appearance of a new Kossel line pattern coming from martensite (step 3)

Fig. 9 :

 9 Fig. 9: Shift of the Kossel lines during loading a) Steps 1 and 4 -b) Steps 1 and 5


Bruno Godart
email: bruno.godart@ifsttar.fr

Jacques Berthellemy
email: jacques.berthellemy@cerema.fr

born Jean-Pierre Lucas
email: jean-pierre.lucas@cg76.fr

Diagnosis of a large steel bridge close to collapse during a fire

Keywords: steel bridge, fire, emergency, diagnosis, assessment, investigations, recalculation, stability, repair, decision

The Mathilde Bridge is one of the six bridges over the Seine River in the city of Rouen and supports heavy traffic. It was built in 1976-79 and is composed of a single metal span 115 m long, consisting of an orthotropic steel deck resting on two girders. On October 29th, 2012, the Mathilde bridge was subjected to a heavy fire and was immediately closed to traffic. After recalling the events, the article describes the decisions that had to be taken in an emergency, then the extended diagnosis which was conducted in order to be able to take a rapid decision on repair or reconstruction.

The diagnosis consisted of a detailed inspection including the examination of cracks and the condition of the paint, extensive investigations carried out on samples of metal, an accurate measurement of the global deformations of the entire span, a recalculation of the stability of the burned areas, and an expertise of damaged support bearings.

Introduction

The city of Rouen (100,000 inhabitants) is in the heart of an urban agglomeration of 450,000 inhabitants. It is organized around a bend of the Seine and its road crossings. Over the decades, the local authorities have decided to drive the transit traffic away from the center of the city without diverting the internal traffic within the agglomeration. Thus, when the Mathilde bridge was opened to traffic in 1979, the mayor of Rouen, also president of the Departmental Authority of the Seine-Maritime asked to the Government of the French State to study a new crossing in the west of Rouen. This new crossing was built 20 years later with the construction of the Flaubert lifting bridge. From a road perspective, the Rouen agglomeration is located at the intersection of flows coming from northern Europe and France and going to the West and to the Iberian peninsula, with flows between the Ile de France and western France.

The Rouen agglomeration having no ring road, the Mathilde Bridge plays a strategic role for the flow of traffic in the city. Before the fire, it supported more than 80 000 vehicles per working day, with nearly 20 % of transit traffic. Moreover, it supported a sewage pipe serving the north of the city, and an optical fiber network.

The Mathilde Bridge was built between 1976 and 1979 according to a SETRA (now CEREMA) design and has in particular two twin 115 m long steel spans; each span consists of an orthotropic deck resting on two girders which are 4 m high, 17.35 m apart and braced every 4 m. Each steel span is resting at one end on an abutment and at the other end on the extremity of a cantilever which is part of a continuous prestressed concrete box-girder bridge passing over the Lacroix Island (see figure 1). On October 29th, 2012, a tanker truck carrying oil caught fire on the Mathilde bridge, by turning over in an access curve to the bridge. Its load spread over the whole width of the deck of the south steel span, then flowed toward the edges and ends of the bridge and set fire to fairground workers trucks that were parked under one extremity of the bridge. This latter was subjected to a heavy fire whose temperature has been estimated to reach between 650 and 800 °C.

The management of the crisis

On October 30th, a specialized consultant was commissioned to make a first diagnosis, a crisis cell was activated by the Prefect, involving all road managers: City, Agglomeration, Department, State, motorway concessionaires, and the gendarmerie and police services. The prefect asked also for the help of the State national services to provide technical assistance to the Department services. A technical committee was formed with experts and met in early November to provide further diagnosis and to propose investigations.

Meanwhile, a new traffic plan was implemented and adjusted each day to take into account the daily specificities. The city of Rouen opened a website in order to know at anytime the status of the traffic in Rouen (http://trafic.rouen.fr/trafic). The Departmental Authority opened also an information website (http://www.inforoute76.fr/) and the Agglomeration reinforced the buses network. The Departmental Authority reinforced the movement of ferries in the west of the city with 3 river crossings. The Agglomeration and Orange Company worked in emergency to restore the sewage system and the optical fiber network by temporary deviations via the Corneille bridge located nearby. Some private companies and public services adapted the work schedules of their employees by allowing them to start early and leave before the evening rush. Some companies offered two work sites, one on the right bank of the Seine river and the other on the left bank.

Communication activities were undertaken by the city and the Departmental Authority to encourage the use of public transport and carpooling. Carpool areas were urgently arranged on the periphery. The Police was mobilized to manage traffic around eight major strategic crossroads. Some works were undertaken to modify the geography of some intersections and delete conflicting movements.

The Firefighters responded very quickly, avoiding a full collapse of the deck, thanks to their action. Fortunately, there were no dead people and only two persons were slightly injured. The bridge was immediately closed to traffic as well as the navigation under the bridge. This decision led to the displacement of some of the boatmen on the Northern pass usually reserved for recreational activities and to the stop of the yacht club activities. These decisions were taken by the Prefect and the president of the Departmental Authority, given the uncertain condition of the bridge. The road traffic on the bridge is still closed today.

2.1

Description of the main disorders of the frame

Before describing the main disorders, it should be noticed that firefighters were able to intervene quickly with water, which helped limit the temperature in the steel frame before it reaches the critical temperature of 723 ° C. If this action, that probably prevented the bridge from collapse, has not been carried out very quickly, the effect of metallurgical quenching on a steel having exceeded the critical temperature would have caused a steel embrittlement.

The deck remained distorted like an accordion over a length of about 40 meters from the south abutment (Fig. 3). The transverse beams supporting the deck were alternately raised and lowered due to the combined effect of heat which lowers the modulus of elasticity, and predominantly of the compression that occurred during the fire in the deck due to the restrained expansion of the deck. The web of the downstream girder was distorted dramatically during the fire (Fig. 4): a tensioned diagonal has formed over a length of 8 meters from the abutment with a wave folding the first intermediate vertical stiffener located at 4 meters from the abutment. -To advise the Department of Seine-Maritime, owner of the bridge, -to specify the conditions for reopening the river navigation in the South channel, -to assess the condition of the structure and its supports, -to evaluate scenarios of repair or replacement, -to give an opinion on the conditions for carrying out the repair work.

Although the bridge did not collapse under its own weight, the TC recommended that no prop be installed. Such a prop profoundly alters the distribution of forces in the structure and represents a serious obstacle for the navigation in the South channel. Therefore, the navigation on the Seine river was restored at the end of December 2012.

A heating of the end of the downstream girder over a length of 10 meters from the abutment was set up in December to avoid any brittle failure of the welds during the cold weather in winter. This precautionary measure was taken because the weld beads located in the deformed areas could be subjected to a steel embrittlement, without knowing the investigations results. Moreover, damaged cornice elements were removed to prevent them from falling.

Specific investigations regarding the effects of the heat on the deck

The investigations carried out on the Mathilde bridge were derived from the emergency procedures established by CEREMA to evaluate the condition of a bridge after a fire accident.

Observation of colors and laser topographic survey

The observation of the residual deformations and colors taken by the corrosion protection containing lead minium indicate that the temperature has reached at least 600 ° C on significant areas (Fig. 5). It could be noticed the white lead oxides of the primer over the entire height of the web and a complete disappearance of the primer in the center of this area. The deformation of the downstream girder was measured using a 3D laser scanner by the company SITES. The laser record allowed to acquire 500 million points including 100 million for the end of the downstream girder. The interpretation of this survey allows the creation of a polygonal mesh with a precision of + / -1 to 2 mm. The main results showed that the girder had a vertical displacement of the order of 210 mm and the transverse deformation of the web of the girder varied between + 18 mm and -28 mm apart from the mean plane (Fig. 6). It can be assumed that the margins of the ultimate strength of the structure at the peak of the fire were likely very low with respect to the buckling of the downstream girder near its support on abutment. This survey also allowed CEREMA to create a mechanical model of the end of the downstream girder.

Examination of fillet welds

A first careful visual examination of thick welds beads on the downstream girder did not show cracking: this observation concerns the web-vertical stiffeners welds (in particular the stiffener at the end of the girder), the web-flange welds, and the web-longitudinal stiffeners welds.

This examination was then supplemented by the SOCOTEC Company with penetrant dye tests on weld beads at the extremity of the downstream girder, on the inside, near the tension diagonals, and with magnetic tests on different types of weld beads in areas exposed to fire. Apart from an area that has a lot of small corrosion craters over a length of 400 mm, and except for some spot areas at the junction of the webs with the flanges that need to be recharged, the welds are generally in good condition.

Moreover, the tests already performed in 1998 (penetrant dye, magnetic and U.S.) have been reproduced on the welds of the additional flanges, according to the same procedure. No defects were detected.

It should be noted that the girders of the Mathilde bridge are fully welded, and this configuration contributed to the survival of the bridge because experience shows that bolted joints can undergo serious damages in a fire. For the weld beads which have not been directly subjected to fire hoses, the high temperature of the fire may even have improved their resistance by relieving residual manufacturing stresses.

Effect of heat on steel: Measure of the Vickers hardness

This measure can be easily conducted at multiple points using a portable instrument with a diamond tip. This is a non-destructive examination intended to highlight the presence of martensite, and thus It was possible to make ten points of replicas that could be interpreted early December 2012. Nine points were chosen on site according to the curvatures of the steel plate and its surface appearance on the inside of the downstream girder (Fig. 7), because it is between the girders of the bridge that the warm air was trapped and that the temperature rose the most. One of these replicas was made in the area directly sprayed by the firefighters. Some points were localized on welds. A reference point was also taken out of the zone subjected to the fire. The results were very positive because the observed structure of the steel was everywhere purely ferrito-pearlitic without any trace of martensite or other structure degraded by an excessive heat. Moreover, the Welding Institute did not noticed any cracking due to creep damage or attack by liquid metals.

Effect of heat on steel : Steel samples for Charpy tests

These samples are much smaller (10 x 10 x 55 mm) than the samples taken for tensile tests. In addition they provide more relevant information on the risk of embrittlement due to tempering of the zones that have exceeded 725 °C. Therefore, SOCOTEC Company collected many specimens for Charpy tests in the highly redundant troughs of the orthotropic decks (Fig. 8) and at the ends of the longitudinal stiffeners of the web of the downstream girder (Fig. 9), with the objective that these samples do not weaken the remaining structure. The samples were taken on the edge and oriented parallel to the stiffener, with a cutting a little more than 1 cm deep. These small samples are cut with a water cooled saw, so that the heat of the cut does not distort the results.

The resilience values Kv (at the temperature of -20 °C) taken from the steel deck after the fire were between 75 and 125 J/cm2 on the normalized specimen. They were all above the average reference value KCV fixed at 50 J/cm2 and above the individual reference value fixed at 35 J/cm2, which eliminates any risk of embrittlement and crack propagation.

Figure 9: Sample taken on a deck trough Figure 10: Sample taken on a web stiffener

Other investigations on steel samples

Others samples were taken by SOCOTEC and tested by LHEM (Laboratoire Havrais d'Essais Mécaniques) for mechanical properties. Among 17 samples, 11 had an elastic limit Re greater than the reference value of the standard NF A 35-501, Re = 355 MPa , and six samples (all coming from the troughs) had an elastic limit less than that value. Concerning the limit strength Rm, 15 samples had a value Rm greater than the reference value Rm = 510 MPa, and two samples (from the troughs) had a Rm value less than 510 MPa. It was thus found that the stiffeners had a steel grade higher than that of the troughs.

The elongation values for all 17 samples were above the reference value fixed at 22 % (for a plate thickness e such that 3 < e < 30 mm). The values of the modulus of elasticity are generally satisfactory and close to 200 Gpa.

The carbon contents measured by the CEREMA laboratory of Nancy are relatively high but still below the limit value in the standard 0.22. The equivalent carbon of the steel is between 0.423 % and 0.462 %, which gives a high susceptibility to cold cracking. The laboratory concluded that the studies qualifying welding procedures should take into account this risk and secure precautions when welding for repair.

Specific investigations regarding the effects of heat on the support bearings

Expertise of damaged bearings

The expansion joints on the abutment subjected to the fire were re-opened after the fire, which seemed to show that the mobile pot bearings were functioning again. But this reopening was perhaps only due to the shortening of the deformed deck after cooling. The bearings are not accessible and the condition of the Teflon was therefore unknown. It was recommended to record the temperature and the elongations of the two girders that have not been exposed to the same fire for comparison purpose.

A comprehensive longitudinal model of the entire bridge was developed by DRIEA (a French governmental office for roads near Paris) as a bar taking into account the thermal inertia of the prestressed concrete spans over the Lacroix island. This model succeeded to explain the observed elongation with the hypothesis of pot bearings with Teflon sheets always sliding with very little friction.

Detailed inspection of the concrete supports

The examination of the concrete by the DREIA under the south bearing showed no visible cracking. This first examination made from accessible areas was completed by the inspection of the bearing from a positive inspection platform. The challenge was to ensure the resistance of the crossbeam of the abutment supporting the bearings.

The succinct hammering of the abutment face showed no degradation of concrete in depth. It seemed that the striated architectural stamping had partially protected the structural concrete that it covered. Some aggregates were observed as turning pink at the surface only. The removal of the deteriorated concrete revealed aggregates having a normal color within 1cm depth beyond the striated architectural stamping. No reinforcement steel appeared because the concrete scales remained of the order of a centimeter and were inferior to the cover depth. It could be concluded that the reinforcement had not been heated to a high temperature SOCOTEC completed this inspection by a physico-chemical analysis of cores to determine the temperature profile that had been reached (see Fig. 11 for the location of the cores taken).

The mechanical properties of the two reinforcement samples that were taken were correct.

Plastic calculation with the new and very deformed geometry

The improved version of Code_Aster 11.3 was then used to show that the structure remained stable until even a load equal to 2 times its own weight (2G). The maximal Von Mises stresses reach 377 Mpa (Fig. 13). At this level of charge, the convergence of the calculation of the structure is obtained by considering both the balance of efforts due to large displacements in the deformed geometry and the plastic behavior of the material. This safety reserve allows a small engine to access on the bridge for repair work, if the company validates these results.

The decision taken and the repair principle

A replacement of the damaged part of the structure respecting its initial geometry was decided. Indeed, a good overall design of the existing structure has made this option more realistic than a full replacement. It generally makes little change to the orthotropic deck, but a simple overweight to meet some minimum thickness requirements.

Furthermore, a removal of the bridge through the waterway was decided because it offers many advantages. The bridge is transported to a dock distant from the site, to be cut so that the damaged parts are replaced with new parts. This principle has several advantages: -Very little interference to navigation except during the two days of operation for the removal and the reinstallation of the entire span, -This method has already been chosen for the construction of the bridge, and therefore it is probably the most economical one, -Less risk than a prop that deeply modifies the mechanical functioning of the bridge in the interim phase, -Simplicity of the ballast of the prestressed concrete cantilever span, -Hazards reduction for repair works because the constraints due to work over the Seine river are avoided, with also, as a consequence, a better quality of execution, -No nuisance at the bridge site near the city, especially during the renovation of the corrosion protection containing red lead. The paint stripping in the city over the Seine river would require the implementation of a scaffold engaging the navigation clearance and a containment to ensure both safety conditions and environmental protection

Conclusions

The implementation of a plan to bypass transit flows via A29 Motorway and the Normandy bridge has significantly reduced the transit traffic. On some secondary roads, particularly in the east of the city, the heavy weight traffic has doubled, resulting in a significant deterioration of pavements. It is the same to the west with an increase in traffic in urban areas through the Cailly valley (Maromme in particular).

Moreover, this situation with essentially only a bridge (Flaubert bridge with 2x3 lanes) to support the North-South traffic shows limits every day. Indeed when an accident occurs at the approach of this bridge, the whole agglomeration is congested very quickly. Some days (exceptional circumstances), it takes almost 2 hours for transit vehicles to cross the Seine river. It has been established that some days, there are more than 60 km of traffic jams. The traffic crossing over the Seine river decreased by 40,000 vehicles per day.

Attendance of public transport has increased such as rail, intercity (about 10% ) and urban (more than 15 %, especially on own site lines). Bicycle use has grown despite a topography and an inauspicious road for practice. Walking has made its comeback for travel over short distances. A survey conducted during the spring 2013 by the Chamber for Trade and Industry shows that about 10 % of economic actors have seen their situation weaken because of accessibility issues.

Today several establishments submit briefs to the court to obtain repair for their disfavors. The Departmental Authority, although pending a judicial decision fixing the amount of damages to the bridge, conducted its rehabilitation. Re-commissioning is expected in the summer of 2014, most likely during the month of August.

Figure 1 :

 1 Figure 1: Longitudinal elevation of the Mathilde bridge (the steel spans are in blue)

Figure 2 :

 2 Figure 2: View of the two fires over and under the bridge

Figure 3 :

 3 Figure 3: Waves of the deck

Figure 5 :

 5 Figure 5: Colors taken by the corrosion protection Figure 6: Transverse deformation of the web

  to reveal embrittlement in the tempered zones that have exceeded 725 ° C. The measurements were done by the Welding Institute of Le Havre. No evidence of embrittlement was identified. The hardness values obtained in the steel of the web and the stiffeners are quite close to that of the reference area. However, the hardness / Rm conversion according to EN ISO 18625 of June 2004 gives values that are lower of just greater than the minimum required for a steel grade E36-4 (Rm = 510 MPa according to NFA35-501 of 1973) and E460 (Rm = 590 MPa according to NF This technique has been implemented by the Welding Institute of Le Havre. It is standardized (NF A04 -154 of December 1985) and allows rapid diagnosis of steel after a fire at multiple points by implementing a lightweight portable equipment. The replica method consists of five steps: 1 . Local grinding to remove surface layers of oxides, paint, or decarburization 2 . Preparation of the area by polishing, using more and more finer abrasive paper 3 . Metallographic attack of the polished area to reveal the microstructure, by a chemical solution based on nitric acid and ethanol . 4 . Taking the impression of the microstructure by placing a film of cellulose acetate 5 . Microscopic examination

Figure 7 :

 7 Figure 7: Location of the replicas Figure 8: Observed structure of the steel on replicas

Figure 11 :

 11 Figure 11 : Location of cores

Figure 12 :

 12 Figure 12: Elastic calculation under 1,35 G + H (scale going from 0 to 366 MPa)


Ouahab Razane
email: razane.ouahab@ensam.eu

Pesci Raphaël
email: raphael.pesci@ensam.eu

Berveiller Sophie
email: sophie.berveiller@ensam.eu

Etienne Patoor
email: etienne.patoor@ensam.eu

Influence of temperature on stress distribution in bainitic steels -Application to 16MND5-A508 pressure vessel steel

Keywords: 16MND5 steel, in situ tests, XRD, phase stress distribution, bainite

In this study, the internal stress evolution of the ferrite phase of 16MND5-A508 has been determined using X-Ray Diffraction (XRD). The results of in situ tests combined with XRD analyses and performed at different temperatures (-150°C and 22°C) exhibit a difference of about 200MPa between the macroscopic stress and the ferrite one. The stress state in the cementite is determined by a mixture law; it reaches very high values up to 9000MPa. These results highlight the need to analyze the stress directly in the cementite phase by using appropriate tools, since its volume fraction does not allow it using XRD.

Introduction

In complex assemblies such as nuclear reactors, sources of radiation can induce changes in the mechanical properties of materials. When considering the resilience curve of the 16MND5-A508 pressure vessel, the ductile-to-brittle transition region is therefore shifted to high temperatures, leading to an increase in the risk of cleavage brittle fracture in case of rapid cooling [START_REF] Renevey | Approche globale et locale de la rupture dans le domaine de transition fragileductile d'un acier faiblement allié[END_REF]. To bring a better comprehension of failure mechanisms in this low alloy steel at different scales (macroscopic, phase, intragranular), especially those involved in the cleavage fracture, a series of experiments have been performed at room and low temperatures (-150°C) to promote this kind of damage; in situ tests combined to laboratory XRD analyses enabled to follow the stress evolution in ferrite phase.

Studied materials

The 16MND5-A508 steel provided by EDF company is a low alloy steel used for the conception of Pressurized Water Reactors (PWR). The steel underwent three stages of heat treatment: two austenitizations at 850°C followed by water quenching, a tempering between 630°C and 645°C to improve the toughness and finally a stress relieving treatment at 610°C. At the end, an upper bainite with two phases was obtained: ferrite (Fe), which forms the matrix and precipitates of cementite (Fe 3 C), which reinforce the matrix [START_REF] Osipov | Génération et calcul de microstructures bainitiques, approche locale intragranulaire de la rupture[END_REF]. Before crystallographic observations, the samples were mechanically polished and etched with 2% Nital. Fig. 1-a shows a typical microstructure of 16MND5 steel obtained with FEG-SEM. The size of the bainite grains is relatively homogeneous (~50µm). the observation of which is difficult Diffraction) technique can be more appropriate to make evident the real size of each element of the microstructure. The examination reveal the former austenitic grain boundaries homogeneously: they can be intragranular structure) or intergranular with performed by neutron diffraction: the The presence of MnS inclusions with a low volume fraction (<1%) mainly elongated in the direction of forging (~60µm) or spherical (~8µm), and can be isolated or grouped in clusters of 3 to 8 inclusi A second steel type XC40 supplied by ARCELORMITTAL was also studied i possible size effect on phase stress distribution and given in table 2.

C Mn 0.4 0.7

Table 2: Chemical composition of

This steel was heat treated in order to obtain cementite (Fig. 2) with a carbide size is smaller than in 16MND5 steel: it is about 15µm. this paper as it's a confidential data to SEM micrograph showing the microstructure of the 16MND5 steel ighlighting the complex structure of the steel relatively homogeneous (~50µm). Ferrite is composed of which is difficult with SEM. The EBSD (Electron BackScattered technique can be more appropriate to make evident the real size of each element of the . The examination reveals a size of lath from 5 to 15µm, oriented tic grain boundaries (Fig. 1-b). Fe 3 C precipitates are distributed non intragranular (located between ferrite laths or with an average carbide size less than 0.1µm. A performed by neutron diffraction: the volume fraction of cementite was estimated presence of MnS inclusions with a low volume fraction (<1%) is also observed mainly elongated in the direction of forging (~60µm) or spherical (~8µm), and can be isolated or grouped in clusters of 3 to 8 inclusions.

supplied by ARCELORMITTAL was also studied i stress distribution and brittle fracture. Its chemical composition Mn Si P et S 0.7 0.10-0.40 <0.035

: Chemical composition of XC40 steel [weight %-iron balance steel was heat treated in order to obtain a two-phased microstructure composed cementite (Fig. 2) with a carbide size of about 1µm and a volume fraction of 16MND5 steel: it is about 15µm. The heat treatment will not be specified this paper as it's a confidential data to the industrial. The carbide size distribution determined by image intergranular carbides are the large For the work presented here, the evolution of internal stress loading has been determined for both steels described above.

Stress analyses by XRD in ferrite phase

XRD allows the determination of stresses analysis, the sin²ψ method is used stress in the ferrite phase considering lattice plane distance as method are given in [START_REF] Totten | Handbook of residual stress and deformation of steel[END_REF]. The experimental work concerns 16MND5 steel is too low to make any S 1 =-1.277*10 -6 MPa -1 and 1 2 S 2 =5 and Poisson's ratio of the material

S 1 =-1 υ E & 1 2 S 2 = 1+υ E
For direct measurements in cementite, the using synchrotron radiation or neutron estimated using a mixture law which takes size distribution determined by image analysis reveals an average size of 0.7 largest while smaller carbides are predominantly intragranular. , the evolution of internal stress in the two phases determined for both steels described above.

Stress analyses by XRD in ferrite phase

allows the determination of stresses in each phase of a polycrystalline is used to determine the macroscopic or pseudo in the ferrite phase considering lattice plane distance as a strain gauge. More details about this s only the ferrite phase, since the volume fraction of cementite in w to make any analyses; we use the following 5.919*10 -6 MPa -1 which are determined from de Young material (Eq.4):

For direct measurements in cementite, the radiation flow or diffracted volume must be increase neutron diffraction. Therefore, the stress state in cementite is a mixture law which takes into account the volume fraction of each phase (Eq is the macroscopic stress, (σ Fe I , σ , since the volume fraction of cementite in ; we use the following elastic constants which are determined from de Young's modulus [START_REF] Libert | Etudes expérimentale et numérique de l'effet des mécanismes de plasticité sur la transition ductile-fragile dans les aciers faiblement alliés[END_REF] volume must be increased state in cementite is first into account the volume fraction of each phase (Eq. 5): [START_REF] Mathieu | Analyse et modélisation micromécanique du comportement et de la rupture fragile de l'acier 16MND5: prise en compte des hétérogénéités microstructurales[END_REF] the stresses in the ferrite and cementite phase, ing, XRD analyses are goniometer (Fig. 3): a/

Fig. 3: Experimental setting up

The influence of temperature on the mechanical behavior of the The small tensile machine is kept and isolated in a thermal enclosure to limit ice formation This allows maintaining intense diffraction peak For all the tests, flat specimens ( determine the evolution of internal stresses at room temperature during loading considering the { (wavelength λ = 2.2897Å).

Results and discussion

The stress results determined in situ The curves give the evolution of internal stress in the ferrite phase during macroscopic an average uncertainty of about ±20MPa. Fig. 4 shows the ferrite stress evolution (isolated obtained for the 16MND5 steel at room temperature and with extensometer; the macroscopic stress and displacement sensors directly attached bainite one and the ferrite stress is a/

Fig. 4: Stress evolution in ferrite

At room temperature, the yield strength of 16MND5 with literature data [START_REF] Libert | Etudes expérimentale et numérique de l'effet des mécanismes de plasticité sur la transition ductile-fragile dans les aciers faiblement alliés[END_REF]. It is observed that the stress state in the ferrite macroscopic one. The difference on the mechanical behavior of the two studied tensile machine is kept at low temperature with a cooling system and isolated in a thermal enclosure to limit ice formation on the samples during cooling diffraction peaks. tests, flat specimens (dimensions in mm: 60x2x1) were loaded determine the evolution of internal stresses at room temperature and -150°C. Analyses were {211} crystallographic planes of ferrite and using

The stress results determined in situ through tensile tests on the two materials are presented below. rves give the evolution of internal stress in the ferrite phase during macroscopic about ±20MPa. shows the ferrite stress evolution (isolated dots) and the macroscopic stress ( at room temperature and -150°C. The specimens were not equipped the macroscopic stress and the strain were calculated from a attached to the tensile machine. The macroscopic one and the ferrite stress is determined by XRD. deformation: it is approximately 150MPa. At low temperature, there is a classical increase in the macroscopic yield strength that is about 620MPa. The ferrite stress also slightly increases. If one compares the ferrite and bainite stresses, the gap between the two is greater than at room temperature since it reached 200MPa. These results are also observed in Mathieu's works [START_REF] Mathieu | Analyse et modélisation micromécanique du comportement et de la rupture fragile de l'acier 16MND5: prise en compte des hétérogénéités microstructurales[END_REF].

The same tests were conducted on XC40 steel that differs mainly to 16MND5 steel by its cementite volume fraction. Fig. 5 shows the curves obtained for this steel at room temperature and -150°C with the evolution of the stress in ferrite. XC40 steel is more ductile than 16MND5; its yield strength is about 420MPa at room temperature and reached 500MPa at -150°C. The evolution of internal stresses in ferrite during loading seems to be similar for both steels. At room temperature, the gap between the macroscopic stress (bainite) and ferrite one is 200MPa for XC40 steel. Unlike 16MND5 steel, this gap seems to be less important at low temperature, since it is only 100MPa. This behavior might be explained by the difference in the volume fraction of cementite that characterizes the two steels. Likewise, the different grain size and carbide distribution may also provide an explanation since they could compensate the macroscopic stress level reached [START_REF] Kim | [END_REF]. However, more works must be performed to confirm these results. On the other hand, XRD analyses reveal that the macroscopic stress in both steels is governed by ferrite phase due to its high volume fraction in both steels (97.9% for 16MND5 steel and 94.3% for XC40 steel).

Considering now the stress in cementite calculated from the XRD measurements in ferrite and mixture law indicated in paragraph 3, the values obtained at 22°C and -150°C for both studied steels are presented in Table 3 with an average uncertainty of about ±930MPa for 16MND5 steel and ±330MPa for XC40 steel. This uncertainty is linked to the mixture law that is very sensitive to errors in main phase stresses when being applied to a material with phases of very low volume fraction. In the case of 16MND5 steel, we observe that cementite stress state increases with increasing plastic deformation; the reached values are about 9000MPa. Cementite stress also increases when the temperature drops: the maximum value is 11000MPa.

In the XC40 steel, the values of stress in cementite are much lower than in the 16MND5 steel; it does not exceed 3700MPa. We note however that in this steel, the cementite stress state appears to decrease with decreasing temperature.

When comparing these values with the macroscopic or ferrite stresses in both steels, we can see that they are much higher due to the low volume fraction of cementite. Therefore, this highlights strong stress heterogeneity by phase. These values are comparable to those predicted by models that take into account even more important volume fractions [7]. But since many people wonder if it is realistic to think that this phase can really stand such loadings, it is crucial to estimate experimentally the level of stress in cementite to validate or refute these various models developed to predict the mechanical behavior of the PWR steel. This underlines the need to determine directly the stress in the cementite phase during loading using diffraction techniques more appropriate than XRD (neutron diffraction and synchrotron radiation).

Conclusion

The present experimental works were carried out in order to characterize the brittle mechanical behavior of 16MND5 steel through in situ tensile tests combined to X-ray diffraction. The obtained results of per phase stress analysis are consistent with literature. The macroscopic strength tensile of 16MND5 steel is about 700MPa at room temperature and 820MPa at -150°C. According to the stress distribution presented above, the breaking strength of the ferrite phase is estimated to 550MPa at 22°C and 600MPa at low temperature. However, more tests are necessary to confirm XC40 steel results. The volume fraction of cementite in the 16MND5 steel (2.1%) and its low carbide size require the use of tools, for stress determination, more powerful than XRD (neutron diffraction and synchrotron radiation). First stress analyses were conducted directly in the cementite phase at the Institute Laue Langevin in Grenoble; they showed that for this low volume fraction of carbides, it is necessary to use a maximum neutron flux in order to achieve useful results. New experimental tests are therefore projected in this way and tests performed at the European Synchrotron Radiation Facility in Grenoble are still under examination.

Fig. 1 :

 1 Fig. 1: a/-FEG-SEM micrograph showing the m mapping highlighting the complex structure of

Fig. 2 :

 2 Fig. 2: SEM micrograph showing the m

  σ I =f Fe σ Fe I +f Fe 3 C σ Fe 3 C I where σ I is the macroscopic stress, and (f Fe , f Fe 3 C ) are the volume fractions of ferrite and cementite. In order to determine the stress carried out in situ using a small tensile SEM micrograph showing the microstructure of the XC40

  Fe 3 C I ) are the stresses in the ferrite and cementite phase, are the volume fractions of ferrite and cementite. stress evolution of the ferrite phase during loading tensile machine placed directly under a Proto goniometer 20µm XC40 steel analysis reveals an average size of 0.73 µm: re predominantly intragranular. in the two phases during macroscopic polycrystalline material. For stress pseudo-macroscopic strain-. More details about this

:

  Experimental setting up -a/ Proto diffractometer -b/ Small tensile situ analyses

Ferrite

  b/ evolution in ferrite phase, 16MND5 steel -a/ T=22°C At room temperature, the yield strength of 16MND5 steel is about 500MPa, which is in ]. It is observed that the stress state in the ferrite . The difference between the two stresses increases slight 15 studied steels is investigated. at low temperature with a cooling system using liquid nitrogen mples during cooling operation. in uniaxial tension to 150°C. Analyses were made and using CrK α radiation tensile tests on the two materials are presented below. rves give the evolution of internal stress in the ferrite phase during macroscopic loading with ) and the macroscopic stress (continuous line) 150°C. The specimens were not equipped calculated from a force and The macroscopic stress represents the b/ °C -b/ T=-150°C steel is about 500MPa, which is in agreement ]. It is observed that the stress state in the ferrite remains close to the increases slightly with the plastic

Fig. 5 :

 5 Fig. 5: Stress evolution in ferrite phase, XC40 steel -a/ T=22°C -b/ T=-150°C

Table 1 : Chemical composition of 16MND5 steel [weight % -iron balance]

 1 Table 1 gives its chemical composition in weight percentage.

	C	S	P	Mn	Si	Ni	Cr	Mo	V. Cu. Co. Al et N
	0.159 0.008 0.005 1.37 0.24 0.7 0.17 0.5	<0.1

Table 3 : Stress distribution in both studied steels at T=22°C and T=-150°C

 3


Congcong Wang
email: congcong.wang@hig.no

Xingbo Wang
email: xingbo.wang@hig.no

Jon Yngve Hardeberg
email: jon.hardeberg@hig.no

A Linear Interpolation Algorithm for Spectral Filter Array Demosaicking

Keywords: Multispectral, Demosaicking, Linear, Residual

Linear interpolation methods have the characteristics of low computational complexity which makes them widely developed in CFA (color filter array) demosaicking. However, the trichromatic nature of colour images enables CFA demosaicking algorithms to take advantage of the luminance-chrominance representation to interpolate the colour planes efficiently and effectively. It seems, however, this does not apply to multispectral images in a straightforward manner. In this paper, we first propose a linear interpolation method for SFA (spectral filter array) demosaicking drawing on the mathematical analysis of mosaicking, demosaicking processes and the idea of residual interpolation. We then compare the performance of the proposed method with that of five other techniques by means of the SSIM index. The result shows that our new algorithm has a good performance with less computing time.

Introduction

Multispectral/hyperspectral imaging is now a promising technology for the capture, analysis and representation of objects' accurate information which is widely used in remote sensing, biological and so on. Inspired by the revolution in color imaging system caused by color filter array (CFA) [START_REF] Bayer | Color imaging array[END_REF], some research work appears and makes progress in the design of SFA (spectral filter array, a mosaic array comprised of more than three types of filters) and the associated demosaicking algorithms, aimed at a low-cost, portable and efficient multispectral imaging system, as shown in Fig. 1. However, introducing the mosaic technique into multispectral system will lose plenty of spectral information which needs to be estimated in a proper way.

Image demosaicking refers to the interpolation of raw data (CFA samples) [START_REF] Li | Image demosaicing: a systematic survey[END_REF] to obtain full resolution color images, which is a similar problem in SFA demosaicking. In the last decades, a large amont of CFA based methods have been proposed. According to whether the inter-channel correlation is utilized or not, they can be divided into two main categories [15]. The first one treats each channel separately without making use of inter-channel correlation. The techniques include nearest-neighbor interpolation, bilinear/bicubic interpolation [START_REF] Yu | Adaptive cubic convolution interpolation and sequential filtering for color demosaicing of bayer pattern image sensors[END_REF], spline interpolation [START_REF] Hou | Cubic splines for image interpolation and digital filtering[END_REF] and so on. This type of interpolation methods are simple but suboptimal. Exploiting the inter-channel information is of great significance for improving the performance of the demosaicking algorithms. The design of Bayer CFA has the properties that G channel contains the most information, and can be regard as the luminance channel which has less aliasing. Therefore, most of the methods interpolate the G channel first, then utilize the interpolated G channel for the recovery of the chrominance channels (R, B). According to the spatialdomain and frequency domain properties, there are many methods proposed, the detail information can be found in [START_REF] Li | Image demosaicing: a systematic survey[END_REF]. There are also some other approaches, such as the discrete wavelet transform (DWT) based interpolation, in which the images are transformed into various frequency bands, the high similarity between the high-frequency bands can help to recover the mosaicked images [START_REF] Chen | Color filter array demosaicking using wavelet-based subband synthesis[END_REF].

To our best knowledge, the earliest work on SFA demosaicking is proposed in 2006 by Miao et al. [START_REF] Miao | Generic MSFA mosaicking and demosaicking for multispectral cameras[END_REF], in which they extend the idea of CFA to multispectral imaging by developing generic mosaicking and demosaicking algorithms based on the binary tree theory. Later some more work has been published which focuses more on the demosaicking methods. Baone and Qi [START_REF] Baone | Demosaicking methods for multispectral cameras using mosaic focal plane array technology[END_REF] explore ways of extending the existing methods to multispectral imaging directly. They treat the demosaicking process as an image restoration task and solve the optimization problem using the gradient descent method which can also reduce the external noise and degradations. Recently, the kernel upsampling and guided filter method is also introduced into multispectral demosaicking in [START_REF] Monno | Multispectral demosaicking using adaptive kernel upsampling[END_REF] [START_REF] Monno | Multispectral demosaicking using guided filter[END_REF]. In [START_REF] Monno | Multispectral demosaicking using guided filter[END_REF], they assume that the guide image can be generated from the most densely sampled spectral component of SFA, then the other spectral components can be interpolated by the guided filter technique. Another example of extending CFA demosaicking method to SFA demosaicking is the idea Wang et al. proposed in [15]. Most of the demosaicking ideas used in SFA come from the mature CFA (especially Bayer CFA) based methods. In addition, a frequency analysis based multispectral demosaicking method is still under development [START_REF] Simon | Single Shot High Dynamic Range and Multispectral Imaging Based on Properties of Color Filter Arrays[END_REF].

Therefore, in this work, we focus on extending some linear methods to multispectral domain. This paper is organized as follows. In section 2, the proposed method is presented in detail. Result and analysis of the new method as well as other algorithms are shown in section 3. Section 4 concludes the work and suggests directions for future research.

Proposed Method

In order to design a generic algorithm that can handle the diversified SFA patterns, we need go back to the mathematical analysis of mosaicking and demosaicking as shown in [START_REF] Chaix De Lavarène | Practical implementation of lmmse demosaicing using luminance and chrominance spaces[END_REF]. Assume Y is a full resolution multispectral image and X is the corresponding SFA image, the mosaicking process can be presented in equation 1:

X = P Y. ( 1
)
where P represents the mosaic which converts the original full resolution image Y to SFA image X. If the superpixels are stacked in a row-wise way, the size of matrix P is fixed [START_REF] Chaix De Lavarène | Practical implementation of lmmse demosaicing using luminance and chrominance spaces[END_REF]. If we can estimate P, the demosaicking process can be written as Ỹ = P -1 X = DX. It is not sufficient to estimate Ỹ from P and X. Therefore, a larger kernel contains the information of the neighborhood pixels is required, we use a n × n window here.

Assume the SFA has 4 channels and we use R channel in Fig. 2 for example. Every R value is estimated by its neighbors in the n × n window. Regardless of the size of the image, the size of matrix D is n 2 ×4×4. n 2 represents the window's size, w ij are the weights of different neighbors within the window. The first '4' means there are four types of R values: R values in position 1, 2, 3 and 4 which is determined by the SFA pattern. The second '4' means we have 4 channels in the SFA. The red rectangles of Fig. 2 show the computational process in recovering R1.

If we divide an original full resolution image Y by the mosaicked image X, we can estimate the parameters of D. D depends on the instinct characters of the SFA pattern, and represents the correlations between one pixel and its neighbouring pixels in the vicinity of n 2 window. Then the trained D can be used to estimate Ỹ for other mosaicked images X. Apparently, D is a global matrix which is sensitive to noise and will introduce large residual for a local particular pixel.

In order to reduce the artifacts, we also propose the second step of this method. In paper [START_REF] Kiku | Residual interpolation for color image demosaicking[END_REF], they propose a residual interpolation idea to replace the color difference interpolation in CFA based demosaicking. The residual idea is suitable for multispectral images without other constraints. In Fig. 3, channel 1 is shown as an example. Firstly, the instinct matrix D is applied to the original mosaicked image, after which the estimated full resolution image Ỹ is obtained by Ỹ = DX, called tentative estimate image. Secondly, each tentative estimated channel is subtracted by the original mosaicked channel in order to get the residuals ∆. Then simple linear interpolation method is performed to get the estimated residuals ∆'. Finally, the demosaicked image is acquired by adding the estimated residuals to the tentative estimate image.

1 2 1 2 3 4 3 4 1 2 1 2 3 4 3 4 3' 3' 3' 3' 3' 3' 3' 3' 3' 3' 3' 3' 3' 3' 3' 3' 2' 2' 2' 2' 2' 2' 2' 2' 2' 2' 2' 2' 2' 2' 2' 2' 1' 1' 1' 1' 1' 1' 1' 1' 1' 1' 1' 1' 1' 1' 1' 1' 4' 4' 4' 4' 4' 4' 4' 4' 4' 4' 4' 4' 4' 4' 4' 4' Step1: X D Y 1 0 1 0 0 0 0 0 1 0 1 0 0 0 0 0 ∆1' 0 ∆1' 0 0 0 0 0 ∆1' 0 ∆1' 0 0 0 0 0 ∆1' ∆1' ∆1' ∆1' ∆1' ∆1' ∆1' ∆1' ∆1' ∆1' ∆1' ∆1' ∆1' ∆1' ∆1' ∆1' 1'' 1'' 1'' 1'' 1'' 1'' 1'' 1'' 1'' 1'' 1'' 1'' 1'' 1'' 1'' 1''

Experimental Result

Due to the practicability of realizing mosaic-mounted image sensors, we conducted experiments on a simulated platform [15]. A set of hyperspectral images are used as approximations to irradiance images which are spatially sampled and contain rich spectral information. Simulated SFAs then spectrally filter the irradiance images, and a simulated sensor with specified spectral sensitivities produces mosaic multispectral images. A replacement of SFAs with the corresponding set of filters will generate full resolution multispectral images with which the performance of demosaicking algorithms can be evaluated by comparing these "original" images with demosaicked images.

The CAVE hyperspectral image database [START_REF] Yasuma | Generalized assorted pixel camera: Post-capture control of resolution, dynamic range and spectrum[END_REF] was used in the experiment, which contains 32 reflectance images of real-world materials and objects. This images comprise 31 bands ranging from 400 nm to 700 nm with an interval of 10 nm, which defines the spectral range and accuracy of the experiments. An image named 'balloons' was employed for training purpose, and another 20 images selected from CAVE database are used for testing purpose. The images are then lit by the CIE D65 illuminant.

In this work, we experimented with two given types of SFAs, i.e., a 4-band SFA and a 5-band SFA, as shown in Fig. 1. The spectral transmittances of the filters are set so that the peak transmittances sample the visible spectrum evenly. In order to simplify the simulation, an ideal sensor with constant spectral sensitivities across the spectrum was utilised.

For the sake of comparison, we also implemented, in addition to the proposed method, another 5 techniques of multispectral demosaicking including bilinear interpolation [START_REF] Yu | Adaptive cubic convolution interpolation and sequential filtering for color demosaicing of bayer pattern image sensors[END_REF], channel difference interpolation extended from the color difference method [START_REF] Ramanath | Demosaicking methods for bayer color arrays[END_REF], LMMSE method [START_REF] Chaix De Lavarène | Practical implementation of lmmse demosaicing using luminance and chrominance spaces[END_REF] based on constant difference, binary tree based demosaicking [START_REF] Miao | Binary tree-based generic demosaicking algorithm for multispectral filter arrays[END_REF] and discrete wavelet transformed based approach [15].

Please note that both of the SFAs were designed artificially. As the binary tree based approach works only with mosaic patterns that the accompanying method generates [START_REF] Miao | Generic MSFA mosaicking and demosaicking for multispectral cameras[END_REF], the patterns for this method is slightly different from the SFAs shown in Fig. 1. And in the channel difference interpolation, the third filter is considered as the reference channel as the peak transmittances lie at about 550 nm for both filter sets used in the two SFAs.

There exists a variety of image quality evaluation methods to assess the performance of interpolation algorithm: PSNR (peak signal-to-noise ratio), SSIM (structural similarity) index, color appearance models and so on. Here, we use the SSIM metric. The SSIM index defines structural information in an image as attributes that represent the structure of the objects in the scene, independent of the average luminance and contrast. The index is based on a combination of luminance, contrast, and structure comparison. A value of 1 means perfect match between the two input images. Detail information and analysis about SSIM is shown in [START_REF] Wang | Image quality assessment: from error visibility to structural similarity[END_REF].

The SSIM values of 20 images for 4-band SFA demosaicking is shown in Table 1(a), the corresponding plot is shown in Fig. 4. Table 1(b) and Fig. 5 present the result of 5-band SFA demosaicking. The overall result of 5-band SFA demosaicking is not as good as 4-band's case mainly because of the coarser spatial distribution.

In general, the binary tree based method performs fairly well, and can be seen as a benchmark. However, it cannot be compared directly with other methods as the mosaics are not the same.

Our proposed method outperformed the binary tree approach and achieved the best performance for 4-band SFA, because it is not sensitive to the contents of different images. Although, the less information contained in a local window can exaggerate the role of noise which results in the decline of SSIM values for 5-band SFA, it outperforms most of the other methods except binary tree method.

In spite of its simplicity, the channel difference based interpolation [START_REF] Ramanath | Demosaicking methods for bayer color arrays[END_REF] yielded promising results especially in the case of the 5-band SFA demosaicking. This suggests that the images tested possess a great deal of smooth chromatic transition.

The inferior results of DWT based approach [15] may indicate that the interchannel correlation in high-frequency region of the images is not sufficiently high. Besides, the different signatures of each band will reduce the smoothness of inter-band edge information, thus resulting in a worse performance for the 5-band SFA. The LMMSE method [START_REF] Chaix De Lavarène | Practical implementation of lmmse demosaicing using luminance and chrominance spaces[END_REF] needs more strict assumption, for example the second-order directional Laplacian filter of Adams and Hailton [START_REF] Adams | Adaptive color plane interpolation in single sensor color electronic camera[END_REF] requires constant color difference in either horizontal or vertical direction.

As expected, bilinear interpolation is the worst in this comparison, however the extremely low SSIM indices are worth further investigations.

Considering the computational efficiency, our proposed method only costs 0.5 seconds on average which shows great improvement compared to all the other methods. Next comes the binary tree method whose time consumption is 2.5 seconds on average, the other methods cost around 5 seconds. Also increasing number of bands led to higher computational cost.

Conclusion

The assumptions on which a variety of CFA demosaicking methods are based are very often not applicable to the multispectral domain [START_REF] Miao | Binary tree-based generic demosaicking algorithm for multispectral filter arrays[END_REF]. As an example, the luminance-chrominance color space used for CFA pattern is senseless for SFA demosaicking, as a result of the difficulties in extracting luminance and chrominance information separately in multispectral images.

In this paper, we propose a linear method that combine linear Wiener estimation and an interpolation on residual channels in the context of multispectral demosaicking. The latter may overcome the error introduced by the former's sensitivity to noise. This method yielded an average SSIM value of 0.994465 for a 4-band SFA and 0.978215 for a 5-band SFA with the running time of about 0.5 seconds.

This method has several advantages. First, its dependency on the mosaic design is not as high as the other methods. Second, the residual interpolation is of high scalability. Third, the computation efficiency is improved greatly in comparison with other methods. Certainly this method has some disadvantages as well. It requires a priori information and it is sensitive to noise.

The SFA patterns were randomly designed in this article. The rules of designing an efficient SFA with high inter-channel correlation merit further development. In turn, these rules can be used to design homologous demosaicking algorithms. From the performance of the binary tree based interpolation, we can conclude that the relation between mosaic design and demosaicking is important. Therefore, for multispectral demosaicking, it is beneficial to consider mosaic design and demosaicking algorithms as a whole as suggested by Miao et al. [9][10].

Fig. 1 .

 1 Fig. 1. Spectral filter arrays (left: 4-band, right: 5-band) with corresponding spectral transmittances of the filter sets.

Fig. 2 .

 2 Fig.2. Illustration of how a full resolution image Ỹ is constructed from a matrix multiplication between mosaicked image X and the matrix D (use R channel for example).

Fig. 4 .

 4 Fig. 4. SSIM index values of the 4-band SFA.

Fig. 5 .

 5 Fig. 5. SSIM index values for the 5-band SFA.

Table 1 .

 1 Framework of the second step of the proposed method. Performance of the algorithms (a) Results of 4-band SFA demosaicking

	Mosaicked images	Tentative estimate
		Residuals ∆	Estimated residuals ∆'	Demosaicked images
	Fig. 3.


Xingbo Wang
email: xingbo.wang@hig.no

Jean-Baptiste Thomas
email: jean-baptiste.thomas@u-bourgogne.fr

Jon Yngve Hardeberg

Pierre Gouton
email: pierre.gouton@u-bourgogne.fr

DISCRETE WAVELET TRANSFORM BASED MULTISPECTRAL FILTER ARRAY DEMOSAICKING

Keywords: Multispectral, imaging, filter array, demosaicking, discrete wavelet transform

The idea of colour filter array may be adapted to multispectral image acquisition by integrating more filter types into the array, and developing associated demosaicking algorithms. Several methods employing discrete wavelet transform (DWT) have been proposed for CFA demosaicking. In this work, we put forward an extended use of DWT for multispectral filter array demosaicking. The extension seemed straightforward, however we observed striking results. This work contributes to better understanding of the issue by demonstrating that spectral correlation and spatial resolution of the images exerts a crucial influence on the performance of DWT based demosaicking.

INTRODUCTION

Man-made image sensors permit the conversion of an optical image to its electronic representation. Nonetheless, they are incapable of distinguishing the incident irradiance at one wavelength from another since all that matters is the number of photons. However, what precisely produces the sensation of colour is the form that the spectral power distribution of radiation takes. Therefore single-sensor based electronic trichromatic imaging systems mostly employ an array of colour filters in order to sense a portion of the spectra selectively on a pixel-by-pixel basis, which permits the acquisition of a colour image by means of a single sensor at one exposure, at the cost of reduced spatial sampling rate. A companion post-processing step, called demosaicking, becomes essential to estimate the lost information thus recovering colour images of full resolution. An example of such a colour filter array (CFA) that has achieved commercially notable success is known as Bayer filter mosaic [START_REF] Bayer | Color imaging array[END_REF], named after its inventor, Bryce E. Bayer. A Bayer-like filter array meant for colour image acquisition normally consists of three types of filters, i.e., red, blue and green, as shown by the leftmost pattern in Figure 1(a). Since one CFA is typically customised for a specific sensor design, each pixel on the sensor corresponds to a filter tile of one particular type. In other words, parts of the incoming spectra pass through the filters and strike the sensor, whereas other parts are either absorbed or reflected by the filter and thus lost. Nonetheless, thanks to the spatial and spectral inter-pixel correlation an image may possess, the lost information about the spectra can be estimated through demosaicking, an operation in the image processing chain carried out on the mosaicked image read from the sensor. Consequently each pixel will comprise three components, thereby resulting in a full colour image.

A large number of CFA demosaicking algorithms have been proposed over the decades [START_REF] Menon | Color image demosaicking: An overview[END_REF]. In general, these methods can be divided into two main categories according to whether the inter-channel correlation is utilised. The former includes the techniques such as nearest-neighbour, bilinear and bicubic [START_REF] Samuel S Rifman | Digital rectification of ERTS multispectral imagery[END_REF] interpolation and more complicated methods take edges into account in order to avoid cross-edge interpolation [START_REF] Robert | Apparatus and method for adaptively interpolating a full color image utilizing luminance gradients[END_REF]. Methods falling into this category estimate unknown colour components by exploiting the spatial correlation between sampled colours in a certain spectral plane, and function plane by plane. Algorithms of the other group bring the inter-channel correlation into play in a number of ways. For instance, some interpolate the difference between one colour plane and another [START_REF] Jr | Interactions between color plane interpolation and other image processing functions in electronic photography[END_REF] or the ratios of one colour plane to another [START_REF] Cok | Signal processing method and apparatus for producing interpolated chrominance values in a sampled color image signal[END_REF], and some utilise the inter-channel differently by observing and filtering the spectra in frequency domain [START_REF] Alleysson | Color demosaicing by estimating luminance and opponent chromatic signals in the fourier domain[END_REF][START_REF] Glotzbach | A method of color filter array interpolation with alias cancellation properties[END_REF]. In addition, discrete wavelet transform (DWT) also stimulates particular research interest, since it transforms an image into various frequency bands, and natural images often possess rather similar high-frequency information among these bands, which provides yet another solution to demosaicking problem.

The idea of capturing a colour image by means of a CFA fits itself for the multispectral image acquisition as well. In recent years, sustained research effort went into designing multispectral filter array (MSFA) and developing associated demosaicking algorithms. The first notable work in this regard was conducted by Miao et al. [START_REF] Miao | A generic method for generating multispectral filter arrays[END_REF] who develop a generic binary tree based generation method of MSFA spatial ar- rangement starting from a chequerboard pattern, and further proposed a generic MSFA demosaicking method following the same binary tree structure [START_REF] Miao | A generic binary tree-based progressive demosaicking method for multispectral filter array[END_REF]. Baone and Qi [START_REF] Gaurav | Demosaicking methods for multispectral cameras using mosaic focal plane array technology[END_REF] treat the MSFA demosaicking problem as an image restoration problem, and another interpolation algorithm introduced by Brauers and Aach [START_REF] Brauers | A color filter array based multispectral camera[END_REF] low-pass filters colour channel differences. Monno et al. [START_REF] Monno | Multispectral demosaicking using adaptive kernel upsampling[END_REF] propose a novel multispectral demosaicking algorithm by extending existing upsampling algorithms and further interpolate each spectral component with guided filters capable of preserving structures [START_REF] Monno | Multispectral demosaicking using guided filter[END_REF]. As an example of adapting CFA demosaicking algorithms to MSFA demosaicking, Wang et al. [START_REF] Wang | Median filtering in multispectral filter array demosaicking[END_REF] investigate median filtering operation in MSFA demosaicking and post-processing with the aim of alleviating colour artefacts. The intrinsic similarity between colour and multispectral images in terms of inter-band spectral correlation makes the utilisation of DWT potentially beneficial to MSFA demosaicking.

In this work, we first present a demosaicking algorithm making use of DWT. It begins by a review of related work in Section 2. A description of the proposed method is explained in Section 3. An introduction of the experiments and the results are shown in Section 4. However, we observed unexpected results and subsequently performed brief analyses on the results in Section 5 which leads to reasonable conclusions drawn in Section 6.

CFA DEMOSAICKING BASED ON DWT

In recent years, DWT is referred to and taken advantage of by a succession of research work on CFA demosaicking. As known, DWT makes use of subband coding in which an image is decomposed into a set of bandlimited components. A commonly seen four-band decomposition splits an image into four quadrants containing an approximation subband, a horizontal detail subband, a diagonal detail subband, and a vertical detail subband, respectively. All subbands, except of the approximation subband, represent high frequency information at a certain scale. It is the subband coding that builds a viable foundations for DWT based demosaicking by taking into account the inter-channel correlation most images bear.

To the best of our knowledge, Gunturk et al. [START_REF] Bahadir | Color plane interpolation using alternating projections[END_REF] first bring the DWT in CFA demosaicking. The algorithm begins by an initial estimation of all channels by means of intrachannel interpolation. It then decomposes the channels with a filter bank into four sub-bands and the high-frequency subbands of the red and blue channels are updated according to some criteria. Next, the ground truth red/blue samples are inserted back to the reconstructed channels. The last two steps are repeated until a stopping criterion is satisfied. The initial estimates of the green samples follow the same procedure as red/blue samples do.

Driesen and Scheunders [START_REF] Driesen | Wavelet-based color filter array demosaicking[END_REF] propose a similar approach. However the initial interpolation is performed on the luminance image derived from the original RGB channels, as the interpolated luminance image possess higher spatial resolution than interpolated R/G/B channels. Then both the luminance image and the original RGB channels are discrete wavelet transformed resulting in wavelet coefficients. Two merging rules are applied singly in order to benefit from the inter-channel correlation. One of them, named "replace" rule, simply replaces at each scale the coefficients of each band of R/G/B band images with the corresponding coefficients in the luminance image, whereas the other, labelled "max" rule, compares the coefficients of each R/G/B band with those of the luminance image, and assigns the larger one to each R/G/B band. The demosaicking is accomplished when the updated wavelet coefficients are inversely transformed.

Chen et al. [START_REF] Chen | Subband synthesis for color filter array demosaicking[END_REF] introduce the concept of downsampled (DS) images. First, this method is initiated by an estimation of missing green pixels with edge-directed interpolation and missing red/blue pixels with bilinear interpolation, from which the DS images are then derived and wavelet transformed. Next, the "replace" rule is applied to the highfrequency sub-bands. The last step aims to reduce colour artefacts by median filtering the low-frequency wavelet subbands and updating high-frequency sub-bands of Red/Blue images with those of the Green images. A simplified version of such an idea was implemented by Courroux et al. [START_REF] Courroux | A wavelet-based demosaicking algorithm for embedded applications[END_REF].

Jeong et al. [START_REF] Gyu | Edge adaptive demosaicking in wavelet domain[END_REF] also use the polyphase-like downsampling prior to DWT. The wavelet coefficients of the low frequency sub-bands are then estimated by means of an edge adaptive interpolation method using high frequency wavelet coefficients as the edge indicators. An estimation of the coefficients of the high frequency sub-bands is performed in accordance with the "replace" rule.

Slightly different from the above approaches, Kim et al. [START_REF] Hyuk | Wavelet-domain demosaicking using linear estimation of interchannel correlation[END_REF] perform DWT on observed DS images to obtain the missing high-frequency sub-bands by linear estimation, and estimate the low-frequency sub-bands by wavelet transforming the DS images that have been interpolated beforehand.

Indicated by the aforementioned literature, DWT based demosaicking methods outperform the conventional counterparts relying merely on intra-channel correlation.

MSFA DEMOSAICKING BASED ON DWT

As mentioned above, DWT decompose images into a series of sub-bands with different frequency components. The spectral correlation, namely the inter-plane similarities, then comes into play. In theory, the idea is also applicable to multispectral images. Inspired by this, we extend the application of DWT into MSFA demosaicking.

The essence of the proposed method may be encapsulated by the concept of DS images, the Haar wavelet (D2), the "replace" rule for the estimation of high-frequency sub-bands and bilinear interpolation for the estimation of low-frequency sub-bands. The workflow is summarised as follows.

1. Estimate high-frequency coefficients. 3. Recombine the low-frequency and high-frequency components and apply inverse discrete wavelet transform.

Reconstruct the demosaicked image of full resolution.

This method is compatible with any MSFAs with regular mosaic patterns regardless of number of channels. For the purpose of evaluation and comparison, we apply this method to a set of RGB images in addition to multispectral images.

EXPERIMENTS AND RESULTS

Two hyperspectral image database were used in this study. 12 hyperspectral reflectance images are from Foster et al. [START_REF] Foster | Frequency of metamerism in natural scenes[END_REF]. The test set consists of a mixture of rural scenes containing rocks, trees, leaves, grass, and earth and urban scenes. The spectral span ranges over 400-720 nm in 10 nm steps yielding 33 bands, while scene 5 has only 32 bands due to the noisy 720 nm band. Another 32 hyperspectral images are from the CAVE project [START_REF] Yasuma | Generalized assorted pixel camera: Post-capture control of resolution, dynamic range and spectrum[END_REF]. Divided into 5 sections, the images are of a wide variety of real-world materials and objects, covering various colourful stuffs, skin, hair, paints, food, drinks and fake stuffs mimicking the colour of real objects mentioned. The spectral span ranges over 400-700 nm in 10 nm steps yielding 31 bands. For the ease of processing and comparison, the spectral range between 410 nm and 700 nm was used in this study resulting in 44 hyperspectral images of 30 bands. A framework that simulates the key elements of an imaging system was built as a testbed.

Also employed are 24 Kodak RGB test images [START_REF]Kodak lossless true color image suite[END_REF] for the sake of evaluation of the proposed methods. There images are widely used in the realm of image related research, and are utilised to evaluate the performance of algorithms in the majority of aforementioned papers about DWT based CFA demosaicking.

In addition, one artificial pattern that includes one circle and one line and resembles the first publicly broadcast test card, was also created and employed as a test image. It has 30 bands as well, but only two reflectance values, namely, 0 and 1. To conduct the test, two steps were followed. In case of hyperspectral image sets, we decided on a set of algorithms and parameters, as described below.

1. Images were rendered with the illuminant of CIE D65; 2. In each case, spectral transmittances were determined so that each of them had a regular Gaussian shape and the centres of them were evenly distributed over the pertinent spectrum with the distance of 2 × σ, as shown in Figure 2; 3. Three filter array patterns were chosen, Bayer type 3band setup, 4-band setup in form of 2×2 moxel 1 and 8band in form of 4 × 4 moxel, the moxels were repeated across the whole image;

4. The filter arrays were designed in accordance with the binary tree approach proposed by Miao et al. [START_REF] Miao | A generic method for generating multispectral filter arrays[END_REF], a perfect binary tree to be exact, with two levels and three levels corresponding to 4-band and 8-band arrangements respectively, as indicated in Figure 1; 5. Two additional demosaicking algorithms, bilinear interpolation and Miao et al.'s binary tree based progressive demosaicking [START_REF] Miao | A generic binary tree-based progressive demosaicking method for multispectral filter array[END_REF], were implemented and compared with the proposed method; 6. To visualise the images, the original hyperspectral images were rendered in the sRGB colour space, and the demosaicked multispectral images were first restored to hyperspectral images with a spectrum reconstruction method exploiting a priori knowledge of the imaged objects [START_REF] Yngve | Acquisition and Reproduction of Color Images: Colorimetric and Multispectral Approaches[END_REF] and then rendered to sRGB.

7. Evaluation of the performance was carried out by means of peak signal-to-noise ratio (PSNR) computed between the original and reproduced multispectral images, so as to avoid the error introduced by the spectrum reconstruction.

In case of Kodak images, we follow step 5 & 7 described above.

Results for Kodak and hyperspectral images are shown in Figure 3-6 respectively. As can be seen, the proposed method outperforms two other algorithms for most Kodak RGB images. Surprisingly and interestingly, the order of performance is almost inverted in case of hyperspectral images regardless of the number of channels.

DISCUSSION

Indicated by the results is a seemingly obscure issue. Nevertheless, further analyses of the cross-correlation between high frequency sub-bands of the DS images partly uncover the mystery. As shown by Figure 7, Kodak images represented in yellow circles possess relatively higher correlation 1 The smallest pattern repeated to form the MSFA. [START_REF] Wang | Median filtering in multispectral filter array demosaicking[END_REF] than the multispectral images plotted in red/green/blue. Further, PSNR difference between the proposed method and bilinear interpolation bears a somewhat linear relationship with the cross-correlation between the DS images. That is to say, the higher the correlation of the images is, the more we may gain from the proposed DWT demosaicking approach. It is consistent with the basic assumption and principles of the method.

Obviously the spectral correlation of a multispectral image depends not only on the original scene and the illumination, but also on the design of filters present on the CFA/MSFA. Tests performed have shown that the correlation increases when degree of overlap between adjacent filters increases. In addition, Kodak images were originally acquired by film cameras and scanned, whereas the hyperspectral images employed in our study were captured by digital cameras and filtered with idealised Gaussian-shaped filters. The discrepancy between the two systems might influence the correlation in a way.

Also notable is the fact that most of the images from the Foster set appear blurred to some extent in comparison with the CAVE set and Kodak images. This might be pertinent to chromatic aberration or vibration in the course of acquisition, and this also influences the spectral correlation. To verify the hypothesis, we blurred Kodak images by means of the convolution of each plane of an image and a Gaussian kernel and applied the same algorithms as in the previous experiment. Figure 8 illustrates the case when the kernel measures 5 × 5 pixels with σ of 3. Clearly the DWT based algorithm turns the worst in contrast to the other two, contrary to the situation shown in Figure 3, which again illustrates the dependence of DWT demosaicking on the intrinsic properties of images.

CONCLUSION

In this work, we first propose to adapt DWT based MSFA demosaicking algorithms to multispectral demosaicking. They work quite well with the Kodak test image set, however an extension of such a method does not perform satisfactorily with the hyperspectral image set employed in the study. Experiments and analyses indicate that the spectral cross-correlation plays an important role in determining the performance of the proposed algorithm. In other words, the "replace" rule depends largely on the assumption that the images are highly correlated, which explains the unexpected results. More indepth and theoretical analyses are expected in order to evaluate the appropriateness of the use of DWT transform in multispectral demosaicking.

Fig. 1 :

 1 Fig. 1: Examples of CFA and MSFA design.

2 .

 2 (a) Construct DS image using polyphase transform. (b) Apply discrete wavelet transform. (c) Estimate the coefficients of the missing DS images at high-frequency sub-bands according to the "replace" rule. Estimate the low-frequency coefficients. (a) Apply bilinear interpolation to the mosaicked image plane by plane. (b) Construct DS image using polyphase transform. (c) Apply discrete wavelet transform. (d) Replace the coefficients of the missing DS images at low-frequency sub-bands with those of the interpolated DS images.

Fig. 2 :

 2 Fig. 2: Filter design for 3-band, 4-band and 8-band set-up.

Fig. 3 :Fig. 4 :

 34 Fig. 3: Comparison of performance on Kodak images

Fig. 5 :Fig. 6 :

 56 Fig. 5: Comparison of performance on 4-band multispectral images

Fig. 7 :Fig. 8 :

 78 Fig. 7: Difference of PSNR between proposed method and bilinear interpolation