Pascal Conan
email: pascal.conan@obs-banyuls.fr

Léna Philip

Eva Ortega-Retuerta

Charlène Odobel

Clélia Duran

Caroline Pandin

Carolane Giraud

Anne-Leila Meistertzheim

Valérie Barbe

Alexandra Ter Hall

Mireille Pujo-Pay

Jean-François Ghiglione

Evidence of coupled autotrophy and heterotrophy on plastic biofilms and its influence on surrounding seawaters

We measured phytoplankton primary production and heterotrophic bacterial activities on microplastics and seawater in the Northwestern Mediterranean sea during two 3-month spring periods over consecutive years. Microorganisms growing on a 5 mm diameter low density polyethylene (LDPE) film (200 µm thick) were faced with two contrasting conditions from one year to the next. Spring 2018 was characterized by consistent nutrient inputs and bloom development. In spring 2019, nutrient inputs and bloom were low. For the first time, we observed a clear coupling between primary production and heterotrophic prokaryote production on microplastics during both years, but with different intensity between years that reflected the crucial role of the trophic environmental conditions (nutrient supply) in shaping microbial activities on plastics. Indeed, we proved that high primary production on plastics could support the whole (net autotrophy) or the majority of the bacterial carbon demand needed for heterotrophic activities, supplemented by other carbon sources if surrounding waters are highly productive. We also propose that microbial activity on plastics influence the microbial community in the surrounding seawaters, especially when the environmental conditions are less favorable. An illustrative image of the role of plastics in the environment could be that of an electrical inverter in a circuit that allows for the attenuation of both positive and negative variations. Our results highlight the potential role played by the plastisphere shaping biogeochemical cycles in a context of an increasing amount of plastic particles in the marine environment.

Introduction

The growing use of plastic worldwide together with waste mismanagement resulted in an estimated of 24.4 trillion plastic particles floating in the world's oceans, which majority has a particle-size <5 mm, the so-called microplastics, discharged mainly from rivers [START_REF] Isobe | A multilevel dataset of microplastic abundance in the world's upper ocean and the Laurentian Great Lakes[END_REF][START_REF] Weiss | The missing ocean plastic sink: Gone with the rivers[END_REF]. Indeed, the introduction of microplastics into the aquatic ecosystems is mainly related to domestic runoff that contains microbeads and fragments of microplastics (used in cosmetics and other consumer products) and also to the fragmentation of large plastic waste. Plastic manufacturing industries and coastal activities (fishing, aquatic tourism...) are also sources of microplastic pollution in the marine ecosystems [START_REF] Subhankar | Microplastics in our oceans and marine health[END_REF].

The emissions of microplastic are estimated to increase, even under the most optimistic scenarios [START_REF] Borrelle | Predicted growth in plastic waste exceeds efforts to mitigate plastic pollution[END_REF]. This pollution is considered "poorly reversible" with potential impacts including changes to the carbon and nutrient cycles, co-occurring biological impacts on endangered or keystone species, ecotoxicity and other related societal impacts [START_REF] Macleod | The global threat from plastic pollution[END_REF].

Once in the marine environment, plastics are rapidly colonized by microorganisms, forming the so-called plastisphere [START_REF] Zettler | Life in the "Plastisphere": Microbial Communities on Plastic Marine Debris[END_REF], and interact with the overall ecosystem depending on their size (PVC panels of 5 × 5 cm or 15 × 3.5 cm for [START_REF] Catão | Shear Stress as a Major Driver of Marine Biofilm Communities in the NW Mediterranean Sea[END_REF]Nanoto Micro-plastics for Ter Halle and Ghiglione, 2021). Most of the studies so far revealed a large diversity and a clear niche partitioning among marine bacteria living on plastics compared to the free-living [START_REF] Crisafi | Bacterial biofilms on medical masks disposed in the marine environment: a hotspot of biological and functional diversity[END_REF][START_REF] Debroas | Plastics in the North Atlantic garbage patch: A boat-microbe for hitchhikers and plastic degraders[END_REF][START_REF] Zettler | Life in the "Plastisphere": Microbial Communities on Plastic Marine Debris[END_REF] and organic particle-attached fractions (Dussud et al., 2018b;[START_REF] Oberbeckmann | Environmental Factors Support the Formation of Specific Bacterial Assemblages on Microplastics[END_REF]. Autotrophic microorganisms such as cyanobacteria and diatoms have been shown to be particularly overrepresented in microplastics, suggesting the potentially high impact on the global pelagic primary production and more generally on the carbon and nitrogen biogeochemical cycles in the surface oceans [START_REF] Amaral-Zettler | Ecology of the plastisphere[END_REF][START_REF] Crisafi | Bacterial biofilms on medical masks disposed in the marine environment: a hotspot of biological and functional diversity[END_REF]Dussud et al., 2018b;[START_REF] Nava | A critical review of interactions between microplastics, microalgae and aquatic ecosystem function[END_REF]. Because plastic debris offer an abundant growth matrix for microalgae while limiting predation pressure by zooplankton (Kong and Koelmans, 2019), it has been proposed that plastic pollution can promote significant proliferations of microalgae, with consequent harmful effects for aquatic ecosystems already disturbed by eutrophication processes (Durrieu de [START_REF] Durrieu De Madron | Marine ecosystems' responses to climatic and anthropogenic forcings in the Mediterranean[END_REF][START_REF] Zhang | How climate change and eutrophication interact with microplastic pollution and sediment resuspension in shallow lakes: A review[END_REF]. Moreover, processes involved in the nitrogen (nitrogen fixation, ammonia and nitrite oxidation, denitrification) and phosphorus cycling (P accumulation, alkaline phosphatase activity) are overexpressed in the plastisphere, which can modify the nutrient concentration in the surrounding seawater and thus indirectly affect primary producers [START_REF] Chen | Changing Biogeochemistry in the South China Sea[END_REF]. For instance, plastics may represent net autotrophic hot spots in the oligotrophic ocean, with high density of chlorophyll a and high oxygen production [START_REF] Bryant | Diversity and Activity of Communities Inhabiting Plastic Debris in the North Pacific Gyre[END_REF]. Despite growing interest on the plastisphere influence on ecosystem functions, direct measurements on microbial activities remain scarce. Some studies found also high heterotrophic prokaryotic production (Dussud et al., 2018b;[START_REF] Odobel | Bacterial Abundance, Diversity and Activity During Long-Term Colonization of Nonbiodegradable and Biodegradable Plastics in Seawater[END_REF] and ectoenzyme activities [START_REF] Odobel | Bacterial Abundance, Diversity and Activity During Long-Term Colonization of Nonbiodegradable and Biodegradable Plastics in Seawater[END_REF] on plastics. It is surprising that primary production in the plastisphere was never recorded in marine ecosystems. Both primary production and heterotrophic activities play a key role in the carbon cycle. The former participating in about half of global primary production [START_REF] Behrenfeld | A consumer's guide to phytoplankton primary productivity models[END_REF], and the latter channeling half of the oceanic primary production through the microbial loop [START_REF] Del Giorgio | Bacterial energetics and growth efficiency[END_REF].

A recent review underlined the need of coupling primary production and heterotrophic activities to evaluate the potential role of the plastisphere on carbon cycling in the oceans [START_REF] Jacquin | Modeling Decreased Resilience of Shallow Lake Ecosystems toward Eutrophication due to Microplastic Ingestion across the Food Web[END_REF]. In oceanic systems, the supply of organic matter from autochthonous (autotrophic) or allochthonous (heterotrophic) primary production is mainly dependent on nitrogen and phosphorus availability and will defined the trophic status of these ecosystems, and in turn, the potential coupling between phytoplankton and bacteria [START_REF] Dodds | Expanding the concept of trophic state in aquatic ecosystems: It's not just the autotrophs[END_REF]. Finally, the characterization of the trophic state is necessary for understanding food web linkages as well as biogeochemical features and subsequently water quality, including human influence.

Our study provides the first measurements to our knowledge of the marine plastisphere primary production together with bacterial heterotrophic activities over time. We hypothesize that natural environmental factors (such as temperature, salinity, nutrient availability…) play a major role in plastisphere activities, thus influencing the net autotrophy vs. heterotrophy budgets over time. More precisely, trophic links within the niche created by the plastisphere would be strengthened when external conditions become less favorable (i.e. meso vs oligotrophic). Polyethylene-based microplastics were incubated in natural seawater for two 3month periods, including a classical spring phytoplanktonic bloom event that occurs every year at the long-term SOLA marine observatory station (NW Mediterranean Sea, France). The experiments revealed a clear link between autotrophic and heterotrophic productions of the plastisphere over time, as well as a possible influence of the microbial activities on microplastics in surrounding seawaters.

Material and methods

Experimental conditions during 3 months in two following years

We incubated 200 circular pieces of polyethylene of 5 mm diameter and 200 µm thickness Borealis,ref. FA6224,Austria) in a 50 L aquarium with direct circulation to the sea.

The incubation was repeated twice at one year interval: the first incubation was performed from 12 February to 7 May 2018 (44 th to 127 th day of the year, here after D44-127 1 ; period of 85 days) and the second incubation was performed from 19 February to 8 May 2019 (D50-133; 1 D1 correspond to the 1 st January with this time scale 7 period of 91 days). Each plastic piece was first sterilized with 70% ethanol and rinsed 3-times with sterile seawater before incubation. The aquarium was placed outside with natural light conditions, and covered by a transparent screen. Seawater circulating through the aquarium was pumped from 4 m depth and 30 m from the coast in the Banyuls bay (NW Mediterranean Sea, France) and the seawater flow rate in the aquarium was ~5 L per hour.

Triplicate plastic pieces were sampled 12 times during each experiment with sterile forceps at each sampling time (i.e. D44, D46, D50, D53, D57, D60, D72, D79, D85, D99, D113, D127 for 2018 and D50, D57, D64, D72, D86, D93, D100, D106, D114, D120, D126, D133 for 2019). In parallel, 3 L of seawater were sampled in a sterilized glass vial at the same sampling days from the aquarium or at the SOLA marine observatory located at 0.5 nautical miles from the coast in the Banyuls Bay (42°29'300 N; 03°08'700 E).

Heterotrophic bacterial production

Heterotrophic bacterial production was measured in triplicate on each plastic and seawater sample at each sampling time by the 3 H-leucine incorporation into proteins method, as previously described in Dussud et al. (2018b). Briefly, the plastic pieces were transferred into 1.5 mL of sterilized seawater and a soft cell detachment pre-treatment was applied for each sample consisting in 3 cycles of 1 minute vortexing followed by 3 minutes in an ultrasound bath. This pre-treatment greatly improves the signal, especially in the case of mature biofilms (Dussud et al., 2018b). Immediately after cell-detachment, 3 H-leucine (specific activity 4.2 x 10 12 Bq.mmol -1 ; Perkin Elmer) was added at a final concentration of 0.9975 x 10 -9 mol.L -1

(completed with cold leucine to 1.49 x 10 -7 mol.L -1 ). A same 3 H-leucine concentration of 3,97

x 10 -9 mol.L -1 (completed with cold leucine to 3.6 x 10 -8 mol.L -1 ) was used for seawater from the aquarium and from the SOLA marine station. No cell detachment pretreatment was used for seawater samples, since Dussud et al. (2018b) showed that did not influence the signal for free-living bacteria in seawater. All samples were incubated in the dark at 18°C for 2-3 h. The theoretical conversion factor of 1.55 ngC.pmol -1 of incorporated leucine was used to calculate heterotrophic bacterial production [START_REF] Simon | Protein content and protein synthesis rates of planktonic marine bacteria[END_REF]. Data were normalized for microplastics with blank values according to Dussud et al. (2018b).

Primary production

Primary production (PP) was measured using a modified protocol of the radioactive 14 C tracer technique [START_REF] Fitzwater | Metal contamination and its effect on primary production measurements[END_REF]. Measurements were carried out in triplicate. One plastic piece was transferred in 10 mL of sterilized seawater. The same volume was taken for seawater from the aquarium and from the SOLA marine station. Each sample was inoculated with Na2H 14 CO3 (final activity of 18.5 kBq.mL -1 ). The introduced quantity was measured by mixing 100 µL ethanolamine to 100 µL of inoculated sample and 10 mL of scintillation cocktail Ultima Gold uLLT. The samples were placed for 5 to 8 h in a thermic and light regulated incubator (14 to 18°C -500 to 2000 µE.m -2 .s -1 according to the season). Plastics were isolated and acidified with 6 N HCl and dried during 8 h at 50 °C. Seawater samples were acidified with 6N HCl (final pH = 2) and agitated, lid open, for at least 12 h at 130 rpm and then mixed with scintillation cocktail during 4 hours before measuring radioactivity with the scintillation counter 300 SL Hidex.

The percentage of carbon originating from the extracellular release (ER) that is theoretically available for bacterial heterotrophic activity was calculated as previously described in Van Wambeke et al. (2002). Briefly, the theoretical ER associated to the autotrophic carbon fixation was calculated for a range of 5 and 20% of the PP to cover at least 90% of the wide range of all values found in the literature [START_REF] Conan | Relationship between Phytoplankton Efficiency and the Proportion of Bacterial Production to Primary Production in the Mediterranean Sea[END_REF]. It has been compared to the minimal and maximal theoretical bacterial carbon demand (BCD) ranging from 1 to 25% of the ER (review in [START_REF] Van Wambeke | Small-scale variability in the coupling/uncoupling of bacteria, phytoplankton and organic carbon fluxes along the continental margin of the Gulf of Lions, Northwestern Mediterranean Sea[END_REF]. Comparison between microbial activities on microplastic and in the seawater was calculated by integrating their respective dynamics over the 3 months sampling periods in 2018 and 2019.

Environmental parameters

Temperature, salinity, nutrients and chlorophyll a were measured in the aquarium with direct circulation to the sea and at the SOLA marine observatory station all along both 3-month incubations. Temperature, salinity and fluorescence were continuously recorded with a CTD probe (Seabird SBE16+) equipped with a fluorimeter (ECO FLNTU WETLab). All chemical measurements were carried out following standard procedures defined by SOMLIT protocols (www.somlit.fr/parametres-et-protocoles/). Briefly, ammonium was detected at nanomolar concentrations by fluorimetric detection according to [START_REF] Holmes | A simple and precise method for measuring ammonium in marine and freshwater ecosystems[END_REF] using a Turner Designs Trilogy fluorimeter. Nitrate, nitrite and phosphate concentrations were simultaneously measured in 10 mL of sample, on a continuous flow Autoanalyser III Seal-Bran&Luebbe [START_REF] Aminot | Dosage automatique des nutriments dans les eaux marines. Méthodes en flux continu[END_REF]. For Chlorophyll a, 250 mL samples were filtered using precombusted 25 mm diameter Whatman® GFF filters (~0.4-0.7 µm porosity). Concentrations were determined by fluorimetry [START_REF] Lorenzen | A method for the continuous measurement of in vivo chlorophyll concentration[END_REF] on a turner design Trilogy fluorimeter.

Statistical analysis

All statistical and graphical analyses as well as application condition checks were performed using StatEL software v3 for excel (www.adscience.fr).

Results

Representativeness of aquarium with circulating seawater compared to in situ conditions

We measured the temporal evolution of several environmental parameters in order to test that the experimental setup in the 50 L aquarium with continuous circulating seawater originating from the bay of Banyuls (flow rate of ~5.0 L per hour) was representative of the in situ conditions classically observed at the coastal observatory marine station (SOLA located in the Banyuls bay). We found significant correlations for all parameters (R>0.78, p<0.05) between our experimental aquaria and the natural seawater in both experiments performed in 2018 and 2019.

In particular, we compared the primary production (PP) and heterotrophic bacterial production (BP) in the aquarium with continuous seawater circulation and the measurements carried out with water sampled at SOLA station (Fig. 1). Analysis of the comparison points showed a linear regression slope of 0.98 ± 0.17 (p<0.00001; n=24, slope is not significantly different from 1 and the origin is not significantly different from 0; p<0.05). The same patterns were obtained with temperature, salinity, chlorophyll a and nutrients thus confirming the absence of an "aquarium effect" during our 3 month-experiments.

Biogeochemical conditions during 2 years experiments in a local temporal context

Our two experiments from 2018 and 2019 were done using the same 50-L aquarium with circulating seawater pumped in the Banyuls Bay, in the vicinity of the SOLA observatory station. Weekly measurements are made at SOLA since 1997 as part of the National Observation Service (SNO-SOMLIT). We compared the biogeochemical conditions of our experiments, i.e. from 13 February 2018 (D44) to 14 May 2019 (D133), to the nearly 24 years of observation at the SOLA station (see www.somlit.fr/mysomlit/). Very contrasted environmental conditions were observed between 2018 and 2019, with higher variations of temperature, salinity, nutrients and chlorophyll a concentrations during the first year (Fig. 2).

Indeed, when comparing the data of the 2 years with respect to the long-term average values represented by the continuous line (Fig. 2A and2B), the 2018 measurements are variable and distributed rather below the normals, while the 2019 measurements are more stable, close to the normals in temperature (Fig. 2A) but largely above for salinity (Fig 2B). This leads to important differences in terms of nutrients, particularly visible on nitrates (Fig. 2E), with 2018 being rich compared to 2019 marked by minimal nutrient concentrations.

Then, the spring conditions in our 2018 experiment were particularly cold between D30 and D90 (Fig. 2A) and marked by two episodes of low salinity water at D72 (36.4) and D105

to D115 (35.3 and even 33 for D156; Fig. 2B). These desalted episodes caused strong variations in nutrient concentrations during 2018, especially phosphate, marked by 2 peaks with nearly 0.1 µM (Fig. 2C). While ammonium concentrations were rather low (~0.1 µM) compared to "usual values", we still observed 2 peaks approaching 0.3 µM each (Fig. 2D). Nitrate concentrations were high throughout the period, with peaks above 3 µM (Fig. 2E). The abundance of nutrients in 2018 resulted in a particularly pronounced spring bloom with chlorophyll a maximal concentrations close to 3.5 and 4.1 mgCHL.m -3 for D72 and D107 respectively (Fig. 2F).

In 2019, the temperatures remained close to the climatologic curve during the experimental period, while the summer period was warmer (Fig. 2A). Throughout the experiment, salinity was high, with values varying around 38 but the time series was marked by 2 weak decreases for D72 (37.7) and D120 (37.5) (Fig. 2B). As a result, all nutrient concentrations were rather low. Phosphate concentrations were relatively stable at about 0.03 µM and marked by 2 maxima (0.05 µM for D86 and 0.11 µM for D114; Fig. 2C). Ammonium concentrations were close to the detection limit, but again, we observed 2 peaks (0.2 µM for D93 and 0.23 µM for D114; Fig. 2D). A similar pattern was observed for nitrate, with very low concentrations throughout the period, and 2 minor peaks at D64 and D100 (Fig. 2E). The low abundance of nutrients in 2019 resulted in a rather weak spring bloom with chlorophyll a maximal concentrations of 1.5 mgCHL.m -3 recorded between D115 and D125 (Fig. 2F).

Primary production and heterotrophic bacterial production in seawater and on microplastics

Significant differences were found when comparing the evolution of both primary and heterotrophic bacterial productions in seawater between the experiments conducted in 2018 and 2019 (Fig. 3A & 3B). In 2018, primary and heterotrophic bacterial productions presented a succession pattern over the 3 months spring period. Production rates varied from low activities between D40 and D80 (on average 5.9 ± 5.0 µgC.L -1 .h -1 for PP [Fig. 3A] and 3.3 ± 1.3 ngC.L - 1 .h -1 for heterotrophic bacterial production [Fig. 3B]) to a high peak observed at D100 for both activities (88.2 ± 6.1 µgC.L -1 .h -1 and 58.3 ± 6.4 ngC.L -1 .h -1 for PP and heterotrophic bacterial production, respectively) followed by a rapid decrease until the end of the incubation (~D130).

In 2019, the evolution of primary and heterotrophic bacterial productions were clearly different compared to 2018, with no synchronicity or succession observed between the 2 types of productions. Primary production varied over time with maximal values at the beginning of the experiment between D50 and D90 (maximum rate at 12.5 ± 0.1 µgC.L -1 .h -1 ) and then remained low and relatively stable until the end of incubation (<5.1 ± 0.1 µgC.L -1 .h -1 ; Fig. 3A). In contrast, heterotrophic bacterial production had a similar evolution to 2018, but with lower values and a low later peak (maximum for D120 with 2.5 ± 0.5 ngC.L -1 .h -1 ; Fig. 3B).

Years 2018 and 2019 showed sharp contrasted spring conditions, reflected in the distinct evolution of primary and heterotrophic bacterial productions in seawater. On the contrary, the productions measured on microplastics showed a surprisingly similar evolution during the two years (Fig. 3C &3D). Primary production increased rapidly after D60 to reach maximum activities of about 20 ± 3 µgC.dm -2 .h -1 (Fig. 3C) and remained variable (from 8 ± 1 to 23 ± 2 µgC.dm -2 .h -1 ) until D100 in 2018 and D115 in 2019 to finally drop rapidly until the end of the experiment. Heterotrophic bacterial production increased during about 80 days to reach a maximum of 86 ± 5 ngC.dm -2 .h -1 in 2018 at D99 and 78 ± 10 ngC.dm -2 .h -1 in 2019 at D114.

During both years, peaks were followed by a rapid decrease and a second phase of increase until the end of the incubation (Fig 3D ). (ngC.dm -2 .h -1 ) on microplastics (µPl). Whiskers represent standard deviations.

Comparison of microbial activities on microplastics and in seawater

We calculated the percentage of carbon originating from the extracellular release (ER) that was available for bacterial heterotrophic activity, based on the theoretical ranges of ER (from 5 to 20% of the PP) and the bacterial carbon demand (BCD) considering a Bacterial Growth Efficiency (BGE) of 10 to 30%. Indeed, BGE is the amount of new bacterial biomass produced per unit of organic C substrate assimilated. The wide ranges used in our calculation are sufficient to cover about all natural and mesocosm situations for phytoplanktonic excretion [START_REF] Baines | The production of dissolved organic matter by phytoplankton and its importance to bacteria: Patterns across marine and freshwater systems[END_REF][START_REF] López-Sandoval | Particulate and dissolved primary production by contrasting phytoplankton assemblages during mesocosm experiments in the Ría de Vigo (NW Spain)[END_REF][START_REF] Moran | Dissolved primary production and the strength of Phytoplankton -Bacterioplankton coupling in contrasting marine regions[END_REF], or BGE (Del Giorgio and [START_REF] Del Giorgio | Bacterial Growth Efficiency in Natural Aquatic Systems[END_REF]Lemee et al., 2002) and match also with the rates found in benthic communities [START_REF] Hubas | Role of the bacterial community in the annual benthic metabolism of two contrasted temperate intertidal sites (Roscoff Aber Bay, France)[END_REF].

During the spring sampling periods, we found that the free-living seawater bacteria might theoretically consume between 2 and 25% of the carbon excreted by autotrophs in the 2018 conditions and between 0.5 and 6% in the 2019 conditions (Tab. I). The situation was quite different on microplastics, where we found that between 14 to 167% of the extracellular release was needed to cover the BCD of the heterotrophic community in 2018 and between 8 to 101% in 2019.

Table I. Comparison between microbial activities on microplastics and in the surrounding

seawater. On top: integrated values of primary production and bacterial heterotrophic production (BP) in seawater (in mgC.L -1 ) and on microplastics (mgC.dm -2 ) over the 88 days of incubation. In the middle: theoretical extracellular release (range for 5 to 20% excretion, as mentioned by [START_REF] Conan | Relationship between Phytoplankton Efficiency and the Proportion of Bacterial Production to Primary Production in the Mediterranean Sea[END_REF] and theoretical bacterial carbon demand (with a BGE of 10 to 30% as mentioned by [START_REF] Van Wambeke | Small-scale variability in the coupling/uncoupling of bacteria, phytoplankton and organic carbon fluxes along the continental margin of the Gulf of Lions, Northwestern Mediterranean Sea[END_REF]. At the bottom: theoretical carbon from extracellular release available for (BP) bacterial heterotrophic activity (see text for more explanation).

Influence of microplastic microbial activities on the surrounding seawater

In the 2018 experiment, we compared measurements of total primary production in the seawater samples that were incubated with or without microplastics (Fig. 4). Primary production in seawater incubated in the presence of plastic is higher than when no plastic is present. The positive linear correlation between samples is estimated using a Spearman's test (p<0.005), and then the regression is calculated using Pearson's regression (PPseawater + Plastic = 2.73 x PPseawater + 3.59 ; R² = 0.95 ; p<0.00001). The intercept of the regression is not different from 0 (p<0.05), and so the increase in the rate of primary production of seawater in the presence of plastic is about 2.8 over the incubation period.

In order to clarify the origin of this significant increase, we separate the data into 2 subsets (Fig. 4), the first grouping the measurements during the development of the platisphere (D44 -D60), and the second including the data in the presence of the "mature platisphere" (D65 -D100). Finally, there was no significant difference between the PP of seawater incubated with

20

or without plastic during the first 26 days (PPseawater + Plastic = 0.97 x PPseawater -0.08 ; R² = 0.99 ; p<0.00001). The slope of the regression model was not different from 1 (Ancova after a Mann-Whitney test, p<0.05) and the intercept was not different from 0 (p<0.05). The situation was quite different for the second period (D65 -D100), when the platisphere was well developed (Fig. 4). In this case, the PP in seawater incubated with plastic increased by 2.5 times compared to PP in seawater that has not been in contact with plastic (PPseawater + Plastic = 2.49 x PPseawater + 12.36 ; R² = 0.95 ; p<0.00085). The slope of this regression model is significantly different from 1 (Ancova after a Mann-Whitney test, p<0.00067) and the intercept is significantly different from 0 (p<0.00085).

DISCUSSION

The impact of plastic pollution on biogeochemical cycles can no longer be considered limited today, since the amount of microplastics found at sea could soon cover more than 4.2 million km 2 of the sea surface [START_REF] Eriksen | Plastic Pollution in the World's Oceans: More than 5 Trillion Plastic Pieces Weighing over 250,000 Tons Afloat at Sea[END_REF][START_REF] Hidalgo-Ruz | Microplastics in the marine environment: a review of the methods used for identification and quantification[END_REF]. To date, most of the research dealing with the plastisphere has been focused on its biodiversity (including putative pathogens or harmful algae), with little documentation on how the plastisphere may influence the biogeochemical cycles [START_REF] Jacquin | Modeling Decreased Resilience of Shallow Lake Ecosystems toward Eutrophication due to Microplastic Ingestion across the Food Web[END_REF]. Here, we present the first data of PP coupled with heterotrophic bacterial production on microplastics. After a month of colonization of polyethylene microplastics in seawater, we observed that the mature biofilm presented high PP levels, up to 20 ± 3 µgC.dm -2 .h -1 compare to the beginning of the experiment. Consistent to our results, phototrophs such as diatoms and cyanobacteria have been commonly described in the plastisphere, at least on plastics that are exposed to sunlight (Amaral- [START_REF] Amaral-Zettler | Ecology of the plastisphere[END_REF][START_REF] Decelle | PhytoREF: a reference database of the plastidial 16S rRNA gene of photosynthetic eukaryotes with curated taxonomy[END_REF], confirming that microplastics floating at the ocean surface provide good substrates for the development of autotrophic microorganisms. For example, [START_REF] Crisafi | Bacterial biofilms on medical masks disposed in the marine environment: a hotspot of biological and functional diversity[END_REF] recently studied the growth of microbial biofilms on medical masks in a coastal area affected by different anthropogenic inputs. The authors observed that after one month in the field, the masks were colonized by a bacterial community significantly different from that detected in natural matrices in the same areas (seawater and sediments), and allowed the growth and enrichment of phototrophic microorganisms such as Oxyphotobacteria, Ruminococcaceae, Gracilibacteria, and Holophageae in particular.

Plastics in the environment are composed by a large variety of organic and inorganic constituents according to their surface properties and mediated by physical forces common to all molecular systems (i.e. van der Waals, electrostatic and others), the so-called eco-corona [START_REF] Monopoli | Biomolecular coronas provide the biological identity of nanosized materials[END_REF][START_REF] Ter Halle | Nanoplastics: A Complex, Polluting Terra Incognita[END_REF]. Together with light and CO2 availability, nutrients such as NO3 2-, NH4 + and PO4 3-are abundant in the biomolecular corona of microplastics [START_REF] Galloway | Interactions of microplastic debris throughout the marine ecosystem[END_REF], thus potentially fueling the intense PP observed in our experimental conditions.

Our results also showed a direct link between PP and heterotrophic bacterial production (BP) within the microplastic biofilms. BP reached 86 ± 5 ngC.dm -2 .h -1 , which was in the same order of magnitude than the few previous studies recording this parameter on microplastics in marine waters (Dussud et al., 2018b;[START_REF] Odobel | Bacterial Abundance, Diversity and Activity During Long-Term Colonization of Nonbiodegradable and Biodegradable Plastics in Seawater[END_REF]. Heterotrophic bacterial activities play a key role in different biogeochemical cycles such as carbon, since it is known that half of the oceanic primary production on average is re-mineralized by bacteria into the microbial loop [START_REF] Azam | The ecological role of water-column microbes in the sea Mar[END_REF][START_REF] Cole | Bacterial production in fresh and saltwater: a crosssystem overview[END_REF], thus transforming the particulate organic matter to dissolved organic matter [START_REF] Grossart | Comparison of cell-specific activity between free-living and attached bacteria using isolates and natural assemblages[END_REF]. Here, we propose for the first time that heterotrophic microbes have the ability to process considerable amounts of organic matter produced on microplastics by phytoplankton activities. Here, we propose for the first time that heterotrophic microbes have the capacity to process considerable amounts of organic matter produced on microplastics by phytoplankton activities. Indeed, photosynthates are known to be particularly labile and rapidly consumed by bacteria [START_REF] Conan | Relationship between Phytoplankton Efficiency and the Proportion of Bacterial Production to Primary Production in the Mediterranean Sea[END_REF][START_REF] Van Wambeke | Small-scale variability in the coupling/uncoupling of bacteria, phytoplankton and organic carbon fluxes along the continental margin of the Gulf of Lions, Northwestern Mediterranean Sea[END_REF]. The authors showed that the lability/quality of this material increased with its freshness (age) and with trophic conditions (i.e. material produced in oligotrophic environments is generally less labile than that of mesotrophic and eutrophic environments). The spatial proximity of the protagonists and their greater number within the plastisphere would be a factor in strengthening the trophic link, both in the autotrophic to heterotrophic direction and in the other direction. Interestingly, our results also indicate that PP can theoretically support most of the heterotrophic activities within the microplastic biofilm. If we assume that taking the extremes, the orders of magnitude cover most values encountered in natural environments and in biofilms, and considering the ranges of ER between 5 and 20% and BGE on labile material between 10 and 30% [START_REF] Baines | The production of dissolved organic matter by phytoplankton and its importance to bacteria: Patterns across marine and freshwater systems[END_REF][START_REF] Conan | Relationship between Phytoplankton Efficiency and the Proportion of Bacterial Production to Primary Production in the Mediterranean Sea[END_REF][START_REF] Del Giorgio | Bacterial Growth Efficiency in Natural Aquatic Systems[END_REF][START_REF] Hubas | Role of the bacterial community in the annual benthic metabolism of two contrasted temperate intertidal sites (Roscoff Aber Bay, France)[END_REF]Lemee et al., 2002;[START_REF] López-Sandoval | Particulate and dissolved primary production by contrasting phytoplankton assemblages during mesocosm experiments in the Ría de Vigo (NW Spain)[END_REF][START_REF] Moran | Dissolved primary production and the strength of Phytoplankton -Bacterioplankton coupling in contrasting marine regions[END_REF][START_REF] Van Wambeke | Small-scale variability in the coupling/uncoupling of bacteria, phytoplankton and organic carbon fluxes along the continental margin of the Gulf of Lions, Northwestern Mediterranean Sea[END_REF], we observed that the carbon requirement of free-living heterotrophic bacteria could be largely covered by PP for both years.

Similarly, ER might have theoretically covered most of the BCD in 2018 (min. 14% and max. 167%, with <100% corresponding to the BCD totally covered by ER) and in 2019 (from 8-101%). Thus, while in 2019 the BCD appeared to be fulfilled by autotrophic production on microplastics (i.e. when PP in surrounding seawater was low), it is possible that in 2018 an additional organic carbon source was required to supplement the needs of attached heterotrophic bacteria. This observation is consistent with the fact that in 2018, PP in the surrounding water was particularly strong and could logically have been a supplemental source. However, our results suggest that the use of such a complementary carbon source by the plastisphere and the depletion of carbon and nitrogen in the surrounding seawater would rather be an exception than a rule. This confirms that microplastic is an ecological niche in itself (Dussud et al., 2018b), where strong phytoplankton-bacteria interactions exist that control nutrient cycling and biomass production at the scale of microplastic pieces. Roughly, plastics appear to be comparatively less autotrophic than surrounding seawater during our experiments.

Another originality of our study was to take into account two contrasted environmental conditions during each of the 2 years. It is indeed very likely that biofilm composition and microbial interactions within plastic surfaces are in turn influenced by environmental perturbations [START_REF] Crisafi | Bacterial biofilms on medical masks disposed in the marine environment: a hotspot of biological and functional diversity[END_REF]. For example, the works of [START_REF] Allgaier | Coupling of heterotrophic bacteria to phytoplankton bloom development at different pCO2 levels: a mesocosm study[END_REF] in natural mesocosms were designed to test whether the predicted change in pCO2 will affect the communities of heterotrophic bacteria during a phytoplankton bloom development. The authors concluded that bacterial abundance and activities were similar among the various treatments but the community structure of free-living bacteria changed with pCO2 on the contrary of that of attached bacteria. These latter were tightly coupled to phytoplankton bloom development.

During spring 2018, we observed conditions favoring a classical phytoplankton bloom in seawater triggered by nutrient inputs from coastal water intrusions [START_REF] Olita | Observations of a phytoplankton spring bloom onset triggered by a density front in NW Mediterranean[END_REF]. Less favorable conditions were observed in spring 2019, where low nutrient concentrations limited the phytoplankton bloom development. Such irregular nutrient supply from year to year is a classical observation in the North Western Mediterranean sea [START_REF] Sánchez-Pérez | Mismatched dynamics of dissolved organic carbon and chromophoric dissolved organic matter in the coastal NW Mediterranean Sea[END_REF], that strongly structures microbial communities by promoting planktonic blooms and by stimulating the growth of certain microbes [START_REF] Céa | An annual survey of bacterial production, respiration and ectoenzyme activity in coastal NW Mediterranean waters: temperature and resource controls[END_REF][START_REF] Conan | Partitioning of organic production in marine plankton communities: The effects of inorganic nutrient ratios and community composition on new dissolved organic matter[END_REF][START_REF] Ghiglione | Spatial and temporal scales of variation in bacterioplankton community structure in the NW Mediterranean Sea[END_REF][START_REF] Lambert | Rhythmicity of coastal marine picoeukaryotes, bacteria and archaea despite irregular environmental perturbations[END_REF]. In contrast, we observed that microplastic biofilms were not influenced by the irregularity of the nutrient supply in the seawater. Indeed, primary and secondary productions were similar on microplastics over the two years, whereas a clear decrease in these activities was observed in spring 2019 in seawater, when the environmental conditions were less favorable as compared to spring 2018. It is classically admitted that sessile microorganisms forming a biofilm on natural or engineered materials have greater access to nutrients and any resources that accumulate and concentrate on the biofilms surface [START_REF] Salgar-Chaparro | Nutrient Level Determines Biofilm Characteristics and Subsequent Impact on Microbial Corrosion and Biocide Effectiveness[END_REF], which are classically diluted in the open ocean and less accessible for planktonic microorganisms [START_REF] Dang | Microbial Surface Colonization and Biofilm Development in Marine Environments[END_REF]. Again, these results strengthen the vision of microplastic biofilm as a niche in itself, with the presence of an eco-corona together with strong phytoplankton-bacteria interactions that will control the exchange and interaction with materials, minerals and other components of the surrounding seawater. We propose that biofilm development on microplastics is a possible strategy for microbial survival in the marine environment, especially in oligotrophic seawaters such as the Mediterranean Sea, which is one of the most microplastic-polluted areas in the world ocean [START_REF] Cózar | Plastic Accumulation in the Mediterranean Sea[END_REF]Dussud et al., 2018a).

Interactions between microplastic biofilms and surrounding seawater could be finally bidirectional, i.e. nutrient and organic matter inputs on microplastics support the phytoplankton development together with heterotrophic activities (see above discussion), and vice versa with the release of materials nutrients from the microplastic biofilm to the seawater. In our study, we also incubated seawater with and without microplastics in spring 2018 and we evidenced a 2-3 fold increase of PP when microplastics were present together with their biofilm. As mentioned earlier, a more thorough assessment of the spatial scale of the different processes reported is necessary. ER and the release of other molecules produced by the microplastic biofilm or even the breakdown of cells may explain the positive effect on PP in the surrounding waters (example with DOC; [START_REF] Romera-Castillo | Dissolved organic carbon leaching from plastics stimulates microbial activity in the ocean[END_REF]. Further experiments are obviously needed to investigate the molecular exchanges related to this positive effect. Comparing oxygen measurements with and without microplastics, a previous study in the North Pacific gyre measured a net autotrophic production of about 30-60 µgC.L -1 .h -1 for communities attached to microplastics harvested by manta net, while they found zero or even net heterotrophic bacterial production in the surrounding waters [START_REF] Bryant | Diversity and Activity of Communities Inhabiting Plastic Debris in the North Pacific Gyre[END_REF]. These authors proposed that microplastic particles create net autotrophic 'hot spots' in the oligotrophic ocean. However, such comparison is difficult because of the method used, as oxygen measurements in their experiment integrated the activities of the microplastic biofilms and also of planktonic organisms surrounding the microplastics during incubation. The use of radioactive carbon incorporation removes this ambiguity and confirms the clear influence of the microplastic biofilm on the increased productivity in the surrounding seawater, despite the contrasted environmental conditions encountered in spring 2018 and 2019. Moreover, it is actually the presence of the platisphere that is at the origin of the increase in production since it is necessary to wait for the installation of the biofilm to observe the stimulation of 2 to 3 times of the PP in the surrounding sea water. Our results are nevertheless limited by the size of the microplastic pieces incubated in 10 mL of seawater. Further studies are needed to evaluate the volume of seawater that could be impacted around the microplastic particles in more realistic conditions.

Interestingly, a previous study estimated that globally up to 23,600 metric tons of dissolved organic compounds (from truly dissolved substances to any polymeric nanoplastic that might have been also fragmented from the plastic surface) are released from the estimated 35,000 metric tons of all the microplastics floating in the ocean [START_REF] Romera-Castillo | Dissolved organic carbon leaching from plastics stimulates microbial activity in the ocean[END_REF].

These numbers consider the release of compounds from the microplastic itself, but do not take into account the biofilm covering its surface. Finally, a funny and illustrative image of the role of plastics in the environment could be that of an electrical inverter in a circuit that allows the attenuation of both positive and negative variations. A better estimation of the carbon fluxes generated by microplastic biofilm exudates may help to refine the balance between direct microplastics leaching and the contribution of their associated biofilm.

CONCLUSION

Our study provides an essential step in understanding the importance of the coupling between phytoplankton autotrophy and bacterial heterotrophic production on microplastics and its relationship with surrounding seawater. The plastic waste quantity entering the oceans is predicted to increase by up to one order of magnitude by 2025 [START_REF] Gewert | Pathways for degradation of plastic polymers floating in the marine environment[END_REF], with potential major consequences for marine microbes and the biogeochemical cycles in the ocean.

Further studies considering the influence of microplastic features and water environmental characteristics are needed. This will enable the development of predicting models for the impact of biofilm activities on the carbon cycle in marine ecosystems.

Figure 1 :

 1 Figure 1: Comparison of phytoplankton primary production (PP = ) and heterotrophic

Figure 2 :

 2 Figure 2: Seasonal evolution at SOLA observatory station for 2018 (D44-127, open dots) and

Figure 3 :

 3 Figure 3: Temporal evolution during the incubation time in 2018 (D44-127, open dots and

Figure 4 :

 4 Figure 4: Comparison of primary production (PP) in seawater without (SW) or with

Acknowledgment: This project was supported by the MITI-CNRS "Défi Plastiques et microplastiques en milieux aquatiques" project BIOGEOPLAST, by the European project JRA-ASSEMBLE+ and the Tara Ocean Foundation project MICROPLASTIC 2019. We are grateful to the sailors of RV Néreis II, to the SNO SOMLIT (Service d'Observation en Milieu Littoral ; www.somlit.fr), to Nolwenn Callac for assistance during experiments and to Guigui PA, VF and JPJS, JJV for insightful comments on the manuscript.


Durix Bastien
email: bastien.durix@irit.fr

Leonard Kathryn
email: kleonard.ci@gmail.com

Morin Géraldine
email: geraldine.morin@irit.fr

Chambon Sylvie
email: sylvie.chambon@irit.fr

Durix et al. / A clean, robust 3D medial axis 1 A clean, robust 3D medial axis

Keywords: Skeleton, structure, mesh, delaunay, voronoï Voronoi

Computing the medial axis of a 3D surface mesh is challenging. Points on the discrete medial axis can be defined as interior Voronoï vertices of the surface mesh, but the resulting medial structure rarely has clean connectivity and consistent geometry. In this paper, we provide a medial axis computation based on the Voronoï diagram able to generate manifold medial sheets with coherent topology and geometry, generating consistent geometric structures similar to those in the continuous setting. Because of the correspondences between the surface mesh and resulting medial mesh, we also provide an efficient method for separating the shape into coherent regions associated to medial structures. This correspondence allows for a medial-axis-based filtration of surface structures to generate a Hausdorff ε-approximation of the surface points based on a simplified medial axis, thereby providing a robust medial representation with guaranteed surface approximation.

Introduction

Medial representations are quite popular in computer graphics because they combine implicit and explicit shape representations, and offer a lower dimensional model of a 3D shape, the skeleton, that captures the geometric information from the boundary, while a radius or distance function describes the interior of the shape. The first such model, the Blum medial axis, was introduced in [Blu73] for 2D shapes. It consists of a skeleton and a radius function defined at each point of the skeleton that gives the distance of the skeleton point to the boundary. A useful discretization of the Blum medial axis relies on a shape's Delaunay triangulation and its dual, the Voronoi diagram.

As described in [TDS * 16], several approaches exist to compute a full skeleton. Voxel-based methods [HR08, ADBS11] which are a 3D version of classical 2D thinning methods, easily maintain the structure of the shape, and can be less subject to noise. In [JST15], the voxel grid is the support of a mass transport process, leading to a multiscale, simplified skeleton. In [START_REF] Yan | Voxel cores: Efficient, robust, and provably good approximation of 3d medial axes[END_REF], the authors use a Delaunay tetrahedralization of the open voxels of a shape boundary to compute a voxel core that converges to the true medial axis as the voxelization of the shape becomes dense. A separate pruning step then removes noisy elements in the voxel core with a guarantee on the approximation and the homotopy equivalence. In these approaches, the shape must first be converted to a voxel grid, and its sampling density is directly linked to the skeleton accuracy, thereby introducing an additional parameter to the model. Also, in the voxel based approaches, no global and topologically coher-ent structure of the skeleton is available. Other methods compute through the scale axis transform that takes into account a scaling of medial balls, but the approach results in some artifacts that require a separate pruning step. This method has generated multiple variants and improvements [LWS * 15, PWG * 19] that produce a simplification of a Voronoi skeleton. However, these methods do not maintain a robust link to the 3D mesh. In contrast, our goal in the current work is to be independent of a pruning step. We both maintain global connectivity consistency in the structure of the medial axis and geometric consistency with the surface mesh, in the sense that the Delaunay tetradhedralization of the output surface mesh is dual to the output Voronoi skeleton.

Many researchers compute a curve skeleton, which is a network of curves, instead of the full skeleton in 3D, [CSM07, SYJT13].

A curve skeleton is appealing because it addresses the previously mentioned instability while capturing most of the major structures of the shape. It can be estimated by mesh contraction [ATC * 08, LW18, ZLZ20], or through the computation of a 3D skeleton [DS06, TAOZ12], or directly from the mesh, using tools such as local separators [BR21]. Curve skeletons necessarily maintain only a weak relationship to the surface mesh, however, and there are some shape structures (funnel shapes, for example) that are not well-suited for description by a single curve.

Depending on particular goals and applications for a skeletal model, different properties of the skeleton may be desired, such as continuity, connectivity, geometric correspondence with the boundary, or a particular level of simplicity. These variations in goals have produced variations in skeletonization algorithms that make different trade-offs, as the work cited in the previous paragraphs demonstrates. In this article, we take as input a Delaunay surface mesh of a shape and compute a medial axis that has a full array of desirable properties. This approach is similar to a 2D skeletonization method by propagation [DCL * 19, DMC * 19], where the skeleton is computed step by step, and where a Hausdorff distance criterion identifies noise so that insignificant parts of the medial axis are not computed. The next section describes this simplified skeletonization.

Simplified skeletonization

In this paper, we present a method to compute a simplified, robust medial axis for a 3D shape from a Delaunay input mesh representing the shape surface. Our approach is based on a propagation of medial balls, or equivalently, Delaunay tetrahedra, which means we do not require a pruning step. A Delaunay tetrahedron is a tetrahedron such that its circumscribed sphere does not contain any other point from the surface mesh. Therefore, each skeletal point is associated to a tetrahedron, and the edges between skeletal points are associated to tetrahedra that share faces. Finally, a Delaunay mesh is a mesh for which each triangle is a triangle from a Delaunay tetrahedron. Local geometry is combined into a global structure on the skeleton, which is then used for the simplification analysis. This In this section, we recall the structure of the medial axis in the 156 smooth setting where the shape surface is differentiable, in order 157 to provide an analogy with the discrete setting we are interested in. In the case of the medial axis for the surface of a 3D shape, such 175 as in Figure 2, each stratum is either a 2D medial sheet (in light blue 176 in the figure), a 1D medial curve (in red -generically, these will be Figure 2: A skeleton structure in the continuous setting: a continuous shape and its skeleton containing three medial sheets (and corresponding strata). Regular skeleton points (in light blue) lie in the interior of a skeleton sheet. They are bounded by singular boundary points (in blue), and intersect at interior singular curves (in black). The medial axis also contains two non-generic medial strata of lower dimension, curves (in red).

2. Singular interior points where two strata intersect, in black.

190

For singular interior points, the medial spheres will be at least point on the other side ("above" and "below"), see left, but other

V 1 V 2 V 3 V 4 N V 1 V 2 V 3 V 4 N (a) Regular nodes V 1 V 1 V 3 V 4 N (b) Singular node V 1 V 2 V 3 V 4 N (c) Boundary node
Figure 3: A sampling of some different (non-exhaustive) configurations for a discrete medial node N, in blue, corresponding to different point type classifications of N. The vertices (V i ) i=1...4 are its corresponding boundary points and tetrahedron. Among the tetrahedral edges, the green edges belong to the surface mesh and the red dashed ones traverse the volume inside the 3D shape. Note that the node N is not necessarily inside the tetrahedron.

configurations are possible such as two points above and two where each oriented sheet corresponds to a different side of the sur-279 face mesh, "above" or "below", see Figures 4a and4b). The upper surface region, however, is interrupted by region associated to the smaller medial sheet that is connected to one side of the main sheet. The singular path separating the two sheets corresponds to the black dashed path on the surface mesh and the resulting dark gray region corresponds to the surface that is above the main sheet. (c) If a medial boundary path is not a closed loop, the surface path will also not be a loop, as can be seen in the surface path generated by the medial boundary of the smaller sheet. If a medial boundary path is a loop, the corresponding path on the surface mesh will also be a loop, as can be seen in the surface path generated by boundary of the main medial sheet. (d) For medial singular paths, in contrast, a medial path that is an interval can correspond to a surface path that is a loop, as can be seen in the path on the surface mesh that corresponds to the closed black medial singular path.

separating two sheets to a closed path on the shape surface that These primal elements have corresponding dual elements in V :

• tetrahedra correspond to Voronoi vertices which are centers of the circumspheres of the tetrahedra;

• tetrahedral faces correspond to Voronoi edges;

• tetrahedral edges correspond to Voronoi faces. Note that a medial face, and more generally a Voronoi face, is a planar polygon since it lies on the medial plane of the dual edge and can have an arbitrary number of nodes.

• medial vertices are a subset of Voronoi vertices and we denote them as nodes;

• medial edges are a subset of Voronoi edges;

• medial faces are a subset of Voronoi faces.

We construct our simplified medial axis by exploring the Voronoi Naturally, an open Voronoi face contains at least one open edge.

These edges do not belong to the medial mesh.

Oriented medial edges and faces

As discussed in Section 3.2, medial sheets comprising the skeleton have two sides; each side corresponds to a region of the shape surface that is either "above" or "below." This two-sidedness also will result in a path that lies on a single side of a sheet.

416

We distinguish between the edges that are above and those be- and edges within a (non-oriented) sheet. Recall that the degree of an 442 edge is defined by the number of medial faces containing the edge.

443

A singular edge is then an edge of degree 3 or more, a boundary 444 edge is an edge of degree 1, and a regular edge is an edge of degree 445 2. These edges may belong to multiple sheets.

446

We now consider the half edges and how they compose paths 447 on a sheet (cf. Figure 8). Since each half edge is associated to a 448 half medial face, each of them will be associated to a unique sheet. Once a sheet has been computed, its singular edges are used to 497 detect the singular paths that separate the current sheet from its Figure 9: A (black) singular path on the skeleton associated to a (black) surface mesh path, whose vertices are the centers of the five Voronoi cells the path is crossing (as each medial face is separating two Voronoi cells). The computed surface path surrounds a surface portion for which we can evaluate an ε-significance.

neighbor sheets. Suppose we are given a tolerance parameter ε. We only wish to compute those neighbor sheets that correspond to a surface region, as described in 3.4.1, that has significance greater than ε, where significance is measured by Hausdorff distance to the original surface.

Consider a singular path contained within a computed sheet. The singular path corresponds both to a set of medial spheres centered at its nodes, and also to the corresponding surface region for the neighbor sheet whose significance we wish to determine. If the Hausdorff distance between the vertices of the surface region and the medial spheres is greater than ε, the neighbor sheet is computed.

Otherwise, the sheet may be ignored: this means that the surface region is entirely covered by existing spheres of the skeleton, with a radius increased by ε (cf. Figure 10). The algorithm proceeds through all sheets in the medial axis of 523 the input mesh, computing those with significance above the thresh-524 old ε. We summmarize the pipeline here, then detail each step in the paragraphs that follow. To begin, we compute a first sheet, starting from a first medial face (cf. 4.2) and aggregating neighboring medial faces into a complete sheet (cf. 4.3). Each singular medial edge is attached to a new sheet. We compute the singular path connecting this sheet (cf. 4.4) and determine if the significance if the sheet is greater than ε (cf. 3.5). If not, we discard the skeleton propagation to the insignificant sheet. If the sheet is relevant, we allow the propagation to compute the sheet. Repeating this operation over each half singular path linking a sheet will produce the full skeleton of ε-significant sheets. We then simplify the skeleton boundaries according to the same ε criterion, removing medial faces when possible (cf. 4.6). This step highly simplifies the resulting skeleton, removing small peaks along sheet boundaries. Finally, we ensure the sheet is manifold, removing problematic paths that create degenerate geometry (cf. 4.7).

Input mesh

The input of this algorithm is a Delaunay triangular mesh bounding a 3D shape. Each triangle of such a mesh is on a Delaunay tetrahedron: thus a Delaunay tetrahedron is either inside or outside the shape. As a consequence, we know which portion of the skeleton is inside, and which is outside, offering a clear labeling of outside and inside skeleton. Furthermore, because the mesh is Delaunay, a direct connection exists between the connectivity of the skeleton and the connectivity of the surface mesh, and between the topology of the skeleton and the topology of the surface mesh.

A Delaunay triangular mesh can be built from a given manifold surface mesh (cf. [DZM07]), using edges flipping and splitting. The vertices of the original mesh are preserved in the Delaunay mesh, and the skeletal structure we compute can still be applied to the non-Delaunay version of a mesh, though the duality will not be preserved. We also note that, since the atomic operations applied to convert a mesh into a Delaunay mesh do not affect the topology of the mesh, topology is preserved between original mesh and Delaunay mesh, and therefore maintains consistency with the skeleton.

Computing the first medial face

The algorithm starts with an initial medial face of the skeleton. We first choose a node of the skeleton. This node will likely belong to multiple medial faces, and we select one.

To compute the first node on the skeleton, we need to find an internal, or volumetric, tetrahedron. We select an arbitrary triangle of the surface mesh. As the mesh itself is a subset of the Delaunay tetrahedralization, this triangle is necessarily a face of a tetrahedron. Once this triangle is chosen, two tetrahedra are candidates to be our first tetrahedron: one interior and one exterior to the shape volume. Using the orientation of the mesh surface, we can determine which one is interior, which gives us a first interior medial node. Finally, to avoid choosing a first medial node that is not significant, we detect the neighboring node with a maximal radius (as each interior tetrahedron corresponds to a medial sphere), and iterate this operation so that that chosen initial node has a locally maximal radius.

Finally, the first medial face is randomly chosen from the medial faces surrounding the chosen node. More precisely, as an medial 578 face is the dual of a volumetric edge, we choose one volumetric 579 edge among the edges of the chosen interior tetrahedron which de-580 termines an medial face. Find pe f , next half edge on same medial face than pe Find peo, opposite half edge of pe f , on the neighbor half medial face Find pe , next half edge on same medial face than peo while pe is not singular;

The result of this operation is sequence of half edges, called a 608 singular path. A given singular path can be tested to identify in-609 significant sheets, according to ε parameter. we do not compute that sheet. In order to maintain the geometric 614 relationship between the skeleton and the mesh, we also update 615 the connectivity of the mesh when we do not compute a sheet (cf. From a to d, we first take the next edge on the same half medial face, b, then take the opposite edge to b, c. Finally, we take the next edge to c, which is the singular edge d. More generally, to go from d to i, the same steps are repeated on each medial face until a singular half edge is found. Here, it is done twice: through half edges e, f and g, h. At this point, the singular path is composed by a, d and i. During this process, each visited half medial face belongs to a unique Voronoi cell with associated dual (green) boundary point. Each new point is stored as the next vertex of the path. When the path closes, the result is a sequence of (green) vertices creating a closed path on the boundary. The main step in not computing a sheet is removing any interior tetrahedra from the object volume. This implies removing some vertices, namely vertices strictly within the region bounded by the corresponding surface path, and remeshing this portion of the surface, creating a new set of edges and faces to fill the hole.

Simplifying boundaries

Boundary paths for sheets are prone to noise. Using the same ε significance measure, we simplify sheet boundaries by removing some medial faces. We define the notion of steepness of a consecutive pair of half boundary edges to be the angle between them. At each step of the simplification (cf. Algorithm 1), we check the steepest consecutive half edges along the boundaries. We then compute the singular path separating those medial faces from the skeleton, and use it to check whether the sheet beyond the path can be removed using the ε criterion. If so, we remove the medial faces using same method as in 4.5.

Handling problematic paths

A computed sheet may contain some non-manifold edges. This happens when skeletal edges are connected to three medial faces that are labeled as belonging to the same sheet, violating the geometry of regular edges contained in a sheet interior. We first label these edges problematic edges, that is, any edge belonging to three different medial faces but labeled as members of the same sheet (see Figure 13). To resolve the problematic edges, we wish to create We require manifold meshes, and so all non-manifold mesh are dis-659 carded. Moreover, our algorithm assumes that we have Delaunay In Figure 18, we compare our approach to Voxel Core [START_REF] Yan | Voxel cores: Efficient, robust, and provably good approximation of 3d medial axes[END_REF] Figure 14: Skeletons computed with our algorithm, with ε = 0.01. To highlight the link between skeletal sheets and mesh portions, we use the same color for a skeletal sheet (right) and its associated mesh portion (left). In Figure 15, we show skeletons of the same object computed 688 with different values of ε. As ε increases from 0.00, to 0.01, and to 0.05, successive sheets are removed, but we can see a larger difference between 0.00 and 0.01 than between 0.01 and 0.05. This may suggest that each input shape has an intrinsically appropriate ε value.

Figure 16 shows skeleton computed with ε = 0. These skeletons have many spurious peaks.

Figure 17 shows effects of noise in the input surface mesh. We observe that even if more skeletal peaks are present on the thin parts, the global skeleton stays clean.

Conclusion

In this article, we present a 3D skeletonization approach to produce a simplified and geometrically consistent medial axis. Our algorithm identifies the primary structures in the medial skeleton, which are medial sheets and curves, and their boundaries, which are singular or boundary structures. These structures are topologically valid, manifold, and cleanly separated, thereby maintaining the Whitney stratified structure of the Blum medial axis in the discrete setting. Furthermore, we maintain the relationship between the surface mesh and the medial axis by associating nodes, edges, and medial faces of the skeleton with tetrahedra, triangles and edges of the Delaunay structure generated by the surface mesh points.

Using these medial structures, we compute a simplified skeleton, excluding insignificant sheets. Using the geometric links between the mesh and the skeleton, we evaluate the relevance of a skeletal sheet by evaluating the deviation from the surface mesh that would result from removing the surface region associated to that sheet. As a result, we do not compute insignificant sheets for which the Hausdorff distance of the points in the associated region of the surface mesh to the remaining surface points is lower than an ε threshold. • As shown on Figure 13, some edges can be problematic, as they Figure 19: Some skeletons computed by our method (ε = 0.01).

29a

  skeleton from boundary samples [RAV * 19, MBC12, LBKP21]. 30 Each boundary sample is characterized by a position and a nor-31 mal, and no additional structure is needed. A set of tangent spheres 32 can then be computed, filtering any non-descriptive ones. Methods 33 based on deep learning have also been proposed. In [YYW * 20], a 34 simplified skeleton is computed and accurately describes the shape 35 (depending on the chosen precision) but they do not capture the 36 shape structure, as the resulting skeleton is typically not a graph. 37 In [CD23] the local structure of the skeleton is preserved in gen-38 eral, but there is no guarantee on the coherence of the connectivity, 39 that is, a tetrahedron may belong to the resulting skeleton. In fact, 40 by construction deep learning methods do not maintain a clear re-41 lationship to the shape boundary. In [RLS * 21], the shape and its 42 skeleton are linked but the representation is implicit, giving neither 43 connectivity nor structure. 44 Other approaches, including ours, are based on the computation 45 of a Voronoi diagram. For simple shapes such as polyhedra, the are based on pruning [DZ03, AL01] prune the Voronoi diagram to obtain a sensible medial structure. [YSC * 16] compute a significance measure: the authors demonstrate that a global measure related to Erosion Thickness, generalized from 2D to 3D, is more efficient than a local measure. The well-known Scale Axis Transform [MGP10] has the advantage of taking into account multiple medial representations. They propose a multi-scale representation

113•••

  global structure consists of sheets, which are surface portions of the 114 skeleton, boundary paths, which are curve boundaries of a single 115 sheet, and singular paths, which are curve portions separating dif-116 ferent sheets (See Section 3 for details). As a result, our approach 117 guarantees the following properties (cf. Figure1): 118 The topological and geometric relationships between the sur-119 face and medial meshes are preserved. Specifically, (a) the re-120 sulting medial mesh is homotopically equivalent to the output 121 surface mesh by construction, and (b) the points on the medial 122 mesh are Voronoi duals of Delaunay tetrahedra generated by the 123 surface mesh. Each vertex, edge, face and sheet of the medial 124 axis corresponds to a well-specified set of vertices, edges, faces, 125 or region of the surface mesh. This means that we maintain cor-126 respondence between the surface and medial meshes. 127 The geometric structures of the medial mesh are clean and 128 coherent. The skeleton is divided into sub-structures of sheets 129 with topologically consistent manifold interiors, which are sep-130 arated and bounded by manifold paths. The skeleton mesh itself 131 is connected for an input shape with a single component due to 132 the homotopy equivalence. 133 The simplified medial axis provides a guaranteed 134 ε-approximation to the original surface points. The simplified 135 medial mesh generated for some choice of ε > 0 corresponds to a 136 subset of the original surface points that are ε-close in Hausdorff 137 distance to the original surface points. All topological features at 138 a scale larger than ε are maintained, though features at a scale 139 smaller than ε may be deleted. 140 In what follows, we first describe the continuous and discrete 141 constructions of the interior Blum medial axis (BMA) in Section 2. 142 In Section 3, we describe the global structures of the discrete BMA 143 and describe how these are related to local structures. Finally, we 144 describe the algorithm for generating the simplified skeleton based 145 on the ε error parameter in Section 4, and present results in Section 146 5. We also include a brief appendix to explain some special cases 147 that arise in this discrete setting, and how we handle them.148 2. Geometry of the interior medial axis 149 The Blum medial axis is structured into parts that offers strong 150 geometric properties in the smooth setting; we seek and provide 151 analogs in the discrete setting. In this section, we describe these 152 properties and the correspondences between smooth and discrete 153 representations.

158

  Given a closed, non-self-intersecting surface S ⊂ R 3 with con-159 tinuous curvature and a finite set of curvature extrema and ridge 160 curves, the interior Blum medial axis can be defined as the locus of 161 centers of maximally inscribed spheres within the surface, together 162 with their radii. This produces a skeleton composed of sphere cen-163 ters that is a Whitney stratified set M with a finite number of strata.

  Figure 1: Global structures of the skeleton (a), and their consistency with the surface mesh (b). The color coding in the figures shows which global skeletal structures in (a) correspond to which global shape structures in (b).

  174

  191 tri-tangent to S, corresponding to the merging of bitangencies 192 on at least two strata meeting along the singularity.

193 3 .

 3 Singular boundary points where the inscribing sphere oscu-194 lates at an extremum of at least one principal curvature, in blue. 195 For singular boundary points, the medial sphere has only one 196 tangency, the point of extremal curvature, corresponding to the 197 merging of at least one bitangency where the points "above" and 198 "below" merge into a single point.

  of our paper is to construct well-defined dis-201 crete versions of the medial structures described for differentiable 202 shape surfaces in the previous section. In applications, the smooth 203 surface S described above is typically replaced by a sampling of its 204 points and connectivity between those points, a surface mesh con-205 sisting of vertices joined by edges and faces. Similarly, the smooth 206 strata of the medial skeleton are replaced by a medial mesh. In what 207 follows, we assume sampled points of a 3D shape surface are ver-208 tices in a Delaunay surface mesh S, with an induced medial mesh 209 M. Points on M can then be computed as interior Voronoi vertices 210 generated by the surface points. Because the Voronoi vertices will 211 be centers of the circumspheres of its dual, the Delaunay tetrahe-212 dralization of the volume enclosed by S, the tetrahedra will provide 213 a discrete analogue of medial balls in the smooth case. Similarly, 214 distance from the Voronoi vertices to the surface points will pro-215 vide a discrete version of the radius function r. As the sampling 216 on the surface becomes dense, converging to the smooth surface, 217 a subset of this discrete medial skeleton converges to the smooth 218 skeleton [DZ03]. 219 With this characterization, much of the structure from the dif-220 ferentiable setting carries over. The resulting discrete M will have 221 three types of points, see Figure 3 for illustration: 222 1. Regular points, in (3a). For regular points, each point of M 223 will typically correspond to a tetrahedron that crosses the in-224 terior of S, with three points of S on one side of M and one 225

  227below [CDE * 00], see right.

228 2 .

 2 Singular interior points, in (3b). For singular interior points, 229 the tetrahedron will correspond to the convergence of tetrahedra 230 of regular points on at least two strata meeting along the singu-231 larity. In this case, there will be at least two sets of points above 232 and below corresponding to at least three disconnected regions 233 of the surface. 234 3. Singular boundary points, but without a notion of tangency, 235 in (3c). For singular boundary points, the points in S composing 236 the tetrahedron will be adjacent and belong to a single connected 237 component, with the two "sides" of M having converged. 238 This framework gives the foundation for the structures we use to 239 generate the strata of the discrete skeleton. Unfortunately, guaran-240 tees of topological and geometric coherence from the smooth set-241 ting do not automatically carry over, as rounding errors and other 242 issues often lead to medial skeletons with more complicated struc-243 tures [TDS * 16]. Our algorithm addresses these issues to produce a 244 geometrically coherent skeleton. 245 3. Hierarchical structure of the discrete skeleton 246 In what follows, we distinguish between local medial structures 247 (e.g., medial points and neighbors, corresponding surface points 248 and neighbors) and global medial structures (e.g., medial sheets, 249 boundaries, singular sets). Our algorithm analyzes local structures 250 to build coherent global structures. 251 The constructions presented in this section are necessary for un-252 derstanding the algorithm in Section 4. We define the global medial 253 structures in Section 3.5, which allow us to segment the surface 254 correspondingly in Section 3.2. In section 3.3, we present the local 255 medial structures, derived from the Voronoi and Delaunay duality 256 linking the surface mesh to the medial mesh, that provide the basis 257 for the global structures and also for the simplification of the me-258 dial skeleton. Section 3.4 describes how the global structures are 259 obtained from the local structures. Finally, section 3.5 describes 260 how we compute the simplified 3D medial axis. 261 3.1. Global structures of the discrete skeleton 262 Medial sheets are typically two-dimensional for a 3D shape, but 263 can also be one-dimensional. Sheet boundaries and singular paths 264 are typically one-dimensional, but can also be zero-dimensional. 265 For simplicity, we present our algorithm for two-dimensional case, 266 but the process is analogous in the one-dimensional or zero-267 dimensional cases. 268 Sheets, the top-dimensional strata described in Section 2, are the 269 fundamental building blocks of the skeleton. We define a discrete 270 sheet of the medial axis to be a connected set of regular medial 271 points for which the corresponding regions on the surface mesh 272 have two connected components, together with any neighboring 273 singular points (corresponding to more than two connected com-274 ponents) or boundary points (corresponding to one connected com-275 ponent) (see Figure 3). Each sheet corresponds to one or more con-276 nected portions of the input surface mesh. Furthermore, each sheet 277 has two sides, and can be therefore viewed as two oriented sheets, 278

280

  Figure 4d, where two medial sheets intersect, or both. The bound-285

Figure 4 :

 4 Figure 4: Correspondences between skeletal structures and surface regions. Solid line boundary paths (darker blue) and singular paths (black) represent paths on the mesh. Dashed paths represent the corresponding paths on the surface mesh. (a) The medial boundary path of the main (horizontal) sheet separates the surface into two regions, above and below, with the lower region shown in darker gray. (b)The upper surface region, however, is interrupted by region associated to the smaller medial sheet that is connected to one side of the main sheet. The singular path separating the two sheets corresponds to the black dashed path on the surface mesh and the resulting dark gray region corresponds to the surface that is above the main sheet. (c) If a medial boundary path is not a closed loop, the surface path will also not be a loop, as can be seen in the surface path generated by the medial boundary of the smaller sheet. If a medial boundary path is a loop, the corresponding path on the surface mesh will also be a loop, as can be seen in the surface path generated by boundary of the main medial sheet. (d) For medial singular paths, in contrast, a medial path that is an interval can correspond to a surface path that is a loop, as can be seen in the path on the surface mesh that corresponds to the closed black medial singular path.

  Figure 4, consider the upper part of the object where there is a 305

  315 3.3. Local structures of the discrete skeleton 316 The global structures described in the previous section can be 317 formed by aggregation of the appropriate local structures. Philo-318 sophically, the aggregation is straightforward -join connected sets 319 of points that are regular, or singular, or boundary -but many lo-320 cal degeneracies arise in practice (some of which are collected in 321 Appendix A) that make the process challenging. We now define the 322 local structures that will allow us to generate global structures. 323 3.3.1. Voronoi and Delaunay duality between the skeleton and 324 the surface mesh 325 Based on a Delaunay triangulation of a surface mesh S, we con-326 struct the Delaunay tetrahedralization T of the volume of the inte-327 rior of S. T is the dual of the Voronoi diagram V of the vertices of 328 S. Each of S, M, T , and V have vertices, edges, and faces, and T 329 also has tetrahedra. 330 All vertices of T are vertices of the surface mesh. Note that while 331 all edges and faces of S belong to the Delaunay tetrahedralization 332 T , only a subset of edges and faces of T belong to S. Because of 333 the number of types of collections of vertices, edges, and faces, 334 we specify vocabulary and color-code mesh elements consistently 335 in the text and figures to distinguish between them. All vertices, 336 edges, faces (triangles in the general setting), and volumes (tetra-337 hedra in the general setting) belonging to the Delaunay tetrahedral-338 ization are colored in red. We identify the edges and faces that cross 339 the volume of the interior of S by appending the word volumetric. 340 The faces of the Delaunay tetrahedralization that are also faces of 341 the surface S are colored in green and carry the word surface. 342 Similarly for V , almost all vertices interior to S belong to the me-343 dial mesh M [DZ03], while only a subset of the edges and faces 344 of V are contained in M. The elements in the Voronoi diagram V 345 are colored in dark blue except for the subset of the elements that 346 belong to the medial axis M, which are lighter blue. We refer to 347 the faces of V as Voronoi faces to emphasize that they are not typi-348 cally triangular. Identifying the faces, edges, and vertices of V that 349 are completely inside the volume bounded by S defines the interior 350 discrete medial axis that is our primary object of study. Note that 351 there are also Voronoi vertices, edges, and faces that are completely 352 outside the volume bounded by S. These define an exterior medial 353 skeleton that we do not study here.

  354

Figure 5 :

 5 Figure 5: Difference between volumetric and surface tetrahedral components. Surface edges and faces (green) lie on the mesh of the object, while volumetric edges and faces (red) cross the volume of the object. The shapes of the drawn tetrahedra refer to Figure 3.

  diagram to determine which elements to include in the medial axis, and then computing the medial axis components according to their importance to the overall shape as measured by the Hausdorff distance between the original corresponding surface points and the simplified version. The process requires both local and global considerations of both primal and dual components. Around a given medial edge, there can be as many Voronoi faces as there are Delaunay edges (including volumetric edges and surface edges) around the associated Delaunay face. This means that edges are not necessarily surrounded by 3 Voronoi faces. A Voronoi face dual to a Delaunay edge that belongs to the surface, that is, dual to a surface edge, is called an open Voronoi face, cf. Figure 6. An open Voronoi face contains a node that is outside the shape boundary. This type of Voronoi face does not belong to the skeleton of the medial axis, because it is not entirely contained in the interior S. This allows us to filter out Voronoi face that are not part of the interior medial axis. Using the same idea, an open edge is associated to a Delaunay face that belongs to the surface mesh. For this type of edge, one of its extremities is a Voronoi nodes on the inner medial axis, and the other one is a Voronoi vertex on the outer medial axis. Like the open Voronoi face, the open edge therefore crosses the surface S.

Figure 6 :

 6 Figure 6: Illustration of the relationship between skeleton and surface components. An open skeletal edge (dashed blue) is associated to its dual, a surface mesh face (green triangle). An open face, containing open edges is associated to its dual, a surface mesh edge (green). This kind of face contains edges that lie on both the internal and the external skeletons. A Delaunay triangle (red) is associated to its dual, a skeletal edge (black), and each of its crossing edges (dashed red) is associated to its dual, a skeletal face.

417Figure 7 :

 7 Figure 7: Representation of the half edges around a (black) singular medial edge, from different views (up left, up right, below). Around this degree 3 edge, there are 6 half edges represented by the black arrows. Each pair of (black arrows) half edges represents a different area on the surface mesh.

449••). 457 A

 457 Figure8: A path on the skeleton is a sequence of half edges. Here, we can see two sides of the same skeleton, where two paths are drawn (yellow and purple), representing the same (black) singular edges. By construction of half edges, these paths have opposite direction. Note that green and purple half paths do not lie on the same medial faces, and therefore are not on the same sheet.

Figure 10 :

 10 Figure 10: Illustration of the ε criterion. A tetrahedron corresponds to a unique medial ball, passing through its four vertices. Adding ε to the radius of the ball, we obtain a new (dashed) sphere, covering other vertices of the surface mesh.

5814. 3 .

 3 Single sheet propagation 582We will assign a label to each skeletal sheet. To compute a sheet, we 583 start from a first medial face, which we label. We then add all the 584 neighbor medial faces sharing a regular edge in a propagation pro-585 cess, and assign the same label to them. We repeat the propagation 586 for each added face, labeling the medial faces until no neighboring 587 unlabelled medial faces sharing a regular edge remain. This pro-588 cess ends because the boundaries of sheets consist of singular or 589 boundary edges which are not regular edges, and a medial sheet for 590 a finite volume discrete surface mesh will have a finite number of 591 medial faces.

Figure

  Figure8).

602

  We start from a half singular edge to form the candidate list ls 603 and identify a neighboring half singular edge (cf. Figure 11) using 604 Algorithm 2. We iterate until the loop of half edges closes, and 605 returns the complete singular path. We then remove every used half 606 edge from the candidate list ls. 607 ALGORITHM 2: Computing next half singular edge. Input : Half singular edge pe Output: Next half singular edge pe pe = pe do

6104. 5 .

 5 Identifying an insignificant sheet beyond a singular path 611 If a singular path separates the current sheet from a sheet that has 612 been determined to be insignificant according to the ε criterion,

  613

616

  Figure12).

Figure 11 :

 11 Figure11: Identifying the next half edge to compute a singular path (here (a, d, i) ). The singular edges are in black. Two examples are shown here. From a to d, we first take the next edge on the same half medial face, b, then take the opposite edge to b, c. Finally, we take the next edge to c, which is the singular edge d. More generally, to go from d to i, the same steps are repeated on each medial face until a singular half edge is found. Here, it is done twice: through half edges e, f and g, h. At this point, the singular path is composed by a, d and i. During this process, each visited half medial face belongs to a unique Voronoi cell with associated dual (green) boundary point. Each new point is stored as the next vertex of the path. When the path closes, the result is a sequence of (green) vertices creating a closed path on the boundary.

ALGORITHM 3 :••

 3 Figure12: Singular path detection and ignoring a sheet. Here, to ignore the sheet, two edges, then three faces, are created on the mesh. The mesh vertices inside the region are removed from the mesh. To ignore the sheet, the mesh vertices within the closed boundary path are removed from the mesh and the corresponding region is re-triangulated. Note that that singular edges bordering the medial path have become regular.

  658

  Figure 13: An example of problematic edges, and their resolution.A problematic edge belongs to three medial faces belonging to the same sheet. Note that in (a), all medial faces belong to the same sheet even though there are some faces connected by singular edges. In (a), the black edges are problematic because they are not cleanly separating two sheets. This situation cannot be characterized locally, but requires the full computation of the sheet to detect. Our solution is to relabel appropriate regular edges as singular edges (new edge in dashed black in the bottom figure) in order to split the affected medial faces into two separate sheets. In (b), the medial faces surrounded by the yellow edges now belong to a separate sheet. This means that singular edges (black edges in the bottom figure) now belong to at most two medial faces belonging to the same sheet.

and

  Scale Axis Transform (SAT) [MGP10]. Our method consis-677 tently retains salient shape features across different types of shape 678 configurations, unlike the other two, while maintaining a similar 679 level of cleanliness. For example, SAT erodes important regions of 680 the body of the bunny. Similarly, Voxel Core erodes important re-681 gions of the tool (bottom left) and the head and the wings of the 682 dragon, as well as legs of the chair. This is because our method 683 explicitly preserves ε-fidelity with the input surface vertices while 684 simultaneously seeking geometric coherence. In other words, we 685 have the additional advantage that we maintain geometric consis-686 tency with the surface mesh.

  687

Future

  work might explore the relationship between the noise level of the surface mesh and optimal ε values for several distributions of noise. Additionally, exploring preprocessing approaches to allow for more general formats for surface data beyond Delaunay meshes would be useful.

Figure 15 :

 15 Figure 15: Comparison of the same skeleton with different ε values. Between 15a and 15b, we can see that most of the noisy peaks were removed. Few sheets were removed between 15b and 15c.

  (a) Skeleton and mesh without noise (b) Skeleton and mesh with noise

Figure 17 :

 17 Figure 17: Effect of the mesh noise on the skeleton (ε = 0.01).

Appendix A: Irregularities of the discrete skeleton Generating the medial mesh from the Voronoi/Delaunay structures of the surface mesh, even a nice one, can produce geometric irreg-

744

• Though it is rare, some "sheets" may be one-dimensional in cer-745 tain places or in their entirety (for example, an arm that is a long 746 circular cylinder will have a curve as its skeleton). We handle 747 these cases separately: instead of propagating medial faces, we 748 compute successive degree 0 edges.

749

• Two singular paths can cross each other: this situation is not 750 generic behavior in the continuous setting, but can happen in the 751 discrete setting.

752

• Some medial nodes can be associated to a sphere outside the 753 mesh (cf. Figure 16). This is because, for some interior boundary 754 tetrahedra, the associated sphere center will be outside of the