domain: sde phys sdu sdu.ocean

text:
Thibaut Tuna
email: tuna@cerege.fr

Yoann Fagault

Lise Bonvalot

Manuela Capano

Edouard Bard
email: bard@cerege.fr

Development of small CO 2 gas measurements with AixMICADAS

Keywords:

After three years of tests and developments, the performances of the AixMICADAS facility have been established for small CO 2 samples with a carbon mass inferior to 100 μg. The instrument shows very good stability and reliability when measuring small samples with its gas ion source. In this configuration, the precision is mainly limited by counting statistics and also the accuracy by contamination effects for the smallest samples (< 20 μgC). A long term variability of 7-8‰ is evaluated with 80-100 μgC NIST oxalic acid 2 samples and modern carbonate samples. Our pretreatment and analytical protocols allow for very low blank levels corresponding to 14 C ages of 53,000 years BP and 48,000 years BP for carbonate samples and organic materials, respectively.

Introduction

AixMICADAS [START_REF] Bard | AixMICADAS, the accelerator mass spectrometer dedicated to 14C recently installed in Aix-en-Provence, France[END_REF] is a new AMS facility used to date paleoclimatic records and to study natural and anthropogenic exchanges of carbon between the atmosphere, the oceans, the biosphere and soils by using radiocarbon as a tracer. Installed in April 2014 in Aix-en-Provence (France) at the CEREGE laboratory, the facility has so far been used to measure more than 2600 CO 2 gas samples and 2300 solid graphite samples. These were measured over three years to develop and test the performances of our protocol for various samples.

Thanks to its gas interface system (GIS, [START_REF] Wacker | A versatile gas interface for routine radiocarbon analysis with a gas ion source[END_REF]) AixMICADAS allows the analysis of small CO 2 samples (< 100 μgC) from carbonates (by acidification with the carbonate handling system, CHS), aerosols, collagen and other molecular compounds (by combustion with an elemental analyzer, EA), molecular compounds and in situ 14 C (CO 2 sealed in closed ampoules) with no need of a graphitization step. Carbonate samples hydrolyzed with the CHS represent almost 50% of the CO 2 gas samples measured so far with AixMICADAS (see companion paper by Fagault et al. in this NIMB issue). Precision, accuracy and residual contaminations have been thoroughly studied, as reported below (all 14 C data are reported as conventional ages or F 14 C following [START_REF] Stuiver | Discussion reporting of 14 C data[END_REF][START_REF] Reimer | Discussion: reporting and calibration of post-bomb 14C data[END_REF]). After a short description of the instrument and the protocols used to measure small gas samples, this paper will report the performances of the AixMICADAS instrument which have been evaluated after three years of operation.

Setup and measurements

AixMICADAS is an updated version of the original compact AMS system developed by the ETH in Zürich [START_REF] Synal | MICADAS: a new compact radiocarbon AMS system[END_REF]. With its gas ion source interface system, it allows 14 C measurement of CO 2 directly for both ultra-small samples and standard mass samples with high stability and reproducibility [START_REF] Wacker | MICADAS: routine and high-precision radiocarbon dating[END_REF][START_REF] Fahrni | Improving a gas ion source for 14C AMS[END_REF]. The design of the MICADAS instrument has been described in detail in various articles [START_REF] Bard | AixMICADAS, the accelerator mass spectrometer dedicated to 14C recently installed in Aix-en-Provence, France[END_REF][START_REF] Synal | MICADAS: a new compact radiocarbon AMS system[END_REF].

The C -beam is produced inside the Cs sputtering ion source with an extraction energy of 40 keV. The low energy 90°magnet associated with the fast beam pulsing system sequentially injects the three isotopes 12 C (30 μs), 13 C (520 μs) and 14 C (40 ms) into the tandem accelerator unit. An offset 12 C -Faraday cup is placed just after the LE magnet to monitor the low energy current. The acceleration voltage of AixMICADAS is typically set to a value of 193.6 kV and the stripping with helium ensures an overall transmission for 12 C of 48.1 ± 0.6% for all measured gas samples ( 12 C -current < 20 μA). Faraday cups are located in the focal plane of the high-energy magnet to measure ion beam currents of 12 C + , 13 C(H) + and 13 C + , where 13 C(H) + represents 13 C + ions fragmented from the 13 CH -beam that is injected along with 14 C -. The 13 C(H) + Faraday cup is used to monitor the contribution of molecules that interfere with 14 C measurement [START_REF] Synal | MICADAS: a new compact radiocarbon AMS system[END_REF][START_REF] Wacker | MICADAS: routine and high-precision radiocarbon dating[END_REF]. Finally, a 90°e lectrostatic deflector energy filter is used just before the ion gas detector which is itself isolated from the vacuum of the accelerator by a 5 mm × 5 mm nitride silicon window of thickness 50 nm [START_REF] Müller | Performance of the ETH gas ionization chamber at low energy[END_REF]. BATS software is used for data reduction [START_REF] Wacker | Bats: a new tool for AMS data reduction[END_REF]. The optimal parameters of the ion source in gas configuration are given in Table 1, where they are compared with those used in solid configuration.

Three devices providing CO 2 for the ion source are connected to the gas interface system (GIS): a cracker for CO 2 sealed in glass ampoules, an elemental analyzer (EA) and a carbonate hydrolysis system (CHS) [START_REF] Ruff | On-line radiocarbon measurements of small samples using elemental analyzer and MICADAS gas ion source[END_REF][START_REF] Wacker | A novel approach to process carbonate samples for radiocarbon measurements with helium carrier gas[END_REF]. We have previously described the configuration of our EA coupled to the GIS [START_REF] Bonvalot | Estimating contributions from biomass burning, fossil fuel combustion, and biogenic carbon to carbonaceous aerosols in the Valley of Chamonix: a dual approach based on radiocarbon and levoglucosan[END_REF]. The main characteristic of the EA is that it works with a combustion tube filled with tungsten oxide heated to 1050 °C to allow the introduction of silver boats. We have tested different metals (Al, Sn, Ag), boat sizes, and cleaning protocols. Optimal blanks are obtained with Ag cups that are baked at 800 °C for 2 h to eliminate organic contamination.

Carbonates are processed with the CHS following a leaching procedure where 30% of the initial CaCO 3 mass is removed with hydrochloric acid (HCl) prior to the full hydrolysis of the remaining mass with phosphoric acid (H 3 PO 4 85%). The CHS device and the pretreatment protocol for various carbonates are described in detail in a companion paper [START_REF] Fagault | Radiocarbon dating small carbonate samples with the gas ion source of AixMICADAS[END_REF].

The CO 2 produced by both EA or CHS is captured in the zeolite trap inside the GIS [START_REF] Wacker | A versatile gas interface for routine radiocarbon analysis with a gas ion source[END_REF]. The CO 2 is then released by heating the zeolite trap to 450 °C and is mixed inside the syringe with a helium flow in order to obtain 5% CO 2 in the gas mixture, which will be fed into the ion source.

In parallel, two cylinders of gas standards are connected to the GIS in order to monitor the blank (5% 14 C-dead CO 2 and 95% He provided by Linde Gas) and oxalic acid 2 (4.5% CO 2 from NIST 4990C OxA2 and 95.5% helium -bottle provided by IonPlus AG) used to perform the measurement normalizations.

The CO 2 is injected from the GIS into the ion source through a small fused silica capillary (SGE analytics ID = 0.060 mm; OD = 0.220 mm; methyl deactivated; 90 cm length) continuously fed by the syringe which is driven by a stepping motor controlled by the GIS software. The carbon flow is software controlled and is set to 2.80 μgC/min to keep the pressure constant inside the syringe (filled to 1300 mbar), allowing the ion source to produce stable currents.

The tuning of AixMICADAS in gas configuration begins by loading the last optical parameters of the instrument used to measure solid targets. When the cesium temperature is sufficiently high and stable, an OxA2 standard sample from the CO 2 gas cylinder is injected to perform small adjustments of the optical elements at the low energy side and especially inside the ion source to maximize beam currents and ion transmission. The optimization process in gas configuration takes no more than 15 min which can be done with a fully filled syringe.

Fig. 1 shows a compilation over one year of the average low energy ion current as a function of sample mass. The current increases quickly with the sample size up to 20 μgC, stabilizing between 10 and 15 μA for samples > 20 μgC. The large scatter of ion currents for samples of the same mass is still poorly understood and may be related to variations introduced by the sputtering process and the variations of the pressure inside the syringe and deviations of the CO 2 -He proportions from their optimal values (1300 mbar and 5% CO 2 , respectively). The overall system can be operated in a fully automated way for samples larger than 10 μgC, since LE 12 C -currents can be maintained above 10 μA for 4-15 min. Measuring smaller samples is more difficult as they need to be manually controlled to guarantee the proper onset of the ion current and an efficient acquisition process of the data during the few minutes of stable currents > 3 μA (e.g. Fig. 2

left panel).

Even if the gas targets are baked at 350 °C for 2 h before being placed in the 40 position magazine, a pre-sputtering process of each new target is performed for about a minute to achieve the complete cleaning of the titanium insert inside the target (until the 12 C -ion current becomes < 110 nA). The CO 2 of the sample is then pushed inside the capillary connected to the hollow target and the 12 C -ion current takes about 100 s to stabilize.

Our standard protocol in gas configuration consists of starting the measurement with two OxA2 standards from the gas cylinder for about ten minutes each (30,000 counts per target). Blank samples are measured after a purge of the system using the blank gas cylinder, which allows to reach operational level for real blank sample measurement (when F 14 C < 4‰). Each gas measurement sequence ends with the analysis of one or two OxA2 samples from the gas cylinder to ensure a proper normalization of all samples.

In gas configuration, the ionization efficiency (number of 12 C -ions divided by the total number of CO 2 molecules in the sample) ranges between 2% and 5%, based on 17 samples with sample sizes between 3 μgC and 55 μgC. Similar efficiencies have been measured on other MICADAS systems [START_REF] Fahrni | Improving a gas ion source for 14C AMS[END_REF][START_REF] Ruff | Gaseous radiocarbon measurements of small samples[END_REF][START_REF] Hoffmann | Status report: implementation of gas measurements at the MAMS 14C AMS facility in Mannheim, Germany[END_REF].

As typical examples, Fig. 2 shows the 12 C + current and the raw isotope ratios 14 C/ 12 C and 13 C/ 12 C recorded during the measurements of a 6 μgC sample and a 117 μgC sample produced with the CHS device.

For the large sample (Fig. 2 right panel), data cycles recorded before the current stabilization are removed and the measurement is manually stopped after 12 min before the degradation of the Ti target.

For the typical small sample shown in Fig. 2 left panel, all data points before the stabilization of the current and after its fall are discarded in the final measurement evaluation and calculation. In this case, the measurement time is limited only by the low mass of the sample and the CO 2 is entirely consumed and ionized in optimal conditions with a single target. In the example shown in Fig. 2, the F 14 C values of both samples are in agreement: F 14 C 6μgC = 1.123 ± 0.020 and F 14 C 117μgC = 1.103 ± 0.009.

In fact, only about a third of the large sample (≈ 30 μgC) was consumed during the 12 min analysis shown in Fig. 2. When the sample is large enough (i.e. > 60 μgC), it is also possible to measure the remaining CO 2 with a second or even a third Ti target in order to improve the overall precision. For instance, the measurement of three replicates of sample Aix-10339 on one target give 14 C ages of 11,890 ± 110 years BP, 11,815 ± 107 years BP and 11,809 ± 108 years BP. The measurement of the same sample material on three targets gives a 14 C age with an improved precision of 11,940 ± 70 years BP.

Table 1

Main operation parameters of the AixMICADAS system in gas and solid configurations. a When the sample mass is higher than 30 μgC. b For standard precision measurement. C For high precision measurement. Sample mass ( Fig. 1. Low energy current dependence on the injected mass of about 800 CO 2 samples measured over the last year using the CHS and the EA devices connected to the GIS.

Contamination effects

The memory effect of one sample onto the next is an issue which is particularly significant when the F 14 C of both samples are different. Even if a trap cleaning procedure is always applied between samples (heating to 450 °C and flushing with helium for a minute), a very small proportion of the sample still remains inside the zeolite trap. Nevertheless, the zeolite trap is probably not the only source of the memory effect. There may be additional CO 2 adsorption in other parts of the system: capillaries, EA tubes, or water traps, for instance. Fig. 3 left panel shows a measurement sequence of modern samples (OxA2; Aix-10000.9) directly followed by the measurement of blank samples (Phthalic anhydride; Aix-10109.2) combusted inside the EA device. Fig. 3 right panel shows an equivalent measurement sequence with carbonates using the CHS device with a modern coral sample (Porites from Moorea, Aix-10327.2) and IAEA-C1 reference material as a blank sample (Aix-10342.1). In both cases, one sample measurement is enough to suppress the memory effect, which is equivalent to 4‰ for the EA + GIS (compatible with the value observed in [START_REF] Ruff | Gaseous radiocarbon measurements of small samples[END_REF]) and 3‰ with the GIS + CHS. This memory effect is larger than the values of 1.5‰ and 1.0‰ measured for mg-size samples with the graphitization system AGE3 connected to the EA and the CHS, respectively. The difference may be linked to the GIS characterized by additional complex surfaces and valves.

To limit or totally suppress the memory effect, three possibilities can be chosen during gas measurement sessions. The first alternative is to inject the CO 2 from the same sample into the zeolite trap, release it inside the syringe and purge the syringe without measuring the sample. This procedure requires that enough material is available and that it is not too precious to afford losing half of it. The second way is to measure samples with similar ages in sequence. This requires some a priori knowledge on ages, which is often the case. For example, in sediment cores, carbonate samples can be measured following the from bottom to top. The third possibility is to correct the activity of the contaminated sample from the previous one by using the memory effect values determined with modern and blank samples.

In addition to the memory effect, the contribution of background contamination exists and must be taken into account when the size of the sample is small. There is no clear identification of the origin of this contamination. It may come from microparticles suspended in ambient air (aerosols), from the various gases used in EA, CHS and GIS, or from any other material used in the laboratory.

As an approximation, this background contamination is assumed to be constant in terms of mass and F 14 C ratio. It is evaluated by measuring two sets of samples: the first one includes 14 C free samples and the second one includes modern samples. Both sets are measured for a mass range between 3 μgC and 100 μgC. Fig. 4 shows the measured F 14 C of the IAEA-C1 blank sample and of modern coral (Porites from Moorea) for different masses combusted in the EA coupled to the GIS. A least squares procedure is used to compute the optimized couple of values for the mass M c and the F 14 C c of the contaminant (in the computation the mass is free to spread between 0 μgC and 20 μgC and the F 14 C c between 0 and 1.7).

Table 2 shows that the calculated F 14 C c of the residual contaminant is about the same for all devices, even for the graphitized samples prepared with the AGE3 system. This contamination activity is very low compared to that evaluated in other 14 C laboratories [START_REF] Ruff | Gaseous radiocarbon measurements of small samples[END_REF][START_REF] Hua | Small-mass AMS radiocarbon analysis at ANTARES[END_REF][START_REF] Salehpour | Small sample Accelerator Mass Spectrometry for biomedical applications[END_REF] and to values previously extracted from our aerosol measurements using the EA coupled to the GIS (F 14 C c = 0.73 and M c = 1.45 μg) [START_REF] Bonvalot | Estimating contributions from biomass burning, fossil fuel combustion, and biogenic carbon to carbonaceous aerosols in the Valley of Chamonix: a dual approach based on radiocarbon and levoglucosan[END_REF]. The difference with our previous evaluation comes from the fact that the measurement of aerosols requires the use of much larger Ag boats (ie 240 mg). This can lead to contamination over and above that determined in Table 2 with small Ag boats for the EA + GIS configuration. Further tests on samples with intermediate activities are still needed to confirm the low activity measured for this constant contamination.

Results and performances

Standard samples

All measurements are normalized with the OxA2 (NIST 4990C) samples from the gas cylinder connected to the GIS. Fig. 5 shows 132 OxA2 samples with a mass of about 100 μgC measured during the last 2.5 years. Each data point corresponds to a measurement time of about 12 min on a single target to reach a counting statistic of around 30,000 14 C + counts. In order to evaluate the level of precision and accuracy of the GIS-AMS instrument, all these individual OxA2 measurements are not used for the normalization calculation and are considered as unknown samples. Statistical analysis of the data gives an average F 14 C of 1.3403 and a standard deviation (SD) of 0.0078 (i.e. 6‰). This arithmetic mean and its associated error (std err = 0.0007) are compatible with the weighted mean (1.3405) and the weighted error (0.0008), and closely agree with the reference value of 1.3407 ± 0.0005 [START_REF] Stuiver | International agreements and the use of the new oxalic acid standard[END_REF].

The scatter value (6‰) probably overestimates the true operational dispersion because the number of OxA2 samples is limited during a gas measurement session and the first OxA2 (used for normalization) is also the first sample measured during a day-session, and might thus exhibit extra variability. We can estimate from these measurements the contribution of an external error of roughly 4‰, which is an indication of the system stability through time. This error is evaluated as the additional contribution to the error coming from pure counting statistics to describe the observed long term scattering of the data (σ long term 2 = σ stat 2 + σ external 2

). In addition, the precision of AixMICADAS is assessed by measuring small samples of a modern coral sample (Porites from Moorea (Aix-10327), while accuracy and precision are also evaluated with a halfmodern carbonate sample from the reference material IAEA-C2 (Aix-10346). Following our standard pretreatment protocol (see the companion paper by [START_REF] Fagault | Radiocarbon dating small carbonate samples with the gas ion source of AixMICADAS[END_REF]), the measured F 14 C value for the modern coral is 1.1105 with a standard deviation of 0.0089 (i.e. 8‰) for 18 measured samples with a mass larger than 80 μgC. The standard deviation for aliquots of the modern coral sample is thus about 2‰ larger than that measured in OxA2 samples. This additional scatter may be linked to the chemistry treatment applied to the carbonate samples, and possibly to heterogeneities among the coral subsamples.

The 23 IAEA-C2 samples give a F 14 C value of 0.4086 with a SD of 0.0031 and a stderr of 0.0006, which are close to the consensus value of 0.4114 ± 0.0003 [START_REF] Kromer | Van der Plicht, The IAEA 14C intercomparison exercise[END_REF]. The measured SD is equivalent to 8‰, which is compatible with that of modern coral. Taken at face value, the results indicate a small difference of 0.0028 between the measured F 14 C value and the consensus value for the IAEA-C2 standard. This small difference may come from the leaching process, as we also find the same difference between untreated and leached samples measured for solid graphite targets: mean F 14 C = 0.4121, SD = 0.0035, n = 7 and 0.4093, SD = 0.0024, n = 10, respectively (see companion paper by [START_REF] Fagault | Radiocarbon dating small carbonate samples with the gas ion source of AixMICADAS[END_REF]).

Blank samples

Table 3 shows the 14 C blanks and the δ 13 C measured AixMICADAS on small samples processed with the different devices connected to the GIS. All samples presented in Table 3 are large enough to avoid the constant contamination effect. Blanks (n = 129) from the CO 2 bottle (Linde Gas) are measured at least once for every gas sample magazine primarily to purge the system before measuring the procedural blanks that represent the different sample types and secondly to check that the system is ready to measure 14 C free CO 2 samples. The second set of blanks (i.e. processing blanks) is the one used to correct the 14 C results of unknown samples. Usually two CO 2 bottle blank samples are measured at the beginning of a gas session before the procedural blanks. In Table 3, the mean value of the CO 2 blanks bottle is based on the second blank measurement (and more if available in the particular magazine). This mean value reflects the starting level just before the measurement of processing blanks.

The IAEA-C1 calcite is our reference blank for measurements with the CHS coupled to the GIS. It shows a very low F 14 C value of 0.0014, SD = 0.0004, n = 51, equivalent to a 14 C age of 52,800 years BP, SD = 2300 years BP. A similar blank level is measured on the SIRI-K calcite, which agrees with results reported in the frame of the SIRI intercomparison program.

Different organic materials have been used with the EA coupled to the GIS. VIRI-K is our reference blank for wood measurement, phthalic For a particular gas measurement session, typically lasting a full day, a background estimation is used to correct the raw results for unknown samples. This background is the average value of representative blanks measured during the same session. As a conservative measure, a variability of ± 30% around this background value is propagated in the final error bar of the unknown sample (to take into account the long term variability of blanks). In rare cases of blank variability greater than ± 30% during a particular session, the measured variability during the session is used instead for error propagation.

13 C/ 12 C ratios and fractionation correction

The sequential injection of the three isotopes allows AixMICADAS to measure the 13 C/ 12 C ratio, which is needed to correct 14 C measurements for fractionation effects using BATS software [START_REF] Wacker | Bats: a new tool for AMS data reduction[END_REF]. For OxA2 samples from the standard CO 2 bottle that are considered as unknown samples, a mean δ 13 C value of -18.0‰ with an SD of 1.8‰ is calculated based on 132 measurements (leading to a standard error of 0.2‰). These gas data agree with our measurements on OxA2 solid targets considered as unknown: mean δ 13 C of -17.5‰ with a SD of 1.0‰ based on 150 analyses (leading to a standard error of 0.08‰). Both δ 13 C averages measured with AixMICADAS agree with the reference value of (-17.8 ± 0.1) ‰ [START_REF] Wb | An international reference material for radiocarbon dating[END_REF] and the direct IRMS measurement provided for the CO 2 bottle (-17.2‰).

By contrast, the δ 13 C of IAEA-C2 based on the gas measurement of 23 samples through the CHS-GIS interface is -10.4‰ with a SD of 0.8‰, which is about 2‰ lower than the consensus value (-8.25 ± 0.31‰ [START_REF] Kromer | Van der Plicht, The IAEA 14C intercomparison exercise[END_REF]).

Similarly, the δ 13 C values measured for the different blank samples through the EA + GIS or CHS + GIS exhibit a negative shift of around -2‰ when compared to precise and accurate results on solid graphite targets. For IAEA-C1, we obtained a mean δ 13 C of 2.5‰ with a SD of 2.2‰ on graphite targets, based on 22 analyses (leading to a standard error of 0.5‰). This mean value is compatible with the reference value of 2.42 ± 0.33‰ [START_REF] Kromer | Van der Plicht, The IAEA 14C intercomparison exercise[END_REF], in contrast to gas samples that are shifted by -2.5‰ (Table 3). A similar bias is observed on the δ 13 C of VIRI-K and Phthalic acid anhydride which provided the following averages with graphite targets: respectively, -20.2‰ with a SD of 1.1‰ based on 64 analyses and -29.0‰ with a SD of 1.9‰ based on 27 analyses.

Collectively, our observation seems to point towards an additional isotopic fractionation linked to the zeolite trap system within the GIS or to a similar fractionation for both EA and CHS systems. Indeed, these steps are not included in the measurements of CO 2 bottle, which lead to δ 13 C in agreement with expectations.

Conclusion

The performances of AixMICADAS for the measurement of small CO 2 gas samples have been evaluated over the last three years. Modern samples can be measured with a precision of 7‰ in a run of 12 min with stable 12 C -ion current typically on the order of 10-15 μA. An additional scattering of about 4‰ has been evaluated from the longterm measurements of modern standards.

Detailed work has been performed to suppress contamination effects for both carbonate and organic material and to survey the residual background for samples ranging between 10 and 100 μgC. These developments allow to date material over the full 50-kyr dating range of 14 C and have enabled us to date ancient collagen [START_REF] Fewlass | Size matters: radiocarbon dates of < 200 µg ancient collagen samples with AixMICADAS and its gas ion source[END_REF] and analyze atmospheric aerosols [START_REF] Bonvalot | Estimating contributions from biomass burning, fossil fuel combustion, and biogenic carbon to carbonaceous aerosols in the Valley of Chamonix: a dual approach based on radiocarbon and levoglucosan[END_REF]. Ongoing work on dating different fractions of aerosols, purified organic compounds and 14 C formed in situ will incite us to perform more tests with the closed ampoule cracker.

We also plan to dedicate more research to improving understanding of the isotopic fractionation problem, notably by measuring OxA2 samples through the EA + GIS system and the cracker to better evaluate the GIS component. Even if natural and procedural isotopic fractionations need to be corrected for in the F 14 C normalization procedure, we will continue to monitor and compare 14 C and 13 C results on the same samples measured with gas and solid targets. With the goal of using similar protocols for standard and unknown samples, we will also test normalization based on OxA2 samples measured through the EA + GIS.

A comparable procedure will be used for the CHS + GIS by using a modern carbonate in-house standard.

Fig. 3 .

 3 Fig.3. figures showing the F 14 C of modern samples (dashed bars) and raw blank samples (blank bars) measured in a direct sequence to estimate the cross contamination effect induced by the GIS + EA on the left and GIS + CHS on the right. The memory effect is evaluated at 4‰ for the GIS + EA configuration and 3‰ for the GIS + CHS configuration.

Fig. 4 .

 4 Fig. 4. F 14 C of IAEA-C1 blank sample (black dots) and modern sample Moorea porites (red squares) for different masses of sample measured on the EA + GIS system. Solid lines and dashed lines represent the least squares fit model with its 95% confidence interval with a constant contamination. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Table 2

 2 Evaluation of F14 C and mass of the constant contamination for the EA and CHS coupled to the GIS and for the CHS coupled to the AGE3 system for graphite target production. Values are calculated using a weighted nonlinear least squares model. Merck) is used to control the blank of graphite production and the C 16 H 32 O ketone (from Fluka) is a standard molecular compound used in HPLC extraction. The three samples show a similar F 14 C level below 0.003, equivalent to a 14 C age of 47800 years BP.

	System used	M c (μgC)	F 14 C c
	GIS + EA	0.55 ± 0.05	0.12 ± 0.03
	GIS + CHS	0.18 ± 0.02	0.14 ± 0.02
	AGE + CHS	1.74 ± 0.42	0.14 ± 0.05
		1.40	oxa2 samples
			consensus value
		1.38	mean value + std dev
		1.36
	F 14 C	1.34
		1.32
		1.30
		1.28
		0 10 20 30 40 50 60 70 80 90 100 110 120 130 140
			Sample
	Fig. 5. F 14 C values of 132 oxalic2 standard material considered as unknown gas
	samples. The red dot shows the weighted mean value and its associated stan-dard deviation of 7.8‰.White circles represent the measured F 14 C of each
	OxA2 sample. The blue line represents the reference value with its associated
	error [17].		Table 3
				Mean values of F 14 C and δ 13 C with the associated standard deviation measured
				for N blank samples measured on the GIS interface.
				Blank (device)	F 14 C	Std. dev. δ 13 C (‰) Std. dev N
				CO 2 LindeGas (bottle)	0.0028 0.0011	-3.9	1.9	129
				IAEA-C1 (CHS)	0.0014 0.0004	-0.2	1.1	51
				Iceland doublespar SIRI-K	0.0016 0.0005	-10.1	2.4	5
				(CHS)
				VIRI-K (EA)	0.0027 0.0010	-21.6	1.2	7
				C 16 H 32 O (EA)	0.0026 0.0005	-44.4	0.6	8
				Phthalic acid anhydride (EA)	0.0026 0.0007	-31.7	1.5	9

Acknowledgements

AixMICADAS was acquired and is operated in the framework of the EQUIPEX project ASTER-CEREGE with additional matching funds from the Collège de France and the ANR project CARBOTRYDH.


Chunyuan Li
email: chunyuan.li@hotmail.commaks@lix.polytechnique.frfrederic.chazal@inria.fr

Maks Ovsjanikov

Frederic Chazal

Persistence-based Structural Recognition

This paper presents a framework for object recognition using topological persistence. In particular, we show that the so-called persistence diagrams built from functions defined on the objects can serve as compact and informative descriptors for images and shapes. Complementary to the bag-of-features representation, which captures the distribution of values of a given function, persistence diagrams can be used to characterize its structural properties, reflecting spatial information in an invariant way. In practice, the choice of function is simple: each dimension of the feature vector can be viewed as a function. The proposed method is general: it can work on various multimedia data, including 2D shapes, textures and triangle meshes. Extensive experiments on 3D shape retrieval, hand gesture recognition and texture classification demonstrate the performance of the proposed method in comparison with state-of-the-art methods. Additionally, our approach yields higher recognition accuracy when used in conjunction with the bag-offeatures.

Introduction

Over the years, the bag-of-features (BoF) model has been extremely popular for the recognition of text, images and shapes [2,7,17,[START_REF] Li | A bayesian hierarchical model for learning natural scene categories[END_REF][START_REF] Sivic | Video google: A text retrieval approach to object matching in videos[END_REF]. In the image domain, this approach corresponds to treating a given image as a collection of unordered local descriptors, extracted from feature points, and quantizing them into discrete "visual/geometric words" [START_REF] Van Gemert | Visual word ambiguity[END_REF]. The distribution of visual words in an image is then summarized by a fixed-sized vector using various pooling techniques [4]. Due to the compactness and informativeness of the resulting representation, BoF is widely used for learning and recognition.

A large number of extensions of the BoF have been proposed with the aim to recover the geometric (spatial) relations between features, which are discarded by BoF. The two most common approaches include explicitly encoding geometric information in the bag of features [3,6,9,[START_REF] Zhang | Descriptive visual words and visual phrases for image applications[END_REF]46] or decomposing the object into spatial subregions and performing pooling in each subregion separately [START_REF] Lazebnik | Beyond bags of features: Spatial pyramid matching for recognizing natural scene categories[END_REF]. These approaches, however, assume explicit parametrization of the spatial information that needs to be captured (e.g. distribution of pairwise offsets as in [6,46] or commute times in [3]) or a pre-defined spatial decomposition to guarantee the stability of the representation under deformations. Thus, devising a compact and informative representation to characterize the structural properties of features while preserving stability, remains a challenging open problem.

Unlike the majority of the previously proposed methods that try to augment the bag-of-features representation to include some notion of spatiality, we propose to use a different approach, which is inherently well-suited to capture structural properties of functions defined on different domains in a stable way. In particular, we propose to complement to the BoF approach, by computing the persistence diagrams (PD) of functions defined on different data modalities, including 2D shapes, textures and triangle meshes.

Intuitively, instead of considering each feature point and its associated descriptor vector independently, we analyze all the points together by considering each dimension of the feature vector as a real-valued function, defined on the entire domain. With the assumption that the descriptors are stable and informative enough in the object space, we compute the persistent homology of the resulting functions. The derived PD provides a simple yet powerful description that captures the structural properties of the object. Importantly, unlike the BoF representation which captures the distribution of the values in the function, PD exploits the connectivity between different points in the domain to characterize the relative prominence of different feature points.

In order to demonstrate the usefulness of this representation, we argue both theoretically and experimentally that it captures information which is complementary to the information in the bag-of-features representation. In particular, we perform extensive experiments on 3D shape retrieval, hand gesture recognition and texture classification, where we demonstrate state-of-the-art performance by using the PDs in conjunction with the bag-of-features. We argue that this is largely due to the fact that persistence diagrams allow to distinguish feature configurations that are structurally different. Schematic comparison between the BoF with PD is shown in Fig. 1. The rest of the paper is organized as follows. In Sect. 2, we discuss some related works. Sect. 3 presents the framework of our proposed algorithm. Experimental results in three applicable scenarios are provided in Sect. 4. Finally, we conclude in Sect. 5.

Related Work

The past decade has witnessed a surge in popularity of BoF and its extensions for solving various computer vision problems. Among them, one particularly successful method is to form a global representation by spatially aggregating the local descriptors pioneered by Lazebnik et al. [START_REF] Lazebnik | Beyond bags of features: Spatial pyramid matching for recognizing natural scene categories[END_REF]. The subsequent research on spatial aggregation get compelling performance because of two seemingly independent advantages: the better design of (1) spatial regions and (2) aggregating operator. The spatial pyramids manually define the multi-scale grid-structured regions over the image space, and many excellent visual recognition methods either directly use them [START_REF] Lazebnik | Beyond bags of features: Spatial pyramid matching for recognizing natural scene categories[END_REF][START_REF] Yang | Linear spatial pyramid matching using sparse coding for image classification[END_REF], or modify the spatial decomposition to fit their data [9,[START_REF] Li | Intrinsic spatial pyramid matching for deformable 3d shape retrieval[END_REF]. Recently, Jia et al. [23] proposed to learn more adaptive regions by the receptive fields. Popular aggregating (pooling) strategies include averaging and max pooling, which have been used by Lazebnik et al. [START_REF] Lazebnik | Beyond bags of features: Spatial pyramid matching for recognizing natural scene categories[END_REF] and Yang et al. [START_REF] Yang | Linear spatial pyramid matching using sparse coding for image classification[END_REF] respectively. Coates et al. proposed to aggregate over multiple features in the context of deep learning [START_REF] Boureau | A theoretical analysis of feature pooling in vision algorithms[END_REF].

Our work relies on the theory of persistent homology, which falls under the umbrella of topological data analysis (TDA).It was first formalized by Edelsbrunner et al. [START_REF] Edelsbrunner | Topological persistence and simplification[END_REF] by building on earlier notions of size functions used by Frosini et al. for shape analysis [41] and later developed in [START_REF] Chazal | Proximity of persistence modules and their diagrams[END_REF][START_REF] Edelsbrunner | Topological persistence and simplification[END_REF]47]. Persistence diagrams appear prominently in TDA, and in particular provide an efficient way to encode topological properties of real-valued functions defined over spaces. PDs have provable stability properties [START_REF] Chazal | Proximity of persistence modules and their diagrams[END_REF]15], and allow to infer robust topological information on the studied data. Persistent homology has been successfully applied to clustering tasks [13], vision tasks such as shape segmentation [37], component detection [30] and recognition [START_REF] Carlsson | Persistence barcodes for shapes[END_REF][START_REF] Chazal | Gromov-hausdorff stable signatures for shapes using persistence[END_REF]20]. Our work is inspired by [START_REF] Carlsson | Persistence barcodes for shapes[END_REF][START_REF] Chazal | Gromov-hausdorff stable signatures for shapes using persistence[END_REF] but is different in that we exploit a given connectivity on the object and use feature functions to build the persistence diagrams rather than using point samples and simplicial complex filtrations. We also show that multiple PDs of different functions as well as the BoF representation can be combined to improve recognition tasks in a variety of scenarios. In this paper we focus on 0-dimensional persistent homology that encodes connectivity information in an intuitive and easyto-compute way.

Topological Object Representation Using Persistence Diagram

Feature Encoding: From BoF to PD

We assume that every object (e.g., image, 2D shape or 3D mesh) is represented as a connectivity graph G = {V, E}, where the set of nodes V of size V are samples on the object, and the set of edges E represent neighborhood relations between samples. For example, considering images as grids of points, we use the 4-neighborhood to define the edges in E. Similarly, for 3D objects, we use the vertices and the edges of the faces in the mesh to define G.

Similarly to the BoF approach we start by associating a D-dimensional feature vector to every sample in V resulting in a V × D matrix:

P (G) =    p 11 • • • p 1D . . . • • • . . . p V 1 • • • p V D    ∈ R V ×D .
While the BoF approach can be thought of as analyzing P by considering its rows, which correspond to point descriptors, we propose to consider P column-wise by interpreting each column as a real-valued function defined on the nodes of G. The persistence diagrams of these functions allow us to capture the structural properties of the object and in particular robustly encode the relative prominence of the different feature points.

Persistence Diagram

Given a real-valued function f defined on the nodes of G, the (0-dimensional) persistence diagram of f encodes the evolution of the connectivity of the 1-parameter family of superlevel-sets F α = f -1 ([α, +∞)) as α goes from +∞ to -∞ in the following way. A node v ∈ V is called a peak if it is a local maximum of f , i.e., if f (v) ≥ f (u) for any u such that (v, u) is an edge of G. For any peak v we say that v The lifespan of peak v is thus determined by its birth f a = f (v) and death f b ≤ f a . This allows us to associate a point (f a , f b ) on a 2D plane to each peak v. Collecting all these points, we obtain the persistence diagram (PD) of f (illustrated in Fig. 2(b)). The difference τ = f af b ≥ 0 is called the prominence (or persistence) of the peak v. Note that the prominence of v is equal to the vertical distance of its corresponding point in the PD to the diagonal. Lastly, the death time of the global maxima of f , i.e. the peaks v that satisfy f (v) = max f , is set to min f , and associated to the point (max f, min f ) in the PD. Overall, the PD encodes the relative prominence of the different peaks of a given function by considering the connectivity information in the domain.

is born at f (v). For α ≤ f (v), let C(v, α) be the connected subgraph in F α ⊆ G that contains v. The infimum of α such that f (v) is the global maximum of f restricted to C(v, α) is called the death of v. See
Computation. In practice the persistence diagram can be simply computed using the Union-Find algorithm [START_REF] Cormen | Introduction to Algorithms[END_REF] by sorting the nodes of G according to their function value and keeping track of the corresponding connected components.

Stability. One important property of the persistence diagram is that it is stable under some perturbations of the feature function. Intuitively, the stability theorem in [START_REF] Chazal | Proximity of persistence modules and their diagrams[END_REF]15] states that if the values of a function f are perturbed by no more than some ε > 0, then the points on the persistence diagrams will also be perturbed by no more than ε. In Fig. 2(a), the blue function f is a noisy version of the red function f . The PDs of f and f are plotted in Fig. 2(b). These two PDs are similar, in the sense that the distance between points that are far from the diagonal is small, even if some extra points with low prominence can appear.

Invariance. Similarly to the BoF representation, PD is invariant to rotations, translations and scaling of the object provided that the feature function remains the same. However, unlike the BoF which is invariant to any permutation of the nodes, the PD is only invariant under continuous deformations. In particular, given two objects represented as graphs G 1 and G 2 and a map T between them, for any function f on G 2 the PDs of f and f • T are the same if T is an isomorphism. Conversely, if T is not an isomorphism then there exists a function f whose PD will be different from the one of f • T . Intuitively this implies that PDs capture local spatial information through the connectivity structure.

Relation to spatial pyramid approaches. Note that PDs can be seen as alternatives to spatial decomposition methods, and espacially Spatial Pyramid Matching SPM [START_REF] Lazebnik | Beyond bags of features: Spatial pyramid matching for recognizing natural scene categories[END_REF] and ScSPM [START_REF] Yang | Linear spatial pyramid matching using sparse coding for image classification[END_REF], that apply BoF approach to each region of a multi-scale decomposition of the object and often use max pooling to characterize the features in each region. Unlike these approaches, PDs do not require an explicit decomposition of the object, and allow to capture structural properties of functions in a robust and compact way.

PDs for object comparison

In this paper we propose to use persistence diagrams of different feature functions as descriptors for object recognition and classification. To illustrate this idea, we compute the PD for several 3D meshes in Fig. 3 where we use the Heat Kernel Signature (HKS) [21,[START_REF] Sun | A concise and provably informative multi-scale signature based on heat diffusion[END_REF] for fixed time as the feature function. The first row displays the HKS value for the four models: two horses, a dinosaur and pliers. The second row shows the corresponding PDs. Note that in this case the PD is invariant to isometric deformations because the underlying HKS feature is an isometry invariant. The PDs allow us to distinguish these 3 classes of objects since the diagrams associated to the horses are more similar to each other to those of the other objects.

Quantitatively, we use the following construction to compare PDs. For two functions f and g with respective PDs D 1 and D 2 , we construct a weighted bipartite graph B between D 1 ∪ Π(D 2 ) and D 2 ∪ Π(D 1 ) where Π is the projection onto the diagonal, i.e. for a point (a, b) ∈ R 2 , Π(a, b) = ( a+b 2 , a+b 2 ). This can guide the points with smaller persistence corresponding to the points on diago-nal, and thus remove their interference in prominent point matching. The weight of an edge

(u, v) ∈ B is defined as v -u ∞ if u or v is in D 1 ∪ D 2 and is set to 0 if both u and v are in Π(D 1 ) ∪ Π(D 2 ). The bottleneck distance d btnk. (D 1 , D 2
) is defined as the minimum value d such there exists a perfect matching in B with all edges having weights less than d. Similarly, the degree-p Wasserstein distance d wsst. is defined as the minimum value d such there exists a perfect matching in B whose the sum of the weights raised to the power p of all edges is less than d p . As a consequence, computing both distances boils down to computing maximum matchings in bipartite graphs. This can be done efficiently using e.g. the Hungarian algorithm for PDs with few points. However, this might become prohibitive for PDs with many points.

For this reason we also use an alternative representation of PDs, the persistent landscape [START_REF] Bubenik | Statistical topology using persistence landscapes[END_REF], which allows us to represent a PD as a point in a high dimensional Euclidean space. Given a PD with points

{(a i , b i )} n i=1 , a i ≥ b i . For each point (a i , b i ), let f (ai,bi) : R → R, f (ai,bi) (t) = min(a i -t, t -b i ) +
, where z + denotes max(z, 0). The persistence landscape is defined as the set of functions

β k : R → R, k ∈ N given by β k (t) = kth largest value of f (ai,bi) (t) n i=1 (1)
with β k (t) = 0 if k > n. Given k we consider the first k landscape functions, and discretize them with m sampling points along t. The derivatives of the sampling points are calculated and form a mk-size vector. The landscape distance d k,m ldsp. is defined as the L p norm between these vectors.

Metric Learning for Multiple Functions

In a bid to achieve higher performance, multiple features are usually used. When these features are designed in similar ways, the final distance can be simply chosen as the sum of their persistence based distances. However, when these features are designed in very different ways, it is more reliable to give different weights to different features. We use labeled data from the dataset to learn these parameters. Suppose we have some set of objects. Each object x i is characterized by D functions as

F (x i ) = {f d (x i )} D d=1
, and d d (x i , x j ) is the persistence based distance of the dth function between two objects x i and x j from the set. Additionally, we are given information that certain pairs of them are "similar": S : (x i , x j ) ∈ S if x i and x j are similar

(2) and certain pairs of them are "dissimilar"

D : (x i , x j ) ∈ D if x i and x j are dissimilar ( 3
)
Our goal is to learn a distance metric to respect that "similar" objects end up with smaller distances to each other, and "dissimilar" objects with larger distances to each other. We consider learning a distance metric of the form

Dis(x i , x j ) = x i -x j A = d(x i , x j ) T Ad(x i , x j ) (4) where d(x i , x j ) = [d 1 (x i , x j ), . . . , d d (x i , x j ), . . . , d D (x i , x j )].
The problem can be formulated as an optimization programming

min A xi,xj ∈S x i -x j 2 A s.t. xi,xj ∈D x i -x j A ≥ 1, A ≥ 0 (5)
This is a standard formulation of distance metric learning, and can be solved by the algorithms in [START_REF] Davis | Informationtheoretic metric learning[END_REF][START_REF] Xing | Distance metric learning with application to clustering with side-information[END_REF]. The learned metric A interprets the contribution of different features. It is further embedded as part of the pipeline to provide a more reliable measure for the rest of testing data in the dataset.

Limitations

While PDs allow to capture the relative prominence of the local extrema of different functions, we note that they do not provide any information about the distribution of other (non-critical) values. Thus, the information contained in PDs is largely complementary to the information captured in the BoF representations. Thus, as we show in the following section, we can often get superior performance by combining the two approaches.

Experimental Results

We present the results of our PD-based object recognition scheme on various datasets of three different types: 3D meshes (Section 4.1), 2D gesture contours (Section 4.2), and texture images (Section 4.3). The method was implemented in MATLAB, and the experiments were performed on a computer with an Intel Core processor running at 1.4 GHz with 4 GB RAM. In each experiment we use 3 randomly selected samples from each class to learn the weights in the metric.

3D Shape Retrieval

Our first set of experiments is on triangulated 3D mesh retrieval. We consider two datasets: SHREC 2010 [START_REF] Lian | track: non-rigid 3d shape retrieval[END_REF] and a robustness dataset [START_REF] Bronstein | robust large-scale shape retrieval benchmark[END_REF]. The former focuses on nearisometric shapes, and consists of 200 watertight shapes equally divided into 10 categories (Figure 4 illustrates a sampling of the shapes from this dataset). The latter has the same data used in the Shape Google [33] with an extra transformation, which is the same as SHREC Robustness benchmark [START_REF] Bronstein | robust large-scale shape retrieval benchmark[END_REF] except for a few transformations. It contains 596 shapes classified into 13 classes, and 456 shapes which do not belong to any class. For both datasets, we use the spectral descriptors as feature functions, including the heat kernel signature (HKS) [21,[START_REF] Sun | A concise and provably informative multi-scale signature based on heat diffusion[END_REF], wave kernel signature (WKS) [START_REF] Aubry | The wave kernel signature: A quantum mechanical approach to shape analysis[END_REF] and scale invariant heat kernel signature (SIHKS) [START_REF] Kokkinos | Dense scale invariant descriptors for images and surfaces[END_REF]. All of them are isometry-invariant. HKS and WKS characterize the macroscopic and microscopic properties of shapes, respectively. SIHKS is a scale-invariant version of HKS. We compare our method to BoF-based approach, spatially-sensitive BoF (SSBoF) [7] and ISPM [START_REF] Li | Intrinsic spatial pyramid matching for deformable 3d shape retrieval[END_REF], which is a spatial pyramid approach on surfaces.

Results on SHREC 2010

On this dataset, we simplify each mesh into 2000 faces, and the retrieval results are evaluated using the measures provided by the organizers, including NN, FT, ST, Emeasure, and DCG (See [START_REF] Lian | track: non-rigid 3d shape retrieval[END_REF] for details). We first illustrate the performance of different methods by considering the HKS and WKS functions for fixed time and energy parameters respectively. DCG and the average running time to compare two PDs are reported in Table 1. The average number of points on PDs ranges around 15 for HKS and 100 for WKS. It can be observed on this example, that bottleneck and Wasserstein distances yield a significantly higher accuracy than the landscape distance, at the price of a higher computation time. Fig. 5 displays an example where PD provides more discriminative information than the histogram of values of the function. The extremities of the objects all correspond to local maxima of the HKS function. This structural information is well preserved in PDs as the prominent points, but lost in the histogram representation that merges function values from different components into the same bins. Based on these experiments, for comparison of 3D shapes, we use the bottleneck distance to compare PDs, as it provides a good trade-off between time and efficiency whenever the diagrams do not contain too many points.

We first compare our PD-based technique with BoF and ISPM methods using multi-dimensional descriptors. In particular, we use 17-dimensional SIHKS, 10-dimensional HKS and 10-dimensional WKS. In this case we do not use the metric learning step as the different dimensions are commensurable. Throughout this experiment with use 32 words to construct the vocabulary for the BoF representation and use soft vector quantization. We summarize the results of SIHKS and HKS in Table 2. As can be seen in this table, the PD-based method consistently outperforms the BoF representation across different feature functions.

We then combine SIHKS, HKS and WKS to form a single high-dimension descriptor. The combined performance of our method again outperforms the BoF approach, in part because PDs allow to capture structural properties of objects which are discarded by the BoF representation.

We also show that the advantages of the BoF and PDbased approaches can be combined by considering the joint representation where d BoF+PD = λ * d BoF + d PD , where λ is the relative weight to normalize the distances. On this dataset, we manually picked λ of SIHKS, HKS, WKS and the high-dimension spectral descriptor to be 0.033, 0.2, 200, 0.1 for BoF+PD, and 0.2, 1, 2000, 0.2 for ISPM+PD, respectively. The combined metric PD+BoF/ISPM always yields the the highest results.

We also compare with the state-of-the-art method, including ShapeDNA [START_REF] Reuter | Laplace-beltrami spectra as 'shape-dna' of surfaces and solids[END_REF] and DM-EVD [START_REF] Smeets | Isometric deformation modelling for object recognition[END_REF], which show the best performance in a larger contest [START_REF] Lian | Shrec'11 track: Shape retrieval on non-rigid 3d watertight meshes[END_REF]. We test the ShapeDNA method on the the same simplified meshes, and take the result of DM-EVD and other methods from [START_REF] Lian | track: non-rigid 3d shape retrieval[END_REF], in which DM-EVD is performed on the fine meshes. As can be seen in Table 2, our combined result is comparable to these methods.

Results on TOSCA-based dataset

On the TOSCA-based dataset [33], we use multidimensional descriptors: HKS and SIHKS with 6 and 5 dimensions respectively. Table 3 summarizes the results obtained using our method compared with the performance of BoF, SSBoF, ISPM and PD in terms of equal error rate (EER) [33]. Overall, the PD-based approach alone shows slightly worse performance than the BoF methods on this dataset. However, we obtain superior performance by combining the two methods. On this dataset, we use the weights: λ = 10, 5, 10 for HKS, SIHKS and HKS+SIHKS respectively in BoF+PD and 50, 10000, 10 for HKS, SIHKS and HKS+SIHKS respectively in ISPM+PD. The EER of ShapeDNA for all transformations is 0.0967, which is worse than state-of-the-art work Shape Google (BoF+spectral descriptors) [33] on this dataset. The proposed method can help Shape Google and its spatial version (ISPM) achieve even better results.

Hand Gesture Recognition

Our second experiment focuses on the recent Kinect sensor-based hand gesture recognition [START_REF] Ren | Robust part-based hand gesture recognition using kinect sensor[END_REF] and is inspired by the work of [41]. This dataset contains 1000 gestures equally categorized into 10 classes. Each gesture is compared against a set of 20 templates (2 samples for each class) and nearest neighbor classification is performed.

Each gesture instance is represented as a sequence of contour vertices, starting from the left side of the wrist (yellow point in Fig. 6(a)). We construct our graph by considering the sequence points as nodes, and their adjacent connectivity as edges. Ren et al. [START_REF] Ren | Robust part-based hand gesture recognition using kinect sensor[END_REF] also provide an informative function to characterize each gesture by first computing the center point, f center , defined as the point with the maximal Distance Transform value (red point in Fig. 6(a)), and computing the normalized Euclidean distance from the contour vertices to the center. We show the corresponding PD in Fig. 6(d). Importantly, due to the stability property, the PD of f center can represent the perturbations of the contour as points close to diagonal, and thus help to alleviate noise brought by the Kinect sensor in data acquisition.

In all of the experiments below we use the Wasserstein distance to compare persistence diagrams of functions in this dataset. With f center alone, BoF achieves its best performance of 72.60% when discretizing the function into 5 bins, while PD gets much higher result of 87.60% (see Fig. 7). This suggests that the information lost by discarding the connectivity in BoF is critical for the recognition of objects with different structural properties.

To improve the performance, we include two more functions by computing the eingenfunctions of the PCA performed on vertex coordinates. We denote the two eigenfunctions as f 1 and f 2 and their absolute values as |f 1 |, |f 2 |. Using f center , |f 1 |, and |f 2 |, together, the BoF reaches its result 83.20% with 15 visual words learned using kmeans. Our method with metric learning solved by [START_REF] Davis | Informationtheoretic metric learning[END_REF][START_REF] Xing | Distance metric learning with application to clustering with side-information[END_REF] gets accuracy of 92.90% and 91.70%, respectively. Furthermore, BoF+PD combined using λ = 12 yields recognition accuracy of 93.50%.

To compare our method to the state-of-the-art, we use f 1 and f 2 directly and compute the distance by testing two sign possibilities i.e., where f i and g i are the i th PCA eigenfunctions of the two gestures. For comparison, we also consider the classification with only one labeled sample per class (10 templates). Recent methods proposed by Ren et al. [START_REF] Ren | Robust part-based hand gesture recognition using kinect sensor[END_REF] and Wang et al. [START_REF] Wang | Perceptually friendly shape decomposition by resolving segmentation points with minimum cost[END_REF] are based on shape segmentation and finger earth mover distance (FEMD) introduced by Ren et al. [START_REF] Ren | Robust part-based hand gesture recognition using kinect sensor[END_REF]. As claimed by the authors, this metric is robust to the fingertouching, which also means that adding more labeled gestures with finger-touching into the training set will not sig- [START_REF] Ren | Robust part-based hand gesture recognition using kinect sensor[END_REF] 93.9 4.0012 Thresholding Decomposition+FEMD [START_REF] Ren | Robust part-based hand gesture recognition using kinect sensor[END_REF] 93.2 0.075 PSD+FEMD [START_REF] Wang | Perceptually friendly shape decomposition by resolving segmentation points with minimum cost[END_REF] 94 nificantly improve accuracy. We validate it using PSD [START_REF] Wang | Perceptually friendly shape decomposition by resolving segmentation points with minimum cost[END_REF] for the settings that 1, 2 or 3 labeled samples are respectively chosen per class. For each setting, 3 trials with different labeled samples (with finger touching) are tested, the mean accuracy of PSD is 94.1%, 93.2%, and 93.1%. We show the comparison with the state-of-the-art in Table 4, the results of other methods are from [START_REF] Ren | Robust part-based hand gesture recognition using kinect sensor[END_REF]. PD can reach higher accuracy with more training samples, without sacrificing efficiency, because it avoids the time-consuming shape decomposition stage. With the same training set 20 templates, persistence based method is 95.40%, while PSD is 93.80%.

Texture Image Classification

Our last experiment is on texture classification. For this we use the standard Outex database [31]. This database consists of 4320 texture images equally categorized into 24 classes. To evaluate our method, we first randomly select 20 images from each class, with one of them acting as a label, and rest are testing samples. We use the CLBP descriptor function [START_REF] Guo | A completed modeling of local binary pattern operator for texture classification[END_REF], which is based on the well-known LBP descriptor [32]. CLBP describes each local region on an image as three discretized components named center (C), sign (S) and magnitude (M). We use CLBP-S, CLBP-MC and CLBP-SMC proposed by Guo et al. [START_REF] Guo | A completed modeling of local binary pattern operator for texture classification[END_REF] as the feature functions since these three functions convey different information. CLBP functions are (1) discrete and severely bounded, (2) have strong local self-repetition. Therefore, the associated PDs contain many points. To efficiently discriminate these PDs, we use persistence landscape method. The dissimilarity is defined as a combination of multiple normalized landscape distances, d = The results are summarized in Table 5. Given these multiple functions, BoF reaches its best result 84.6491% using 100 words. PD with metric learning gives 75.8772%, while the manually chosen relative weights for PD with different functions result in the accuracy 77.4123%. Although higher overall accuracy is always provided by BoF, PD shows its advantage by yielding much higher accuracy in some classes. We show one such example with CLBP-SMC function in Fig. 8. The top two images have larger histogram distance, but their PDs are more similar: fewer points appear on the top-right area, and points in the middle are with higher density. When combining BoF and PD, λ for CLBP-S, CLBP-MC, CLBP-SMC and multiple functions is set as 800,100,165,100, BoF+PD always gives the best performance. Furthermore, we test PD with CLBP-SMC on the whole database, and get accuracy 84.1406%. Its accuracy by BoF is 96.5625%, which is also the reported best in the same settings [START_REF] Guo | A completed modeling of local binary pattern operator for texture classification[END_REF]. By setting λ = 225, we find that BoF+PD gives an even better result 97.0313%.

Conclusion and Future Work

In this paper, we present a method for object recognition using topological persistence. Our experiments on a variety of applicable scenarios demonstrate the effectiveness of this approach. PD and BoF capture different information of a function: PD describes it structural properties, while BoF describes the quantity statistic of the function values.

One interesting future direction is to create a principled theoretical framework to combine the PD and BoF for better recognition, and to extend the persistence-based approach to other applications, such as range scans or RGB-D data.

Figure 1 .

 1 Figure 1. Schematic comparison of BoF with PD.

Figure 2 .

 2 Sketch of persistence computing: (a) a smooth function f maps the nodes of G to R, f is a noisy version of f ; (b) superimposition of PDs of f (red) and f (blue), showing the one-to-one correspondence between the prominent peaks of f and f .

  Fig. 2 for an example where G represents a line-graph (in green) and the node p of G is a peak given the function f shown in red in Fig. 2 (a). A new connected subgraph is born in the superlevel-set F α when α = f (p), and it dies when α = f (s).

Figure 3 .

 3 Figure 3. HKS function and its persistence diagrams.

Figure 4 .

 4 Figure 4. Sample shapes from SHREC 2010.

Figure 5 .

 5 Figure 5. An example that PD discriminates different shapes, while BoF fails.

Figure 6 .

 6 Figure 6. Persistence computing on gestures. (a) a gesture with its center point in red and starting point in yellow; (b) distanceto-center function fcenter; (c) gesture colored by fcenter; (d) PD of fcenter.

Figure 7 .

 7 Figure 7. The confusion matrix of (a) BoF, and (b) PD. The gesture classes are displayed on top.

  di = min {dwsst. (D(fi), D(gi)) , dwsst. (D(fi), D(-gi))} , i = 1, 2.

5 i=1

 5 (k i /k 1 ) * d ki,n ldsp. ,where k i = [100, 200, 500, 800, 1000] and n = 21.

Figure 8 .

 8 Figure 8. The example of PD and BoF for texture classification.

Table 1 .

 1 Comparison between different distances.

	Metrics		HKS	WKS
		DCG	time (s)	DCG	time (s)
	Bottleneck	0.8746	0.0020	0.5988	0.0315
	Wasserstein (p = 2)	0.8774	0.0060	0.6190	0.5354
	Wasserstein (p = 3)	0.8781	0.0061	0.6117	0.6069
	Landscape (K = 10, N = 100)	0.8553	0.0046	0.4935	0.0071
	Landscape (K = 10, N = 50)	0.8471	0.0026	0.4578	0.0037

Table 2 .

 2 Comparison of PD, BoF, SSBoF and ISPM on SHREC 2010[START_REF] Lian | track: non-rigid 3d shape retrieval[END_REF].

		Methods	NN	1-Tier	2-Tier	e-Measure	DCG
		BoF	0.9100	0.4811	0.6374	0.4492	0.8061
		SSBoF	0.9150	0.4737	0.6321	0.4412	0.8040
		ISPM	0.9300	0.5745	0.7018	0.5018	0.8597
	HKS	PD	0.9600	0.5811	0.7034	0.4971	0.8677
		BoF + PD	0.9500	0.6095	0.7382	0.5237	0.8781
		ISPM + PD	0.9700	0.6321	0.7500	0.5335	0.8869
		BoF	0.9700	0.7145	0.8308	0.6020	0.9191
		SSBoF	0.9700	0.7097	0.8297	0.5967	0.9164
		ISPM	0.9700	0.7750	0.8734	0.6359	0.9315
	SIHKS	PD	0.9850	0.8532	0.9697	0.7045	0.9740
		BoF + PD	0.9850	0.8534	0.9697	0.7051	0.9737
		ISPM + PD	0.9900	0.8784	0.9724	0.7114	0.9778
		BoF	0.9700	0.7426	0.8758	0.6275	0.9367
		SSBoF	0.9750	0.7318	0.8511	0.6102	0.9292
	WKS+	ISPM	0.9750	0.8021	0.8979	0.6533	0.9408
	HKS+	PD	0.9850	0.8532	0.9697	0.7045	0.9740
	SIHKS	BoF + PD	0.9850	0.8561	0.9705	0.7061	0.9743
		ISPM + PD	0.9900	0.8779	0.9742	0.7118	0.9778
	ShapeDNA [35]	0.9850	0.7974	0.9203	0.6653	0.9536
	BOF-dSIFT-ERC-Tree	0.9850	0.9092	0.9632	0.7055	0.9763
	DM-EVD [38]	1.0000	0.8611	0.9571	0.7012	0.9773
	Canonical Forms	0.9200	0.6347	0.7800	0.5527	0.8781

Table 3 .

 3 Comparison of PD, BoF, SSBoF and ISPM on TOSCAbased dataset.

	Descriptors	Transformation	BoF	SSBoF	ISPM	PD	BoF+PD	ISPM+PD
		isometry	0.0352	0.0307	0.0284	0.1464	0.0369	0.0274
		isometry+topology	0.0394	0.0320	0.0341	0.1654	0.0397	0.0307
	HKS+	noise	0.1540	0.1464	0.1036	0.2087	0.1462	0.1019
	SIHKS	null	0.0412	0.0394	0.0299	0.1561	0.0428	0.0291
		partiality	0.0507	0.0466	0.0456	0.2630	0.0521	0.0468
		topology	0.0561	0.0572	0.0350	0.2173	0.0563	0.0362
		triangulation	0.0520	0.0495	0.0374	0.2049	0.0562	0.0396
		All	0.0535	0.0519	0.0417	0.1727	0.0527	0.0380
	HKS	All	0.0650	0.0712	0.0595	0.2477	0.0638	0.0581
	SIHKS	All	0.0670	0.0531	0.0507	0.1728	0.0658	0.0507

Table 4 .

 4 Comparison on hand gesture database[START_REF] Ren | Robust part-based hand gesture recognition using kinect sensor[END_REF].

	Methods	Accuracy (%) Time (s)
	Shape Context without bending cost	83.2	12.346
	Shape Context with bending cost	79.1	26.777
	Skeleton Matching	78.6	2.4449
	Near-convex Decomposition+FEMD

Table 5 .

 5 Comparison between PD and BoF on texture database.

	Methods	Accuracy (%)
		BoF	76.0965
	CLBP-S	PD	54.8246
		BoF + PD	79.1667
		BoF	74.1228
	CLBP-MC	PD	55.2632
		BoF + PD	77.1930
		BoF	83.9912
	CLBP-SMC	PD	57.8947
		BoF + PD	85.9649
	CLBP-S+	BoF	84.6491
	CLBP-MC+	PD	75.8772
	CLBP-SMC	BoF + PD	87.5000

Acknowledgements

The authors acknowledge the Marie Curie CIG-334283-HRGP, the CNRS chaire d'excellence, the Google Faculty Research Award, the ANR project Top-Data ANR-13-BS01-0008 and the ERC project Gudhi.