Julien Heuvingh
email: julien.heuvingh@espci.fr

; Bonneau

Asymmetric Oxidation of Giant Vesicles triggers Curvature-associated Shape Transition and Permeabilization

Keywords: Photosensitizer, Lipids, Budding, GUV, PDT, Liposome

Oxidation of unsaturated lipids is a fundamental process involved in cell bioenergetics as well as in cell death. Using giant unilamellar vesicles and a chlorin photosensitizer, we asymmetrically oxidized the outer or inner monolayers of lipid membranes. We observed different shape transitions such as oblate to prolate and budding, which are typical of membrane curvature modifications. The asymmetry of the shape transitions is in accordance with a lowered effective spontaneous curvature of the leaflet being targeted. We interpret this effect as a decrease in the preferred area of the targeted leaflet compared to the other, due to the secondary products of oxidation (cleaved-lipids). Permeabilization of giant vesicles by light-induced oxidation is observed after a lag and is characterized in relation with the photosensitizer concentration. We interpret permeabilization as the opening of a pore above a critical membrane tension, resulting from the budding of vesicles. The evolution of photosensitized giant vesicle lysis tension was measured and yields an estimation of the effective spontaneous curvature at lysis. Additionally photo-oxidation was shown to be fusogenic.

INTRODUCTION

The oxidation of unsaturated lipids is of great interest, both from the biological and medical points of view [START_REF] Halliwell | Free radicals, reactive oxygen species and human disease: a critical evaluation with special reference to atherosclerosis[END_REF][START_REF] Girotti | Photodynamic lipid peroxidation in biological systems[END_REF][START_REF] Ross | The pathogenesis of atherosclerosis: a perspective for the 1990s[END_REF][START_REF] Cadenas | Understanding the process of aging: the roles of mithochondria, free radicals and antioxidants[END_REF][START_REF] Girotti | Role of lipid hydroperoxides in photo-oxidative stress signaling[END_REF]. It may be generated in enzymatic or non-enzymatic reactions involving short-lived activated chemical species known as "reactive oxygen species" (ROS) [START_REF] Paillous | Interest of photochemical methods for induction of lipid peroxidation[END_REF]. In cells, ROS are generated during the normal respiration process involving oxygen, oxidases and electron transport in mitochondria or the endoplasmic reticulum. Peroxidation of unsaturated lipids may be responsible, in vivo, for pathological processes such as drug-induced phototoxicity, arthrosclerosis and aging [START_REF] Ross | The pathogenesis of atherosclerosis: a perspective for the 1990s[END_REF][START_REF] Cadenas | Understanding the process of aging: the roles of mithochondria, free radicals and antioxidants[END_REF]. At cellular level, oxidation of lipids is involved in dysfunctions such as enhanced permeability, changes in membrane fluidity or release of lysosomal enzymes. In addition, the presence of oxidized phospholipids in lipidic membranes induces changes in their physical properties [START_REF] Borst | Oxidation of unsaturated phospholipids in membrane bilayer mixtures is accompanied by membrane fluidity changes[END_REF][START_REF] Ayuyan | Lipid peroxides promote large rafts: effects of excitation of probes in fluorescence microscopy and electrochemical reactions during vesicle formation[END_REF].

The photochemical induction of oxidation is an effective way of inducing oxidation processes [START_REF] Paillous | Interest of photochemical methods for induction of lipid peroxidation[END_REF]. It is supported by the ability of certain molecules, called photosensitizers, to generate ROS upon light irradiation. The specificity and preferential retention of certain photosensitizers by tumors, as compared to normal surrounding tissues, are the basis of an anti-tumoral therapy, the photodynamic therapy (PDT) [START_REF] Dougherty | Photodynamic therapy[END_REF]. At high level, such light-induced molecular damage leads to the targeted cell's death. More recently, photosensitizer-induced lipid oxidation has been used to deliver macromolecular therapeutic agents to their intracellular targets by an approach called Photochemical Internalization (PCI) [START_REF] Selbo | In vivo documentation of photochemical internalization, a novel approach to site specific cancer therapy[END_REF]. After the uptake by endocytosis, the degradation of the macromolecules in lysosomes is greatly reduced by the photodynamic destabilization of the endoctytic vesicles membrane, increasing their biological activity.

The short half-life and limited diffusion length of the photo-induced ROS necessitate the close association of photosensitizers with the target site. For example, chlorin-generated subcellular singlet oxygen lifetime has been experimentaly measured between 4.50.5 µs and 172 µs [START_REF] Kuimova | Singlet Oxygen in a Cell: Spatially Dependent Lifetimes and Quenching Rate Constants[END_REF]. ROS typicaly diffuse less than 0.1 µm in a biological environment [START_REF] Moan | On the diffusion length of singlet oxygen in cells and tissues[END_REF][START_REF] Dairou | Interaction of sulfonated anionic porphyrins with HIV glycoprotein gp120: photodamages revealed by inhibition of antibody binding to V3 and C5 domains[END_REF]. Due to the localized action of photosensitizers, the characteristics of their interaction with lipidic membranes are an important parameter controlling the effects of the photosensitizer-induced lipid oxidation [START_REF] Piette | Cell death and growth arrest in response to photodynamic therapy with membrane-bound photosensitizers[END_REF]. For tetrapyrrole photosensitizers, the ability to cross membranes is governed by the charge of their lateral chains [START_REF] Mojzisova | Structural and physico-chemical determinants of the interactions of macrocyclic photosensitizers with cells[END_REF][START_REF] Bonneau | Tetrapyrrole photosensitisers, determinants of subcellular localisation and mechanisms of photodynamic processes in therapeutic approaches[END_REF][START_REF] Bonneau | Tetrapyrrolephotosensitizers vectorization and plasma LDL: a physico-chemical approach[END_REF]. The chlorin e6 (Ce6), a second generation photosensitizer is not able to cross the biological membranes [START_REF] Mojzisova | The pH-dependent distribution of the photosensitizer chlorin e6 among plasma proteins and membranes: a physico-chemical approach[END_REF][START_REF] Mojzisova | Cellular uptake and subcellular distribution of chlorin e6 as functions of pH and interactions with membranes and lipoproteins[END_REF]. This allows an asymmetric labeling of the membrane, where the photosensitizer interacts only with the monolayer in contact with the photosensitizer solution. We thus labeled model membranes, Giant Unilamellar Vesicles (GUV) composed of dioleoylphosphatidylcholine (DOPC), which is an unsaturated lipid. Under light-induced oxidation, we observed morphological transitions and permeation of the GUV.

Morphological transitions in GUV have previously been observed in response to pH difference between compartments [START_REF] Hope | Phospholipid asymmetry in large unilamellar vesicles induced by transmembrane pH gradients[END_REF][START_REF] Farge | Shape changes of giant liposomes induced by an asymmetric transmembrane distribution of phospholipids[END_REF], change in temperature [START_REF] Dobereiner | Mapping vesicle shapes into the phase diagram: a comparison of experiment and theory[END_REF], grafting of polymers on the membrane [START_REF] Nikolov | Behavior of giant vesicles with anchored DNA molecules[END_REF], ion adsorption on the lipid heads [START_REF] Petrov | Coupling chemical reactions to membrane curvature: a photochemical morphology switch[END_REF] or photoizomerisation of azobenzene-containing amphiphiles [START_REF] Hamada | Reversible photoswitching in a cell-sized vesicle[END_REF]. One of the first models to account for various morphologies in vesicles was the spontaneous curvature model introduced by Helfrich [START_REF] Helfrich | Elastic properties of lipid bilayers: theory and possible experiments[END_REF]. Increasing the spontaneous curvature induces membrane budding outside the GUV, whereas a decrease of the spontaneous curvature down to negative values induces a stomatocyte shape or budding inside the vesicle. Refinement of the theoretical comprehension of vesicle shape transitions led to the area-difference elasticity (ADE) model which takes into account the area difference between the two monolayers [START_REF] Miao | Budding transitions of fluid-bilayer vesicles: the effect of area-difference elasticity[END_REF][START_REF] Seifert | Configurations of fluid membranes and vesicles[END_REF]. Any vesicle can then be defined by an area/volume ratio and a term of effective spontaneous curvature, which contains the area difference between the monolayers and the spontaneous local curvature. These two parameters give a phase diagram of vesicle shape that has been explored [START_REF] Dobereiner | Mapping vesicle shapes into the phase diagram: a comparison of experiment and theory[END_REF]. For area/volume ratio corresponding to nearly spherical vesicles, an increase of the effective spontaneous curvature leads to shape transitions from an oblate ellipsoïd to a prolate ellipsoïd, and then to a pear and to a budding vesicle. Reducing the effective spontaneous curvature leads to a stomatocyte shape or inside budding. DOPC membranes are essentially water permeable (40 µm/s (29)) and not permeable to polar solutes such as sugar at the time length involved in these experiments (6 10 -5 µm/s [START_REF] Faure | Modeling leakage kinetics from multilamellar vesicles for membrane permeability determination: application to glucose[END_REF]). Due to osmotic equilibrium, the very slow permeability ensures a constant volume for vesicles. A much higher solute permeability is possible via pores, holes in the membrane allowing free flow. These pores have been observed in GUV [START_REF] Sandre | Dynamics of transient pores in stretched vesicles[END_REF][START_REF] Rodriguez | Transition from long-to short-lived transient pores in giant vesicles in an aqueous medium[END_REF]. They are energetically unfavorable because of the high cost of the exposition to water of hydrophobic lipid tails at the rim of the pore [START_REF] Karatekin | Cascades of transient pores in giant vesicles: line tension and transport[END_REF][START_REF] Idiart | Rupture of a liposomal vesicle[END_REF].

In this work, we photochemically induce the lipid oxidation in DOPC-GUV. The photosensitizer used, Ce6, allows a symmetric as well as an asymmetric targeting of the membrane bilayers and a fine control of the location of the oxidation. The induced morphological transitions are corelated to the targeted leaflet and show a decrease in the spontaneous curvature of the targeted leaflet. The eventual permeabilization of the membrane has been measured and can be linked to the tension due to the budding of vesicles.

MATERIALS AND METHODS

Chemicals.

All chemicals were purchased from Sigma (USA), except dioleoylphosphatidylcholine (DOPC) and dipalmitoylphosphatidylcholine (DPPC) from Avanti Polar Lipids (USA), and chlorin e6 from Porphyrin Products (USA). Chlorin stock solution (5 mM) was prepared in ethanol and keept at -18 °C. The experimental Ce6 aqueous solutions were prepared, used without delay and handled in the dark. The osmolarity of the solutions was checked with an osmometer (Roebling, Germany).

GUV formation. GUV were formed by the electroswelling method [START_REF] Angelova | A mechanism of liposome electroformation[END_REF]. DOPC in chlorophorm was deposed on ITO-covered glass plates. A chamber was made from two such glass plates and a Teflon spacer of 4 mm; the solvent was dried in vacuum. The chamber was filled then with a solution of 300 mM sucrose and an AC field of 1 Volt and 8Hz was applied between the plates for 4 hours. For DOPC versus DPPC comparison, electroformation took place at 50°C, above the transition temperature of DPPC. For observation, the GUV were mixed with a 300 mM glucose solution. The density difference between sucrose and glucose caused the GUV to sediment to the bottom of the chamber. The difference in optical index between sucrose inside and glucose outside the vesicle allowed phase contrast microscopy observation.

Chlorin labeling. Giant Vesicles were asymmetrically labeled with chlorin. The chlorin molecules were present outside the vesicles, inside the vesicles or both outside and inside. The concentrations of chlorin used were 2.5, 5, 12.5, 25, 50 and 125 M. The measured pH of the solutions was 5.1  0.2. For chlorin present outside the vesicles, chlorin was diluted in the glucose solution before mixing with GUV. For chlorin present inside the vesicles, GUV were prepared with a sucrose solution containing chlorin in the chamber and subsequently rinsed. Rinsing was achieved by mixing the GUV with the glucose solution and carefully centrifuging them twice (25g for 20 min.). At the typical lipid concentration in GUV solution (~10 -6 M), the majority of the chlorin is unbound. This fact, combined with the high exchange rate of the chlorin with the bulk medium [START_REF] Mojzisova | Cellular uptake and subcellular distribution of chlorin e6 as functions of pH and interactions with membranes and lipoproteins[END_REF] ensures the exit of the sensitizer from the membrane outer leaflet and its rinsing. GUV with symmetrical chlorin distribution were prepared with a sucrose solution containing chlorin in the chamber and then mixed with a chlorin-containing glucose solution.

Observation and illumination. GUV were observed under a Nikon Eclipse TE2000-U inverted microscope equipped with a 1.3NA 100x oil objective. Illumination was provided by a 100 W Hg-arc lamp with a 465-495 nm bandpass filter. Images were acquired with a Basler A602f digital camera using Labview (National Instruments, USA). Images were analyzed using Image J (NIH, USA) and Scilab (INRIA, France, www.scilab.org).

Tension measurement. Micropipettes were made from borosilicate glass capillary GC100-15 tubing (1.0-mm outside diameter x 0.58-mm wall thickness x 7-cm length, Harvard apparatus Ltd., Kent, UK) using a pipette puller (Sutter instruments, model P-2000). A homemade microforge was used to tune their inner diameter from 2.3 to 5.3 µm. Pipettes were coated with bovine serum albumin and rinsed extensively. Prior to experiments, the pipettes were filled with the glucose solution. A suction pressure was applied in the pipette by hydrostatic pressure. The suction pressure P produces a tension  in the membrane such as P.D P / 4(1-D P /D V )

(1) where D P and D V are respectively the diameters of the pipette and of the vesicle [START_REF] Karatekin | Cascades of transient pores in giant vesicles: line tension and transport[END_REF].

Statistics. Correlation between the radius and permeabilization time of vesicle were analyzed by a t test on the regression analysis with a statistical significance of 0.01. When the two parameters correlated, the time was "corrected" i.e. deduced from the regression using a radius of 3.98 µm, which is the mean GUV radius of our samples. When no correlation was found, the average over the vesicles was simply used.

RESULTS

Shape Transitions

We prepared giant vesicles with different localizations of chlorin e6 regarding the leaflets. As chlorin molecules have a high affinity for lipids and do not flip-flop at the time scale involved in these experiments [START_REF] Mojzisova | The pH-dependent distribution of the photosensitizer chlorin e6 among plasma proteins and membranes: a physico-chemical approach[END_REF][START_REF] Mojzisova | Cellular uptake and subcellular distribution of chlorin e6 as functions of pH and interactions with membranes and lipoproteins[END_REF], Ce6 labels only the leaflet directly in contact with the photosensitizer solution. Thus, GUV labeled with Ce6 in the outer, the inner and both leaflets were studied. Upon illumination, all vesicles containing photosensitizers underwent major shape transitions within a few to a hundred seconds after illumination start (see Fig. 1). Vesicles with photosensitizer in the inner leaflet showed shape transitions from oblate to prolate ellipsoïds, pear shape deformations and budding of small vesicles outside the GUV. Vesicles in the prolate shape fluctuated during tens of seconds until the budding of vesicles put them out of prolate shape. Vesicles with photosensitizer in the outer leaflets typically showed deformation for a short time (< 2 seconds) followed by an invagination or budding of a small vesicle inside the GUV. The budded vesicles were visible as light dots contrasting with the GUV inner medium. Vesicles with Ce6 in both leaflets show all of the above-mentioned shape transitions. Finally, vesicles with no Ce6 showed none of these shape transitions for over 15 minutes. A quantification of these shape transitions is presented in Fig. 2. These results are in accordance with a lowered spontaneous curvature of the leaflet in which photo-oxidation occurs.

Permeabilization

Vesicles with chlorin in the outer or both leaflets undergo permeabilization. The contrast between the vesicle and the surrounding solution gradually fades away until there is no phase difference between inside and outside medium and the vesicle is barely observable by its membrane. This is a signature of the sucrose diffusion from the inner medium to the outer medium and its replacement by glucose from the outer medium [START_REF] Rodriguez | Indirect evidence of submicroscopic pores in giant unilamellar vesicles[END_REF]. A small portion of the vesicles (<10%) burst or leaked abruptly in a non-gradual fashion. We quantified the contrast fading, which typically varies like a decreasing exponential (Fig. 3 left). c=c 0 exp(-(t-t 0 )/   where c is the concentration of sucrose in the inner medium, c 0 its initial concentration, t 0 the initial time of the experiment and  the characteristic time of the diffusion process. In accordance with Eq. 2, we extracted a starting and a characteristic leaking time. The exponential fit is in good agreement with the data (R 2 >0.95 for over 87% of the vesicles and R 2 >0.9 for over 96% of the vesicles). The characteristic and starting times were compared with the vesicle radius. The starting time is statistically independent of the vesicle size, except at the highest concentration of Ce6 (50 and 125 µM outside the vesicle). The characteristic time depends on the vesicle radius at all concentrations. The dependence is stronger for high concentrations than for low (R 2 =0.69 for 125 µM, R 2 =0.22 for 2.5 µM). The starting and characteristic permeabilization times decrease when the quantity of chlorin is increased for GUV with photosensitizers in the outer medium (Fig. 3 right). A plateau is evidenced over 25 µM. The permeabilization is slower for vesicles photosensitized on both leaflets. In the case of vesicles with photosensitizer only in the inner medium, vesicles are not permeabilized after 15 minutes, except for the higher concentration of 125 µM. At this concentration, permeabilization is 3 times longer for vesicles bearing Ce6 on their inner leaflet than for vesicles bearing Ce6 on their outer leaflet. All vesicles had a spherical shape (showing tension) before the start of permeabilization. In order to verify that the permeabilization was mainly due to the oxidation of the lipid unsaturation, we compared the photosensitization effect on DOPC and DPPC, which is a saturated phospholipid. The experiment was conducted at 50°C, above the transition temperature of DPPC. The vesicles were in presence of 25 µM Ce6 and illuminated by a 100 W Hg-arc lamp during two seconds without any filter. Contrary to the DOPC-GUV, the DPPC-GUV did not permabilize for more than 5 minutes (see Fig. S1 in the Supporting Material). The DOPC-GUV lost half of their contrast in 44 s ± 25 s, whereas more than 70% of the DPPC-GUV retained more than half of their contrast 300s after illumination. No systematic shape changes were detected with DPPC-GUV.

Lysis tension

Micropipette experiments were conducted to estimate the lysis tension of photosensitized DOPC-GUV. Vesicles in presence of 50 µM Ce6 in the outer medium were held by a micropipette and a suction pressure between 50 and 500 Pascal was applied, corresponding to a membrane tension from 0.05 to 0.5 mN/m (see materials and methods section). We kept the vesicle's tension constant and illuminated the sample. After a few seconds of illumination the vesicle's integrity was compromised, i.e. it disappeared inside the pipette. This is interpreted as the opening of a hole in the membrane, allowing the inside solution to leave the vesicle and the membrane to be sucked inside the pipette. The time between illumination and permeabilization were recorded and is presented in Fig. 4. It can be thought as the evolution of lysis tension over time. The time at which the vesicles permeabilize in experiments without pipettes is plotted on the same graph. This time corresponds to the time at which vesicles tensed at 0.15 mN/m leaked. Vesicles tensed in presence of Ce6 without illumination did not break for several minutes.

Fusion

Fusion of giant vesicles in contact with each other occurred frequently for GUV of more than 10 µm radius at the highest concentrations (125 µM) in outer leaflets. The contacts between these GUV lead to fusion in 71% of the cases (35 fusion events). See Movie S1 in the Supporting Material. We observed less than 5% of fusion between vesicles of less than 10 µm radius at 50 µM Ce6 and below. A more precise quantification of fusion proved difficult as the number of contacts is highly dependant on the size and density of the vesicle preparation. After micropipette experiments, when the vesicles were gradually aggregated by the evaporation in the experimental chamber, fusion occurred at a dramatically accelerated rate when illuminated.

DISCUSSION

Photo-oxidation products

The chlorin e6 is a photosensitizer: it interacts very efficiently with light to produce reactive species (singlet oxygen and radicals) from its long-life triplet state. In ethanol, its quantum yield for 1 O 2 production is important (around 0.65 [START_REF] Zenkevich | Photophysical and photochemical properties of potential porphyrin and chlorin photosensitizers for pdt[END_REF]). In unsaturated lipids, excitation of a photosensitizer generates peroxides [START_REF] Castano | Mechanisms in photodynamic therapy: part twocellular signaling, cell metabolism and modes of cell death[END_REF], which is highly unstable in presence of any trace of transition metal [START_REF] Buettner | Catalytic metals, ascorbate and free radicals: combinations to avoid[END_REF]. In our microscopy experiments, it will spontaneously decay to a free radical lipid, which pulls out a hydrogen from another unoxidized unsaturated lipid, creating a new free radical and a hydroxylated lipid. The combination of molecular oxygen with the new free radical leads again to the removing of a hydrogen from another lipid and produces again free radical and lipid peroxide. The lipid peroxidation is thus a radical chain reaction leading to the formation of intermediate hydroperoxide (Fig. 5). Hydroperoxyl group induces hydrophilicity in the chains of lipids.

If it remains in the hydrocarbon region of the membrane bilayer, hydroperoxyl group should drastically change the membrane architecture by increasing the cross-section area of its lipid tails. The peroxide lipid would therefore have a packing parameter above one and would increase the relaxed area of the leaflet in which they are present. However, indirect evidences on polyunsaturated lipids have suggested that the peroxide group may more likely be near the water/membrane interface [START_REF] Buettner | The pecking order of free radicals and antioxidants: lipid peroxidation, alpha-tocopherol, and ascorbate[END_REF]. In this case, the peroxide lipid will have a packing parameter below one and will still increase the leaflet relaxed area. Additionally, in presence of trace amounts of catalytic transition metals (e.g. Fe 2+ ), the photo-oxidation of biological lipidic systems results in a myriad of secondary products [START_REF] Buettner | Catalytic metals, ascorbate and free radicals: combinations to avoid[END_REF][START_REF] Domingues | Mass spectrometry analysis of oxidized phospholipids[END_REF]. For monounsaturated fatty acids (like are the hydrocarbon chains of DOPC), these major final products correspond to the cleavage of the carbon chain near the initial position of the double bound and give an alkene or an aldehyde [START_REF] Domingues | Mass spectrometry analysis of oxidized phospholipids[END_REF]. Such cleaved-lipids present a strongly modified geometry as compared to DOPC. They correspond to a packing parameter below one and to a decrease of the relaxed area in the leaflet in which they are present. They are also known to favor pore formation [START_REF] Karatekin | Cascades of transient pores in giant vesicles: line tension and transport[END_REF].

Changes in effective spontaneous curvature

The light-induced shapes observed in the vesicles containing photosensitizer in the inner leaflet (prolate, pears, external budding) correspond to shapes of higher effective spontaneous curvature in the ADE model. The large and slow fluctuations observed in this case are also typical of an increase of the effective spontaneous curvature near the budding transition [START_REF] Dobereiner | Spinodal fluctuations of budding vesicles[END_REF]. The internal budding observed when the photosensitizer is in the outer leaflet corresponds to a lower effective spontaneous curvature. The variety of shape transitions observed is therefore qualitatively in accordance with a lowered effective spontaneous curvature of the leaflet which is photosensitized.

More precisely, the effective spontaneous curvature in the ADE model is composed of two terms which cannot be experimentally separated. The first is the spontaneous curvature of the lipids linked to their packing parameter; the second is the difference in area between the two leaflets. Increasing the area of the external leaflet or changing its lipids towards cone shapes will increase the effective spontaneous curvature, whereas decreasing the area of the external leaflet or changing its lipids towards inverted cone shapes will decrease the effective spontaneous curvature. The effect on the membrane effective spontaneous curvature will be inversed if the internal leaflet is changed. The products of lipid peroxidation discussed above will have contradicting effects on the effective spontaneous curvature of the leaflet. Lipid peroxides, while having a negative packing spontaneous curvature, will raise the leaflet effective spontaneous curvature by increasing its area. Inversely, cleaved-lipids have a positive packing spontaneous curvature but will lower the leaflet effective spontaneous curvature by decreasing its area. We need to estimate the magnitude of these effects on GUV membranes. The effective spontaneous curvature can be expressed in the ADE model as:

A A C C 0 0 0 h      (3)
where  is the ratio between the local and non-local bending rigidity, h is the thickness of a leaflet, C 0 * and C 0 are respectively the effective spontaneous curvature of the membrane and the spontaneous curvature, and A 0 is the optimal area difference between the two monolayers and A is the area of the membrane [START_REF] Seifert | Configurations of fluid membranes and vesicles[END_REF]. To compare the effect of lipid geometry on leaflet area increase, we use a simple geometric model, in which the cross section area is increased from A/N to (A + A 0 )/N at the position of the lipid double bond (N is the number of lipids in a leaflet). The spontaneous curvature of this lipid will then be equal to

A A C 0 0 h 1   ( 4
)
and the effective spontaneous curvature to

  A A C 0 0 h 1      (5)
This curvature is positive if A 0 is positive for DOPC as  =1.2 [START_REF] Dobereiner | Mapping vesicle shapes into the phase diagram: a comparison of experiment and theory[END_REF]. The area increase is therefore dominant over the lipids packing spontaneous curvature.

The formation of lipid peroxide consequently increases the leaflet effective spontaneous curvature. This is true whether the peroxide is located at the water/membrane interface or in the membrane bulk. Likewise, the formation of cleaved-lipids will decrease the leaflet effective spontaneous curvature. Although both of these chemical products will likely be present in the photosensitized membrane, our results indicate that the cleaved-lipids have the dominant effect on membrane spontaneous curvature, triggering the morphological transitions described here.

Permeabilization

Permeabilization of photosensitized GUV was observed via the contrast fading as sucrose left the vesicle interior. The sucrose flux through a circular opening in the membrane is:

 v =2Dsc (6)
where D is the diffusion coefficient of sucrose, s the radius of the aperture, and c the concentration in the inner medium (see Eq. 2) [START_REF] Idiart | Rupture of a liposomal vesicle[END_REF][START_REF] Castano | Mechanisms in photodynamic therapy: part twocellular signaling, cell metabolism and modes of cell death[END_REF]. We assumed the external sucrose concentration to be zero, as the internal volume is negligible compared to the outer volume. The concentration in the inner medium obeys dc =  v dt /V (V is the volume of the vesicle) and decreases exponentially with a characteristic time of V/2Ds. The experimental data from the permeabilizations of individual GUV are indeed well fitted by such a decreasing exponential as shown in Fig. 2. The fact that permeabilization occurs suddenly, after 20 to 100 seconds of exposition, and the following exponential decay in sucrose concentration are both in good agreement with a pore opening scenario. Based on the above model, the diameter of a pore is estimated to be between 16 nm and 43 nm for a typical vesicle ( between 10.4 s and 27.5 s for a chlorin concentration above 12.5 µM; R = 4 µm). These pore sizes are in the same range as the ones measured in electroporation of red blood cell ghosts [START_REF] Chang | Changes in membrane structure induced by electroporation as revealed by rapid-freezing electron microscopy[END_REF] or the ones calculated for stretched giant vesicles [START_REF] Brochard-Wyart | Transient pores in stretched vesicles: role of leak-out[END_REF].

Permeabilization and tension

Permeabilization and formation of a pore can be triggered by membrane tension. Membrane tension can be described as the combination of two contributions: an entropic tension due to the damping of thermal fluctuation modes, and an elastic term coming from the increased distance between the lipids. When measuring the area variation of a GUV while increasing the membrane tension (via micropipettes), these two contributions are visible successively. First the membrane entropic "ruffles" are smoothed, corresponding to a soft exponential rise in area, then the membrane stretches, corresponding to a linear increase of area [START_REF] Rawicz | Effect of chain length and unsaturation on elasticity of lipid bilayers[END_REF]. Eventually the stretching will break the membrane above a critical stress. The lysis stress for DOPC-GUV was measured to be 9.9 ± 2.6 mN/m (29). Unsurprisingly, photo-oxidation lowers the critical stress of DOPC vesicles as shown in our experiments. We observe that the mean time of membrane rupture during photo-oxidation decreases when the tension is increased, which is interpreted as a reduction of lysis tension with the photo-oxidation duration. The starting time of vesicle permeabilization when no suction is externally applied corresponds to a lysis tension of 0.15 mN/m. Therefore we hypothesize that when no external tension is applied, the vesicle tenses itself up to a tension of 0.15 ± 0.05 mN/m at which it lyses. We will show in the next paragraph how this tension can be reached through shape transition and budding. The fact that it takes longer for a vesicle to lyse when an external tension below 1 mN/m is applied can be explained by a partial suppression of budding due to this preexisting tension [START_REF] Devaux | How lipid flippases can modulate membrane structure[END_REF]. Interestingly, the measured lysis tension of 0.15 mN/m corresponds to the transition between thermal smoothing and stretching (see Fig. 2 in (45)), i.e. lysis occurs at the slightest stretching.

Budding and Tension

In our experiments, the lipid oxidation on a GUV's external leaflet triggers a shape transition from a prolate form to a closed stomatocyte or internal budding. The volume/area ratio of the vesicle after internal budding will be that of a sphere. The ADE model, describing shape transitions due to curvature modifications, assumes the area to be constant and does not take into account any stretching of the membrane. Such description is perfectly valid for moderate changes in the curvature. However above a certain level of curvature, the energy released from the budding will become important compared to the energy in the thermal fluctuations of the membrane. The vesicle would then be able to bud by tensing itself and taking area from the ripples of the membrane. We compare these two energies, in a manner similar to [START_REF] Glassinger | Influence of thermally driven surface undulations on tethers formed from bilayer membranes[END_REF], in order to estimate the minimum effective spontaneous curvature required to tense the vesicle up to its lysis tension .

The energy gained when budding a vesicle of radius r from a vesicle of radius R (r<<R) is:

H b =8k c (1-C 0 * r) (7) (see appendix)
where k c is the bending rigidity of the membrane (see detailed calculation in appendix). The energy necessary to entropically stretch a membrane from a tension  0 to a tension  is:

  0 c B S k 8 T k A H      ( 8
)
where A is the area of the giant vesicle, and k B is the Boltzmann constant.

According to our hypothesis, the area gain from thermal fluctuation will be equal to the area of the budding vesicle:

2 0 c B fluct r 4 Ln k 8 T k A A                (9) Thus              2 2 B c 0 R r T k k 8 exp (10)
For small buds (r<<R (8k c /k B T) 1/2 ~250 nm), the stretching energy simplifies to :

H s  2  (11) The total energy is then given by:

H =H b +H S  2  + 8 k c (1-C 0 *r) (12
) It will be minimum for r= k c C 0 * /at which

H k c (1-C 0 * 2 k c /2) (13)
Budding will therefore be favorable for C 0 *> (2 k c ) 1/2 Therefore, budding can be responsible for the vesicle's tension increase up to the observed lysis tension of 0.15 ± 0.05 mN/m, if the effective spontaneous curvature is above 42 µm -1 (±19 µm -1 ). Expressed in terms of area difference, this corresponds to a decrease of 2.3% (±1.1%) of one of the monolayer area as compared to the other.

It is tempting to explain the different permeabilization behavior between GUV photosensitized on their outer or inner leaflet by the difference between external and internal budding. Due to external budding, GUV photosensitized on their inner leaflet will loose both area and volume, while GUV photosensitized on their outer leaflet will loose area and gain volume due to internal budding. This will result in more stretching when the outer leaflet is targeted than when the inner leaflet is. However, taking into account the size difference between the budding and the original vesicle (typically 1/5 th as in Fig. 1), the change in volume is small compared to the change in area.

Recent publications showed that the GUV electroformation method used in these experiment can lead to a small degree of lipid peroxidation [START_REF] Ayuyan | Lipid peroxides promote large rafts: effects of excitation of probes in fluorescence microscopy and electrochemical reactions during vesicle formation[END_REF]. The presence of peroxidized lipids in our vesicles prior to photosensitization should have no effect on our results as long as it is symmetric regarding the leaflet localization.

Conclusion

The asymmetrical shape transitions observed in photosensitized GUV reveal changes in their membrane spontaneous curvature. These modifications are in accordance with the presence of cleaved-lipids by-products of oxidation. Permeabilization and a decrease of lysis tension were also characterized. We developped a model linking the budding due to the spontaneous curvature change to a tension of the membrane up to the lysis level where membranes are permeated. These findings might shed a new light on some membrane permeation phenomenon involved in biomedecine photodynamic approaches and in cell oxidative stress.

Aknowledgments

The authors thank S. Cribier for the use of her micropipette device and C. Gourier for her microforge device. R. Santus, P. Sens, N. Puff, N. Coq for useful discussions.

APPENDIX

We calculate in this appendix the energy gained from budding a vesicle. The ADE model Hamiltonian is:

    2 0 2 c 2 0 2 1 c A A C C C H Ah 2 k dA k 2 1          (14)
Were C 1 and C 2 are the local curvatures of the vesicle, C 0 the spontaneous curvature of the membrane, A is the area difference between the outer and the inner leaflet, and A 0 is the prefered area difference between the leaflets. Oxidation will change lipid area from the outer leaflet from A to A+A 0 , and the spontaneous curvature of the outer membrane lipids from essentially 0 to C 0 .

For a sphere of radius R:

  2 0 2 c 2 0 c 1 A C H Rh 8 Ah 2 k A R 2 k 2 1              (15)
For a sphere of radius R' plus an internal bud of radius r<<R (we keep the total area constant so that R' 2 +r 2 =R 2 ):

    2 0 2 c 2 2 0 c 2 2 2 0 c 2 A C C H rh 8 h ' R 8 Ah 2 k r 4 r 2 k 2 1 r R 4 ' R 2 k 2 1                           (16)
Assuming 1/R<<C 0 , the energy difference when budding is then:

       2 0 2 0 2 c 2 0 2 0 2 c 1 2 A A C C H H H Rh 8 rh 8 h ' R 8 Ah 2 k r 2 r 4 k 2 1                                 (
                ( 18
)              A r h r 1 k 8 A C H 0 0 c (19)   r 1 k 8 C H 0 c      (20) with A h A C C 0 0 0     
the effective spontaneous curvature.

Time sequences of giant vesicles with photosensitizers targeted to the outer leaflet (upper row) or to the inner leaflet (lower row). The bar is 10 micrometers; the time lapse between two images is 1 second. In the first sequence of the upper row (phase contrast microscopy), a GUV photosensitized in its outer leaflet endures a short time deformation (second image) followed by a budding towards the inside (white arrow). In the second sequence of the upper row (fluorescence of chlorin) a GUV photosensitized in its outer leaflet endures a deformation (second image) followed by multiple budding towards the inside (white arrows). In the sequence of the lower row (phase contrast microscopy), three giant vesicles photosensitized in their inner leaflet show deformations: the upper vesicle exhibits a transition from oblate to prolate (due to the projection, oblate appears as circle and prolate as ellipse) whereas the two other vesicles are deformed into pear shapes. These shape transitions are followed by a budding to the outside on each GUV (black arrows). The morphology transitions observed when the outer leaflet is targeted denote a decrease of the membrane's equivalent curvature, while those observed when the inner leaflet is targeted denote an increase of the membrane's equivalent curvature. The mid-line between long-time and short-time fluctuations materializes the separation between events associated with an increase of the effective spontaneous curvature (above the line) and those associated with a decrease of the effective spontaneous curvature (below the line).

Figure 3

Permeabilization of vesicles. Left: typical evolution of the contrast between the inside and outside of a photosensitized vesicle at 25 µM Ce6 in the outer medium. The decay of the contrast shows equilibrium between the inside solute (sucrose) and the outside solute (glucose). The experimental points are fitted by a decreasing exponential, from which are extracted a starting and a characteristic times of permeabilization. Center and right: starting time (t 0 ) and characteristic time () of permeabilization for 50-120 vesicles for each concentration of Ce6. Upward triangles are vesicles with Ce6 in the outer medium, downward triangles are vesicles with Ce6 in the inner medium, and diamonds are in both media. The dependence on the vesicle radius is corrected (for  at all concentrations and for t 0 at 50 and 125 µM outer medium)

Evolution of lysis tension over time during photodamage. The time after the start of the illumination at which lysis occurs was recorded for different membrane tensions (113 vesicles total). The boxes boundaries are the upper and lower quartiles (median 50% inside the box). The whiskers boundaries are the upper and lower deciles (median 80% inside the whisker).

Figure 5

Diagram of the lipid photo-oxidation processes in the vesicle membrane. The lipid oxidation, initiated via singlet oxygen, is a chain reaction.

Figure 2

 2 Figure 2


Xavier De Tinguy

Claudio Pacchierotti

Maud Marchal

Anatole L Écuyer

Toward Universal Tangible Objects: Optimizing Haptic Pinching Sensations in 3D Interaction *

Keywords:

Tangible objects are a simple yet effective way for providing haptic sensations in Virtual Reality. For achieving a compelling illusion, there should be a good correspondence between what users see in the virtual environment and what they touch in the real world. The haptic features of the tangible object should indeed match those of the corresponding virtual one in terms of, e.g., size, local shape, mass, texture. A straightforward solution is to create perfect tangible replicas of all the virtual objects in the scene. However, this is often neither feasible nor desirable. This paper presents an innovative approach enabling the use of few tangible objects to render many virtual ones. The proposed algorithm analyzes the available tangible and virtual objects to find the best grasps in terms of matching haptic sensations. It starts by identifying several suitable pinching poses on the considered tangible and virtual objects. Then, for each pose, it evaluates a series of haptically-salient characteristics. Next, it identifies the two most similar pinching poses according to these metrics, one on the tangible and one on the virtual object. Finally, it highlights the chosen pinching pose, which provides the best matching sensation between what users see and touch. The effectiveness of our approach is evaluated through a user study. Results show that the algorithm is able to well combine several haptically-salient object features to find convincing pinches between the given tangible and virtual objects.

INTRODUCTION

Haptic sensations can greatly improve the immersiveness and illusion of presence during Virtual Reality (VR) and Augmented Reality (AR) experiences. There are many different ways to provide users with such sensations, including using actuated devices known as kinesthetic or cutaneous feedback devices [START_REF] Hayward | Haptic interfaces and devices[END_REF][START_REF] Meli | Combining wearable finger haptics and augmented reality: User evaluation using an external camera and the microsoft hololens[END_REF][START_REF] Scheggi | Touch the virtual reality: using the Leap Motion controller for hand tracking and wearable tactile devices for immersive haptic rendering[END_REF], exploiting perceptual phenomena with cross-modal effects [START_REF] Ban | Modifying an identified curved surface shape using pseudo-haptic effect[END_REF][START_REF] Lécuyer | Simulating haptic feedback using vision: A survey of research and applications of pseudo-haptic feedback[END_REF], or using passive props also known as tangible objects [START_REF] Billinghurst | The magicbook-moving seamlessly between reality and virtuality[END_REF][START_REF] Cheok | Touchspace: Mixed reality game space based on ubiquitous, tangible, and social computing[END_REF][START_REF] Insko | Passive haptics significantly enhances virtual environments[END_REF][START_REF] Ohshima | Ar2 hockey system: A case study of collaborative mixed reality system[END_REF].

In this respect, tangible objects, also known as passive haptic or physical props, have been proven to be an effective -yet simplesolution for conveying global and distributed shape sensations in VR/AR. However, for the illusion to work, the haptic features of the tangibles should match those of the corresponding virtual objects, i.e., there should be a good correspondence between the, e.g., size, local shape, texture, mass of the tangible object with respect to its virtual counterpart. One way to match what users see in the virtual environment to what they touch in the real world is creating tangible versions of all the virtual objects in the scene. However, this solution may be neither feasible, e.g., if users can modify the virtual objects, nor desirable, e.g., if the virtual environment comprises several objects. It is therefore important to study solutions enabling the use of few tangible objects to render many virtual ones.

Toward this objective, we present an algorithm which analyzes different tangible and virtual objects to find the grasping strategy best matching the resultant haptic pinching sensation. Starting from the meshes of the considered objects, the algorithm guides users towards the grasping pose which best matches what they see in the virtual scene with what they feel when touching the tangible object. By selecting different grasping positions according to the virtual object to render, it is possible to use few tangible objects to render multiple virtual ones (see this idea in Fig. 1). The main contributions of our work can be summarized as follows:

• a novel algorithmic approach for using few tangibles to render multiple virtual objects, maximizing haptic fidelity;

• a preliminary user study assessing the effectiveness of the proposed approach for a set of representative objects.

Figure 1: Illustration of our approach through a carousel of virtual objects that can be grasped using a single "universal" tangible object.

The user is able to turn the virtual carousel and manipulate the three virtual objects using the suggested pinch poses (in green). These poses are proposed by our algorithm to best match the corresponding haptic pinching sensations on the tangible object.

RELATED WORK

Mesh matching. Matching two meshes over their geometry [START_REF] Shapira | Consistent mesh partitioning and skeletonisation using the shape diameter function[END_REF] or topology [START_REF] Tierny | Topology driven 3d mesh hierarchical segmentation[END_REF] is a common task in mesh processing, and it is usually performed by extracting representative features that can be compared. Shapira et al. [START_REF] Shapira | Consistent mesh partitioning and skeletonisation using the shape diameter function[END_REF] address the problems of partitioning and skeleton extraction of a set of 3D meshes. They use a volumebased shape-function mostly invariant to pose changes of the same object and that retains a similar value in analogue parts of different objects. It uses a measure of the diameter of the object's volume in the neighborhood of each mesh point. Using this approach, it is possible to partition and create skeletons among multiple meshes. Kortgen et al. [START_REF] Körtgen | 3d shape matching with 3d shape contexts[END_REF] present a method to evaluate 3D shape similarity. It captures the distribution over relative positions of other shape points to match different global shapes, even if they possess local differences, topological asperity, or a different scale. Objects can also be segmented by evaluating their similarity to basic shapes. A common approach to locally fit primitives has been presented by Zhou et al. [START_REF] Zhou | Generalized cylinder decomposition[END_REF], who introduce a quantitative measure of "cylindricity" for a shape and develop an optimization algorithm for generalized cylinder decomposition. Similarly, Yan et al. [START_REF] Yan | Variational mesh segmentation via quadric surface fitting[END_REF] segment meshes by fitting quadric surfaces, while Vidal et al. [START_REF] Vidal | Mechanical Mesh Segmentation and Global 3D Shape Extraction[END_REF] present a method for segmenting 2-manifold meshes by decomposing them into local shape primitives such as planes, spheres, or cylinders. A similar approach has been also presented in [START_REF] Li | Globfit: Consistently fitting primitives by discovering global relations[END_REF][START_REF] Schnabel | Completion and reconstruction with primitive shapes[END_REF]. Descriptors can also be obtained by evaluating the surface properties. In particular, Cazals et al. [START_REF] Cazals | Estimating differential quantities using polynomial fitting of osculating jets[END_REF] propose a method to compute mesh differential properties by fitting a polynomial to a neighborhood of points for extracting, e.g., curvature directions. This method is implemented in the Computational Geometry Algorithms Library, CGAL1 . Similarly, Szilvasi et al. [START_REF] Szilvási-Nagy | B-spline patches fitting on surfaces and triangular meshes[END_REF] propose a method for the construction of quadratic polynomial B-spline patches fitting analytical surfaces and triangle meshes in a neighborhood of points. These simpler patches can then be used for re-triangulation, and mesh simplification.

Grasping and manipulation. Finding feasible grasping poses on known and unknown objects is a common problem in robotics [START_REF] Prattichizzo | Springer Handbook of Robotics[END_REF], where researchers want to use robotic hands to autonomously pick and place objects in unstructured environments. This issue is quite similar to ours, as we start our algorithm by finding feasible pinching poses on the available virtual and tangible objects. GraspIt! [START_REF] Miller | Automatic grasp planning using shape primitives[END_REF] is a popular software able to generate feasible grasps on an object. It starts from known grasps on the object shape's primitives, which are then combined and tested in simulation for feasibility. More recently, researchers have focused their attention on finding feasible grasps from point cloud data [START_REF] Fischinger | Learning grasps for unknown objects in cluttered scenes[END_REF][START_REF] Gualtieri | High precision grasp pose detection in dense clutter[END_REF][START_REF] Pas | Grasp pose detection in point clouds[END_REF]. For this purpose, ten Pas et al. [START_REF] Pas | Grasp pose detection in point clouds[END_REF] find grasps for a 2-fingered manipulator, without any prior knowledge of the environment, using only a point cloud obtained from a depth camera. Bonilla et al. [START_REF] Bonilla | Grasp planning with soft hands using bounding box object decomposition[END_REF] address the issue of planning grasps for soft hands. As soft hands can easily adapt to known basic shapes, the algorithm decomposes the target object into Minimal Volume Bounding Boxes proven to be graspable by the hand in use. Hand poses are then generated using geometric information extracted from such bounding boxes. Björkman et al. [START_REF] Bjorkman | Enhancing visual perception of shape through tactile glances[END_REF] present a probabilistic approach for learning object models based on visual and tactile perception through physical interaction with an object. This model can then be used to retrieve feasible grasp candidates. Generating suitable grasping poses is also a common problem for animation. For example, Li et al. [START_REF] Li | Data-driven grasp synthesis using shape matching and task-based pruning[END_REF] present a datadriven approach to grasp synthesis, starting from a database of captured human grasps. To identify grasp candidates on a new object, they match hand and object shape by identifying features having similar relative placements and surface normals. Similarly, Lui et al. [START_REF] Liu | Dextrous manipulation from a grasping pose[END_REF] describe a physics-based technique taking an initial grasping pose and partial object trajectory as input, and generating believable hand graspings as output. To address unexpected dynamic situations during manipulation, the algorithm generates a range of possible hand configurations, taking into account changes in joint configurations, contact points, and grasping force.

Tangibles within Virtual and Mixed Reality. Interacting with tangible objects has been proven to improve the immersiveness of VR and AR systems [START_REF] Billinghurst | The magicbook-moving seamlessly between reality and virtuality[END_REF][START_REF] Harley | Tangible vr: Diegetic tangible objects for virtual reality narratives[END_REF][START_REF] Hettiarachchi | Annexing reality: Enabling opportunistic use of everyday objects as tangible proxies in augmented reality[END_REF]. Billinghurst et al. [START_REF] Billinghurst | The magicbook-moving seamlessly between reality and virtuality[END_REF] present an AR tangible book. Users can turn its pages, look at the pictures, and read the text as in a normal book. However, if they use an AR display, they can also see 3-dimensional virtual models popping out of the pages. Users can then change the models by physically turning the book pages. Cheok et al. [START_REF] Cheok | Touchspace: Mixed reality game space based on ubiquitous, tangible, and social computing[END_REF] develop a mixed-reality game carried out in a real environment. Players walk around a large room-size area and pick up real objects, as if they were playing a traditional non-virtual game. However, these real objects are augmented with superimposed virtual objects and figures, e.g., a real box is opened and inside it is found a virtual treasure. More recently, Harley et al. [START_REF] Harley | Tangible vr: Diegetic tangible objects for virtual reality narratives[END_REF] present a system for diegetic tangible objects in VR. They develop four tangible objects prototypes, including a cube, a stuffed animal, a treasure chest, and a wooden boat, providing passive and active haptics. For example, the stuffed animal is a furry, hollow raccoon toy. It has a skeletal cage to give structural integrity to its body and a heartbeat that can be either calm or accelerated. Finally, Dalsgaard and Halskov [START_REF] Dalsgaard | Tangible 3d tabletops[END_REF] combine tangible tabletop interaction with 3D projection so that any tangible on a table can be augmented with visual content corresponding to its physical shape, position, and orientation.

As already mentioned, many work in the literature have argued that a mismatch between virtual and physical objects may severely affect the user's experience and illusion of presence [START_REF] Hershberger | Touch dominates haptic estimates of discordant visual-haptic size[END_REF][START_REF] Lacey | Visuo-haptic multisensory object recognition, categorization, and representation[END_REF][START_REF] Simeone | Substitutional reality: Using the physical environment to design virtual reality experiences[END_REF]. Thus, it is important to well match what users see in the display and what they feel when interacting with the tangible object. In this respect, Hettiarachchi and Wigdor [START_REF] Hettiarachchi | Annexing reality: Enabling opportunistic use of everyday objects as tangible proxies in augmented reality[END_REF] introduce a wearable system able to scan the surroundings of the user and find a matching between the available virtual and tangible objects. However, it only takes into account the objects' size, primitive shape, and orientation. HaptoBend [START_REF] Mcclelland | Haptobend: Shapechanging passive haptic feedback in virtual reality[END_REF] is a self-reconfigurable device able to change its shape to match the target virtual object. A similar result has been also achieved using small robots which can assemble into different shapes [START_REF] Zhao | Robotic assembly of haptic proxy objects for tangible interaction and virtual reality[END_REF]. Sait et al. [START_REF] Han | Evaluating remapped physical reach for hand interactions with passive haptics in virtual reality[END_REF][START_REF] Sait | Physical hand interaction for controlling multiple virtual objects in virtual reality[END_REF] map a single tangible prop to multiple virtual objects distributed throughout a virtual environment. They combine a redirection approach, that uses rotational adjustments to align the user during virtual locomotion, and a resetting approach, that introduces a discrete rotational update when the user virtually approaches a target for interaction. Also Azmandian et al. [START_REF] Azmandian | Haptic retargeting: Dynamic repurposing of passive haptics for enhanced virtual reality experiences[END_REF] use a single tangible prop to provide passive haptics for multiple virtual objects. They achieve this result by dynamically and gradually aligning tangible and virtual objects as the user interacts with the environment. Finally, Eckstein et al. [START_REF] Eckstein | Towards serious games and applications in smart substitutional reality[END_REF] create multiple virtual version of the same real room, with a varying degree of mismatch. They analyze the influence of these environments on the perceived sense of presence and motivation of users.

Our work lies along this line of research and it aims at improving haptic pinching sensations when manipulating a tangible object. With respect to ten Pas et al. [START_REF] Pas | Grasp pose detection in point clouds[END_REF] and Hettiarachchi and Wigdor [START_REF] Hettiarachchi | Annexing reality: Enabling opportunistic use of everyday objects as tangible proxies in augmented reality[END_REF], our approach focuses on the local haptic sensations felt by the user at the grasping location. Although we could retrieve the objects' meshes in a similar fashion to [START_REF] Hettiarachchi | Annexing reality: Enabling opportunistic use of everyday objects as tangible proxies in augmented reality[END_REF][START_REF] Pas | Grasp pose detection in point clouds[END_REF], the proposed approach to generate target grasping poses is substantially different. In this paper, we aim at matching a series of haptically-salient features at the users fingertips, without considering any global shape characteristic.

DESCRIPTION OF THE ALGORITHM

The objective of our algorithm is to provide users with coherent sensations, matching what they see on the display to what they touch in the real environment. Given a set of virtual and tangible objects' meshes, the algorithm identifies two grasping/pinching poses, one on the virtual object and one on the tangible object, maximizing their similarity in terms of haptic perception. The computation is done during the pre-processing phase, and the generated grasping matches are used to provide guidance to the user in VR.

Starting from the meshes of all considered objects, our algorithm behaves in the following way. First, it identifies feasible 2-fingers pinching poses on the considered tangible and virtual objects. Then, for each pose and for each object, the algorithm evaluates a series of haptically-salient characteristics: (i) distance between the two fingertips, (ii) distance of the pinch from the center of mass, (iii) relative orientation between the two fingertips, (iv, v, vi, vii) local surface orientation and curvature. These features have been identified by the authors as a representative -yet non exhaustive -set of important grasping information. Considering a broader range of feature is of paramount importance and will be addressed in the near future (see Sec. 5). Next, the algorithm identifies the two most similar pinching poses, one on the tangible and one on the virtual object, evaluated in terms of the above metrics. Finally, the chosen pinching pose is highlighted in the virtual scene for the user to grasp. It guarantees the best match between what the user sees on the display and what he feels when grasping the tangible object. Of course, the perceptual result highly depends on the intrinsic characteristics of the available tangible and virtual objects. The algorithm is only able to find the best solution given the available objects in the scene, which is not guaranteed to lead to a good/believable interaction. Obviously, as expected, if a tangible replica of every virtual object is available, the algorithm will make the user grasp the virtual and tangible objects in the same place. A more detailed discussion about this point can be found in Sec. 5, together with an analysis of the current limitations and assumptions of the algorithm (e.g., we consider only 2-fingers pinches, we assume a standard size for the fingertip, we consider the meshes of all objects available).

The remaining part of the Section describes in detail the abovementioned process, which is summarized by Algorithm 1.

Algorithm 1:

Input: objects' meshes foreach object do

Step #1: find all feasible pinching poses foreach pinching pose do

Step #2: compute haptic characteristics Retrieve the two criteria 7-tuples

[ζ 1 ζ 2 ]
Find all combinations of poses [V, T] such that one pose is on a virtual object and one on a tangible object.

foreach set of poses [V, T] do

Step #3: choose the matching pinch Evaluate the error between current set of poses ∆ζ Evaluate mean error ∆ζ = ∑ 14 i=1 α i ∆ζ 14

Choose the pose set [V, T] minimizing ∆ζ

Step #1: finding the set of feasible grasping poses

"A grasp is every static hand posture with which an object can be held securely with one hand [START_REF] Feix | The grasp taxonomy of human grasp types[END_REF]." In this work, as a starting point, we only consider thumb-index fingers precision grasps, as defined by Cutkosky [START_REF] Cutkosky | On grasp choice, grasp models, and the design of hands for manufacturing tasks[END_REF]. We believe that considering a broader range of grasps is important, and it will be addressed in the near future (see Sec. 5).

As only two fingers are involved, we refer to this type of grasp as a "pinch," and we then refer to the pose of the two fingertips during the grasp as a "pinching pose." The contact area between each fingertip and the object's surface is referred to as a "contact patch." For simplicity, we consider circular patches, centered in a point p ∈ R 3 on the mesh and including all points on the mesh closer than 8 mm. Once the considered tangible and virtual objects are identified and their mesh is retrieved, we use CGAL to compute the set of all feasible thumb-index fingers precision grasps. Each feasible grasp is composed of two fingertip poses, which in turn define two contact patches. Variables related to pinches on the tangible and virtual objects are indicated with T and V subscripts, respectively. Similarly, the two contact patches on the same object are indicated with 1 and 2 subscripts.

Step #2: computation of the haptic characteristics

To measure the haptic similarity of two pinching poses, for each contact patch, we evaluate seven representative haptically-salient characteristics, ζ = [w d φ ψ ρ κ M κ m ] ∈ R 7 . They take into account both global (w d φ ) and local (ψ ρ κ M κ m ) haptic information. As mentioned at the beginning of this section, we plan to consider more criteria in the near future. This study presents a proof-of-concept of our approach and its implementation. Table 1 shows an example of each criterion. The blue triangles indicate the two pinching poses p 1 , p 2 , the blue line the vector --→ p 1 p 2 connecting them, the red line the object's first principal component axis, and the red dot its center of mass.

Criterion #1: pinch size

Hershberger and Misceo [START_REF] Hershberger | Touch dominates haptic estimates of discordant visual-haptic size[END_REF] found out that, between vision and haptic information, neither modality inherently dominates the perceived size of an object. Discordant haptic information biases visual size estimates by as much as discordant visual information biases haptic size estimates. Since neither modality captures the other one completely, a discordant stimulus will be perceived as discordant. Flanagan and Mandomir [START_REF] Flanagan | Coming to grips with weight perception: effects of grasp configuration on perceived heaviness[END_REF] also found that the width of the grasp affect the object's perceived weight. Therefore, it is very important to well match the size of the tangible object with respect to its virtual counterpart.

To address this important point, we consider the perceived size of the object as our first criterion. For each pinching pose, we measure the distance w between the two contact patches.

Criterion #2: distance from the center of mass

Although an object's weight can be already estimated when it rests on a stationary hand [START_REF] Lederman | Haptic perception: A tutorial[END_REF], its perceived weight and size when wielded is determined by its resistance to rotational forces of the user's limbs. Amazeen and Turvey [START_REF] Amazeen | Weight perception and the haptic sizeweight illusion are functions of the inertia tensor[END_REF] provides a physical measure of this property by means of the inertia tensor.

To take into account this property, we included in our criteria a measure of the distance between the pinching pose and the object's center of mass. Specifically, we evaluate the distance d between the object's center of mass and the projection of the center of vector --→ p 1 p 2 onto the principal component axis.

Criterion #3: pinch tilt

Proprioception provides humans with information about the relative position of their limbs as well as the effort being employed in movement. Therefore, although it is known that absolute position retargeting is possible [START_REF] Azmandian | Haptic retargeting: Dynamic repurposing of passive haptics for enhanced virtual reality experiences[END_REF], it is paramount to well match the relative position of the fingers during the grasp. Proprioceptive accuracy has been measured to be around 20 mm [START_REF] Crowe | Proprioceptive accuracy in two dimensions[END_REF][START_REF] Stelmach | Aging and proprioception[END_REF], increasing with the length of the movement.

For this reason, we considered the relative orientation between the two pinching locations as our third criterion. Specifically, we measure the angle φ between vector --→ p 1 p 2 and its projection onto the plane orthogonal to the principal component axis.

Criteria #4,5,6,7: local surface orientation

Several studies in the literature suggest that the orientation of the contact surface is an important source of information for curvature perception [START_REF] Chinello | A modular wearable finger interface for cutaneous and kinesthetic interaction: control and evaluation[END_REF][START_REF] Pacchierotti | Wearable haptic systems for the fingertip and the hand: taxonomy, review, and perspectives[END_REF], a dominant source of information both in static [START_REF] Pont | Similar mechanisms underlie curvature comparison by static and dynamic touch[END_REF] and dynamic [START_REF] Wijntjes | Local surface orientation dominates haptic curvature discrimination[END_REF] conditions. Gordon and Morison [START_REF] Gordon | The haptic perception of curvature[END_REF] found the 75% discrimination threshold of 2-cm-long curved objects to be approximately 2 m -1 . Pont et al. [START_REF] Pont | Similar mechanisms underlie curvature comparison by static and dynamic touch[END_REF] found the 84% discrimination threshold of 20-cm-long curved objects to be approximately 0.5 m -1 , suggesting that the threshold may not depend directly on the curvature, but rather on the total variation of the surface orientation. In other works, Pont et al. [START_REF] Kappers | Haptic unilateral and bilateral discrimination of curved surfaces[END_REF][START_REF] Pont | Haptic curvature discrimination at several regions of the hand[END_REF] tested static haptic discrimination of curved strips. Results show that the effective stimulus for curvature discrimination is the difference in local surface attitude.

An infinite number of curvatures exists at a given point on a 3-dimensional surface. Principal curvatures are defined as the maximum and minimum of the normal curvature, and they provide information about the maximum and minimum bending of a surface at one point. Similarly, we can also define the normal vector as the vector perpendicular to the normal curvature. As the fourth, fifth, sixth, and seventh criteria, we consider the maximum principal curvature κ M ; the minimum principal curvature κ m ; the angle ρ between the maximum principal curvature direction and the principal component projected onto the patch; and the angle ψ between the pinching vector --→ p 1 p 2 and the normal vector at the surface.

Table 1: Examples of how criteria are evaluated. The rows show a representative example of how each criterion is used to find a pinch match between two different shapes, i.e., one tangible and one virtual objects. Each column shows one criterion and how the matching pinch would be chosen if considering only that criterion. For each object, the blue triangles indicate the two pinching positions p 1 , p 2 , the blue line the vector --→ p 1 p 2 connecting them, the red line the object's first principal component, the red dot its center of mass, and the green line a perpendicular to the principal component. To choose the best matching pinching pose between two objects, i.e., one virtual and one tangible, we perform an exhaustive search between all the feasible poses on the two objects. For each couple of poses, one per object, we calculate the error vector between the haptic criteria, i.e., ∆ζ

= [ζ V,1 ζ V,2 ] -[ζ T,1 ζ T,2 ] ∈ R 14 .
Referring to our example above, 50 feasible pinching poses on each object will result in 50 × 50 = 2500 ∆ζ comparisons. To be able to consider all criteria at once and retrieve a single number describing the haptic similarity of two pinching poses in two different objects, we calculate a weighted sum of the error metric ∆ζ . Then, we compute ∆ζ = ∑ 14 i=1 α i ∆ζ 2 i for each comparison, where α i are weighting factors. These weights can (and should) be adjusted according to their perceptual importance (see Sec. 5). In our implementation, for the sake of simplicity, we scale each element of ∆ζ to have them varying in the same range. Completing our example above, 2500 ∆ζ ∈ R 14 comparisons will lead to the same number of ∆ζ ∈ R. We are finally ready to choose our best matching pinch as the one showing the smallest composite error ∆ζ .

Of course, we are not limited to choosing the best matching pose only between two objects. The proposed algorithm is quite flexible and can take into account a larger number of objects. Imagine an experimental setup composed of n tangible objects and m virtual ones, with m ≫ n. Our algorithm can analyze all the scene at once and then find the best pinch for each virtual object using the available set of tangible ones.

USER STUDY

We conducted a preliminary user study to evaluate the user perception when he is manipulating a tangible object (TO) and visualizing virtual objects (VO) that do not always match the manipulated tangible object. The participants were able to manipulate three basic tangible shapes: a cone, a cube and a cylinder (see Table 2). The virtual objects were designed as a combination of these three different shapes. The participants were asked to rate the similarity between the manipulated object and the seen object. The suggested grasping positions of our approach were compared to grasping positions suggested by a global registration of the tangible and the virtual objects. The main hypotheses of our user experiment are:

• H1: when the tangible object and the virtual object are the same, the user perceives the objects as similar.

• H2: when the tangible object and the virtual object are different, the pinching positions suggested by our approach improves the perceived similarity between the two objects.

• H3: whatever the user is just touching the tangible object or lifting it, the perceived similarity remains identical.

Experiment Description

Participants

Twelve participants (10 males, 2 females, M = 26.67, SD = 5.31) took part to the experiment, all of whom were right-handed.

Experimental Apparatus

The setup is shown in Figure 2. Users wore an HTC Vive headset which displays the virtual scene with the virtual objects superimposed with the tangible one. A Bonita Vicon system combined with the Blade software was used to track their thumb and index fingers, at the back of which we positioned 3D-printed attachments holding a constellation of markers. We made sure that the pad of each finger remained free from the attachments, and to ensure that these did not move we secured them with wig tape. Using this tracking system, participants were able to control index and thumb avatars of their fingers in the virtual environment. This minimalist hand representation [START_REF] Argelaguet | The role of interaction in virtual embodiment: Effects of the virtual hand representation[END_REF] has been chosen to avoid occluding the virtual object from the user's point of view (see inset of Figure 2). The virtual scene is composed of an instruction panel and a pedestal onto which one VO is standing. We chose not to track the tangible object for not affecting its perception. To know its position, we built a 3D-printed support, fixed to the table, which kept the tangible object at a known location.

To make sure that the TO was correctly replaced at the beginning of each trial, the support and the TO base had complementary shapes and magnets. This support also served as a reference point in the real environment. The virtual scene was then centered on this point using a Vive Tracker. Since Blade had also its own reference frame, we calibrated it using a third constellation. Therefore, the position of the TO was always known, standing on the calibrated support, while the VO was generated onto the TO at the beginning of the interaction. The virtual fingers were calibrated through the tracked constellations by touching a tangible prop at a specific known location. This allowed us to know where the contact between the fingers and the TO was with respect to the reference. In this way, we ensured that, whenever users were touching the TO, they were also touching the VO accordingly. This also resulted in the VO always moving together with the TO.

Visual feedback was used to guide users toward the target grasping pinches. Two virtual green cones indicated where the user was supposed to grasp the virtual (and tangible) objects. This pose was generated following the algorithm presented in Sec. 3. Visual feedback was also used to let participants know when they arrived close enough to the targeted pinch: whenever the users fingers were closer than 5 mm to their target pinch location and their orientation error w.r.t. the pinch main axis was lower than 14 • , the grasping was considered successful and the VO turned red [START_REF] Prachyabrued | Visual feedback for virtual grasping[END_REF]. Holding this position for 1 s validated the pinch.

Procedure

During the experiment, participants were asked to perform two tasks: a static task, where they simply grasped the objects; and a dynamic task, where they grasped and lifted the objects.

First, participants were asked to grasp the objects as indicated on the screen, matching their two virtual fingers with the pose suggested by the algorithm via the green virtual cones (see first inset of Figure 2). When the pinch is validated, the object either turns green to confirm the end of the task (static task), or it turns blue to instruct the user to lift it (dynamic task). In this latter case, participants were asked to lift the tangible object up 10 cm and tilt it on each side. After that, participants were asked to put the object down and the task was considered completed. Thanks to the magnets on the support, it was easy to put back the tangible object on its original location. At the end of each trial, participants were asked to answer the following question: "How much does the tangible object correspond to the virtual one?" (see second inset of Figure 2) using a 7-item Likert scale.

The experiment started with an explanation of the procedure and the signature of the consent form. Then, a calibration of the virtual fingers was performed for each participant. Participants went first through all the trials of the static task, and then through the ones of the dynamic task. Before each task, 3 random practice trials were performed to ensure that the user had understood the interaction, the task, and that the calibration was done properly. For each task, participants went through all the pinch pose registrations of the TO with the 6 VOs, either with or without our algorithm, in a random order. Since the algorithm allows a translation along the principal component axis, some of the computed registrations brought the VOs to be partially below or above the TO. The height of the pedestal was modified accordingly. The experimenter also faked the change of the tangible object between each trial.

Design

We chose a unique TO composed of three basic shapes: a cone, a cube, and a cylinder (see the representation in Table 2). The three shapes have the same height and the same width in the middle. The VOs correspond to a combination of these three different shapes. As a result, we ended up with 6 VOs (see Table 2).

In this experiment, we only considered three possible pinch poses on the TO, located at the middle of each shape (see Table 2, for their localization). Then, we used our algorithm to compute the best match between these poses and one potential pinch pose for each VO. We decided to take a set of 17 potential valid pinch poses per VO among the initial set of infinite pinch poses. The matching positions provided by our algorithm are shown in Table 2.

To better evaluate the influence of Criterion #2 (distance to the center of mass) on the suggested pinches, we divided the experiment in two identical parts: a first one where the participant had just to pinch the object (static task), not manipulating the tangible object and thus feeling any inertia, and a second one where the participant had to lift the object (dynamic condition). This order was not counterbalanced since the dynamic task could bring some additional cues about the properties of the tangible object.

In order to assess the performances of our approach, we chose to compare the registration provided by our algorithm with a reference corresponding to a simple global superimposition of the VO with the TO. For each task (static/dynamic), the participants had to evaluate the registration between the three pinch poses on the TO (cone, cube, cylinder) and three pinch poses on the VO (middle of the object for the Ref algorithm, generated following the description in Table 2 for our algorithm). Thus, for 6 different VOs, we obtained a block of 18 pinch poses x 2 algorithms = 36 trials. The different poses were presented in a randomized order within a block. The participant had to do each block three times, for a total of 108 trials for a task, and therefore 216 trials in total for the whole experiment, which lasted 1.5 hours.

Conditions

Three conditions are considered in our experimental design:

• C1 is the registration algorithm between the TO and the VO: the algorithm of reference Ref corresponds to a superimposition of both objects, without using any information on the object properties; our algorithm is using the cost function described in Section 3 for finding the matching pose on the VO, given a pose on the TO.

• C2 is the shape manipulated on the TO: cone, cube or cylinder (see Table 2).

• C3 corresponds to the virtual object. The 6 VOs are represented in Table 2.

Collected Data

For each trial, we collected the participant's answers on the correspondence between the VO and the TO using a 7-item Likert scale. Participants also completed a subjective questionnaire at the end of the experiment, following a 7-item Likert scale:

• Q1: It felt like grabbing a real object.

• Q2: It felt like lifting a real object.

Table 2: Tangible object (left) and the 6 virtual objects (right). All pinch poses are in the frontal plan. The three pinch poses of the TO are: 0 in blue at the middle of the cylinder, 1 in green at the middle of the cube and 2 in magenta at the middle of the cone. The associated pinch poses computed by our algorithm on each virtual object are displayed with the same color code. The "F" means that the registration was performed by flipping the VO upside down. • Q5: The shape allows me to tell if both objects correspond to each other. • Q6: The curvature allows me to tell if both objects correspond to each other. • Q7: The width allows me to tell if both objects correspond to each other. • Q8: The perceived inertia allows me to tell if both objects correspond to each other. • Q9: I felt tired at the end. • Q10: It felt as if the displayed objects were not all positioned at the same height. We also asked them the following question at the end of the questionnaire "How many different tangible objects did you interact with in the experiment?".

Model of the

Results

Static Task

The answers of the participants concerning their perceived correspondence between the TO and the VOs are summarized in Table 3.

To study the perceived correspondence between the TO and the VO when they are identical, we first analyzed the ratings for the answers concerning VO #1 only (first column of the table). We combined answers from both algorithms since the pinch poses are identical for VO #1. We found a significant effect of the shape using a linear mixed model on the ratings (F(2, 202) = 10.02, p < 0.001). We performed a post-hoc analysis on the condition C2 using a Tukey test. Bonferroni correction was used for all post-hoc analysis. We found that the correspondence between the cones of the TO and the VO was less perceived than for the cube (Z = 4.4, p < 0.001) and for the cylinder (Z = 2.93, p = 0.009). We did not find any significant effect between the cube and the cylinder for VO #1.

To study the perceived correspondence in the general case, we used a linear mixed model on the collected data to model the probability of perceived correspondence with respect to the three independent variables C1, C2 and C3 defined in the experimental design. The participants are considered as a random effect in the model. We performed an analysis of variance of the model and we found a significant effect for all conditions C1 (F = (1, 1274) = 466.48, p < 0.001), C2 (F = (2, 1274) = 30.39, p < 0.001), and C3 (F = (5, 1274) = 51.62, p < 0.001) as well as an interaction between C1 and C2 (F = (2, 1274) = 17.83, p < 0.001).

We performed a post-hoc analysis on C1 using a pairwise comparison and we found that the participants rated higher the correspondence between TO and VO for the poses proposed by our algorithm compared to the reference algorithm (Z = 21.6, p < 0.001), whatever the object (p < 0.001 for all with Z = -8.06 for the cone, Z = -16.47 for the cube and Z = -12.88 for the cylinder). We performed also a post-hoc analysis on C2 knowing C1 using a pairwise comparison. For the reference algorithm, we found that the correspondence between the cylinder of the TO and the corresponding VO was rated higher than the cone (Z = 2.87, p = 0.01) and the cube (Z = 3.08, p = 0.005). For our algorithm, the participants rated lower the correspondence between the cone of the TO and the corresponding pose on the VO compared to the cube (Z = -8.21, p < 0.001) and the cylinder (Z = -7.70, p < 0.001). Finally, we performed also a post-hoc analysis on C3 and we found that VO #1 was rated higher than the other VOs (p < 0.001 for each VO) and than VO #4 was rated lower than the other VOs (p < 0.001 for each VO, except with VO #5 with p = 0.007). We found also that VO #2 was rated higher than VO #5 (p = 0.04). Table 3: Ratings (Means and SD) for the static task using a 7-item Likert scale on the correspondence between the TO (first column, three shapes: cone, cube and cylinder) and the six different VOs, for the two compared algorithms (Reference and our approach). Ratings are in bold when TO shape and VO shape were identical.

Dynamic Task

The answers of the participants concerning their perceived correspondence between the TO and the VOs are summarized in Table 4.

To study the perceived correspondence between the TO and the VO when they are identical, we first analyzed the ratings for the answers concerning VO #1 only. We combined answers from both algorithms since the pinch poses are identical for VO #1. We found a significant effect of the shape using a linear mixed model on the ratings (F(2, 202) = 5.18, p = 0.006). We performed a post-hoc analysis on the condition C2 using a Tukey test and we found that the correspondence between the cones of the TO and the VO was less perceived than for the cylinder (Z = 3.15, p = 0.004). We did not find any significant effect between the cube and the cylinder as well as between the cube and the cone for VO #1.

As for the static task, we used a linear mixed model on the collected data to model the probability of perceived correspondence with respect to the three independent variables C1, C2 and C3 defined in the experimental design for the general case. We performed an analysis of variance of the model and we found also a significant effect for all conditions C1 (F = (1, 1274) = 422.07, p < 0.001), C2 (F = (2, 1274) = 48.76, p < 0.001), and C3 (F = (5, 1274) = 61.09, p < 0.001) as well as an interaction between C1 and C2 (F = (2, 1274) = 29.68, p < 0.001). We performed a post-hoc analysis on C1 using a pairwise comparison and we found that the participants rated higher the correspondence between TO and VO for the poses proposed by our algorithm compared to the reference algorithm (Z = 20.54, p < 0.001), whatever the object (p < 0.001 for all with Z = 5.70 for the cone, Z = 16.03 for the cube and Z = 13.85 for the cylinder). We performed also a post-hoc analysis on C2 knowing C1 using a pairwise comparison. For the reference algorithm, we found that the correspondence between the cylinder of the TO and the corresponding VO was rated higher than the cone (Z = 2.43, p = 0.04). For our algorithm, the participants rated lower the correspondence between the cone of the TO and the corresponding pose on the VO compared to the cube (Z = -10.62, p < 0.001) and the cylinder (Z = -10.59, p < 0.001). Finally, we performed also a post-hoc analysis on C3 and we found that VO #1 was rated higher than the other VOs (p < 0.001 for each VO) and than VO #4 was rated lower than the other VOs (p < 0.001 for each VO). Table 4: Ratings (Means and SD) for the dynamic task using a 7-item Likert scale on the correspondence between the TO (first column, three shapes: cone, cube and cylinder) and the six different VOs, for the two compared algorithms (Reference and our approach). Ratings are in bold when TO shape and VO shape were identical. Participants mostly felt that they were seeing their own fingertips (Q4, M = 5.17, SD = 1.77). Figure 3 shows the answers to our subjective questionnaire for the questions regarding the properties of the object (Q5 to Q8). Among the different properties proposed for rating to the participants, they mostly felt that the shape allowed them to tell if both objects correspond to each other (Q5, M = 5. Figure 3: Bar-graph of the answers for questions 5 to 8 concerning the properties of the object that helped the participants to find a correspondence between the TO and the VO. Q5: shape, Q6: curvature, Q7: width, Q8: perceived inertia.

DISCUSSION

In this paper, we propose to use few tangible objects to enable 3D interactions with multiple virtual ones. To do so, we devised an algorithm which performs an exhaustive search on all the feasible pinching poses on the available objects, to find the best match in terms of haptic sensations. The algorithm is designed to provide the best available match between what the user sees in the virtual environment and what he feels when grasping the tangible object.

To evaluate the effectiveness of our approach, we carried out a user study. Twelve participants were asked to grasp objects of different shapes in virtual reality, all rendered by the same tangible, and evaluate how much the former corresponds to the latter. We made three hypotheses, described at the beginning of Sec. 4. First, we wanted to understand if participants are actually able to recognize when they are touching the same object in the real and virtual environments (H1). Results show that, indeed, they can. Values in bold in Tables 3 and4 show users' ratings when provided with the same tangible and virtual shape. As expected, in both conditions (our algorithm and simple superimposition) ratings are very high. The second question we raised is whether, when tangible and virtual objects are different, using our approach improves the perceived grasping sensations with respect to using a standard superimposition (H2). Ratings show that our algorithm was able to well combine the considered haptic features and find convincing pinches between the given tangible and virtual objects. We registered improvements in ratings of 48.8% and 45.1% with respect to a standard superimposition matching technique in the case of static and dynamic grasping tasks, respectively. From Tables 3 and4 we can see that ratings when using our algorithm are significantly closer to the ground-truth than when using simple superimposition. We also hypothesized (H3) that the above results hold when grasping an object with no subsequent motion (static task) as well as when lifting it up (dynamic task). Results show indeed little difference between the two tasks. At the end of the experiment, participants were also asked to guess how many tangible objects were used during the trials. Answers spanned from 2 to 20, with an average of 7.5 objects, and no one guessed that only one tangible object was used! Although we have achieved promising results, our work still shows some limitations. First, our algorithm does not consider that humans have fingertips of different size and stiffness, which can significantly affect how they perceive a surface. Moreover, we only considered a reduced number of criteria for our metric. Of course, for future work, it would be important to include other perceptuallyrelevant properties, such as the object's texture. We also consider pinch grasping only, but extending our approach to other types of grasping should be possible. For example, extending our method to precision grasps with more than 2 fingers would require extending the description of our poses to more contact patches and the computation of the criteria to account for the increased number of contact couples. Also the interaction techniques used to grasp the objects can be improved, to ensure that the user always grasps the objects at the right location. Finally, another limitation of our approach is the tracking. Although we used a state-of-the-art tracking system, it is not easy to consistently and robustly track the fingertips at all time, and any discrepancy between the motion of the user's fingers and the virtual avatar will degrade the experience. However, when the same shapes were registered between the TO and VO, the experiment still gave the best scoring, highlighting the fact that the tracking issues did not hinder the perception of the users.

CONCLUSIONS

We presented an innovative haptic approach which enables the use of a reduced number of tangible objects for interacting with multiple different virtual ones. The need for such an approach stems from two conflicting requirements. To achieve a compelling illusion, there should be a good correspondence between what users see in the virtual environment and what they touch in the real world. In other words, the haptic features of the tangible and virtual objects should match. However, this calls for the creation of tangible replicas of all the virtual objects in the scene -which is often not possible. Therefore, it is important to devise approaches maximizing tangible/virtual objects matching, even when these are different. The proposed algorithm addresses this problem. It analyzes the provided tangible and virtual objects to find the best pinch in terms of matching haptic sensations. It starts by identifying several suitable pinching poses on all objects. Then, for each pose, it evaluates a series of haptically-salient characteristics. Next, it identifies the two most similar pinching poses, one on a tangible and one on a virtual object. Finally, it highlights the chosen pose, which provides the best matching sensation between what users see and touch.

We tested our approach in a user study. Twelve participants were asked to grasp different virtual objects, all rendered by the same tangible one. For every virtual object, our algorithm found the best pinching match on the tangible one, and guided the participant toward that grasp. Results show that our algorithm was able to well combine several haptically-salient object features to find corresponding pinches between the given tangible and virtual objects. At the end of the experiment, participants were also asked to guess how many tangible objects were used during the experiment. No one guessed that we used only one, proof of a convincing experience.

This work then opens several interesting prospects: we could scan the real environment to exploit the real physical surroundings of the user. Our technique would then enable to provide precise pinching sensations. However, of course, this raises the challenge of real-time tracking algorithms. Moreover, using our algorithm, we could analyze the haptic properties of a given virtual scene, and we could try to automatically generate and 3D print, one (or more), universal tangible objects able to provide the best possible match in terms of haptic sensations.

Finally, although the results of our user study are promising and statistically significant, we believe that more experiments could be conducted, considering more objects, different tasks, different contexts.

  #3: choosing the best matching pinch At the end of the previous step, each feasible pinching pose identified in Sec. 3.1 is assigned two 7-tuples [ζ 1 ζ 2 ], one per contact patch. Each 7-tuple contains the seven criteria metrics discussed in Sec. 3.2. For example, if we found 50 feasible pinching poses on a given object, we will end up with 100 7-tuples (two per pose) and 700 metrics (fourteen per pose). However, it is important to notice that the first three criteria [w d φ ] provide pose-specific information while the last three [ψ κ M κ m ] provide patch-specific information. Therefore, as we generate one 7-tuple per patch (and two per pinching pose), the first three components of [ζ 1 ζ 2 ] will be the same, i.e., [w 1 d 1 φ 1 ] = [w 2 d 2 φ 2 ].

Figure 2 :

 2 Figure 2: Setup for the experiment. Insets show the virtual environment during the task.

6 •

 6 Q3: It was hard to grab the object at the right location. • Q4: It felt like I was seeing my own fingertips.

4. 4 . 3

 43 QuestionnaireRegarding Q1 and Q2, most of the participants felt like grabbing (M = 5.67,SD = 0.75) or lifting (M = 5.58 ; SD = 0.86) the real object during the experiment. Regarding the difficulty to grab the object (Q3), the feeling is split between the participants (M = 4.00, SD = 1.35) with rates from 2 (2 participants) to 6 (2 participants).

  92, SD = 1.03) as well as the curvature (Q6, M = 5.42, SD = 1.44). The answers were more mitigated for width (Q7, M = 3.83, SD = 1.77) and inertia (Q8, M = 4.75, SD = 1.88). Finally, most of the participants reported overall medium levels of fatigue (Q9, M = 5.33; SD = 1.37). Most of the them also reported the difference of height between the displayed objects (Q10, M = 5.33, SD = 1.80). The answer to the question regarding the number of different tangible objects they were interacting with was between 2 to 20 (M = 7.5, SD = 5.34).

* This research has received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement No 801413 for project "H-Reality."† e-mail: {xavier.de-tinguy, claudio.pacchierotti, maud.marchal, anatole.lecuyer}@irisa.fr.

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