Laurent Zimmer
email: laurent.zimmer@em2c.ecp.fr

Seiji Yoshida

Feasibility of Laser Induced Plasma Spectroscopy for measurements of equivalence ratio in high-pressure conditions

Keywords: Plasma spectroscopy, Laser induced plasma, High-pressure, Equivalence ratio measurements

In this paper, experimental results obtained with Laser Induced Plasma Spectroscopy to retrieve local compositions are presented for an ambient pressure up to 5.0 MPa in a still cell. Well controlled mixtures of gases are introduced and plasma is obtained with the fundamental emission of a pulsed Nd:YAG laser. Simultaneously, plasma shape and spectrally resolved data are taken with a temporal resolution down to 2 ns. First, the temporal evolutions of a high-pressure nitrogen plasma are analyzed as function of spark energy. It is shown that plasma changes orientation from an elongated shape parallel to the laser line to a perpendicular one in a very short time. Results are reported for both spatial and spectral variations. Afterwards, the effects of increased carbon concentration are discussed in both shape and spectra. It is seen that strong intensity due to the atomic carbon emissions appear for the high-pressure case. From those experiments, calibration strategies are proposed to get equivalence ratio under high pressure conditions with a ratio of carbon versus nitrogen and oxygen. The delay between plasma and measurements is set to 2000 ns and the signal is integrated for 5000 ns, so as to yield a good signal to noise ratio and a good sensitivity of the technique to changes in mixture fraction. Calibration curves are reported for equivalence ratio up to 1.00 and for pressure from 1.0 to 5.0 MPa. It is shown that typical uncertainties are limited to 7.5% regardless the equivalence ratio in a single shot approach using a spectral fit procedure,whereas it accounts to two times more in a more classical peak ratio approach. Increasing the pressure tends to increase the precision as lower pressure had higher uncertainties.

Introduction

Measurements of local composition are of great interest for combustion diagnostics. Whereas different strategies may exist for atmospheric conditions, an increase in pressure tends to induce some difficulties. Using gas sampling techniques requires to have sealed working conditions while keeping the possibility to use traversing systems to have multiple points diagnostics. Alternative techniques to the gas sampling using laser techniques have been reported. One may apply a tracer LIF approach, like reviewed in [START_REF] Schulz | Tracer-lif diagnostics: quantitative measurement of fuel concentration, temperature and fuel/air ratio in practical combustion systems[END_REF] to relate the intensity of a specific molecular fluorescence to the actual fuel to air ratio. This requires the precise knowledge of the amount of the naturally present tracers with respect to overall mass of the fuel or to have a precise control of the injection of the seeded tracers. However, some cautions have to be put when applying this strategies to higher pressures, as reported by [START_REF] Devillers | Investigation of toluene lif at high pressure and high temperature in an optical engine[END_REF] who studied toluene fluorescence for pressures up to 2.0 MPa. Other approaches, based on different molecules, like vapor Kerosene [START_REF] Orain | Simultaneous measurements of equivalence ratio and flame structure in multipoint injectors using plif[END_REF]) proved to work for pressures around 2.2 MPa. However, no discussion on single shot uncertainties were presented. Alternatives to tracer LIF may be absorption approaches, based on C-H stretching vibration for instance [START_REF] Hanson | Applications of quantitative laser sensors to kinetics, propulsion and practical combustion system[END_REF]). This kind of approach is leading generally to line of sight measurements, which may not reveal some inhomogeneities in the process. But this kind of technique provides a good overview of the combustion chemistry. Alternative techniques like Raman and Rayleigh may be possible even at high pressures for gaseous fuel but are not possible choices when dealing with liquid fuel, due to much lower threshold in energy for breakdown to occur. Some results were presented with hydrogen for pressures up to 30 bar, but the single shot uncertainty was estimated to be of the order of ±10% [START_REF] Kojima | Measurement and simulation of spontaneous raman scattering in high-pressure fuel-rich h2-air flames[END_REF]. Due to higher density, maximum fluence before gaseous breakdown tends to decrease; spark's occurrence increases. This is further amplified when using liquid fuel, due to the presence of droplets. One may take advantage of that phenomena to use the spectra from the plasma in order to determine the local concentration of fuel versus air. This technique is called Laser Induced Plasma Spectroscopy and was first introduced in [START_REF] Schmieder | Combustion applications of laser induced breakdown spectroscopy[END_REF] for combustion applications and since that used in many different cases (see for instance [START_REF] Phuoc | Laser-induced spark for measurements of the fuel-to-air ratio of a combustible mixture[END_REF], [START_REF] Sturm | Laser-induced breakdown spectroscopy of gas mixtures of air, co2, n2, and c3h8 for simultaneous c, h, o, and n measurement[END_REF], [START_REF] Stavropoulos | Quantitative local equivalence ratio determination in laminar premixed methane-air flames by laser induced breakdown spectroscopy (libs)[END_REF]). Different strategies may be applied when measuring at atmospheric pressure. The most commonly used consists in using the ratio of hydrogen (Hα at 656 nm or H β at 486 nm) over an atomic emission of oxygen (usually 777 nm) or nitrogen (around 744 nm) like discussed in [START_REF] Phuoc | Laser-induced spark for measurements of the fuel-to-air ratio of a combustible mixture[END_REF]. Other strategies have been proposed where the atomic emission of carbon at 711 nm is used to depict the concentration of fuel whereas oxidizer is determined using the triplet of nitrogen emission around 742 -746 nm (see [START_REF] Ferioli | Measurements of hydrocarbons using laser-induced breakdown spectroscopy[END_REF]). Traditional approaches use spark energy for classification; meaning that the same spark energy is assumed when doing experiments and calibration. An alternative approach has been presented by [START_REF] Zimmer | Laser induced plasma spectroscopy for local equivalence ratio measurements in an oscillating combustion environment[END_REF] in which the actual excitation of the plasma is used when applying the calibration curves. This consists in using different emission lines of the same atom (typically hydrogen) to determine a mean temperature of the plasma and then using appropriate calibration curves to link the different atoms to their original concentration. Single shot uncer-tainty for mixtures around equivalence ratio of 0.5 was found to be around 7% in a spark energy classification scheme and dropped to 3% for the excitation scheme. It was also shown that ambient temperature had no effect on the calibration, nor a change in the laser source. Further applications to different gaseous mixtures (hydrogen-air and helium-air) as well as some theoretical aspects have been recently proposed by Zimmer et al. (2007). It was shown that N/O can also be used for getting an insight in the excitation of the plasma, even though less sensitive when compared to the original hydrogen ratio. Whereas those measurements were done in atmospheric conditions, the present paper deals with cases under higher pressures, from 0.1 to 5.0 MPa for different mixture of gases. Experiments of LIPS applied to high-pressure cases were already reported but for solid targets [START_REF] Arp | Feasibility of generating a useful laser-induced breakdown spectroscopy plasma on rocks at high pressure: preliminary study for a venus mission[END_REF]. It was for instance shown that typical radii of plasma were smaller when increasing the pressure by a factor a cube root of the ratio of the pressure. Another issue that was demonstrated is the problems of self-absorption when increasing the pressure. Self-absorption could be detected through the broadening of the emission and the resulting decrease of intensity in the middle of the emission profile for a specific line. The limited effects of broadening was also shown by [START_REF] Vors | Laser-induced breakdown spectroscopy of carbon in helium and nitrogen at high pressure[END_REF] who studied carbon emissions in nitrogen gases. The pressures tested were ranging up to 8.0 MPa. The carbon emission considered was the 247.86 nm and only time integrated data (over 20 µs) were presented. The lifetime of the plasma was shown to decrease with pressure. However, in none of the previous investigations, a detailed description of the dynamics of those high pressure cases was described. Furthermore, no quantitative evaluation of the recorded spectra were proposed. The relative emission of the carbon was considered by integrating the signal around the emission peak and neighbourtgh emissions exceeding tenth of peak height. Interestingly, no self-absorption was noticed with the selected carbon emission. The purpose of the present paper is to investigate the dynamics of high-pressure laser induced plasma and to propose strategies to get quantitative data (in the present case in terms of equivalence ratio) even at 5.0 MPa. The second section of the article presents the details of the experimental setup. In the third section, the spatial evolution of the plasma as function of plasma energy, gaseous composition and pressures is presented and discussed. The fourth section deals with the spectroscopic data while the fifth shows the two different strategies for getting quantitative data. Finally conclusions are drawn for those high pressure cases and typical applications of the technique are suggested.

Presentation of the setup and processing techniques

To conduct high-pressure tests, a specific test-cell has been built. It is equipped with three optical accesses, two for the laser beam and one perpendicular for imaging purpose. The high-pressure test cell has a length of 198 mm and is equipped with a fan and a vacuum pump. Before each test, vacuum is made down to 0.001 MPa. To reach the desired mixture, four different gases are used: N 2 , O 2 , CH 4 and CO 2 . To achieve a homogeneous mixture, gases are introduced according to their partial pressure (lowest being first introduced) and final temperature is around 293K. Thermocouple as well as pressure measurements are performed while making the measurements to ensure that the conditions stay stable within the experimental campaign. Equivalence ratio is computed on a basis of atomic ratio between carbon and oxygen, assuming the fuel may be represented by CnH 2n+2 . If applied to practical flames, the technique is not sensitive to the actual molecules composing the fuel as all molecules will be dissociated into atoms. The fundamental emission from a Nd:YAG (Quanta Ray Spectra Physics, 1064 nm) is used to create a plasma. Its beam is focussed through a series of concave (f=-30 mm) and convex lenses (f=120 mm and f=250 mm) into the test cell without damaging the windows. Two energy meters are used to measure incident and outgoing laser energy (Ophir Head). To calibrate the energy meter, vacuum is created inside the cell so that plasma can not be achieved, even when passing through the different optics with a high energy. A convex lens (f=120 mm) is used after the cell to converge the laser beam to the energy meter. A linear correlation is obtained and is used to measure the spark energy. It is assumed that the difference detected on the two energy meters comes only from the creation of the spark and therefore the spark energy is attributed this difference. A mirror for 1064 nm is used between the laser and the cell to allow emission analysis in a backward configuration (see Figure 1). This ensures that even for fluctuations in the position of the plasma along the axis of the laser beam, spectra are always measured completely and with the same efficiency with the spectrometer.

The signal from the plasma is provided to a spectrometer through an optical fiber. This spectrometer is equipped with a grating of 150 lines/mm and its slit is set at 50 µm and connected to an ICCD (Andor, 1024x1024) used in full vertical binning mode. The single shot measurements can encompass a spectral band of 320 nm. To achieve fully spectrally resolved measurements, two different series are taken. In the first series, the center wavelength is set at 470 nm, while in the second one, it is set at 740nm. Finally, the region accessible with this setup is therefore 300-910 nm. To monitor both plasma shape and position, an ICCD (Roper Scientific, 1024x1024) equipped with an objective of 105 mm focal length is used. No filters are used so as to measure complete natural emission of the plasma. Integration time for this camera was always adjusted so as to avoid any damages to the detector even with the high photon flux. This will allow discussing the initial growth and afterwards cooling process as function of spark energy, gaseous composition and pressure. Time resolved measurements are obtained by varying the time delay between plasma and measurements and by adjusting the integration time. This is done by synchronizing all measurements with a delay generator (BNC555). For each conditions, a series of 200 shots are taken. Gains for both ICCD are adjusted so as to not saturate the signal while keeping a good signal to noise ratio. Exact initial timing is set for each Q-switch settings on both ICCD's. This has to be done so as to capture the initial creation of the plasma as delays may change due to changes in the laser pulse. Typical temporal analysis may take 45 minutes for a complete description of the statistical changes of the plasma with respect to time. Over the entire duration of a time series, pressure and temperature is monitored in the cell and no fluctuations are observed.

Spatial evolution of laser induced plasma under different pressures and gaseous composition

The first measurements deal with the geometrical evolution of the laser induced plasma as function of time, pressure, gaseous composition and spark energy.

Influence of pulse energy on the evolution of the shape

To quantify the effects of pressure and spark energy on the shape of the laser induced plasma, several conditions are used. Keeping the laser energy constant, different plasma are created as function of gaseous pressure. The list of conditions are summarized in Table 1. Temporally resolved measurements are averaged over more than 150 shots to discuss the influence of those different parameters on the actual shape of the plasma and its temporal evolution. One can see that for very low pulse energy, it is not possible to create a plasma for pressures below 2.5 MPa with the selected optical components. One has to recall that a series of three lenses were used to focus the plasma. This is mainly to avoid creation of plasma on the optical access. Another interesting feature is that the plasma energy is not changing for pressures encompassed between 0.5 and 2.5 MPa, whereas cases for the highest pressure always present a plasma energy about 10% higher than cases at 2.5 MPa and that only one case at 0.1 MPa leads to a plasma energy 10% lower (for a pulse energy of 106 mJ). One can already conclude that potential differences may not be attributed to differences in plasma energy.

To retrieve a mean temporal evolution, single images are analyzed individually. It is checked that plasma energy falls within the desired boundaries, ±1 mJ around the mean value reported in Table 1. Typical images are integrated over the entire wavelength and therefore are very bright all over the time. The aperture of the objective is set to 32 and gain remains equal to zero all over the tests as no interferential filter is used. Typical examples of an instantaneous image for a low energy and for a high energy plasma case are shown in Figure 2 and Figure 3. Those images have been obtained for a delay of 1000 ns and an integration time of 100 ns. One can see some differences in the instantaneous shape of plasma, even though ambient pressure and temporal settings are identical. The intensity has been normalized between 0 and 1 and only levels from 0.1 are represented, as intensity below 0.1 is considered as being noise. The low energy case has an aspect ratio relatively large, whereas the plasma becomes already more spherical for the spark of 160 mJ case. In both cases, the laser beam comes from the left. The exact position of the center of the plasma changes with the energy. In the present case, for clarity reason, the mean position of the plasma at the earliest delay is set to 0 mm in both directions. This allows a discussion on the evolution of the length and height as well as the changes in the center as function of time. One can notice a region with a quite uniform emission above 0.8 which also has a different shape as function of plasma energy. To better analyze the shape of the plasma as function of the various parameters, the mean evolution is computed. An intensity threshold is first used to define the boundary of the plasma on each single shot. The threshold is a simple binarization process as background noise remains limited for the different cases under study. It is taken as being the intensity corresponding to 25% of the maximum intensity, as providing the best signal to noise. The length and height of the plasma are determined and typical fluctuations in position and aspect ratio are discussed. The two dimensional probability density function (2D PDF) to find plasma at given location is also computed based on the analyses of all shots within one series. This gives an overview of the reproducibility of the plasma, while keeping all other parameters constant.

Two examples of such PDF distribution are presented for the case of a plasma of 27 mJ in Figure 4 for earlier delays and in Figure 5 for later delays. The x-axis and y-axis are set at zero for the highest probability at the earliest time (which is in the present case obtained at a delay of 0 ns).

One can notice that the two shapes are very different. Whereas initial delays tend to offer an elongated plasma with several discontinuities (suggesting multiple plasma), later delays show a more compact shape. Furthermore, one can notice a small shift in the central position towards the direction of the laser (as laser comes from left to right). This shift in time is enhanced with an increase of plasma energy. This has some direct consequences on the LIPS technique as the probe volume is not only changing in size and shape but also in space. Therefore, LIPS technique without an ICCD camera to measure the location of the formation of the plasma may not be very accurate as far as position of the probe volume is concerned. To allow a better analysis, the width and height are then defined as the sizes corresponding to a probability higher than 10% on those two dimensional probability density function. This threshold offers the best compromise between accuracy and resolution. A lower threshold tends to include more noise and a higher clearly limits the measurement volume. Furthermore, with a low threshold, the multiple plasma situation of Figure 4 would lead to an overestimated size of the plasma. For this case, the length is considered to be the maximum length of the continuous plasma, ranging from -1.5 to about -0.5 mm for the case presented in Figure 4. Results are presented in Figure 6 where both dimensions of X (along the laser axis, hence referenced as width) and Y (perpendicular to the laser, hence referenced as height) are represented as function of time delay with respect to the time of creation of the plasma. The width and the height are computed as being the longest distance between two points of the plasma. This is required as in some cases two distinct regions are detected, separated by a small region having lower probability. One can first clearly identify the initial effects of plasma energy, as the width of the plasma increases with the plasma energy. The width is about two times longer when creating a plasma of 160 mJ compared to a spark of 27 mJ. One has a longer ionized region. The influence on the height is less important. This is due to the fact that the initial height is fixed by the optical setup.

X(27mJ) Y(27mJ) X(77mJ) Y(77mJ) X(125mJ) Y(125mJ) X(160mJ) Y(160mJ)
Fig. 6 Temporal evolution of both dimensions for N 2 at 5.0 MPa for different spark energies.

After an initial growth, the plasma tends to contract itself for all energies (for time delays around 100 ns). This contraction is not due to an expansion in the other direction as shown by the measurements of the height of the plasma. After this contraction, there is a phase of expansion in the transverse direction. The plasma also changes in shape, from an elongated in the horizontal direction, through a circular shape to an elongated plasma in the vertical direction (a height bigger than the width) for the low energy cases. In this evolution, one can also notice the influence of plasma energy. Low energies tend to delay the vertical growth and limit its expansion, as shown by the results obtained for a plasma energy around 27 mJ. On the other side, little effects are noticed for plasma energies higher than 77 mJ. The three cases have a vertical expansion relatively similar, while having strong differences in the horizontal direction. Through those results, it is clear that the horizontal direction is always influenced by the plasma energy. Higher the energy, longer will be the plasma for delays up to 7000 ns. However, in the vertical axis, no differences are noticed. It is also important to understand that pressure tends to concentrate the time axis, as typical results obtained for atmospheric pressures would not show such a vertical expansion for the plasma emission within the delays investigated here. It is important to recall here that no flow is imposed to the gas. The third lobe generation mechanism (see [START_REF] Bradley | Fundamentals of high-energy spark ignition with lasers[END_REF] for instance) is however not seen with the plasma direct emission. Those results have some direct consequences for LIPS measurements. If the spatial resolution has to be restricted to the smallest size possible, delays around 100 to 500 ns should be privileged. On the other hand, if one can accept a bigger probe volume, it seems more interesting to take measurements for delays higher than 1500 ns to have a more spherical probe volume. A compromise will also include spectral constraints as one may expect more ionized atoms at early stages. Assuming a cylindrical shape for the plasma and a third dimension identical to the height, it is possible to compute the volume of the plasma in terms of cubic millimeter. Results are represented in Figure 7 for the four different energies at the highest pressure (5.0 MPa). One can see that at most, the volume is around 2 mm 3 for this high pressure case. Between time delays lower than 1000 ns, it is limited to values lower than 1 mm 3 . An increase in energy creates a bigger probe volume. Another interesting feature of low energy plasma is the relatively constant volume over the time. Actual probe volume may be a bit bigger if one takes long integration time due to the shift of the plasma towards the laser source as function of time, as discussed earlier.

3.2 Influence of pressure on the evolution of the shape Conducting similar analyses for different gaseous pressures, the evolution for the lowest energy of laser pulse that leads to a creation of plasma for all pressures is investigated. Results are obtained for both the longitudinal evolution (along the laser path, labelled length) as well as for the transverse evolution (labelled height). Results are summarized in terms of probe volume, assuming a cylindrical symmetry and presented in Figure 8.

For lower pressures, the plasma tends to expand in both directions up to delays of the order of 2000 ns. For this delay, the typical length is of the order of 6 mm whereas the height reaches 3.25 mm. For this case, the LIPS measurement is integrated over this entire volume, which, assuming cylindrical revolution along the axis of the laser results in a probe volume of 15 mm 3 at most. Higher pressures lead to initially smaller probe volume. For pressures below 2.5 MPa, a maximum volume is reached for delays around 1000 ns whereas the volume increases for the two highest pressure cases investigated here even for delays around 2000 ns. It is also interesting to recall that the lowest pressure case was leading to a plasma energy of only 58 mJ whereas the highest pressure had a mean plasma energy of 77 mJ. This clearly shows that pressure is a key factor for the volume and as already noticed by [START_REF] Vors | Laser-induced breakdown spectroscopy of carbon in helium and nitrogen at high pressure[END_REF], the plasma is smaller for a higher pressure. The tendency discussed in [START_REF] Arp | Feasibility of generating a useful laser-induced breakdown spectroscopy plasma on rocks at high pressure: preliminary study for a venus mission[END_REF] for which the volume was a linear function of pressure is not found in the present investigation. One of the reason is that in [START_REF] Arp | Feasibility of generating a useful laser-induced breakdown spectroscopy plasma on rocks at high pressure: preliminary study for a venus mission[END_REF], plasma was created on a solid target whereas in the present case, it is free to evolve along the three directions.

Effect of gaseous composition

To discuss the temporal evolution of plasma shape as function of gaseous composition, a procedure similar to the one presented before is used. The four different gaseous compositions are listed in Table 2 and presented as function of their partial pressures. The partial pressure of nitrogen is always adjusted so as to recreate the proper ratio of air with respect to oxygen. Oxygen is not inserted in the cell in presence of fuel, so as to avoid ignition of the mixture. The global equivalence ratio (φ(C)) can be determined based on the ratio of carbon to oxygen and assuming a fuel of typical composition CnH 2n+2 .

The temporal evolution of both length and width for the different mixtures is represented in Figure 9 for a spark energy of 27 mJ and an ambiant pressure of 5.0 MPa.

One can distinguish two regions. For small delays (less than 500 ns), there is little influence of gaseous composition on the plasma expansion. Both dimensions are similar regardless the presence of oxygen, carbon or hydrogen and the resulting probe volume is limited to about 0.1 mm 3 . However, for delays higher than 500 ns, the length of the plasma obtained in presence of carbon clearly increases to reach a maximum of 2.5 mm for delays around 1500-1700 ns. This is seen for both mixtures containing carbon. In parallel to this, a radial expansion can also be noticed, resulting in an overall increase of the probe volume. At most, there is a factor 5 for the probe volume with and without carbon at delays of 1000 ns. For longer delays, there is little difference induced by carbon concentration and typical probe volume are of the same order for delays after 5000 ns. Again, this emphasizes the need to have a simultaneous measurement of the probe volume has its size is influence not only by pressure and laser energy but also by gaseous composition.

Label

It is interesting to see if this increase in the probe volume is accompanied by an overall increase in all the emissions are more specifically to some emissions only. Therefore, the spectral analysis is proposed in the next section to see if similar effects are detected as increasing carbon content. 4 Spectral measurements

Processing of the spectral data

To get an absolute evolution at different wavelengths, the quantum efficiency of the system is first determined. This is done by getting the efficiency at some discrete points using well calibrated lamps. To determine the equivalence ratio from spectral data, three different strategies are proposed. The first one is a classical approach in LIPS. One integrates all the intensity under the selected emission lines. This is done by determining the background level at the selected wavelengths assuming it can be linearly interpolated between two neighboring points (one on the left and the other on the right) that have no interference with any emission lines. This takes into account the possible multiple emission like the triplet of nitrogen around 742, 744 and 746 nm. Another strategy consists in fitting the best simulated spectra with the experimental spectra. This strategy is similar to the one presented in Zimmer et al. (2007) where simulations were reported for atmospheric cases. In the present case, simulations should include effects of pressure for the different atomic emissions concerned: nitrogen, oxygen and carbon. Therefore, an alternative strategy is proposed. It is based in first determining experimentally an averaged spectra for a given mixture fraction with specific time delay and integration. Averaged spectra are then used to create a database. All instan-taneous spectra are correlated with this database and the best fit can be determined. The actual value of the single shot equivalence ratio may be further interpolated between two averaged spectra assuming a gaussian profile for the correlation and using a three-points scheme. This strategy aims at using all the spectral information available and not only few discrete lines as it would be the case for the classical approach. The uncertainties will be discussed in the following sections for both air cases and air-dioxide carbon cases. In this section, individual emission lines are discussed following a classical approach to explain each specific atomic emissions, whereas the global fitting will be used in the fifth section when retrieving quantitative concentration from single shot spectra.

Temporal evolution of the spectral emissions

To discuss the overall evolution, mean spectra at different time delays are taken. Those instantaneous spectra (a typical example being represented in Figure 10) are averaged over each delays. It is extremely difficult to analyze such a single shot as the gate width is limited to 100 ns, but it is possible to distinguish some emissions induced by oxygen around 777 nm and other emissions around 820 nm, induced by both nitrogen and oxygen. Then, to describe the typical spectral evolution and to take into account the exponential decrease of the intensity with time, it is decided to plot the relative spectral evolution at each time step. After averaging the spectra and taking into account the quantum efficiency, the averaged spectra are normalized so as to have a maximum of unity. Afterwards, repeating this operation at different time delays and for different gaseous compositions, a relative temporal evolution of spectra can be obtained. As the purpose of the present investigation is to see the feasibility of Laser Induced Plasma Spectroscopy to measure equivalence ratio at high pressure, the relative evolutions of nitrogen-oxygen and nitrogen-carbon dioxide are compared in Figure 11 When analyzing first the nitrogen-oxygen case, one can see an initial phase for which a continuum is relatively important. This continuum is mainly induced by some ionic emissions of nitrogen around 400-450 nm. Further delays reveal the emission of atomic lines of both nitrogen and oxygen. It is interesting to notice that the emission of the oxygen remains the most intense emission for delays after 200 ns up to delays aroud 6000 ns. One has however to recall that the absolute intensity drops exponentially with time. Later delays also reveal different emission regions, even though no clear emission from molecules are visible. The triplet of nitrogen around 742 -746 nm as well as the emission around 820 nm from oxygen are visible. This graph also reveals that broadening remains limited even at 5.0 MPa and that the spectral width remains constant with time. The accuracy of the present spectral measurements is estimated be 1 nm. From these analyses, it is possible to conclude that averaged spectra can be used to discuss the relative emissions of nitrogen-oxygen. Adding carbon (nitrogen-carbon dioxide conditions) reveals a strongly different spectral evolution and shape. Together with the apparition of new specific emissions, a huge difference is seen in the dynamics of the spectra. Whereas the peak around 400 nm was not predominant for the nitrogen-oxygen case, its strength comes clearly out in the carbonate atmosphere when compared to the emission at 445 nm for delays bigger than 100 ns. Furthermore, this coincides with an increase of the peak around 385 nm. There are two possible origins for the peak around 385 nm, both being molecular emission either of CN (violet system) or C 2 (Deslandres-D'Azambuja system) (see [START_REF] Acquaviva | Simulation of emission molecular spectra by a semi-automatic programme package: the case of c2 and cn diatomic molecules emitting during laser ablation of a graphite target in nitrogen environment[END_REF]). The high-pressure environment may tend to ease molecular formation for smaller delays as compared to atmospheric pressure. The spectral accuracy as well as pressure broadening effects make a clear identification of molecular spectroscopy within the present framework impossible. Another clear peak attributed to carbon can be distinguished around 833 nm compared to the nitrogen-oxygen emission around 820 nm. The peak from carbon is coming from the transition between 61981.82 cm -1 and 73975.91 cm -1 . The relative strength of carbon increases with time whereas nitrogen and oxygen decreases. It is interesting to note that this peak is a bit delayed compared to the peaks of nitrogen and oxygen. A possible explanation resides in the ionization levels of respectively carbon, oxygen and nitrogen. The ionization energy for carbon being the smallest (11.26 eV compared to 13.61 eV and 14.53 eV), it may lead to a formation of delayed atomic emissions.

From those considerations, it seems possible to take into account carbon emission as fuel marker. It is interesting to notice that no self-absorption could be shown in the present experiments. One possible reason is the spectral resolution chosen, which may be too limited to see such effects. From this comparative analyses, it seems possible to retrieve concentrations but at delays higher than 800 ns, to have the best signal to noise ratio possible for carbon emission. The ultra-violate emissions of carbon are difficult to use as one needs for a quantitative equivalence ratio measurement a line representing also oxygen (or nitrogen). Given the spectral resolution, it is impossible to encompass in a single shot both emissions around 400 nm and 780 nm. Therefore, the following analyses are based on the relative evolution of emissions between 700 nm and 850 nm. The spectral evolution for different plasma energy is not proposed in the present context as the induced probe volume would far increase the typical probe volume required in treating turbulent combustion situations and only the lowest energy case is presented.

Specific evolution of N(I), C(I) and O(I)

In order to propose calibration curves, the specific temporal evolution of N, C and O are shown for a mixture of N 2 CO 2 . For each emission line, the true quantum efficiency is used together with the temporal gate width in order to compute relative emission independent of temporal integration. For each line, the local background is subtracted. Furthermore, all lines are normalized with respect to the overall maximum of the different lines; which in this case is from the nitrogen-oxygen around 820 nm at a delay of 275 ns. For carbon atoms, two different lines are considered. The strongest emission is around 833 nm. For this case, the emission between 828 nm and 838 nm is taken to represent its emission and the local background is computed on regions from 792 to 797 nm on the left side and from 852 to 857 nm on the right side. A second emission line is used around 711 nm, which consists in a triplet between higher state around 69700 cm -1 and lower state around 83700 cm -1 . To represent nitrogen emissions, the triplet around 742 -746 nm is used whereas oxygen atoms are only tracked around 777 nm. Finally, the emission around 820 nm is also checked as coming from a combination of nitrogen-oxygen emission lines.

Results are plotted in Figure 13 in a logarithmic scale for the relative intensity to take into account the exponential decrease of intensity with time.

One can distinguish clearly a different behavior between the atoms of nitrogen and oxygen and the atoms of carbon. N(I) emission is decreasing sharply with time after a peak at 275 ns. There is a little difference between pure nitrogen cases and other compositions, due to the fact that the partial pressure of nitrogen is higher for the pure case. Those typical behavior are also found at smaller pressures. initial increase is mainly due to a cooling process in which initial ions are retrieving the missing electrons to give raise to atomic emission again. The further decrease is still induced by the cooling process in which the temperature decreasing the probability to have electronic transitions. On the other side, the carbon-linked emissions do present a different temporal behavior. After a constant initial drop, the intensity of the carbon-linked emissions stay almost constant with time, especially at delays between 1000 ns and 4000 ns. This behavior is independent on the gaseous composition; there is no influence of oxygen on this trend as similar behavior is found for N 2 CO 2 and N 2 CO 2 CH 4 . C(711 nm) emission shows a similar trend as the one shown by C(833 nm), with however an emission intensity almost one order of magnitude lower. Therefore, to retrieve local equivalence ratio at pressures up to 5.0 MPa, strategies involving carbon emission around 833 nm and nitrogen-oxygen around 820 nm may be interesting. After identifying typical lines to represent the fuel, it is also important to optimize time delays to yield the highest signal to noise ratio possible and check if other ratio may present higher signal to noise ratio or higher sensitivity.

Strategies for equivalence ratio measurements

With the increase of pressure, carbon emissions are becoming stronger in the short delays as shown in the previous section. Therefore, different calibration strategies may be proposed. A first one is based using the ratio of atomic emission of carbon around 833 nm versus the emisson of nitrogen-oxygen around 820 nm. Another one is to use the ratio of carbon emission around 711 nm versus the nitrogen emission (from 742 to 746 nm), but as presented in the previous section, the relative intensity of this line is weaker than C(833) and therefore, for simplicity, its evolution is not represented here.

Temporal considerations

An important parameter for calibration is the timing between the plasma creation and the acquisition of the spectra as well as duration of the measurements. For that purpose, the time resolved measurements previously depicted are used to simulate the typical spectral emissions one would obtain when using different integration time. Results are summarized in Figure 14. Five different initial delays have been chosen: 0, 500, 1000, 1500 and 2000 ns from the initial plasma. As spectra have been taken up to delays of 7000 ns, integration time of 5000 ns can be obtained for an initial delay of 2000 ns and an integration time of 7000 ns if the initial delay is fixed at 0 ns. To obtain such an evolution, a central difference scheme is used. For each delay, the relative C(833) over N(744) ratio is computed for different overall integration time. Results clearly show an increase of C(833)/N(744) with an increase of the initial delay. It is worth noting that taking the spectra from 0 ns is clearly not the best choice. Therefore, for such measurements, a gated detector is required to get a good signal. From this plot, it turns out that the higher the initial delay, the best would be the signal for detecting fuel. However, the overall signal is decreasing (as shown in figure 13) and therefore a trade-off has to be found. In the present work, calibrations are reported for delays of 2000 ns and an integration time of 5000 ns. In the following, the spectra shown are taken with those timing and therefore are not reconstructed from previously used time resolved measurements. A typical single shot spectra taken with those timing is shown in figure 15 for an equivalence ratio of 0.50. One can recognize on this single shot spectra the different peaks of carbon, nitrogen and oxygen. For each single spectra, the camera noise is first subtracted using spectra obtained without plasma but with identical settings of gain and integration time. Then, the emission due to carbon is computed as being the intensity emitted by the region 833 ± 5 nm minus the single shot background (due to broadband emission of plasma) which is computed as being the averaged intensity emitted by the region 800 ± 5 nm and 860 ± 5 nm. The emission of nitrogen is computed in a similar way but with 745 ± 5 nm as representing the signal and the averaged of 735 ± 5 nm and 763 ± 5 nm for the local background. Finally the band 777 ± 5 nm is used for the emission of oxygen and the average of 763 ± 5 nm and 788 ± 5 nm is used for its local background.

Calibration curves

Analyzing more than 150 single shots for each pressure, it is possible to determine calibration curves for the different pressures investigated. Results shown in Figure 16 are compiled over the highest range of pressure investigated for C(833) emissions to characterize fuel concentration whereas oxidizer is depicted by the N(I) triplet around 742 -746 nm or by the oxygen emission (777 nm). One can see that a higher pressure leads to a more sensitive ratio as the typical ratio of C/N is of the order of 7 at 5.0 MPa whereas it is around 4 at 2.5 MPa. The ratio of C/O is less sensitive as the slope is about half the slope of C/N, regardless the gaseous pressures. Data obtained at 1.0 MPa do not show a good behavior as carbon emission around 833 nm stays weak.

Uncertainties

Treating the 150 shots for each imposed equivalence ratio and using the previously obtained calibration curve, it is possible to compute the standard deviation of each single shots. Those standard deviation results are plotted in Figure 17 for the three different pressures. It is interesting to notice that a low imposed equivalence ratio leads to a relatively low absolute precision whereas the absolute uncertainties tend to increase with equivalence ratio up to 0.7. Then, one can notice again a decrease in the absolute precision of single shots. One can also notice that an increase in pressure tends to increase the precision of the technique as the absolute uncertainties are the lowest for the highest pressure. If one deals with relative uncertainties, the present technique has 20% accuracy on the lean side when analyzing single shot data and typical 5% close to stoichiometric conditions. To determine the spatial distribution of fuel at 5.0 MPa, and discuss possible temporal changes, one would need to take a trade-off between uncertainties and temporal resolution, keeping in mind that each individual measurement is performed within 7 µs.

Alternative strategies

Based on the previous results, typical precision is of the order of 0.05 in terms of absolute equivalence ratio measurements. Whereas those values are relatively low when dealing with near stoichiometric conditions, the use of lean conditions lead to a high relative uncertainty. Alternative solution to the peak to peak ratios may be used to overcome this limitation. A possible alternative is to simulate the spectra, including collisional effects and use fitting function to retrieve absolute concentration. This leads to some problems as the complete simulation of the physical processes involved in Laser Induced Plasma Spectroscopy may not be straight forward. Another alternative is to compute the correlation between the instantaneous spectra with the experimentally obtained mean data at different equivalence ratio and to use a second order degree polynomial approximation to retrieve the position of the peak. This polynomial fit will then provide the value of the equivalence ratio. This second alternative offers in principle a better efficiency as all the different optical parameters and efficiencies of the collecting system are taken into account. When applying this procedure to the spectra presented in Figure 15, one obtains a correlation coefficient with respect to the different equivalence ratio used for the constructing the spectral data base. The typical values of the different coefficients are presented in Figure 18 with the database obtained at some discrete values depicted by the circles. The correlation coefficient is 0.973 for the case of the spectra obtained at an equivalence ratio of 0.50, 0.944 at 0.33 and 0.946 at 0.66. The interpolated value of the equivalence ratio is 0.498±0.01, which is in very good agreement with the average spectra for an equivalence ratio of 0.50. The correlation does not reach a value of unity due to single shot noise. Applying the same correlation procedure to all the database, it is possible to obtain the averaged values but also the standard deviation. Whereas averaged values do match with the imposed one, by definition, the values obtained for the standard deviation present interesting lower values as compared to peak ratio analyses. Results are summarized in Figure 19. One can see several interesting features. The first is that the absolute uncertainty remains limited for a pressure of 5.0 MPa as for lean mixture, it only accounts to 0.025 compared to 0.05 when using a peak ratio approach. One can also notice again that an increase in pressure increases the accuracy of the technique when using the carbon emissions around 711 nm and 833 nm. The accuracy for an intermediate pressure is for some cases half the accuracy at the highest pressure; but this is due to some erroneous data that bias the standard deviation. For clarity, those have not been removed as in actual case, it would be difficult to remove data as the range of equivalence ratio may not be really assessed. Another feature is that the absolute uncertainty increases with imposed equivalence ratio, but the relative uncertainty remains fairly constant around 7.5% as single shot accuracy at 5.0 MPa. Data obtained at the lowest pressure (1.0 MPa) do not show very consistent tendency and again, as in the peak ratio approach, do not seem to be useful. For lower pressures, an approach based on hydrogen Hα and H β with respect to the triplet of nitrogen around 742 -746 nm and oxygen around 777 nm remains certainly the best choice as typical uncertainties reported in [START_REF] Zimmer | Laser induced plasma spectroscopy for local equivalence ratio measurements in an oscillating combustion environment[END_REF] were less than 4% for single shot measurements.

Conclusions

Laser Induced Plasma Spectroscopy measurements were performed for pressures up to 5.0 MPa. A study on plasma shape showed the influence of carbon in the development of the induced plasma. Its length tended to become longer in presence of carbon which increases the probe volume. Furthermore, several spectroscopic features were determined when adding carbon inside the test cell. A study of their temporal evolution showed a distinctive behavior as compared to typical other atomic emissions. A strategy for measuring mixture composition a 5.0 MPa could be proposed based on the ratio of C around 833 nm to the emission of the triplet of N(I) around 742 -746 nm. A good sensitivity was obtained for pressures above 2.5 MPa, whereas calibration curves for lower pressures were less sensitive. Using a correlation based calibration clearly improves the precision of the technique and typical single shot uncertainties are limited to 7.5% at 5.0 MPa when using the spectral information between 700 nm and 850 nm. This technique has some limitations. The probe volume depends on different parameters and therefore should always be measured. This quite long probe volume may be an issue in cases with strong gradients. One may limit the probe volume by adjusting the laser energy. It requires a relatively long exposure time (5 µs) and therefore may only be applicable to cases for which this integration time is smaller than the typical time scales. However, it is easily applied to liquid fuel which is not completely prevaporized. The next steps consist in applying such strategies in real combustor to show spatial and temporal distribution of kerosene in actual gas turbine environment as LIPS technique is not sensitive to molecular content of kerosene [START_REF] Zimmer | Measurements of mixture fraction with laser induced plasma spectroscopy[END_REF]) or initial temperature. This technique would also work for richer mixtures for which carbon related peaks have stronger emissions. It was not investigated in the present studies because the focus was put on the lean side only.

Acknowledgments

The first author wishes to acknowledge the financial support of the CNRS for providing the grants to travel through the Programme International de Collaboration Scientifique, number 4979.

Fig. 1

 1 Fig. 1 Detailed sketch of the optical layout.

Fig. 2 Fig. 3 LengthFig. 4 LengthFig. 5

 2345 Fig.2Typical plasma shape for N 2 at 5.0 MPa, a delay of 1000 ns and an integration of 100 ns for a spark of 27 mJ.

Fig. 7

 7 Fig. 7 Temporal evolution of the probe volume for N 2 at 5.0 MPa for different spark energies.

Fig. 8

 8 Fig.8Temporal evolution of the probe volume as function of pressure for a laser pulse of 106 mJ.

Fig. 9

 9 Fig.9Temporal evolution of plasma shape for different compositions at 5.0 MPa and a spark energy of 27 mJ.

Fig. 10

 10 Fig.10Typical single shot spectra for a delay of 1500 ns an integration time of 100 ns for a plasma energy of 27 mJ and N 2 CO 2 mixture at 5.0 MPa.

Fig. 11

 11 Fig.11Temporal evolution of the relative spectra for an energy of 27 mJ, a pressure of 5.0 MPa and N 2 O 2 gaseous mixture.

Fig. 12

 12 Fig.12Temporal evolution of the relative spectra for an energy of 27 mJ, a pressure of 5.0 MPa and N 2 CO 2 gaseous mixture.

Fig. 13

 13 Fig. 13 Temporal evolution of N(I), C(I) and O(I) for a spark energy of 27 mJ at a pressure of 5.0 MPa.

Fig. 14

 14 Fig. 14 Temporal evolution of C(833) over N(744) as function of initial delay for a spark energy of 27 mJ, for an overall equivalence ratio of 1.50 and a pressure of 5.0 MPa

Fig. 15

 15 Fig.15Single shot spectra for a delay of 2000 ns, an integration time of 5000 ns, a pressure of 5.0 MPa and an equivalence ratio of 0.50.

Fig. 16

 16 Fig. 16 Calibration curves for different pressures of C(833) / N(744) and C(833) / O(777) at pressures above 1.0 MPa.

Fig. 17

 17 Fig.17Single shot uncertainties at different pressures for the ratio C(833) / N(744).

Fig. 18

 18 Fig.18Evolution of the correlation coefficient as function of the imposed equivalence ratio for a spectra obtained with an imposed equivalence ratio of 0.50 at 5.0 MPa

Fig. 19

 19 Fig.19Evolution of the absolute standard deviation obtained by the spectral fitting procedure as function of imposed equivalence ratio pressures ranging between 1.0 MPa and 5.0 MPa

Table 1

 1 Plasma energy (mJ) as function of laser pulse energy and gaseous pressure

	Pressure	0.1 MPa 0.5 MPa	1.0 MPa	2.5 MPa 5.0 MPa
	Laser Pulse (060 mJ)	-	-	-	18	27
	Laser Pulse (106 mJ)	58	65	66	68	77
	Laser Pulse (160 mJ)	111	109	112	115	125
	Laser Pulse (195 mJ)	142	145	147	148	160

Table 2

 2 Partial pressure of the different gases for the four mixtures investigated at 5.0 MPa.

		N 2	O 2	CO 2	CH 4	φ(C)
	N 2	1.00	0.00	0.00	0.00	0.00
	N 2 O 2	0.79	0.21	0.00	0.00	0.00
	N 2 CO 2	0.79	0.00	0.21	0.00	1.50
	N 2 CO 2 CH 4	0.78	0.00	0.20	0.02	1.65


Boris Mailhé

Daniele Barchiesi

Mark D Plumbley

INK-SVD: LEARNING INCOHERENT DICTIONARIES FOR SPARSE REPRESENTATIONS

Keywords: Sparse coding, Dictionary learning, Coherence, K-SVD

This work considers the problem of learning an incoherent dictionary that is both adapted to a set of training data and incoherent so that existing sparse approximation algorithms can recover the sparsest representation. A new decorrelation method is presented that computes a fixed coherence dictionary close to a given dictionary. That step iterates pairwise decorrelations of atoms in the dictionary. Dictionary learning is then performed by adding this decorrelation method as an intermediate step in the K-SVD learning algorithm. The proposed algorithm INK-SVD is tested on musical data and compared to another existing decorrelation method. INK-SVD can compute a dictionary that approximates the training data as well as K-SVD while decreasing the coherence from 0.6 to 0.2.

INTRODUCTION

In the method of sparse representations, a signal is expressed as a linear combination of a few vectors named atoms taken from a set called a dictionary. A good dictionary must obey several criteria. First it has to be adapted to the data being represented. Good pre-constructed dictionaries are known for common classes of signals, but sometimes it is not enough and the dictionary has to be learned from examples of the data to represent [START_REF] Elad | Image denoising via sparse and redundant representations over learned dictionaries[END_REF]. Second, even when the dictionary is known, finding the sparsest representation of the data is in general an NP-Hard problem. However several polynomial-time algorithms have been proven to be optimal if the dictionary is sufficiently close to orthogonal [START_REF] Tropp | Greed is good: algorithmic results for sparse approximation[END_REF]. Coherence is one measure of this proximity. The coherence µ(Φ) of a dictionary Φ This work was supported by the EPSRC Project EP/G007144/1 Machine Listening using Sparse Representations, the EU FET-Open project FP7-ICT-225913-SMALL and the Queen Mary University of London School Studentships.

is the maximal correlation of any two different atoms:

µ(Φ) = max ϕ i ,ϕ j ∈Φ,i =j ϕ i ϕ i 2 , ϕ j ϕ j 2 (1)
The µ function is valued between 0 and 1. The minimum is reached for an orthogonal dictionary and the maximum for a dictionary containing at least two collinear atoms. This work aims at merging those two criteria: learn an incoherent dictionary that is adapted to the training data.

There have already been a few early attempts at this task, but they either restrict the dictionary to a parametric form [START_REF] Yaghoobi | Parametric dictionary design for sparse coding[END_REF] or use a relaxed constraint penalization method that makes it harder to tune the exact value of the coherence [START_REF] Ramirez | Universal priors for sparse modeling[END_REF] while the results on exact recovery provide hard bounds [START_REF] Tropp | Greed is good: algorithmic results for sparse approximation[END_REF]. Therefore we propose a new algorithm called INcoherent K-SVD (INK-SVD) based on the addition of a decorrelation step to the K-SVD algorithm [START_REF] Aharon | K-SVD: An algorithm for designing overcomplete dictionaries for sparse representation[END_REF]. This step iteratively selects highly correlated pairs of atoms in the dictionary and decorrelates them until the desired coherence is reached.

Section 2 recalls the grounds of the K-SVD algorithm. Section 3 details the decorrelation method we present. Section 4 evaluates our method on musical audio data.

DICTIONARY LEARNING

Dictionary learning problem

Let S be a matrix of N training signals {s n } N n=1 ∈ R D . Dictionary learning consists in finding a dictionary Φ of size D × M with M ≥ D and sparse coefficients X such that S ≈ ΦX. For example, if the exact sparsity level K is known, the problem can be formalized as minimizing the error cost function f (Φ, X) defined as

f (Φ, X) = S -ΦX 2 FRO (2) under the constraints ∀m ∈ [1, M ], ϕ m 2 = 1 (3) ∀n ∈ [1, N ], x n 0 ≤ K (4)
with ϕ an atom (or column) of Φ and x n 0 the number of non-zero coefficients in the n th column of X.

K-SVD algorithm

Many dictionary algorithms follow an iterative scheme that alternates between updates of X and Φ to minimize the cost function [START_REF] Tropp | Greed is good: algorithmic results for sparse approximation[END_REF]. Although this work only presents the combination with K-SVD, our decorrelation method can be combined with any alternate algorithm. K-SVD iterates two steps:

• the sparse approximation step: knowing S and Φ, we estimate X, using a sparse approximation algorithm such as Orthogonal Matching Pursuit,

• the dictionary update step: jointly re-estimate each atom and its non-zero coefficients to minimize the cost function [START_REF] Tropp | Greed is good: algorithmic results for sparse approximation[END_REF]. The reader can refer to the original article for more details [START_REF] Aharon | K-SVD: An algorithm for designing overcomplete dictionaries for sparse representation[END_REF].

Incoherent dictionary learning problem

The low coherence of the dictionary can be enforced by adding another constraint to the dictionary learning problem.

The problem is still one of minimizing the cost function f described in Equation ( 2) under the constraints ( 3) and ( 4), and another low-coherence constraint is added:

µ (Φ) ≤ μ (5)
with μ a fixed coherence threshold. The proposed INK-SVD algorithm solves that problem by inserting a decorrelation step in the K-SVD loop after the dictionary update.

DICTIONARY DECORRELATION

Previous work

Methods for constructing incoherent over-complete dictionaries include an alternate projection technique initially proposed by Tropp et al. in [START_REF] Tropp | Designing structured tight frames via an alternating projection method[END_REF] and modified by Elad in [START_REF] Elad | Sparse and Redundant Representations[END_REF], which we will use in Section 4 as a benchmark for INK-SVD.

In the cited work, the goal is to realize a Grassmannian tight frame, that is, an over-complete dictionary with minimal mutual coherence µ min = (M -D)/D(M -1), by iteratively decorrelate an initial dictionary Φ.

The alternate projection consists in iteratively optimizing the Gram matrix G = Φ H Φ, so that its off-diagonal values are shrunk towards µ min and that its spectrum has nonnegative eigenvalues with rank smaller or equal than the ambient dimension D. This way, we obtain an updated gram G new , that can be in turn factorized into the product G new = Φ H new Φ new . This factorization is not unique and does not take into account the sparse approximation objective of dictionary learning. While we are planning to fill this gap in future work, this paper presents an alternative method for dictionary decorrelation that follows a greedy strategy.

Decorrelation problem

This section proposes a general method that attempts to find the closest dictionary Φ to a given dictionary Φ with a coherence lower than a given μ. Formally, Φ is defined as:

Φ = argmin Φ∈Γ Φ -Φ 2 FRO (6) with Γ = {Φ| µ(Φ) ≤ μ ∧ ϕ m 2 = 1, m ∈ [1, M ]}
The original dictionary Φ is the unconstrained minimum of the cost function [START_REF] Tropp | Designing structured tight frames via an alternating projection method[END_REF]. However, either µ( Φ) ≤ μ and it does not need to be decorrelated or Φ is not in the admissible set Γ of Problem [START_REF] Tropp | Designing structured tight frames via an alternating projection method[END_REF]. In that case Φ is a good candidate for a starting point for the dual problem that minimizes the coherence of the dictionary while staying close to Φ:

Φ = argmin Φ∈Γ µ(Φ) with (7)
Γ = Φ| Φ -Φ 2 FRO ≤ ρ ∧ ϕ m 2 = 1, m ∈ [1, M ]
with ρ the unknown minimum value reached by the criterion (6). We will rather address the dual problem (7).

Decorrelation of two atoms

Let the initial dictionary Φ be composed of only two unitary atoms φ1 and φ2 with a correlation higher than μ. In that simple case we can directly express the optimum of Problem [START_REF] Tropp | Designing structured tight frames via an alternating projection method[END_REF]. Let us assume without loss of generality that φ1 , φ2 > 0 (the opposite case can be derived by considering the couple ( φ1 , -φ2 )) and let θ be the half-angle between φ1 and φ2 . We are looking for the solution Φ = φ1 φ2 of Problem [START_REF] Tropp | Designing structured tight frames via an alternating projection method[END_REF]. The problem only has two degrees of freedom because of the normalization constraint. We choose the half-angle θ between φ1 and φ2 and the angle α between the sums φ1 + φ2 and φ1 + φ2 for parameters as shown on Figure 1. In the orthonormal basis

u 1 u 2 = φ1 + φ2 φ1 + φ2 2 φ1 -φ2 φ1 -φ2 2
all the considered vectors have a simple expression:

Φ = φ1 φ2 = cos θ cos -θ sin θ sin -θ (8)
Φ = φ1 φ2 = cos(α + θ) cos(α -θ) sin(α + θ) sin(α -θ) (9)
Fig. 1. Decorrelation of two atoms. For the optimal decorrelation we would have α = 0 and the pair ( φ1 , φ2 ) would be symmetric with respect to u 1 .

We can then express the different constraints:

| φ1 , φ2 | = | cos 2 θ| ≤ μ (10) φ1 -φ1 2 2 = 2 -2 cos( θ -θ -α) (11) φ2 -φ2 2 2 = 2 -2 cos( θ -θ + α) (12) Φ -Φ 2 FRO = 4 -4 cos( θ -θ) cos(α) (13)
If we assume without loss of generality that cos( θθ) > 0, then the cost function ( 13) is minimal for α = 0 and θ as close to θ as possible: Problem ( 6) is solved by rotating ϕ 1 and ϕ 2 symmetrically with respect to their mean until their correlation reaches μ. The angle θ is the angle that reaches the equality in Equation (11):

cos 2 θ = μ (14) θ = arccos μ 2 (15)
and the dictionary Φ is given by Equation (9).

General case

In the general case, the previous method provides the steepest descent direction if only one pair of atoms reaches the maximal correlation. However one can easily prove that the coherence function is non-convex with respect to Φ so following a steepest descent does not guarantee to find a global minimum. Instead of a descent method, we chose to decorrelate the dictionary by iterating decorrelations of pairs of atoms. The core idea is simple: as long as there are any atoms with correlation higher than μ, select a pair of them and decorrelate them with the method explained in Section 3.3. However, decorrelating two atoms can potentially change correlations with other atoms in the dictionary, so finding the next pair would require to update the correlations after each pair decorrelation. We speed up the process by decorrelating some pairs in parallel. Instead of selecting one pair of atoms at a time, we partition the whole dictionary into high correlation pairs (and single atoms that do not need to be modified), decorrelate all those pairs and only then update the correlations. This is detailed on Algorithm 1.

Algorithm 1 Φ = decorrelate( Φ,μ) Φ ← Φ while µ(Φ) > μ do E = partition(Φ, μ) for ∀(ϕ i , ϕ j ) ∈ E do decorrelate_pair(ϕ i , ϕ j ) end for end while
The partitioning is performed in a greedy way. We start with the whole dictionary, group the pair with the highest correlation together and remove it from the set of considered atoms until there are no pairs left with correlation higher than μ. It is detailed in Algorithm 2.

Algorithm 2 E =partition( Φ, μ)

Φ ← Φ E ← ∅ while µ(Φ) > μ do (i, j) = argmax Φ H Φ -I Φ ← Φ \ {ϕ i , ϕ j } E ← E ∪ {(ϕ i , ϕ j )} end while

NUMERICAL EXPERIMENTS

We tested the incoherent dictionary learning algorithm in order to assess if it can provide us with a dictionary for sparse representation that exhibits minimal coherence and good approximation quality. The test signal we used is a 16 kHz guitar recording that is part of the test data included in SMALLbox [START_REF] Damnjanovic | SMALLbox -an evaluation framework for sparse representations and dictionary learning algorithms[END_REF] 1 , a Matlab toolbox for testing and benchmarking dictionary learning algorithms that we used in our evaluation. A musical audio signal was chosen because previous informal experiments resulted in K-SVD learning a highly coherent dictionary for this type of data.

We divided the recording into 50% overlapping blocks of 256 samples (corresponding to 16ms) with rectangular windows and arranged the resulting vectors as columns of the training data matrix S. Then, we initialized three twice overcomplete dictionaries for sparse representation using respectively 1) randomly chosen subset of the training data S, 2) over-complete DCT and 3) over-complete Gabor frames. We run the K-SVD dictionary learning algorithm for 20 iterations, allowing for 12 non-zero coefficients in each representation (which corresponds to about 5% of active elements if compared with the ambient dimension D). We included the proposed INK-SVD decorrelation algorithm and compared it with the Grassmannian method detailed in section 3.1, using the implementation presented in [7, p.30].

Figure 2 depicts the results of the experiment. The first plot on the top presents the SNR achieved when the dictionary is initialized with random examples from the training data. We can note that, while K-SVD achieves a good approximation quality, it does it at the expense of high coherence µ ≈ 0.95. On the other hand, INK-SVD is able to achieve a lower coherence µ = 0.5 while maintaining a SNR> 20dB and, after this value, the approximation quality drops linearly with the mutual coherence. The Grassmannian method achieves a correlation µ ≈ 0.45, but with a worst SNR ≈ 8dB.

The other two plots corresponding to DCT and Gabor initializations display overall a poorer approximation quality. In these cases, Grassmannian fails to significantly decorrelate the dictionaries and achieves a very poor SNR, while INK-SVD is able to decorrelate the dictionaries up to µ = 0.2 with a small loss in approximation accuracy.

CONCLUSION

We provided an algorithm to learn a dictionary with fixed coherence from training data. The coherence itself is a param-eter of the algorithm that can be tuned to fit the needs of an application. Experiments on musical sound have shown that our algorithm can significantly reduce the coherence of the dictionary while almost preserving the approximation quality.

The proposed decorrelation method is generic and can be used in other contexts. In our experiments it even proved better than the Grassmannian method for coherence minimization. Yet there are still many theoretical questions to be answered, the biggest one being obtaining convergence guarantees depending on the coherence threshold.

It might also be possible to improve the approximation quality by specializing the decorrelation for the learning task. In the decorrelation problem [START_REF] Tropp | Designing structured tight frames via an alternating projection method[END_REF], one could replace the distance to the original dictionary by the dictionary learning error (2).

Fig. 2 .

 2 Fig. 2. Signal to noise ratio as a function of the coherence value for different choices of dictionary initialization and decorrelation functions. The levels µ max = 1 and µ min = (M -D)/D(M -1) indicate the maximum and minimum coherence attainable by a D × M dictionary.

http://small-project.eu/software-data/smallbox


Birgit Müller
email: birgit.mueller@tu-berlin.de

Arne Dahms

Dirk Müller

Henrik N Knudsen

Alireza Afshari

Pawel Wargocki

Bjarne Olesen

Birgitta Berglund

Olivier Ramalho

Joachim Goschnick

Oliver Jann

Wolfgang Horn

Daniel Nesa

Eric Chanie

Mika Ruponen

A EUROPEAN PROJECT SysPAQ

The European research project Innovative Sensor System for Measuring Perceived Air Quality and Brand Specific Odours (SysPAQ) is started under the VIth framework programme under the work programme "New and Emerging Science and Technology" (NEST PATHFINDER "Measuring the Impossible"). The Kick-off of the project was in September 2006. Ten partners (3 Companies, 4 Universities, 3 research Institutes) from 5 countries are involved. The main goal of this project is to develop an innovative system to measure indoor air quality as it is perceived by humans to be used as an indicator and a control device for the indoor air quality.

Le projet de recherche européen, Système de

Capteurs Innovant pour la Mesure de la Qualité de l'Air Perçue et Odeurs Specifiques (SysPAQ) a démarré en Septembre 2006. Il dépend du 6 ème programme cadre de recherche et développement technologique et plus spécifiquement de l'appel à propositions « Sciences et Technologies Nouvelles et Emergentes » (NEST PATHFINDER « mesurer l'Impossible »). Dix partenaires (3 entreprises, 4 universités, 3 organismes de recherche) de 5 pays sont impliqués. L'objectif principal de ce projet est de développer un système novateur pour mesure la qualité de l'air intérieur telle qu'elle est perçue par l'homme, qui servirait d'indicateur et d'outil de contrôle de la qualité de l'air intérieur.

INTRODUCTION

This innovative sensor system is highly demanded by the European society considering that humans spend about 90% of their time indoors, either at work, at home or when commuting between work and home. Recent data shows that improved indoor air quality will result in fewer complaints, increased comfort, less health problems and higher productivity. Consequently, quality of life will be improved.

Up to now, the indoor air quality has been quantified applying three different measurement methods separately. The three methods are based on the human perception of indoor air quality, chemical measurements and sensors for specific odours. Considering measuring perceived air quality human assessments are still being superior to chemical measurements because of the unmatched sensitivity to many odorous indoor air pollutants. One of the reasons is that in most cases the chemical measurements or signals from chemical sensors designed to detect special odours could not be correlated with the assessments made by humans. They do obviously not measure the relevant indoor air pollutants triggering human sensory response. This project will build upon current knowledge on the perceptual effects of indoor air pollutants and on the experience gained in using chemical measurements and sensors for specific odours. The approach of the project will be to enhance the present state of the art of sensor systems, the perceptual methods and the software tools for modelling human response, and integrate them into one innovative system for measuring indoor air quality as it is perceived by humans. A bridge will consequently be created between the previous works in this area and progress will be achieved by integrating measurements, sensors and modelling by a holistic approach.

The main challenges are the finding of a perceptual space by using different reference odours, the measuring procedure for perceived air quality, and a characterisation method of brand specific odours. This work requires advanced knowledge in perception psychology and technical excellence in sensor system design and indoor environment research, knowledge and experience which the project partners contribute. It is still an open question how to mimic the human perception of odours and air quality. A certain risk of failure is involved in this project's unsolved problem of how to define an adequate system of reference odours for human perception of odours. In any case the project will provide new insight to the human perception of odours and indoor air quality, and will deliver a new approach to assess the perceived air quality and brand specific odours by an advanced sensor system.

PROJECT OBJECTIVES

The main goal of this project is to develop an innovative system to measure indoor air quality as it is perceived by humans based on perception modelling combining measurements of sensors and assessments of perceived air quality by sensory panels. The innovative sensor system can be used as an indicator, monitor and control device for the indoor air quality in buildings and vehicles. Furthermore, the system will be able to detect brand specific odours and it will serve as a novel interior odour design tool for the vehicle industry. The main objectives of the project are:

1. Definition of a method for measuring the perceived air quality and perceived odour intensity in buildings and vehicles. This method will be used by all different labs using sensory panels.

2. To find an advanced perception model for indoor air assessments. The model will be the major input to the software design for the innovative sensor system and it provides new insight to the human reaction to odours.

3. Development of an innovative sensor system for measuring, correspondingly, the perceived air quality and brand specific odours quality.

4. Calibration and test of the innovative sensor system for measuring, correspondingly, the perceived air quality and brand specific odours. The final version of the system is intended for the following applications :

• Monitoring of the ambient air within buildings and vehicles. • Monitoring of the quality of the inlet air to buildings to ensure health and comfort for occupants. • Labelling of materials based on emissions from buildings and vehicles. • Control of the production process of building and vehicles materials.

Along with human activities, emissions from building materials, furnishing and equipment are main contributors to air pollution indoors. To reduce indoor air pollution loads it has been recommended to use low-polluting materials and to increase outdoor air supply rates. The new EU Energy Directive requires substantial reduction of energy use which may lead to reduction of ventilation rates and increased indoor air pollution enhancing the need of low-emission materials. In addition to measurements of indoor air quality as perceived by humans the system developed in the project can be used to control the emission rates from building materials already at the production stage. At present, manufacturers generally reduce the emissions from building materials by monitoring the emission rates of a few compounds in practice, but not necessarily the most relevant odour active compounds for perceived air quality. The pollution mixture affects the perceived air quality indoors. It is not taken into account so far. The proposed system can be used by the producers of building materials, furnishing and equipment materials to ensure that the emissions from their products would not negatively affect the perceived air quality indoors. In many countries labelling systems of building materials exist, so that the end-users can select the materials with reduced emissions. The suggested system for measuring the perceived air quality can also be used to quantify whether a material can get a label.

The selection of interior materials is a very important factor for the vehicle and transportation industry (train, car, boat, airplane etc.). The goal of the selection process is to create a high-standard perceived air quality in vehicles combined with a brand specific odour impression. To meet this goal the system measuring the perceived air quality seems indispensable.

The interdisciplinary structure of the project consortium will enable innovative research and it will provide new insights to the human perception of air quality and brand specific odours. The project management will ensure a strong interaction between new perception models, hardware development as well as software design for the innovative sensor system.

APPROACH AND METHODOLOGY

The overall approach of the project and the priorities of the different work-packages are shown in figure 1.

The figure illustrates the parallelism of the human perception of air quality and brand specific odours on the left hand side and the sensor system development on the right hand side.

The two arrows in the centre of figure 1 indicate the coupling of sensory panel experiments and the sensor system hardware and software development. The innovative sensor system has to detect all relevant odour active substances. Based on the knowledge of the project partners and experiments of the project a list of relevant substances is submitted to the two sensor specialists. The second arrow indicates the input of the new perception model of the perception specialist to the software development. The mathematical model of the new software for the sensor system will apply a reference odour system to reproduce an odour space that covers perceived air quality and brand specific odours in buildings and vehicles.

WORKPLAN OF THE PROJECT

Introduction -general description

The work of the project is divided into two main tasks. The first task is focused on human perception of odours. This part includes the definition of a method to measure perceived air quality and odour intensity by sensory panels. The used assessment methods can have a major influence on the assessment results itself. Besides the assessment method the main challenge of the human perception task is the definition of an odour space using reference odours.

The multidimensional odour space has to account for all interior odours in buildings and vehicles. This new odour space is the major input to the second main task of this project. This part is focused on the hard-and software development for the innovative sensor system. The contribution of two sensor specialists enables the application of the most advanced sensor technology. The sensitivity and selectivity of the sensors is adapted to odour active substances and the new odour space. The combination of different sensor types leads to novel sensor system that covers considerable more substances than any present stand alone system. The data processing for this sensor systems combines advanced statistical pattern recognition methods with the novel reference odour systematic. The link between the sensor signals and the perceived air quality consists of a classification process in terms of the reference odours followed by a regression method calibrated by the experimental data of the project. This two main tasks human perception and system development break down to 4 work-packages (human perception and odour space, sensor system, data processing and pattern recognition, system calibration). The management process and the dissemination activities control and link these four work-packages, see figure 2. The human perception task of the project is mainly handled in WP2. Starting point of WP2 is the definition of a method for all sensory panel assessments of perceived air quality and odour intensity. The method is based on the knowledge of the experimentally experienced project partner and the input of new multidimensional psychophysical perception models. This common standard for the assessment method guarantees comparable measurements at all labs for the innovative sensor system calibration data. The second and the most important part of WP2 is the definition of a new odour space based on reference odours. The new odour space has to cover all relevant odours of indoor air related environments. All project partners of WP2 have been working on the characterisation of odours in the past. The work-package leader Karolinska Institute (KI) will provide a human perception model as a theoretical base for the new odour space. First experiments of KI will examine the useability of the new odour space based on reference odours to reproduce indoor air odours from building and vehicle environments. The quality of the odour space is the critical factor for the innovative sensor system and it will influence the software design process. A continuous knowledge transfer between WP2 and WP4 will insure a fast translation of new findings into the software system. WP3 covers all activities related to hardware improvements for the innovative sensor system. This work-package includes the improvement and adaptation of single multi-gas sensors as well as the combination of different multi-gas sensor technology. The adaptation process and the selection of sensors are based on literature review and some analysis of indoor air environment of buildings and vehicles in terms of odour active compounds. The sensor development and sensor combination is handled by two experienced sensor specialist Alpha MOS and Forschungszentrum Karlsruhe (AM, FZK). The two partners can provide gas chromatography/mass spectrometry combined with sniffer experiments to prove the response of the multi gas sensor system to odour relevant substances. The check and minimisation of cross sensitivity (to other chemical compounds, relative humidity and temperature of sample air) of the sensors provides reproducible measurements. The sensor specialists (AM, FZK) manufacture three similar measurement devices for the calibration and validation measurements. The three systems will be jointly used by all partners. Additionally, the partner AM und FZK will provide all necessary hardware information and prototypes to the partner Centre Scientifique et Technique du Bâtiment (CSTB) and Technical University of Berlin (TUB) to insure a parallel development of the data processing and pattern recognition software (WP4).

The set-up of the general software layout for the data processing will be handled in WP4. The work package leader Technical University of Berlin (TUB) contributes many years of experience in pattern recognition methods and software development to calculate the perceived odour intensity based on multi gas sensor systems. The separation of software and hardware (WP3 and WP4) enables a company independent and extendable mathematical modelling of low-concentration odour mixtures as regards sets of critical odours for building and vehicle materials and products and their combination. The task of the computational data processing is to model the human odour perception of air samples. The method shall consider existing theories of perception and it shall link the measurements with psychophysical aspects of odour sensing. The response values of the sensor device will act as the stimuli and the software shall mimic the human perception and evaluation process. The core of the data processing method will be the "memory", the calibration database. This database will have a major influence on the performance of the method, especially on the classification of the investigated odour sample. Therefore the data shall span the whole odour space which is developed in WP2. The principle components of this odour space will be the basic odours. The data of the database consists on a combination of sensor response values and air quality assessments according to the methods of WP2 for a specific odour sample and concentration. The database shall contain for each odour class data for different stimuli concentrations. The system shall be adaptive which requires the database to be expandable and open for new data of odour investigations. The calculation models will always refer to the current data and therefore the algorithms will be adjusted to the extended data set. The unknown investigated odour shall be positioned inside the odour space which is a classification process for "odour recognition". Unknown odours shall be expressed by a combination of the basic odours. Once the sample is classified the quantitative and qualitative parameters of the perceived air quality shall be estimated. This step of the data processing uses the calibration data of the basic odours, the regression and calculation algorithms for this odours as well as weighing algorithms which consider the position of the unknown odour sample in the odour space. Data of odour samples other than the basic odours is needed to develop, test and optimize the weighing algorithms and to estimate the perceived air quality of different odours. Most of the odour space and combinations of basic odours should be covered.

During the validation process of the software the new collected data will be included in the database which may further improve the performance of the data processing method.

The innovative sensor system has to identify all reference odours and their combination in indoor environments. Additionally, the system shall link the pattern recognition method for the identification process to the sensory panel results. Due to the multidimensional character of the problem a large effort is needed to calibrate the innovative sensor system. WP 5 handles the collection of the calibration data. It consists of simultaneous measurements of perceived air quality as well as the perceived odour intensity by a sensory panel and measurements with the sensor systems using the most relevant sensory methods selected in WP2 and the most promising innovative system/technical device selected in WP3. All data of WP5 iterates back to the software enhancement of WP4. The odours for calibration measurements of WP5 are produced in emission chambers. The air flow through the chambers is polluted by a series of building and cabin materials. The concentration of air pollutants will be varied within an indoor realistic range to ensure and test if the system works appropriate. The variation in concentration will be achieved by varying the material loading and the ventilation rate and by selecting high and low emitting materials. Measurements will be performed for individual materials in a laboratory setting, for combinations of materials in a full-scale setting, and in real buildings and vehicles. All data of WP5 is stored in a database. A final test run of the innovative sensor system after calibration will show the ability to predict perceived air quality and to characterise brand specific odours. During the sensor development WP5 will provide preliminary test setups of building materials in order to secure reasonable sensitivity and discrimination power of the novel sensor system.

The dissemination task of WP6 provides the publication of all project results. The main communication medium of the project is an advanced internet portal that offers a public and a non public information area (www.syspaq.eu). The public area publishes all non confidential findings of the project partner during the project life time. The non public area handles the data exchange of all project partners. All project related findings will be open for the public after the completion of the project. The project will provide a final workshop in order to discuss and disseminate all findings of the project. The consortium will prepare a brief project presentation in English which is written in a style which is accessible to nonspecialists, avoiding technical language, mathematical formulae and acronyms as much as possible.

Publication will be done via the NEST www page.

More detailed information on the major work package interdependencies is given in figure 4.

Figure 1 :

 1 Figure 1: Parallelism of the human perception of air quality and brand specific odours on the left hand side and the sensor system development on the right hand side

Figure 2 :

 2 Figure 2: Work-package structure Work planning All overall management activities are included in WP1. The duration of the management activities covers the whole project life time. The main management activities are directed to the organisation of the project meetings and the knowledge transfer between the different work-packages. The management of the project insures a punctual delivery of all reports to the European Commission and it checks milestones and deliverables of all workpackages. The management also handles the global risk management of the project. The risk management includes the call for extra expert meetings in case of unsolved project tasks and the adjustment associated work-packages in the case of missing milestones or deliverables.The human perception task of the project is mainly handled in WP2. Starting point of WP2 is the definition of a method for all sensory panel assessments of perceived air quality and odour intensity. The method is based on the knowledge of the experimentally experienced project partner and the input of new multidimensional psychophysical perception models. This common standard for the assessment method guarantees comparable measurements at all labs for the innovative sensor system calibration data. The second and the most important part of WP2 is the definition of a new odour space based on reference odours. The new odour space has to cover all relevant odours of indoor air related environments. All project partners of WP2 have been working on the characterisation of odours in the past. The work-package leader Karolinska

Figure 3 :

 3 Figure 3: The software will handle the development/calibration process and it provide the final version for diagnostics of unknown odours

Figure 4 :

 4 Figure 4: Major work-package interdependencies

ACKNOWLEDGEMENT

SysPAQ is partly sponsored by the European Community in the Nest programme (NEST-28936) under the management of Mrs. P. Lopez. The coordination is done by the Technical University Berlin. Other participants are: Danish Building Research Institute, Aalborg University; Technical University of Denmark; Karolinska Institute; Centre Scientifique et Technique du Bâtiment; Forschungszentrum Karlsruhe; Federal Institute for Materials Research and Testing; REGIENOV; Alpha MOS; Halton Oy.