C Listowski

F Lefèvre

SOLAR OCCULTATION WITH SPICAM/UV ON BOARD MARS EXPRESS: RETRIEVING AEROSOL AND OZONE PROFILES

Introduction:

We present the first results of solar occultation performed with the UV channel of SPICAM, on board Mars Express (spectral range of 120-320 nm and resolution of 0.51 nm [START_REF] Bertaux | SPICAM on Mars Express: Observing modes and overview of UV spectrometer data and scientific results[END_REF])). We have analyzed observations from the 21st of April 2004 to 21st of August 2008. In Martian Years, this corresponds to MY27&28, and the beginning of MY29. This dataset of 2.25 Martian years allows interannual and seasonal comparisons of the detected species: ozone (column density and local density) and aerosols (slant opacity, extinction coefficient and effective radius).

Occultation is a powerful technique for the observation of planetary atmospheres because it is selfcalibrated, requiring no radiometric registration and being nearly insensitive to instrument aging: at every observation, a reference spectrum of the source is collected to which spectra observed through the atmosphere are compared.

Stellar occultation in UV with SPICAM have already been presented in [START_REF] Quémerais | Stellar occultations observed by SPICAM on Mars Express[END_REF], Montmessin et al. (2006b), [START_REF] Lebonnois | Vertical distribution of ozone on mars as measured by SPICAM/Mars Express using stellar occultations[END_REF], and [START_REF] Forget | The density and temperatures of the upper martian atmosphere measured by stellar occultations with Mars Express SPICAM[END_REF]. These studies present results on CO2, ozone, and aerosols that can be directly used to constrain climate models.

The Martian climate is in great part controlled by the presence and the radiative effect of suspended particles; i.e. mineral dust aerosols and icy condensates. The near-surface winds lift dust from the surface in particular during storms and/or through small-scale vertical phenomena such as dust devils. Dust storms occur mostly when Mars is at perihelion between Ls=220 and Ls=300 [START_REF] Smith | Interannual variability in TES atmospheric observations in Mars during 1999-2003[END_REF], corresponding to southern spring/summer. These storms may have a global extent or be regional [START_REF] Cantor | Martian dust storms: 1999 Mars Orbiter Camera observations[END_REF].

SPICAM nadir observations were used to map ozone column densities during more than one Martian Year resulting in the first complete climatology of O3 with a good temporal and spatial resolution [START_REF] Perrier | Global distribution of total ozone on Mars from SPICAM/MEX UV measurements[END_REF]. SPICAM stellar occultations were used to derive O3 vertical profiles [START_REF] Lebonnois | Vertical distribution of ozone on mars as measured by SPICAM/Mars Express using stellar occultations[END_REF]) that revealed the existence of two ozone layers, a superficial one below 30 km, and another, seasonally and spatially highly variable one, located at 30-60 km. In this work, SPICAM solar occultations are used to complement the existing datasets on the Martian ozone. A parallel work on SPICAM IR solar occultation measurements of water vapor profiles [START_REF] Maltagliati | Vertical Profiles Of Water Vapor In Mars' Atmosphere By Spicam/mex Solar Occultations[END_REF][START_REF] Maltagliati | Discoveries on the vertical distribution of water vapor in Mars' atmosphere as observed by the SPICAM-IR spectrometer[END_REF] will help to constrain the GCM dynamics and chemistry through a comparison of the well-understood ozone-water vapor anticorrelation [START_REF] Lefèvre | Three-dimensional modeling of ozone on Mars[END_REF].

Methods:

The CCD has 288 lines, so that theoretically 288 spectra may be resolved. However, only 5 spectra (summation over several CCD lines) are transmitted per second because of telemetry constraints. Thus, the CCD effectively consists of 5 bands (the regions that are summed up) so that a whole set of observation contains 5 tempo-spectra. Each of the 5 bands resolves the vertical profile with a 1 km resolution resulting from the 1 Hz frequency of monitoring the Sun signal. Merging the observed profiles of the 5 bands increases the resolution to 0.2 km on average.

The data processing leading to vertical profiles of aerosols, O3 and CO2 involves three steps (see [START_REF] Quémerais | Stellar occultations observed by SPICAM on Mars Express[END_REF], Montmessin et al., 2006b): -Determination of atmospheric transmission; -Spectral inversion leading to column densities and slant opacities; -Vertical inversion leading to local atmospheric properties (e.g. densities and extinction coefficients).

Figure 1 shows an example of transmission spectra with a color scale related to the altitude of the LOS. The strong ozone absorption (Hartley band) appears around 250nm. The aerosols affect the entire spectrum (slope of the spectra) Resulting transmission spectra can then be fitted with the simple Beer-Lambert law, allowing the retrieval of absolute quantities (local density or extinction coefficient) from relative (spectral ratio) measurements. The spectral behavior of aerosols is modeled by the -model (e.g. [START_REF] O'neill | Extraction of bimodal aerosol size distribution radii from spectral and angular slope (Angstrom) coefficients[END_REF] The parameter α gives access to, via Mie theory, the effective radius of particles.

While CO2 can be easily deduced from the occultation of bright UV stars, the Sun is a poor emitter below 200nm and the CO2 retrieval yields very large uncertainties. Thus we use EMCD (European Mars Climate Database v.4.3, Millour et al., 2008) profiles that offer a smoother and more realistic estimation than given by our retrieval, and thereby reduce potential biases affecting the other quantities retrieved.

The vertical inversion process allows to find n(z) and kext(z), respectively the local density and the extinction coefficient, at an altitude z along the local vertical passing by the MNP, using an onion peeling method which models the atmosphere in discrete concentric layers in which n(z) and kext(z) are constant, varying only from a layer to the next one.

We define here the general name "orbits X000" all the occultations performed during orbits with number X*1000 ≤ N ≤ X*1000+1000. In each group of orbits, the occultations are performed continuously within a period so that ∆Ls ≤ 70•.

Results for aerosols:

Among the 720 occultations 450 profiles were analyzed. All the orbits #3000 were rejected because of anomalous behavior: the transmissions computed displayed spurious oscillations. Some other profiles were not considered because of CCD intensifier anomalies.

The hazetop In order to investigate the seasonal and interannual behavior of the dust we define here two cases of aerosol vertical distribution based on observed profiles: 1) The well-mixed layer of dust that starts at the surface extending up to a certain altitude, 2) The detached layers that form above this well-mixed "haze".

Thus a first approach is to undertake the mapping of the depth of the persistent dust haze that is continuously observed in the lowest atmosphere. [START_REF] Jaquin | The vertical structure of limb hazes in the Martian atmosphere[END_REF] and more recently Montmessin et al. (2006b) have presented the seasonal variations of this aerosol layer. We define hazetop as the altitude where the opacity of aerosols decreases from its surface value to τaer = 1 (see Fig. 4): any detached layer above this level is ignored and analyzed separately. In our study the spatial coverage (in one year) is not as high as in Montmessin et al. (2006b). However, combining the 2.25 Martian Years of observations (MY27-29), we do observe a seasonal signa-ture of the hazetop. Our observations of orbits #0000 (MY 27), and orbits #5000 (MY 29), overlap in the southern hemisphere near 65°S, and display roughly the same average hazetop values of 22-23 km. Thus we can state that the altitude of the hazetop in southern fall is in general around 20-30 km.

Dust storm Another interesting interannual variation in this dataset is the much higher hazetops observed during and after the southern spring equinox in MY 28 (orbits #4000) than those of the southern summer of MY27 (orbits #2000). Both sets of orbits are located in the [70•S, 50•S] latitude band and are close in Ls. On average, the hazetop in this season of the MY28 is more than 90% higher than the MY27 one observed only slightly later in the season. The most likely reason is the global dust storm observed at this season during MY28 (also observed with THEMIS: [START_REF] Smith | THEMIS observations of Mars aerosol optical depth from 2002-2008[END_REF].

Detached layers In several profiles, clear layers of high opacity at high altitudes have been observed: we have analyzed such layers with statistics performed on the different Martian years of observations. A detached layer may consist of ice condensates or may be a simple dust plume. However, in occultation mode we cannot distinguish between these cases. We systematically looked for the most clearly defined atmospheric structures. Figure 6 shows an example of detection of a layer (using change of sign in the derivative of the profile (z)). About 36 % of the profiles contain at least one clearly identifiable detached layer. Table 1 summarizes the properties of the detected layers. The maximum layer altitude is observed during the dust storm (90 km) and the maximum vertical extent of a layer is 41 km, during southern summer.

Table 1: Mean characteristics of the observed detached layers: the peak layer altitude Zpeak, the vertical extension of the layer H, and the increase of the opacity within the layer in percentage ∆τ/τmin.

The average Zpeak in orbits #4000 (global dust storm) is 71 km, at least 15 km higher on average compared to the orbits 2000 closest in latitude and season.

The lowest detached layers (see Fig. 7) were found in the northern set of the orbits 1000, which were monitored above polar regions during late summer with Zpeak reaching 18-22 km. Although the layers observed with SPICAM are not in the PBL, we interestingly find the lowest layers in the same latitudes and during the same part of the northern summer where Whiteway et al. (2009) detected the formation of PBL clouds.

Radius: haze and detached layers The vertical distribution of aerosols is a result of a complex interplay between gravitational settling, vertical transport and interactions between condensates and dust aerosols. In general, because of gravitational settling, larger particles are found at lower altitudes (Fig. 8). Nevertheless, large particles may be observed at high altitudes in the case of cloud formation, when an ice shell forms a dust grain. Some layers show radii 150% larger than the values observed just below the layer. Figure 9 shows a profile where the radius increases from 60nm below the layer up to of 290nm inside the layer. Conversely to the near-surface haze that shows a decreasing trend of radius with altitude, the radii in the detached layers do not show a similar trend (Fig. 10). Considering only orbits 1000 (light blue on Fig. 10) the bigger mean radii are observed in the lowest layers at high northern latitudes.

First results for ozone:

Ozone layer Ozone profiles were derived and compared with the modeled climatology of O3 [START_REF] Lefèvre | Three-dimensional modeling of ozone on Mars[END_REF]). Figure 11 shows three examples of Slant column of O3 (molecule.cm-2). We observe the nocturnal ozone layer at an altitude around 30-60 km, observed by [START_REF] Lebonnois | Vertical distribution of ozone on mars as measured by SPICAM/Mars Express using stellar occultations[END_REF] and predicted by models. Since we observe the atmosphere at the terminator (during sunrise or sunset) we can conclude that the nocturnal ozone layer is already (still) present during sunset (sunrise). Comparison with the LMD-GCM Comparisons were done with the profiles obtained from the LMD-GCM [START_REF] Lefèvre | Three-dimensional modeling of ozone on Mars[END_REF], and the first results show good agreement (Figure 12).

Conclusion and perspectives:

We present a large dataset of 2.25 Martian Years (MY27-29) of SPICAM/UV solar occultations. We derived aerosol and ozone vertical profiles that can extend to very low altitudes (~10 km). We investigate the aerosol vertical distribution and its seasonal variations through monitoring the haze adjacent to the surface and the detached layers that may form above it. The multiannual dataset allowed us to monitor the dust seasonal cycle. We also observed the consequences of the MY28 dust storm (higher hazetop and higher detached layers) in the southern hemisphere. The lowest layers are observed at high northern latitudes, with large particles of a mean radius between 200 and 300 nm. The ozone profiles confirm the previously observed nocturnal layers [START_REF] Lebonnois | Vertical distribution of ozone on mars as measured by SPICAM/Mars Express using stellar occultations[END_REF]) at sunrise and sunset during the same period. First comparisons with ozone profiles from the LMD-GCM show good agreement. Further comparisons with model profiles will be done for ozone and the water vapor profiles of SPICAM/IR [START_REF] Maltagliati | Vertical Profiles Of Water Vapor In Mars' Atmosphere By Spicam/mex Solar Occultations[END_REF][START_REF] Maltagliati | Discoveries on the vertical distribution of water vapor in Mars' atmosphere as observed by the SPICAM-IR spectrometer[END_REF]): this will allow to better constrain the water-ozone anticorrelation in the LMD-GCM [START_REF] Lefèvre | Three-dimensional modeling of ozone on Mars[END_REF]). Since the solar occultation mode observes at sunrise or sunset at a given location, comparisons of ozone profiles at morning and evening terminator (where atmospheric chemistry might drastically change) would be interesting.

Figure 1 :

 1 Figure 1: Example of transmission spectra at different altitudes (blue: low altitudes; red: highest altitudes). The transmission is plotted versus the wavelength. The signal is very noisy below 200 nm because of low emission of the sun shortward of 200nm.

Figure 2 :

 2 Figure 2: Two examples of slant opacity profiles with a detached layer having a slant opacity greater than 1. The hazetop is shown by the blue arrow, and the red arrow shows the detached haze.

Figure 3 :

 3 Figure 3: Seasonal variation of the hazetop. We observe higher hazetop values in the warmer regions and seasons, i.e., toward the equator and during southern spring and summer (180°≤ LS ≤ 360°)

Figure 4 :

 4 Figure 4: a detection of a layer at Lat=-7°, Long=310°, Ls=43, Lt=17h, during sunset.

Figure 5 :

 5 Figure 5: mean radius (m) VS altitude (m) for orbits 1000 and orbits 4000. The red dashed lines show the variance of the averaged profiles

Figure 6 :

 6 Figure 6: Example of an increase in radius (in m, in red) in a detached layer (slant opacity in black).

Figure 7 :

 7 Figure 7: layer altitude (km) VS mean radius in the detached layers (m). The different colors are related to the different sets of orbits.

Figure 8 :

 8 Figure 8: Three Slant column of O3 (molecule.cm-2). Different colors are related to the different bands of a single observation. Two profiles with an ozone layer (~30 km at sunset and ~40 km at sunrise) are shown. Note the different altitude ranges.

Figure 9 :

 9 Figure 9: Two comparisons of local density (cm-3) profiles of the LMD-GCM (dotted line) with our observations.


Joel Agnus

Ioan Alexandru Ivan

Samuel Queste

DRY ETCHING OF SINGLE CRYSTAL PMN-PT PIEZOELECTRIC MATERIAL

During the last decade, the applications of PMN-PT spread significantly. Unlike PZT, the appropriate microtechnologies for PMN-PT Piezo-MEMS aren't fully documented in the literature. This paper deals with the PMN-PT etching by inductively coupled plasma (ICP) technique, also known as DRIE. The paper quantitatively presents the etching parameters of PMN-PT by the Ar/C 4 F 8 gas combination and reports some related useful experience.

INTRODUCTION

The ability to fabricate well-defined structures on the surface of the substrates, with near vertical sidewalls, a high aspect ratio, and a relatively low surface roughness, is a key factor for many applications, like MEMS, NEMS and integrated optical devices. High depth, high aspect ratios, good uniformity over the wafer, vertical wall profiles and reasonable etching selectivity, i.e. the ratio of the etch rates of the substrate and the mask are the actual challenges.

Plasma-based reactive ion etching (RIE) is a very controllable dry process exploiting both chemical and physical processes to remove solid material locally [START_REF] Rossnagel | Handbook of Plasma Processing Technology -Fundamentals, Etching, Deposition, and Surface Interactions[END_REF]. The plasma is generated with a powerfull RF electromagnetic field under low pressure. Accelerating ions colide with the wafer surface and eventually react with it. The effect is a combination of chemical reaction and atomic sputtering, whose balance depends on the material and method (gas concentrations, powers, substrate temperature).

Deep reactive ion etching (DRIE) using inductively coupled plasma (ICP) RIE system has demonstrated its potential in batch fabrication of deep anisotropic microstructures in Silicon. The technology is also used for highly directional and precise micro-scale etching of materials including piezoelectric. The interest is in highly integrated piezoelectric and piezo-on-silicon microsytems (from resonators and transducers to actuators and energy harvestors) [START_REF] Tadigadapa | Piezoelectric MEMS sensors:state-of-the-art and perspectives[END_REF] where traditional methods fail (e.g. mechanical lapping and trenching, ultrasonic etching).

Dry etching of piezoelectric materials is not new issue, researches being reported in quartz [START_REF] Schreiter | A new plasma-etching technique for micromechanical structuring of quartz[END_REF], AlN [START_REF] Khan | High rate etching of AlN using BCl3/Cl2/Ar inductively coupled plasma[END_REF], LiNbO 3 and PZT ceramics [START_REF] Efremov | Etching characteristics and mechanism of Pb(Zr,Ti)O3 thin films in CF4/Ar inductively coupled plasma[END_REF] where etching is mainly obtained by a rather physical sputtering with insignificant chemical contribution. Other newer materials with outstanding piezoelectric properties such as PMN-PT and PZN-PT are not documented in the literature.

The paper is dedicated to the investigation of dry etching of PMN-PT material and is organized as follows. After the introductory part, the Section II presents some information about the PMN-PT material and its applications in microsystems. Then, the experimental plan and the related facilities for mask deposition and dry etching are presented. The fourth section called Experimental results is the most important, presenting the graphical and numerical results. Finally, a Conclusions section is dedicated for several remarks.

PMN-PT BASED MICROSYSTEMS

In the recent years there has been an increasing interest in growing and characterizing xPb(Mg 1/3 Nb 2/3 )O3-(1x)PbTiO 3 (PMN-PT) [START_REF] Wang | Complete set of elastic, dielectric and piezoelectric constants of orthorhombic 0.71Pb(Mg1/3Nb2/3)O3-0.29PbTiO3 single crystal[END_REF] and xPb(Zn 1/3 Nb 2/3 )O 3 -(1x)PbTiO 3 (PZN-PT) [START_REF] Zhang | Complete set of material constants of 0.93Pb(Zn1/3Nb2/3)O3-0.07PbTiO3 domain engineered single crystal[END_REF]. Initially used as ceramics, they were grown into single crystals by modified Bridgman or solid-state single crystal growth methods and have become commercially available. Those materials exhibit outstanding piezoelectric properties (e.g. d 33 =3500 pC/N, d 13 =-1200 pC/N, k 33 =0.95 etc.) that considerably surpass the PZT ceramics by a factor of 4 to 5 [START_REF] Wang | Complete set of elastic, dielectric and piezoelectric constants of orthorhombic 0.71Pb(Mg1/3Nb2/3)O3-0.29PbTiO3 single crystal[END_REF].

During the last decade, the applications of PMN-PT increased dramatically, including high frequency bulk and surface resonators and filters, sensors and actuators, ultrasonic transducers, energy harvesters. Most applications that are not presented here apply to the classical mechanical machining (saw dicing, lapping etc.) of bulk material while others document recent MEMS micro-devices.

For instance, one of the early-patented micromachined devices is a PMN-PT composite ultrasound transducer (PC-MUT). The fabrication technology is shown in [START_REF] Jiang | Microfabrication of Piezoelectric Composite Ultrasound Transducers (PC-MUT)[END_REF] but with undisclosed quantitative parameters regarding DRIE method. In [START_REF] Hur | Fabrication and characterization of PMN-PT single crystal cantilever array for cochlear-like acoustic sensor[END_REF] arrays of etched cantilevers were reported to serve as acoustic sensors; equally lacking parametric details. Finally, in the recent topic review article [START_REF] Tadigadapa | Piezoelectric MEMS sensors:state-of-the-art and perspectives[END_REF] that reveals the state of the art in piezoelectric MEMS, the dry etching characteristics of most piezoelectric materials are summarized, except for the ones of PMN-PT.

The present paper is related to the authors mainstream activities in piezoelectric micro-actuators and tweezers for micromanipulation/microassembly.

More recently, integrated PMN-PT piezo-actuators raised our interest, given that simulations predict improved performance [START_REF] Ivan | Comparative material study between PZT ceramic and newer crystalline PMN-PT and PZN-PT materials for composite bimorph actuators[END_REF]. Unlike PZT, the appropriate micro-technologies for PMN-PT based Piezo-MEMS aren't fully documented in the literature; this paper attempts to fill a gap by quantitatively reporting the etching parameters of PMN-PT.

ETCHING EXPERIMENTS

Material samples

The material used for these experiments was TRS-X2C from TRS Technologies of 68% PMM and 32% PT concentration, whose cut and poling was performed in <001> direction for an optimised out-of-plane operation. This type of plate possesses a longitudinal piezoelectric coefficient d 33 between 2300 and 3300 pC/N. The k 33 coupling factors range from 0.90 to 0.97. The transverse coefficients are isotropic for this type of cut, varying from -700 to -1200 pC/N according to several intrinsic or external parameters such as material concentration, crystallographic defects and operating temperature.

A number of ~30 samples of roughly 5 x 10 x 0.2 mm 3 were initially prepared, of which 18 were successfully tested under different conditions. The DRIE mask was of a regular type: Nickel on a thin Chrome buffer. Nickel is known as performing well under intensive etching energies. Other materials such as Cr, Al or Cu were studied for etching masks but were dismissed due to low selectivity, etching chamber contamination or limited deposition thickness.

Mask deposition and dry etching facilities

The sputter deposition machine was a Plassys MP500 model while the evaporation one an Alliance concept Eva 450. Provided that the nickel is known for inducing residual stress during deposition, its thickness must be limited. For the sputtering technique the maximum layer should be under 0.7 μm. The maximum limit for evaporated Ni on PMN-PT is larger, of around 1.5μm. Finally, electrodeposited layers can be much greater, like 15 μm. Under special conditions we reached an electroplated 25 μm layer, but the sample became too constrained. On the other hand, as will be seen, etching time for such a layer would take too many hours for a realistic process. A related graphical representation is provided in the figure 4.

The ionic etching was carried out using a commercial multiplex AOE MPO 562 reactor from STS with an ICP RF generator of 3 kW and a RIE generator of 1.5 kW. The maximum size of the wafer is 6 inches, and the gas available are SF6, CF4, C4F8, He, O2 and Ar.

The temperature of the substrate holder was varied between -20 and 60°C. The gas mixture we selected was a combination Ar/C 4 F 8 (64sccm /5sccm). The PMN-PT samples were fixed on Si carriers with thermal grease for an optimal cooling of the sample. Step heights in μm were obtained progressively with a Tencor Alpha-step profilometer. Final profiles were also measured with a scanning electron microscope (SEM) and with a MSA-500 MEMS analyzer from Polytec. This specific tool based on Michelson interferometry may offer nanometer-scale resolution in out-of-plane direction.

Experimental design

Several combinations of etching parameters were chosen in the attempt of assessing as much data as possible from the 18 samples. Input parameters were: -RF source power (500, 1200 W), -bias power (100, 300, 400, 500 W), -temperature (-20, 0, 60 o C) and -internal pressure (2, 5, 10, 15 mTorr).

Etching times varied from 30 minutes to 280 minutes.

EXPERIMENTAL RESULTS

Mask Type dependence

One of the predominant challenges in PMN-PT DRIE is the mask sufficiency, which is limited by the etching selectivity between PMN-PT and the mask and by the thickness of the mask. Ni masks can be used as mask materials because of their reasonably high growth rate, typically 17µm/h for electroplated Ni. However, as the thickness of the mask is increased to several micrometers, some practical challenges or disadvantages are encountered. Nickel does not react with fluorine compounds to form volatile products. In addition to the micro-masking caused by re-deposition of the non-volatiles on the area to be etched due to mechanical erosion, electroplated metals are often non-uniform in thickness. Also, thicker hard masks may introduce severe stress on the PMN-PT wafer leading to possible breakage of the mask and the PMN-PT substrate during processing. The etch mask selectivity was studied for electroplated, evaporated Ni and sputtered masks at 500W powers. It was shown that sputtered Ni gives better selectivity (12:1 compared to 8,3:1 for electroplated Ni). But the limiting factor of Ni sputtered mask was the difficulty to deposit layers thicker than 1.5μm using the lift-off.

Bias power dependence

The etch rate decreased with the drop of bias power, from 0,14μm/min for 500W to 0.06μm/min for 100 W and conversely the selectivity raised from 7,85:1 to 9:1 (Fig. 5, Ni evaporated mask).

Figure 6. Effects of increased bias power (right): larger microtrenches, larger lateral mask erosion, presence of nanometer slits (arrows)

Microtrenches and nanometer slits

For the electroplated Ni, some micro-trenches at the bottom edge of the sidewall appeared and was attributed to a higher electron density at the edges during etching, leading to higher etching speed (Fig. 6,7,9). This phenomenon was already observed for Quartz by Abe et al. [START_REF] Abe | One-chip multichannel quartz crystal microbalance (QCM) fabricated by Deep RIE[END_REF] and was attributed to a bias voltage too high. Dahm et al. have also reported this effect [START_REF] Dahm | Quartz etching for phase shifting masks[END_REF] and interpreted this to a high flow of precursor gas. The main contribution for the appearance of this phenomenon is the bias voltage value.

Nanometer slits were also observed at the interface between very close structures. The phenomenon was also reported in the case of PZT [START_REF] Wang | Deep Reactive Ion Etching of Lead Zirconate Titanate Using Sulfur Hexafluoride Gas[END_REF] and is not fully understood, being attributed to material ferroelectric properties.

Pressure dependence

After several tries, we have found that the lowest achievable pressure (2 mTorr) of the DRIE STS machine give improved results in terms of etching speed (by 10%). The associated selectivity is however noticed to decrease quite significantly (from 10:1 to 8:1) at lower pressure values (Fig. 10).

Temperature Dependence

As noticed from Fig. 11 there seems no linear correlation between the etching speed and the process temperature, but the selectivity increase by several percents with the increase of the substrate holder temperature. This must be the result of a small release of the stress in the mask layer. However, due to the low Curie temperature of PMN-PT material we don't recommend high temperatures etching

Lateral mask etching

The etching anisotropy in terms of sidewalls angles is affected mainly by lateral mask shrinking (see Fig. 6 and9) with lateral speeds slightly superior to the vertical ones. Because of the difficulty to asses reliable results for shallow etched surfaces, data for only electroplated mask profiles were reliable. As noticed from the figures lateral erosion increase with the bias power (18 nm/min at 300W and 22 nm/min at 500W).

CONCLUSIONS

As most piezoelectric materials, the reactive component of PMN-PT etching represents an insignificant part of the process, which mainly relies on the physical sputtering of the surface atoms due to energetic ions collisions. Therefore, we used an Ar composition, knowing that the inert gases sputter yield is higher than that of other elements or molecules (Cl 2, C 2 H 6 etc.). An experience plan was set-up in the attempt to test various parametric conditions. The material was PMN-PT [001] lapped at 200μm thickness. The Nickel shadowing masks were electroformed at ~5.5μm or sputtered and evaporated at 0.3-0.7μm thickness.

Preliminary experience indicated the gas combination Ar(92%) + C 4 F 8 (8%). We used an STS ICP machine, a couple dozens charts were obtained by varying different parameters (gas pressure, RF source and bias powers and the substrate temperature).

Best achieved etching rate was 0.19 μm/min with a selectivity ratio of 8:1 for 1200W source and 300W bias powers at 2 mTorr pressure. Sidewalls angles lie at ~70 0 , mainly due to erosion of the Ni mask in lateral (horizontal) direction (Fig. 6, 9.b, 9.d). Lateral mask shrinking increased with the raise of the bias power. Expected micro-trenches at the bottom edge of the sidewall were noticed and quantified, being attributed to high local ion density at the edges during etching. Nanometer slits were also observed at the edges between structures. Compared to PZTs [START_REF] Efremov | Etching characteristics and mechanism of Pb(Zr,Ti)O3 thin films in CF4/Ar inductively coupled plasma[END_REF][13], (selectivity ratios of up to 35:1, etching speeds of 0.14 ÷ 0.4 μm/min, and profile angles of ~75 0 ), we conclude that PMN-PT is slightly more renitent to dry etching than PZT. However, according to our experience, profiles of ~100 μm are achievable, with the cost of extended etching times.

Figure 1 .

 1 Figure 1. a) PMN-PT 5x10 mm initial samples, b) Ni electroformed mask sample.Finally, Ni sample masks were realised using three different technologies: sputtering, evaporation and electroplating. The mask features (Fig. 1.b) were developed using lift-off for the sputtering-deposited samples, wet etching for evaporated ones and electroforming for the latter

Figure 2 .

 2 Figure 2. Feature on an electroplated sample.

Figure 3 .

 3 Figure 3. 3D topography picture (measurement software TMS 3.1 installed on MSA-500 analyzer)

Figure 4 .

 4 Figure 4. Comparative PMN-PT etching results for electroplated, evaporated and sputtered Ni masks. ICP parameters: 500/500W, 10 mTorr, 0 0 C, Ar/C 4 F 8 (64/5).

Figure 5 .

 5 Figure 5. Etching speed and selectivity vs. bias power.

Figure 7 .

 7 Figure 7. a,b) Electroformed test structures before and after DRIE tests. Numerical quantities for evaluating the etching rate, selectivity, sidewall angles, mask erosion and microtrenching c) SEM image of an etched feature inclined at 30 0 .

Figure 8 (

 8 Figure 8 (up). Etchching profiles and speed diagram. Figures 9 a,b). Zoomed SEM images taken at 130': lateral Ni mask erosion is in progress, profile is 26 um deep. Figures 9 c,d). Same features etched after 280': profile is 47 um deep, notice the complete erosion of Ni mask (fig 9.d, except for the intersections) Sidewalls rugosity and microtrenches are increased. Ni lateral erosion. Ni redeposition.

Figure 10 .

 10 Figure 10. Selectivity vs. pressure (different param.)

Figure 11 .

 11 Figure 11. Etching speed and selectivity vs. temperature.


Fernand Meyer

Adjunctions on the lattice of hierarchies

published or not. The documents may come

Adjunctions on the lattice of hierarchies

Introduction

Hierarchies are the classical structure for representing a taxinomy. The most famous taxonomy, the Linnaean system classi…ed nature within a nested hierarchy, starting with three kingdoms. Kingdoms were divided into Classes and they, in turn, into Orders, which were divided into Genera (singular: genus), which were divided into Species (singular: species). Below the rank of species he sometimes recognized taxa of a lower (unnamed) rank (for plants these are now called "varieties").

Hierarchies are also useful in the domain of image processing, as they represent in a condensed way nested partitions obtained through image segmentation. Hierarchies appear quite naturally in the …eld of morphological segmentation, which uses as tool the watershed of gradient images. As a matter of fact, the catchment basins of a topographic surface form a partition. If a basin is ‡ooded and does not contain a regional minimum anymore, it is absorbed by a neighboring basin and vanishes from the segmentation. A hierarchy is hence obtained by considering the catchment basins associated to increasing degrees of ‡ooding. They structure the information on the image by weighting the importance of the contours : the importance of a contour being measured by the level of the hierarchy where it disappears [START_REF] Meyer | An overview of morphological segmentation[END_REF].

After an axiomatic de…nition of a hierarchy, we show that hierarchies are characterized by an ultrametric ecart (also called ultrametric half-distance [START_REF] Schwartz | Analyse : topologie générale et analyse fonctionnelle[END_REF]). This ecart permits quite simply to de…ne and analyze the complete lattice structure of hierarchies. They also permit to de…ne two adjunctions between hierarchies. Some examples of hierarchies met in morphological segmentation are given.

Hierarchies appear as natural generalizations of partitions. A hierarchy is characterized by an ultrametric half-distance, whereas a partition is characterized by an ultrametric binary half-distance. The algebraic structure of partitions has been studies by Serra, Heijmans, Ronse ( [START_REF] Heijmans | Connected morphological operators for binary images[END_REF], [START_REF] Serra | A lattice approach to image segmentation[END_REF], [START_REF] Ronse | Adjunctions on the lattices of partitions and of partial partitions[END_REF]). The same set may be partitioned into distinct partitions according the type of connectivity one adopts. Serra has laid down the adequate framework for extending the topological notion of connectivity by de…ning connective classes ( [START_REF] Serra | Image Analysis and Mathematical Morphology. II: Theoretical Advances[END_REF], [START_REF] Serra | A lattice approach to image segmentation[END_REF]). We extend this construction to hierarchies by introducing taxonomy classes.

De…nition of hierarchies 2.1 Axiomatic de…nition of hierarchies

The axiomatic de…nition of hierarchies is due to Benzecri [START_REF] Benzécri | L'analyse des données 1. La taxinomie[END_REF].

De…nition of a partition

Let E be a domain whose elements are called points. Let be a subset of P(E). is a partition of E if the following conditions are veri…ed:

S B2 B = E B 1 ; B 2 2 imply B 1 \ B 2 = ? or B 1 = B 2

De…nition of a dendrogram and its elements in P(E)

Let X be a subset of P(E), on which we consider the inclusion order relation.

X is a dendrogram if the following axiom is veri…ed :

Axiom 1 (Dendrogram axiom) A; U; V 2 X : A U and A V ) U V or V U An example : if the tiger is simultaneously a mammal and an animal, then any mammal is an animal or any animal is a mammal. A; U and V belong to three di¤erent levels of the hierarchy. Partitions on the contrary consider only one level, and distinct elements have an empty intersection. As a matter of fact, the dendrogram axiom is weaker than the axiom de…ning partitions. if U; V are classes of a partition and there exists a set A included in both U and V then U = V:

If X is a dendrogram, we may de…ne : -the summits :

Sum(X ) = fA 2 X j 8B 2 X : A B ) A = Bg -the leaves : Leav(X ) = fA 2 X j 8B 2 X : B A ) A = Bg -the nodes : Nod(X ) = X Leav(X )
X is a hierarchy, if the two following axioms are veri…ed:

Axiom 2 (Intersection axiom) : two elements of X which are not comparable for the inclusion order have an empty intersection: A; B 2 X : A \ B 2 fA; B; ;g Axiom 3 (Union axiom ) Any element A of X is the union of all other elements of X contained in A: 8A 2 X : S fB 2 X j B A ; B 6 = Ag = fA; ;g Proposition 4 The intersection axiom implies that A is a dendrogram for the inclusion order.

Proof.

If A 6 = ;, A U and A V , then U \ V 6 = ;, implying that U \ V = U or U \ V = V; that is V U or U V showing that the dendrogram axiom is satis…ed.
A series of partitions ( i ) i2I is said to be nested, if each region of j is the union of regions of i for j > i: Considering all tiles belonging to a series of nested partitions ( i ) i2I ; obviously yields a hierarchy X .

Strati…ed hierarchies, ultrametric distances and nested partitions

X is a strati…ed hierarchy, if it is equipped with an index function st from X into the interval [0; L] of R which is strictly increasing with the inclusion order:

8A; B 2 X : A B and B 6 = A ) st(A) < st(B):
Strati…cation o¤ers the possibility of thresholding a hierarchy: the elements A of a hierarchy verifying st(A) are all coarser than : Given a strati…ed hierarchy X , verifying st(A) = 0 for each A 2 Leav(X ), a distance between the elements of P(E) is de…ned by: 8C This last inequality is called ultrametric inequality, it is stronger than the triangular inequality. It expresses that the index of the smallest set containing C and D is smaller or equal than the index or the smallest set containing all three elements B; C and D; and whose diameter is max fd(C; B); d(B; D)g.

For X 2 P(E) the closed ball of centre X and radius is de…ned by Ball(X; ) = fD 2 P(E) j d(X; D) g :

Hierarchies as balls of an ultrametric distance

Inversely, given an ultrametric distance index ; the closed balls of radius form a partition. For increasing values of these partitions are nested and become coarser and coarser. Hence we obtain like that a strati…ed hierarchy. In order to establish it, we prove the three following lemmas. Proof. Suppose that B is an element of Ball(A; ). Let us show that then B also is centre of this ball. Ball(B; ) and Ball(A; ) have the element B in common, hence, according to the preceding lemma they are identical.

Lemma 7

The radius of a ball is equal to its diameter.

Proof. Let Ball(A; ) be a ball of diameter ; that is the maximal distance between two elements of the ball. Hence : Let B and C be two extremities of a diameter in Ball(A; ) :

= (B; C) (B; A) _ (A; C) = : Hence = :
Since two closed balls of same radius are either identical or disjoint, they form a partition. For increasing values of , the balls are also increasing, hence we obtain nested partitions.

For this reason, given an ultrametric distance d; the closed balls of radius form a partition. For increasing values of ; these partitions are nested, become coarser and coarser and form a strati…ed hierarchy. Partitions as particular hierarchies Partitions are hierarchies with only two strati…cation levels. For all elements A of a partition , we have st(A) = 0: The only element for which we have st(B) = 1 is the domain E itself. The associated ultrametric distance also is binary. For two distinct tiles of a partition B 1 ; B 2 ; we have (B 1 ; B 2 ) = 1; whereas (B 1 ; B 1 ) = 0 2.2 Extending the hierarchies to the points of E

Hierarchies

Consider a strati…ed hierarchy X to which is associated an ultrametric distance : The hierarchy X is a collection of subsets of P(E); each of them containing points of E: We designate with the letters p; q; r::: the points of E: If E belongs to the hierarchy, then each point p of E is contained in a unique leave A = Leav(X ) of X .: As this leave is unique we call it Leav(p):

We now extend the ultrametric distance on X into an ultrametric ecart on the points of E : for p; q 2 E : (p; q) = (Leav(p); Leav(q)) Its properties are :

for p 2 E : (p; p) = (Leav(p); Leav(p)) = 0 : re ‡exivity for p; q 2 E : Leav(p) = Leav(q) ) (p; q) = 0 for p; q 2 E : (p; q) = (Leav(p); Leav(q)) = (Leav(q); Leav(p)) = (q; p) : symmetry for p; q; r 2 E : (p; r) = (Leav(p); Leav(r)) (Leav(p); Leav(q)) _ (Leav(q); Leav(r)) = (p; q) _ (q; r) : ultrametric inequality Hence is an ultrametric ecart but not a distance, as the antisymmetry is not veri…ed, since distinct points p and q belonging to a same leave have an ecart equal to 0: On the other hand, the ultrametric inequality is stronger than the triangular inequality characterizing metric distances and ecarts. Laurent Schwartz called these ecarts half distances in [START_REF] Schwartz | Analyse : topologie générale et analyse fonctionnelle[END_REF]. His de…nition of half-distances and half-metric spaces is given below. The open half balls B i;o (a; R) (resp. closed B i (a; R) of a center a 2 E; of radius R and index i are all x of E such that d i (a; x) < R (resp. R).

Half distances and half metric spaces

A half metric space is then a topological space de…ned as follows : a subset O of E is open, if for each point x 2 O; there exists a half ball B i (x; R) centered at x; with a positive radius entirely contained in O.

If the triangular inequality is replaced by the ultrametric inequality, we call it ultrametric half-distance or ultrametric ecart. To any hierarchy X to de…ned on subsets of P(E) is thus associated an ultrametric half-distance or ultrametric ecart.

Equivalence relations and partitions

Consider now the case of a binary ultrametric ecart de…ned on the points of E: This ecart permits to de…ne the binary relation R:

For p; q 2 E : p R q , (p; q) = 0
This relation is re ‡exive and symmetrical as is itself. Consider now p; q; r 2 E verifying p R q and q R r: This is equivalent with (p; q) = 0 and (q; r) = 0: But then (p; r) (p; q) _ (q; r) = 0 indicating that p R r: Hence the relation R is also transitive: it is an equivalence relation. The classes of this equivalence relation precisely are the leaves of the ecart and form a partition of E:

Partial partitions and partial hierarchies

Consider a subset A of P(E). A hierarchy X (resp. a partition ) on A is called partial hierarchy (resp. partial partition) on E: The domain A is called support of the partial hierarchy (resp. partial partition) and is written supp(X ) (resp.

supp( ))

The notion of partial partition has been introduced and studied by Ch. Ronse in [START_REF] Ronse | Adjunctions on the lattices of partitions and of partial partitions[END_REF] ; Ronse denotes (E) the class of partial partitions of E: Likewise let us denote X (E) the class of partial hierarchies.

Partial partitions Consider a partial partition in (E):

is a normal partition on supp( ). So the binary ultrametric ecart (p; q) is perfectly de…ned for p; q 2 supp( ): In order to completely characterize the partial partition without the need to specify its support, we extend in order to get de…ned on E E :

(pp1) : for p; q 2 supp( ) : (p; q) = (p; q) is an ultrametric ecart (pp2) for p = 2 supp( ); 8q 2 E : (p; q) = 1
This last relation is also true for p itself : for p = 2 supp( ) : (p; p) = 1 The support supp( ) is characterized by fp j (p; p) = 0g : We call cl(p) the closed ball of centre p and of radius 0 associated to : Relation (pp2) implies that for p = 2 supp( ) the class cl(p) is empty. Consider now p; q 2 E such that q 2 cl(p). This shows that p; q 2 supp( ) and (p; q) = 0: If r 2 cl(p); then (p; r) = 0 and (q; r) (q; p) _ (p; r) = 0 showing that r 2 cl(q): Similarly r 2 cl(q) ) r 2 cl(p): Hence for any p; q 2 E ; q 2 cl(p) ) cl(q) = cl(p): But these are precisely the criteria given by Ronse for de…ning partial partitions:

(P1b) for any p 2 E ; cl(p) = ? or p 2 cl(p) (P2a) for any p; q 2 E ; q 2 cl(p) ) cl(q) = cl(p)
As shown by Ronse, to partial partitions correspond partial equivalence relations, which are symmetric and transitive but are not re ‡exive. The support of a partial equivalence relations R being the set of all points p 2 E for which there exists a point q 2 E verifying p R q:

Partial hierarchies Consider a subset A of P(E). A hierarchy X on a subset A of P(E) is called partial hierarchy E: The domain A is called support of the partial hierarchy (resp. partial partition) and is written supp(X ). We call X (E) the family of partial hierarchies on E:

Consider a partial hierarchy X represented by its ultrametric ecart : The ecart can then be extended to an ultrametric mapping :

(xp1) : for p; q 2 supp(X ) : (p; q) = (p; q) is the ultrametric ecart on A

(xp2) for p = 2 supp(X ); 8q 2 E : (p; q) = L
This last relation is also true for p itself : for p = 2 supp( ) : (p; p) = The support supp( ) is characterized by fp j (p; p) = 0g :

Aliens and singletons Partial hierarchies and partial partitions have thus two domains, the support domain and its complementary set.

We call "aliens" the points verifying 8q 2 E : (p; q) > 0 ; that is the points outside the support.

Aliens should not be mixed up with the singletons, which duly belong to the support. The singleton fxg is a set of P(E) reduced to the point x: Singletons are characterized by: 8q 2 E ; p 6 = q; : (p; q) 0 and (p; p) = 0:

Transforming a set into a partition Given a set A of P(E); we have three ways to complete it, in order to create a partition: Backg(A) = A [ A : the set A and its complement A form a partition singl(A) is the partition where A is pulverized into singletons alien(A) is the partition where A is pulverized into aliens

Dilations and erosions associated to partitions and hierarchies

An adjunction associated to a partition

To any partition on E we may associate a dilation . For a point p 2 E; one de…nes (p) = cl(p): One then de…nes (X) = S f (x) j x 2 Xg = S fC i 2 j X \ C i 6 = ;g The properties of are the following : is increasing and commutes with union : it is indeed a dilation obvioulsy x 2 (x) ; hence is extensive it is also a closing. The fact that also is a closing seems at …rst sight strange, as the class of invariants of a closing is stable by intersection. But the invariants of are unions of classes of the partition : Hence their class is stable by intersection. It is easy to check that is a dilation-closing : Let us now study the erosion " adjunct to : Y "(X) , (Y ) X:

(Y ) = (Y ) ) (Y ) " (Y )
Obviously "(X) = S fY j Y "(X)g = S fY j (Y ) Xg : But since is extensive and idempotent S fY j (Y ) Xg = S f (Y ) j (Y ) Xg = S fC i 2 j C i Xg :
By duality " is increasing, anti-extensive, idempotent and commutes with intersection, it is an erosion-opening: " = " Adjunctions associated to a hierarchy X . The closed balls Ball(p; ) of radius form a partition, for which we may apply the results of the previous paragraph and de…ne the adjunction ( ; " ) de…ned by:

(X) = S fBall(x; ) j x 2 Xg " (X) = S fY j Y "(X)g = S fY j (Y ) Xg = S fBall(x; ) j x 2 E ; Ball(x; ) Xg
Applications : interactive segmentation The following examples have been developed within a toolbox for interactive segmentation ( [START_REF] Zanoguera | An interactive colour image segmentation system[END_REF]).

Intelligent brush An intelligent brush segments an image by "painting" it: it …rst selects a zone of interest by painting. Contrary to conventional brushes, the brush adapts its shape to the contours of the image. The shape of the brush is given by the region of the hierarchy containing the cursor. Moving from one place to another changes the shape of the brush, when one goes from one tile of a partition to its neighboring tile. Going up and down the hierarchy modi…es the shape of the brush. In …g.1, on the left, one shows the trajectory of the brush ; in the centre, the result of a …xed size brush, and on the right the result of a self-adapting brush following the hierarchy. This self adapting brush is nothing by the dilation r by a ball associated to the hierarchy, centered at the position of the mouse and of a radius, also easily modi…ed through the mouse. This method has been used with success in a package for interactive segmention of organs in 3D medical images.

Mouse trajectory

Fixed size brush Intelligent brush Mouse trajectory Fixed size brush Intelligent brush

Magic wand

The magic wand in a conventional computer graphics toolbox consists in extracting the region which touches the position of the mouse and whose colour lies within some prede…ned limits from the coulour at the mouse position. The next step consists in replacing this set by the smallest set of the hierarchy which contains it. This operation is a closing, described by Ch. Ronse in [START_REF] Ronse | Adjunctions on the lattices of partitions and of partial partitions[END_REF]. The result is shown in …g.2

The lattice of hierarchies

It is often interesting to combine several hierarchies, in order to combine various criteria or merge the information obtained from diverse sources (colour or multispectral images for instance). We …rst de…ne an order relation between hierarchies which structures them into a complete lattice.

Order relation

Complete hierarchies Let A and B be two strati…ed hierarchies, with their associated half-distances : A and B : The following relation de…nes an order relation between the hierarchies: B < A , 8p; q 2 E A (p; q) B (p; q) It follows that 8p 2 E : Ball B (p; ) Ball A (p; ) With this order relation the hierarchies of P(E) form a complete lattice. The maximal element is the hierarchy having E as only element and the smallest hierarchy contains only singletons fxg : Partial hierarchies This order also holds for partial hierarchies. Let A and B be two strati…ed partial hierarchies, with their associated half-distances : A and B : The following relation de…nes an order relation between the hierarchies: B < A , 8p; q 2 E A (p; q) B (p; q) : For each p = 2 supp(A) : A (p; p) = L; which implies that B (p; p) = L; indicating that supp(A) supp(B); or equivalently supp(B) supp(A)

The smallest partial hierarchy contains only aliens, i.e. points p verifying 8q 2 E ; (p; q) = L: Particular case of partitions Let A and B be two partitions, with their associated binary half-distances : d A and d B : The partition B is …ner than the partition A i¤ 8p; q 2 E d A (p; q) d B (p; q)

It follows that 8p 2 E : Ball B (p; ) Ball A (p; ) But the balls of a partition Ball B (p; ) are the tiles of this partition. Hence the tiles of the …ner partition B are included in the tiles of the coarser partition A which is coherent with the usual de…nition of the order between partitions.

In…mum of two hierarchies

Complete hierarchies The in…mum of two hierarchies A and B is written A ^B and is de…ned by its ultrametric half-distance

d A^B = d A _ d B .
It is easy to check that it is indeed a half-distance. It is symmtrical and half-positive. Let us check the ultrametric inequality:

(d A _ d B ) (p; r)_(d A _ d B ) (r; q) = (d A (p; r) _ d A (r; q))_(d B (p; r) _ d B (r; q)) > (d A (p; q) _ d B (p; q)) = d A _ d B (p; q)
Its balls are de…ned by : 8p 2 E : Ball A^B (p; ) = Ball A (p; ) ^Ball B (p; ) Partial hierarchies If A and B are partial hierarchies, their supremum is de…ned as for the hierarchies. The aliens of a partial hierarchy X are characterized by 8p; q 2 E : (p; q) = L: Hence the aliens of A ^B are the union of the aliens of A and of B, i.e. supp(A ^B)=supp(A)_supp(B) or equivalently supp(A ^B) = supp(A)^supp(B):

In…mum of two hierarchies

The subdominant ultrametric half-distance The supremum of two hierarchies A and B is written A _ B and is the smallest hierarchy larger than A and B.

As d A ^dB is not an ultrametric distance, we chose for d A_B the largest ultrametric distance which is lower than d A ^dB : This distance exists: the set of ultrametric distances lower than d A ^dB is not empty, as the distance 0 is ultrametric ; furthermore, this family is closed by supremum, hence it has a largest element. Let us construct it.

Consider a series of points (x 0 ; x 1 ; ; x n ): As d A_B should be an ultrametric distance, we have for any path x 0 ; x 1 ; :::

; x n d A_B ((x 0 ; x n ) d A_B (x 0 ; x 1 ) _ d A_B (x 1 ; x 2 ) _ _ d A_B (x n 1 ; x n ): But for each pair of points x i ; x i+1 we have d A_B (x i ; x i+1 ) [d A ^dB ] (x i ; x i+1 ): Hence d A_B (x 0 ; x n ) [d A ^dB ] (x 0 ; x 1 )_[d A ^dB ] (x 1 ; x 2 )_ _d [d A ^dB ] (x n 1 ; x n ):
There exists a chain along which the expression on the right becomes minimal and is equal to the maximal value taken by [d A ^dB ] on two successive points of the chain. This maximal value is called sup section of the chain for d A ^dB : For this reason, the chain itself is called chain of minimal sup-section. This valuation being an ultrametric ecart necessarily is the largest ultrametric ecart below d A ^dB : Let us verify the ultrametric inequality.

For p; q; r 2 E there exists a chain between p and q along which [d A ^dB ] (p; q) takes its value and another chain between q and r along which [d A ^dB ] (q; r) takes its value. The concatenation of both chains forms a chain between p and q which is not necessarily the chain of lowest sup-section between them, hence:

[d A ^dB ] (p; r) [d A ^dB ] (p; q) _ [d A ^dB ] (q; r):
We write d A ^dB for the subdominant ultrametric associated to d A ^dB :

Partial hierarchies If A and B are partial hierarchies, their in…mum is de-…ned as for the hierarchies. The aliens of a partial hierarchy X are characterized by 8p; q 2 E : (p; q) = L: Hence the chains characterizing the subdominant ultrametric distance associated to A _ B avoid the supports supp(A) and supp(B).

Geometrical interpretation Suppose that (x 0 ; x 1 ; ; x n ) is the chain for which d

A ^dB (x 0 ; x n ) = [d A ^dB ] (x 0 ; x 1 )_[d A ^dB ] (x 1 ; x 2 )_ _[d A ^dB ] (x n 1 ; x n ) is minimal with a value : Then [d A ^dB ] (x i ; x i+1 )
means that the ball Ball A (x i ; ) or the ball Ball B (x i ; ) contains the point x i+1 : If it is Ball A (x i ; ); then x i+1 also is center of this ball. Hence a series of points x k ; x k+1 ; x k+2; all belong to the same ball Ball A (x i ; ), they are all centers of this ball and it is possible to keep only one of them and suppress all others from the list. Like that we get a path where the …rsts two points x 0 ; x 1 belong to one of the balls, say Ball A (x 0 ; ); the couple x 1 ; x 2 belong to the other Ball B (x 2 ; ); and so on. The successive ovelapping pairs of points belong alternatively to balls Ball A or Ball B :

The necessity of chaining blocks for obtaining suprema of partitions is well known [START_REF] Serra | Image Analysis and Mathematical Morphology. II: Theoretical Advances[END_REF] ; Ronse has con…rmed that it is still the case for partial partitions [START_REF] Ronse | Adjunctions on the lattices of partitions and of partial partitions[END_REF].

Illustration If A , B and A _ B are the partitions obtained by taking the balls of radius in each of the three hierarchies, then the boundaries of A _ B are all boundaries existing in both A and B . The in…mum and supremum of two hierarchies are illustrated in …g.3

Lexicographic fusion of strati…ed hierarchies

Let A and B be two strati…ed hierarchies, with their associated distances d A and d B : In some cases, one of the hierarchies correctly represents the image to segment, but with a too small number of nested partitions. One desires to enrich the current ranking of regions as given by A; by introducing some intermediate levels in the hierarchy. The solution is to combine the hierarchy A with another hierarchy B in a lexicographic order.

One produces the lexicographic hierarchy Lex(A; B) by de…ning its ultrametric distance ; it is the largest ultrametric distance below the lexicographic A hierarchy X 2 X (E) is a collection of sets X i 2 P(E): Through the translation by a vector ! t ; these sets X i ! t form a new hierarcy X ! t : If is the ultrametric ecart associated to X , the ultrametric ecart associated to X ! t will be written ! t :

As the hierarchies form a complete lattice X (E), we may use the same mechanism for constructing an erosion and a dilation on hierarchies. We de…ne two operators operating on a hierarchy X . For showing that the …rst X

B = V x2B X !
Ox is an erosion and the second X B = W x2B X ! xO a dilation, we have to show that they form an adjunction.

We have to prove that for any two hierarchies X ; Y 2 X (E) : X B < Y , X < Y B:

We will prove the adjunction through the half distance associated to the hierarchies X and Y.

We have the following correspondances between the hierarchies and the ultrametric ecarts :

X $ Y $ Y B = V x2B Y ! Ox $ W x2B ! Ox X B = W x2B X ! xO $ V x2B ! xO X B < Y , X < Y B $ V x2B ! xO > , > W x2B ! Ox
Let us now prove the adjunction. For two arbitrary ultrametric ecarts and :

X < Y B , > W x2B ! Ox , 8x 2 B : > ! Ox , 8x 2 B : ! xO > , V x2B ! xO >
Remains to establish :

V x2B ! xO > , V x2B ! xO > :
Initial partition

Eroded partition

Figure 7: Erosion of a partition

V x2B ! xO > ) V x2B ! xO > since V x2B ! xO is the largest ultrametric ecart below V x2B ! xO Suppose now V x2B ! xO > : Since is an ultrametric ecart below V x2B ! xO ; it is smaller or equal to the largest ultrametric ecart below V x2B ! xO ; that is V x2B ! xO :
This completes the proof :

X < Y B , > W x2B ! Ox , V x2B ! xO > , V x2B ! xO > , X B < Y
The erosion of a partition by a square structuring element (8 connexity) is illustrated in …g.7

Remark 11 The adjunction de…ned for hierarchies is de…ned in a similar fashion for a partial hierarchy.

Decomposition and recomposition of hierarchies

Thresholding

Consider a hierarchy X with its associated ultrametric ecart : By thresholding the ultrametric at level one obtains a binary ultrametric ecart :

T ( ) = 1 if > 0 if T ( ) characterizes a partition.
The hierarchy can be recovered from its thresholds by = W T ( ):

Remark 12 If X is a partial hierarchy, then T ( ) is a partial partition

Reconstructing a hierarchy

Consider a strati…ed hierarchy X .

Inf-generation Consider a hierarchyX , union of a family (A i ) of P(E). If a set A i belongs to X , then we construct a partition by adding to A i its complement. We have de…ned this operator earlier Backg(A i ) = A i [ A i to which we associate the binary ultrametric ecart i :

The ultrametric ecart associated to X is sup-generated and equal to

W i st(A i )
i and the hierarchy is inf-generated by the partitions Backg(A i ) weighted by their strati…cation level.

This decomposition helps understanding how the erosion of ultrametric hierarchies work. To X B is associated

B = W i st(A i ) i B = W i [st(A i ) i ] B For each set A i ; the partition A i [ A i is eroded, that is the new partition A i [ (A i =A i B) [ A i =A i B [ A i is created.
This partition keeps the same strati…cation level as A i itself. This new collection of partitions creates the eroded hierarchy.

Remark 13 In the particular case of a partition, the sets forming this partition are eroded, and the space left by the erosion is …lled by the intersection of all partial partitions

(A i =A i B) [ A i =A i B .
The result is illustrated by …g.7. This di¤ ers from the de…nition given by J.Serra in [START_REF] Serra | Morphological operators for the segmentation of colour images[END_REF], where he …lled the spaces left empty by singletons.

Sup-generation Consider a hierarchy X , union of a family (A i ) of P(E). We associate to the set A i the following half-distance i :

For any p; q 2 A i : i (p; q) = st(A i ). For p = 2 A i ; and any q we have i (p; q) = L: We have thus associated to the set A i a partial partition equal to A i on A i ; with a strati…cation level st(A i ) and containing only aliens on A i :

The half-distance is then inf-generated and equal to

V i i , corresponding
to the sup-generation from the partial hierarchies c A i associated to the i : This decomposition helps understanding how the dilation of ultrametric hi-

erarchies work. To X B = W i c A i B is associated the half-distance ^ V i i B

Illustration

We illustrate the erosion and the opening of a one dimensional hierarchy, …rst by a structuring element reduced to two pixels, then by a structuring element made of three pixels. In the …rst case, the erosion and the dilation have to use the structuring element for the erosion and its transposed version for the dilation. Ouverture par 3 points

Image de départ

Dendrogram of an initial image and its opening by a segment of 3 points.

Examples of hierarchies

Hierarchies associated to a dissimilarity index A series of nested partitions (X i ) ; and hence a hierarchy, may easily be generated from an initial …ne partition X 0 = [R i , i = 1; : : : ; n on which a dissimilarity index is de…ned between a subset G of all couples of tiles. For a couple of tiles which do not belong to G; we de…ne a dissimilarity equal to 1:

If we now take the union of all tiles of X 0 with a dissimilarity index below a given threshold ; we obtain a coarser partition with a strati…cation index equal to . For increasing values of we obtain a series of nested partitions, forming a hierarchy A. The ultrametric distance d associated to this hierarchy is precisely the the subdominant ultrametric distance associated to , that is the largest ultrametric distance below (see below the supremum of two hierarchies, where the subdominant ultrametric distance also appears) For two tiles A and B of X 0 ; the subdominant ultrametric distance will be the lowest level for which A and B belong to the same tile (if it does not happen, their distance is 1)

Case of the watershed tesselation If the tessellation is the result of the watershed construction on a gradient image, the dissimilarity measure can be de…ned as the altitude of the pass point separating two adjacent regions. The ultrametric half distance between two minima is then the " ‡ooding distance" : the ‡ooding distance between two points p and q is the altitude of the lowest ‡ooding for which p and q both belong to a common lake.

Other possible measures are color distances, various measures of local contrast, or even motion or texture dissimilarity between adjacent catchment basins.

Connectivity and taxonomy classes

The notion of a connected set in E is well de…ned if E is a topological space. In [START_REF] Serra | Image Analysis and Mathematical Morphology. II: Theoretical Advances[END_REF], Serra generalized this concept by the introduction of a connectivity class. Connectivity classes de…ne the subsets of E which are connected. Hence they help decomposing every set X 2 P(E) into its connected components. Connectivity classes have been extensively studied by Serra and Ronse ([8], [START_REF] Ronse | Adjunctions on the lattices of partitions and of partial partitions[END_REF]).

We extend here the notion of connectivity classes and de…ne taxonomy classes. We present in parallel the notions related to partitions and to hierarchies. For our presentation of the binary case, we largely follow Henk Heijmans, who gives a clear presentation of the developments linked to connectivity in [START_REF] Heijmans | Connected morphological operators for binary images[END_REF].

6.1 Connectivity and taxonomy classes This de…nition is compatible with the de…nition of a connectivity class, in the case where H contains only one element : if J contains only one index l; then H l is a connectivity class, as the axioms (C1) and (C2) are veri…ed. This shows that any taxonomy class is a series of nested connectivity classes. Inversely, it is obvious that a series of nested connectivity classes is a taxonomy.

Adjacency relations

Connectivity classes

An important subclass of connectivity classes is based on adjacency.

De…nition 16 A binary relation on E E is called an adjacency relation if it is re ‡exive (x x for every x) and symmetric (x y i¤ y x).

Given an adjacency relation on E E; we call x 0 ; x 1 ; :::; x n a path between x = x 0 x ::::

x n = y: De…ne C P(E) as the collection of all C 2 E such that any two points in C can be connected by a path that lies entirely in C:

Proposition

Taxonomy classes

For de…ning a taxonomy class we need a series of nested adjacency relations (the adjacency between a cat and another cat, or between a cat and a tiger, or a tiger and a mammal cannot be the same). Example Consider a grey tone image f de…ned on a grid with a neighborhood relation. We de…ne the adjacency relation p e i q by the following conditions: p and q neighbors on the grid jf p f q j i For the value i; the connected components are the lambda- ‡at zones of slope i: For increasing values of i; this slope increases and so do the lambda ‡at zones.

Connectivity openings

Connectivity classes

Serra in [START_REF] Serra | Image Analysis and Mathematical Morphology. II: Theoretical Advances[END_REF] has shown that any connected class C is equivalent with the datum of a connected opening, de…ned through its invariance domain. If C x denotes the subclass of C 2 C that contains a given point ;

C x = fC : x 2 C Cg
then the union of each non-empty family of sets of C x ; all containing x still belongs to C x ; because of (C2). Hence Inv( x ) = C x [ f?g is the invariant set of an opening x ; called connected opening of origin x: Its expression is The principal interest of connection openings lies in the following corollary of [START_REF] Serra | Image Analysis and Mathematical Morphology. II: Theoretical Advances[END_REF] Corollary 22 Openings x partition any X E into the smallest possible number of components belonging to the class C.

Given a set X E; every connected component x (X) of X is called a grain of X: The next result ( [START_REF] Heijmans | Connected morphological operators for binary images[END_REF]) says that every connected subset of X is contained within some grain of X Proposition 23 Given a connectivity on E and a set X E: If C X is a connected set, then C is contained within some grain of X:

Another useful property ( [START_REF] Serra | Image Analysis and Mathematical Morphology. II: Theoretical Advances[END_REF]), shows that x plays no particular role in x (X):

Corollary 24 For all x; y 2 E and all X E we have y 2 x (X) , x (X) = x (X) and in particular y 2 x (X) , x 2 y (X)

And …nally the link between connective classes and partitions.

Proof. We have to verify that both the intersection axiom and union axiom are satis…ed (Intersection axiom) Consider two sets of the hierarchy P i (X)(q) and P j (X)(p) for i j a) q 2 P j (X)(p) : then P j (X)(p) P j (X)(q) and since P i (X)(q) P j (X)(q); the result P i (X)(q) P j (X)(p) is proved b) q = 2 P j (X)(p) then P j (X)(p) \ P j (X)(q) = ?: Since P i (X)(q) P j (X)(q) we also have P i (X)(p) \ P j (X)(q) = ? (union axiom) Consider a set A = P j (X)(p) of the hierarchy. For a point q 2 P j (X)(p) we have also P j (X)(q) = P j (X)(p): Since P i (X)(q) P j (X)(q) for i j; it shows that P j (X)(q) P j (X)(p) Hence P j (X)(p) = S q2A;i<j P i (X)(q) is indeed the union of the elements of the hierarchy it contains. The family (P i (X)(x)) i2I;x2E forming a hierarchy, we may apply to it all results we have established in the …rst part of the paper, in particular we may associate to it an ultrametric half-distance (X)

Connected operators

De…nition 31 An operator on P(E) is connected if the hierarchy X H ( (X)) is coarser than X H (X) for every set X E This means that the identity operator from the half-metric space E with the half distance H (X) into the half-metric space E with the half distance H ( (X)) is Lipschitz, as for any two points p; q 2 E; we have H ( (X))(p; q) H (X)(p; q)

Conclusion

We may now give a summary of the results, which are all linked to the properties of the ultrametric half distances.: each point is centre of a ball two balls with the same radius are either disjoint or identical two balls with a di¤erent radius are either disjoint or one is included in the other Finally, rather than starting with a hierarchy and de…ning an ultrametric half-distance, we rather start with an ultrametric half-distance which helps deriving all other concepts: taxon at level i: subset A 2 P(E) with a diameter equal to i or empty set two taxons A and B have an empty intersection or they form a taxon with a diameter equal to diam(A) _ diam(B) the connected opening : i p (X) = i x (X) = S fC : x 2 C H i and C Xg

The grain at level i of the hierarchy associated to X :

P i (X)(p) = i p (X) if p 2 X i p (X c ) if p = 2 X
Adjacence relations : the ultrametric half distance may be de…ned if one knows the distance (p; q) for a collection of pairs of points (p; q). From them it is possible to construct all taxonomy classes and obtain the hierarchy by adding the singletons and the empty set.

  ; D 2 P(E), d(C; D) = inf fst(A) j A 2 X : C A and D Ag : Properties : d is an ultrametric distance : 8A; B 2 X d(A; B) = 0 ) A = B 8C; D 2 P(E) d(C; D) = d(D; C) 8B; C; D 2 P(E) d(C; D) max fd(C; B); d(B; D)g

Lemma 5 Lemma 6

 56 Two closed balls Ball(X; ) and Ball(Y; ) with the same radius are either disjoint or identical. Proof. Consider two closed balls Ball(X; ) and Ball(Y; ) with a non empty intersection and let A be an element in this intersection. Then necessarily Ball(X; ) = Ball(Y; ): Let us show for instance the inclusion Ball(X; ) Ball(Y; ): Let B 2 Ball(X; ); then (Y; B) (Y; A) _ (A; X) _ (X; B) ; showing that B 2 Ball(Y; ) Each element of a closed ball Ball(X; ) is centre of this ball

Remark 8

 8 Instead of closed balls, we could have taken open balls. The results are the same.

De…nition 9 A

 9 half-distance on a domain E is a mapping d from E E into R + with the following properties: 1) Symmetry : d(x; y) = d(y; x) 2) Half-positivity: d(x; y) 0 and d(x; x) = 0 3) Triangular inequality: d(x; z) d(x; y) + d(y; z)De…nition 10 A half metric space is a set E with a family (d i ) i2I of halfdistances verifying the following condition: the family (d i ) i2I is a "…ltering family", i.e. for any …nite subset J of I; there exists an index k 2 I such that d k d j for all j 2 J

  by adjunction ; but " being anti-extensive, we have (Y ) " (Y ) (Y ); hence (Y ) = " (Y ): By duality, we have " = ":

Figure 1 :

 1 Figure 1: Comparison of the drawing with a …xed size brush and a self adaptive brush.

Figure 2 :

 2 Figure 2: Left: initial image Center: result of the magic wand Right ; smallest region of the hierarchy containing the magic wand.

Figure 3 :

 3 Figure 3: Two hierarchies HA and HB and their derived supremum and in…mum

Figure 4 :

 4 Figure 4: Supremum of two hierarchies.

Figure 5 :

 5 Figure 5: Two hierarchies HA and HB and their derived lexicographic combinations.

Figure 6 :

 6 Figure 6: Lexicographic fusion of two hierarchies

H k 2 C

 2 De…nition 19 A family ( e i) i2I of adjacency relations is nested if x e i y implies x e j y for j > i:To each adjacency relation e i we associate its connectivity class C e i Proposition 20 If the family ( e i) i2I of adjacency relations is nested, then the family H = (C e i ) i2I is a taxonomy class.Proof. (H1) is trivially veri…ed. (H2) is veri…ed as x e i y implies x e j y for j > i; hence C e i C e j : Let us prove (H3). Suppose that for i 2 J : H k 2 C e i and T k2K H k 6 = ?: If l = max(J) is the maximal index of J, we have H k 2 C e i C e l : And as C e l is a connectivity class, T k2K H k 6 = ? implies S k2K e l :

  x (X) =SfC : x 2 C C and C Xg Since any x 2 E belongs to a connected set of C, we haveC = S x2E Inv( x )Proposition 21 Assume that C is a connectivity on E, then the following conditions are satis…ed: (O1) every x is an opening(O2) x (fxg) = fxg (O3) x (X) \ y (X) = ? or x (X) = y (X) (O4) x = 2 X ) x (X) = ? Conversely if x; x 2 E;is a family of operators satisfying (O1)-(O4) then C = S x2E Inv( x ) de…nes a connectivity.

  Alternatively, we say that C de…nes a connectivity on E: An element of C is called a connected set. This de…nition is "generative" : larger connected sets are generated from elementary ones with a non empty intersection. Let E be an arbitrary nonempty set. H = (H i ) i2I is called a taxonomy class if H i P(E) satisfy (H1) for i 2 I : ? 2 H i and fxg 2 H i for x 2 E (H2) H i H j for j > i (H3) if for i 2 J I; k 2: H k 2 H i and T

	6.1.1 General de…nition
	Connectivity classes
	De…nition 14 Let E be an arbitrary nonempty set. A family C called a connectivity class if it satis…es	P(E) is
	(C1) ? 2 C and fxg 2 C for x 2 E (C2) if C i 2 C and T i2I C i 6 = ?; then	S i2I	C i 2 C
	Taxonomy classes
	De…nition 15 k2K	H k 6 = ?; then	S k2K	H k 2 H max(J)
	H is called a taxonomy, each H i a taxonomy class and H k 2 H i a taxon of level i:

  17 If is an adjacency relation on E E, then C is a connectivity class.There exists two indices in I such that x 2 C i1 and x 2 C i2 : There exists a path linking x with z in C i1 and a path linking z with y in C i2 : The path between x and y is obtained by concatenating both paths.De…nition 18 C is a strong connectivity class if there exists an adjacency relation on E E such that C ~and E is connected. We say that E possesses a strong connectivity.

	T i2I C i can be connected by a path that lies entirely in C i ; we have to show that any Proof. (C1) is obvious. If C i 2 C and z 2 two points x; y in S S
	i2I

i2I

C i :

Ouvert

Erosion and opening by a pair of pixels: intermediate steps

Ouverture par un bi-point

Image de départ

Dendrogram of an initial image and its opening by a segment of 2 points.

5.2.2

Erosion and opening by a centered segment of 3 pixels.

x 2 E (ii) P (x) = P (y) or P (x) \ P (y) = ?; for x; y 2 E If E is endowed with a connectivity C and if P (x) 2 C for every x 2 E; then we say that the partition P is connected.

Given a connective class, every binary image (i.e.set) X E can be associated with a connected partition P (X) where the zones of P (X) are the grains of X and X c : The zone of P (X) containing a point p is :

Corollary 26 For all x; y 2 E and all X E we have y 2 P (X)(x) , P (X)(x) = P (X)(y) and in particular y 2 P (X)(x) , x 2 P (X)(y)

Corollary 27 For all x; y 2 E and all X E we have y = 2 P (X)(x) , P (X)(x) \ P (X)(y) = ?

Proof. If x 2 X and y = 2 X; or vice-versa, then the implication is obvious. Consider the case where x; y both belong to X or both belong to X c : Suppose that there exists a point z 2 P (X)(x) \ P (X)(y) ; this would imply that P (X)(x) = P (X)(z) = P (X)(y) which contradicts the hypothesis

Connected operators

De…nition 28 An operator on P(E) is connected if the partition P ( (X)) is coarser than P (X) for every set X E

Taxonomy classes

Consider a taxonomy class H = (H i ) i2I . Each H i is a connectivity class to which is associated a connection opening i

x : H i also segments every binary image (i.e.set) X E into a connected partition P i (X): The grains of this partition are the sets P i (X)(x); for x 2 E. Lemma 29 For j > i; we have P i (X)(x) P i (X)(x) for x 2 E Proof. i x (X) = S fC : x 2 C H i and C Xg and H i H j for j > i; it follows that i x (X) j x (X) and P i (X)(x) P j (X)(x) for j > i:

Proposition 30 The family (P i (X)(x)) i2I;x2E [ f?g forms a hierarchy. We call it X H (X) and write H (X) for its associated ultrametric half-distance


Dr Paul Mcnally
email: paul.mcnally@olchc.ie

Peter Greally

Use of post-exercise laryngoscopy to evaluate exercise induced dyspnea

Keywords: Asthma, Exercise, Laryngomalacia Exercise Induced Laryngomalacia Asthma, exercise, laryngomalacia, flexible laryngoscopy Exercise Induced Laryngomalacia Asthma, exercise, laryngomalacia, flexible laryngoscopy

We present the case of a child with asthma who continued to have marked exercise induced dyspnea despite appropriate treatment, and in the face of adequate control of all other asthma symptoms. Spirometry showed a marked truncation of inspiratory flow, and laryngoscopy performed immediately after exercise showed laryngomalacia with dynamic, partial inspiratory obstruction. Exercise induced laryngomalacia (EIL) is a rare cause of exercise induced dyspnea which is diagnosed by post exercise flexible laryngoscopy and may require supraglottoplasty.

Case Report

A 9 year old boy was referred with troublesome exercise induced dyspnea and noisy breathing. He had doctor diagnosed asthma and was taking combination high dose inhaled corticosteroids and long acting bronchodilators with additional pre-exercise albuterol. Despite adequate resolution of resting symptoms, and a modest improvement in exercise symptoms, he continued to have noisy breathing and dyspnea with exercise, which resulted in premature cessation of activity. Prior to treatment with asthma medications, recovery time after exercise induced dyspnea was up to 10 minutes. This reduced to less than five minutes on treatment. As an infant he had always had 'noisy breathing' but this had largely resolved as he grew older. He had normal phonation.

), however on maximal inspiration there was a truncated

provocation test was subsequently performed with flexible nasolaryngoscopy performed immediately post-exercise. Laryngoscopy was normal initially but after vigorous exercise showed marked laryngomalacia with prolapse of the corniculate cartilages inferomedially on inspiration, causing partial obstruction. These changes reverted to normal during tidal breathing. In addition to these findings, it was also demonstrated that this child had exercise induced bronchoconstriction with a drop in FEV1 of >15% on exertion, in keeping with his history of reactive airways disease (Figure 1C). The child subsequently underwent staged supraglottoplasty over a period of 18 months, resulting in incremental improvements in exercise tolerance and stridor during exertion. Follow up spirometry, shown in Figure 1D, reflects this clinical improvement demonstrating improved (but still not normal) inspiratory flows.

Discussion

Laryngomalacia occurs when flaccid supraglottic tissue prolapses inferomedially during the increase in negative airway pressure that is observed during inspiration. It is a common cause of stridor in infants, where it is normally a benign self limiting condition usually peaking by around six months of age and ceasing in most children by the age of two. However, some children with more severe symptoms continue to have stridor into childhood and may need operative intervention in the form of a supraglottoplasty.

Further, some older children who had laryngomalacia in infancy, despite resolution of symptoms, continue to have detectable extrathoracic obstruction to inspiration on spirometry (11,12). Asthma, as manifested by recurrent episodes of reversible airway obstruction, is the commonest chronic medical condition in children. Common triggers for bronchoconstriction include inhaled allergens such as dust mite, pollens and molds, environmental factors such as cigarette smoke, and exercise. The diagnosis of asthma is a clinical one, and is usually clear when a concise history is taken. A good response to asthma medication supports this. In some circumstances, however the diagnosis can be difficult, and response to treatment disappointing. In these cases, investigations may help to confirm the diagnosis. The measurement of total and allergen specific serum IgE or skin prick testing can detect atopy and allergic sensitization in an individual. Spirometry before and after bronchodilator may also be helpful, and in some situations bronchial challenge with histamine or methacholine followed by the administration of albuterol may also be necessary (1). Exhaled nitric oxide (FeNO), a marker of airway inflammation, is elevated in steroid naïve asthmatics and may also be of some use (2).

In the case of asthma like symptoms only during exercise where a clear diagnosis cannot be made, and the response to treatment is disappointing, exercise bronchial provocation with spirometry before and after can often be very informative. A decrease in FEV1 of >15% after exercise confirms this diagnosis (3). If correctly diagnosed, EIA usually responds to treatment with asthma medications. Failure to respond adequately to asthma EIL is a rare cause of exercise-induced dyspnea. There are six case reports in the literature (5)(6)(7)(8)(9)(10) in patients ranging in age from 10-32yrs. Only one child had either stridor or documented laryngomalacia in infancy (this infant had laryngomalacia and subglottic stenosis requiring a tracheostomy) (10). All of these patients had been unsuccessfully treated medically for exercise induced asthma (EIA) for between 1 and 10 years. Two patients had laser supraglottoplasty with improvement of symptoms (7,8) and a further patient underwent staged supraglottoplasty with resolution of exercise symptoms (10).

Our patient presented a diagnostic difficutly in that he had demonstrable exercise-induced bronchoconstriction. Although, when the complaint of 'breathlessness' on exertion was further elucidated, it involved mainly noisy breathing on inspiration and not the diffuse expiratory wheeze one would expect in purely exercise induced bronchoconstriction. The

Case Report

A 9 year old boy was referred with troublesome exercise induced dyspnea and noisy breathing. He had doctor diagnosed asthma and was taking combination high dose inhaled corticosteroids and long acting bronchodilators with additional pre-exercise albuterol. Despite adequate resolution of resting symptoms, and a modest improvement in exercise symptoms, he continued to have noisy breathing and dyspnea with exercise, which resulted in premature cessation of activity. Prior to treatment with asthma medications, recovery time after exercise induced dyspnea was up to 10 minutes. This reduced to less than five minutes on treatment. As an infant he had always had 'noisy breathing' but this had largely resolved as he grew older. He had normal phonation.

A chest X-ray was normal. A normal tidal inspiratory flow volume loop was observed during spirometry (figure 1A), however on maximal inspiration there was a truncated showed marked laryngomalacia with prolapse of the corniculate cartilages inferomedially on inspiration, causing partial obstruction. These changes reverted to normal during tidal breathing. In addition to these findings, it was also demonstrated that this child had exercise induced bronchoconstriction with a drop in FEV1 of >15% on exertion, in keeping with his history of reactive airways disease (Figure 1C). The child subsequently underwent staged supraglottoplasty over a period of 18 months, resulting in incremental improvements in exercise tolerance and stridor during exertion. Follow up spirometry, shown in Figure 1D, reflects this clinical improvement demonstrating improved (but still not normal) inspiratory flows.

Discussion

Laryngomalacia occurs when flaccid supraglottic tissue prolapses inferomedially during the increase in negative airway pressure that is observed during inspiration. It is a common cause of stridor in infants, where it is normally a benign self limiting condition usually peaking by around six months of age and ceasing in most children by the age of two. However, some children with more severe symptoms continue to have stridor into childhood and may need operative intervention in the form of a supraglottoplasty.

Further, some older children who had laryngomalacia in infancy, despite resolution of symptoms, continue to have detectable extrathoracic obstruction to inspiration on spirometry (11,12). Asthma, as manifested by recurrent episodes of reversible airway obstruction, is the commonest chronic medical condition in children. Common triggers for bronchoconstriction include inhaled allergens such as dust mite, pollens and molds, environmental factors such as cigarette smoke, and exercise. The diagnosis of asthma is a clinical one, and is usually clear when a concise history is taken. A good response to asthma medication supports this. In some circumstances, however the diagnosis can be difficult, and response to treatment disappointing. In these cases, investigations may help to confirm the diagnosis. The measurement of total and allergen specific serum IgE or skin prick testing can detect atopy and allergic sensitization in an individual. Spirometry before and after bronchodilator may also be helpful, and in some situations bronchial challenge with histamine or methacholine followed by the administration of albuterol may also be necessary (1). Exhaled nitric oxide (FeNO), a marker of airway inflammation, is elevated in steroid naïve asthmatics and may also be of some use (2).

In the case of asthma like symptoms only during exercise where a clear diagnosis cannot be made, and the response to treatment is disappointing, exercise bronchial provocation with spirometry before and after can often be very informative. A decrease in FEV1 of >15% after exercise confirms this diagnosis (3). If correctly diagnosed, EIA usually responds to treatment with asthma medications. Failure to respond adequately to asthma EIL is a rare cause of exercise-induced dyspnea. There are six case reports in the literature (5)(6)(7)(8)(9)(10) in patients ranging in age from 10-32yrs. Only one child had either stridor or documented laryngomalacia in infancy (this infant had laryngomalacia and subglottic stenosis requiring a tracheostomy) (10). All of these patients had been unsuccessfully treated medically for exercise induced asthma (EIA) for between 1 and 10 years. Two patients had laser supraglottoplasty with improvement of symptoms (7,8) and a further patient underwent staged supraglottoplasty with resolution of exercise symptoms (10).

Our patient presented a diagnostic difficutly in that he had demonstrable exercise-induced bronchoconstriction. Although, when the complaint of 'breathlessness' on exertion was further elucidated, it involved mainly noisy breathing on inspiration and not the diffuse expiratory wheeze one would expect in purely exercise induced bronchoconstriction. The Pediatric Pulmonology

  loop (Fig 1B). Bronchoscopy under anesthesia, performed in another center, had shown normal upper and lower airways. An exercise bronchial

  diagnosis is made should always prompt one to carefully re-examine the history. The key feature in this case was the history of stridor during exercise. The history of stridor during exercise should always prompt one to consider vocal cord dysfunction (VCD) and exercise induced laryngomalacia. VCD is a benign condition that usually responds well to simple exercises (4). Exercise induced laryngomalacia on the other hand is unlikely to respond significantly to either asthma treatment or breathing exercises, given the anatomic basis for the symptoms. Endoscopy during or shortly after exercise should reliably differentiate VCD from EIL (4).

  of noisy breathing in infancy most likely represents laryngomalacia that subsequently resolved to a degree that, at rest, it caused no symptoms. However the increased work of breathing associated with exercise and the additional burden of exercise induced bronchoconstriction was likely sufficient to result in significant inspiratory effort, with very negative inspiratory pressures causing dynamic upper airway obstruction. Endoscopy under general anaesthesia with tidal respiration is insufficient to demonstrate the dynamic upper airway obstruction that occurs with the marked negative intraluminal pressures seen during exercise. Dynamic obstruction on inspiration during forced maneuvers or exercise increases the likelihood of this diagnosis, however laryngoscopy during or after exercise, while difficult to perform, is the gold standard diagnostic test. Staged laryngoplasty can be very helpful in patients where the degree of exercise symptoms is intolerable.

Abstract

  We present the case of a child with asthma who continued to have marked exercise induced dyspnea despite appropriate treatment, and in the face of adequate control of all other asthma symptoms. Spirometry showed a marked truncation of inspiratory flow, and laryngoscopy performed immediately after exercise showed laryngomalacia with dynamic, partial inspiratory obstruction. Exercise induced laryngomalacia (EIL) is a rare cause of exercise induced dyspnea which is diagnosed by post exercise flexible laryngoscopy and may require supraglottoplasty.

  loop (Fig 1B). Bronchoscopy under anesthesia, performed in another center, had shown normal upper and lower airways. An exercise bronchial provocation test was subsequently performed with flexible nasolaryngoscopy performed immediately post-exercise. Laryngoscopy was normal initially but after vigorous exercise

  diagnosis is made should always prompt one to carefully re-examine the history. The key feature in this case was the history of stridor during exercise. The history of stridor during exercise should always prompt one to consider vocal cord dysfunction (VCD) and exercise induced laryngomalacia. VCD is a benign condition that usually responds well to simple exercises (4). Exercise induced laryngomalacia on the other hand is unlikely to respond significantly to either asthma treatment or breathing exercises, given the anatomic basis for the symptoms. Endoscopy during or shortly after exercise should reliably differentiate VCD from EIL (4).

  breathing in infancy most likely represents laryngomalacia that subsequently resolved to a degree that, at rest, it caused no symptoms. However the increased work of breathing associated with exercise and the additional burden of exercise induced bronchoconstriction was likely sufficient to result in significant inspiratory effort, with very negative inspiratory pressures causing dynamic upper airway obstruction. Endoscopy under general anaesthesia with tidal respiration is insufficient to demonstrate the dynamic upper airway obstruction that occurs with the marked negative intraluminal pressures seen during exercise. Dynamic obstruction on inspiration during forced maneuvers or exercise increases the likelihood of this diagnosis, however laryngoscopy during or after exercise, while difficult to perform, is the gold standard diagnostic test. Staged laryngoplasty can be very helpful in patients where the degree of exercise symptoms is intolerable.

  Spirometry: A Tidal breathing loop showing normal flow volume characteristics. B Resting forced spirometry showing truncation of the inspiratory loop. C Pre-and post-exercise trace showing more marked obstruction on exhalation (arrow) reflective of exercise-induced bronchoconstriction. D Postsupraglottoplasty spirometry showing improvement in inspiratory flow limitation. 458x191mm (72 x 72 DPI)

John Wiley & Sons, Inc.Pediatric Pulmonology