Diogo C Barros
email: diogo.camello-barros@univ-amu.fr

Yanchong Duan

Daniel R Troolin
email: dtroolin@tsi.com

Ellen K Longmire

Air-filled soap bubbles for volumetric velocity measurements

Keywords:

The use of air-filled soap bubbles for volumetric velocimetry in air flows is demonstrated experimentally. The tracers are produced by a novel system that seeds high number density soap bubble streams for particle image velocimetry applications. Particle number density considerations, spatial resolution and response time scales are discussed in light of current seeding techniques for volumetric measurements. The micro soap bubbles are employed to measure the 3D velocity field in intermediate size flow volumes of 75 -500 cm 3 , which are difficult to measure with existing tracers such as liquid droplets or relatively large helium-filled soap bubbles, while resolving small scale flow variations. The mean volumetric concentration of approximately 50 tracers per cm 3 provides a potential for highresolution measurements. Finally, the methodology is demonstrated for velocity measurements in the wake of a sphere immersed in a turbulent boundary layer. Both the wake structures and the boundary layer statistics are characterized successfully with a reasonable spatial resolution.

sure 3D velocity fields accurately. In particular, particle image velocimetry (PIV) and particle tracking velocimetry (PTV) are now widely employed in various flow configurations [START_REF] Discetti | Volumetric velocimetry for fluid flows[END_REF]. All of these techniques rely on imaging of discrete seeding tracers suspended in the flow. While larger seeding particles scatter more light from the illuminating source, smaller ones follow the flow more accurately and can resolve smaller scale variations. This trade off is inherent to particle-based velocimetry [START_REF] Melling | Tracer particles and seeding for particle image velocimetry[END_REF].

Adrian and Westerweel give a detailed discussion of possible tracer particles for gases and their performance [START_REF] Adrian | Particle image velocimetry[END_REF]. Current seeding tracers for air flows include micron-size oil and di-ethyl-hexyl-sebacic acid-ester (DEHS) droplets [START_REF] Kähler | Generation and control of tracer particles for optical flow investigations in air[END_REF] as well as soap bubbles filled with a low-density gas such as helium. While droplets of size 1µm typically can follow fluid elements faithfully and remain suspended in the flow, the maximum size of the volumetric domain considered is usually limited to smaller than 50 cm 3 [START_REF] Scarano | Tomographic PIV: principles and practice[END_REF]. On the other hand, large heliumfilled soap bubbles (HFSB) ranging from 1 mm to 5 mm allow only qualitative investigation of larger domains, as they do not faithfully follow the flow like neutrally buoyant bubbles [START_REF] Kerho | Neutrally buoyant bubbles used as flow tracers in air[END_REF]. By contrast, sub millimeter (300µm) helium filled soap bubbles have been utilized successfully in both low and high-speed tomo-PIV configurations for large-scale aerodynamic applications [START_REF] Scarano | On the use of helium-filled soap bubbles for large-scale tomographic PIV in wind tunnel experiments[END_REF] in domains larger than 4500 cm 3 .

A compilation of volumetric measurements in air flows using these seeding strategies is shown in figure 1. There is a noticeable gap between volumetric domains in the range between nearly 50 cm 3 and 5000 cm 3 . This is an effect of the large difference between the mean seeding diameters of DEHS droplets and HFSB bubbles. The purpose of the current work is to demonstrate the viability of smaller size airfilled soap bubbles as seeding tracers in gas volumes of intermediate size. The bubbles are generated by the rapid de-pressurization of a pressurized mixture of liquid and surfactant. The depressurization causes the gas to leave the liquid media. This controlled degasification inside a reservoir produces a large number of soap bubbles with mean diameter of 15µm. The bubbles can be imaged easily either by low or high-speed light sources, yet they are small enough to resolve turbulent flow structures in intermediate-size domains. Fig. 1 Compilation of volumetric velocimetry investigations in air flows: acquisition frequency of the PIV set-up and the imaged volume. References: [START_REF] Atkinson | The accuracy of tomographic particle image velocimetry for measurements of a turbulent boundary layer[END_REF], [START_REF] Kühn | Large-scale tomographic particle image velocimetry using helium-filled soap bubbles[END_REF], [START_REF] Cafiero | Flow field topology of submerged jets with fractal generated turbulence[END_REF], [START_REF] Caridi | Hfsb-seeding for large-scale tomographic PIV in wind tunnels[END_REF], [START_REF] Boushaki | Volumetric velocity measurements (V3V) on turbulent swirling flows[END_REF], [START_REF] Pröbsting | On the estimation of wall pressure coherence using time-resolved tomographic PIV[END_REF], [START_REF] Ghaemi | Counter-hairpin vortices in the turbulent wake of a sharp trailing edge[END_REF], (Schneiders et al, 2016;Schneiders and Scarano, 2016), [START_REF] Elsinga | Tomographic particle image velocimetry[END_REF], [START_REF] Schröder | Investigation of a turbulent spot and a tripped turbulent boundary layer flow using time-resolved tomographic PIV[END_REF], [START_REF] Humble | Three-dimensional instantaneous structure of a shock wave/turbulent boundary layer interaction[END_REF], [START_REF] Terra | Aerodynamic drag of a transiting sphere by large-scale tomographic-PIV[END_REF][START_REF] Terra | Drag resolution of a PIV wake rake for transiting models[END_REF], [START_REF] Scarano | On the use of helium-filled soap bubbles for large-scale tomographic PIV in wind tunnel experiments[END_REF], [START_REF] Violato | Lagrangian and eulerian pressure field evaluation of rod-airfoil flow from time-resolved tomographic PIV[END_REF], [START_REF] Michaelis | Reconstructing the vortex skeleton of the desert locust using phase averaged POD approximations from time resolved thin volume tomographic PIV[END_REF] and [START_REF] Staack | 3D-3C-coherent structure measurements in a free turbulent jet[END_REF]. The squares represent volumetric measurements using seeding particles with diameter of about 1µm, while the circles indicate investigations where HFSB were used. The crosses represent the two volumetric domains considered in the present work.

The paper is organized as follows. In section §2, we discuss spatial and image considerations for volumetric velocimetry in air flows, focusing on the seeding size. The methodology to generate the soap bubbles and their characteristics are outlined in §3, where a full description of the response time of the tracers is demonstrated theoretically and experimentally. A demonstration of the use of micro soap bubbles in volumetric wind-tunnel measurements is provided in section §4. Finally, our concluding remarks and perspectives on these new seeding tracers are discussed.

Considerations of seeding size

Consider a volume V = L × H ×W imaged by a camera with a pixel resolution l * , pixel pitch ∆ pix equipped with an optical system with numerical aperture f # by the use of an illuminating source with wavelength λ . In the volume V , W represents the depth of the domain.

The depth of focus reads

∆ Z = 4.88 f # 2 λ 1 + 1 M 2 . (1)
Here, M is the optical magnification M = l * ∆ pix /L. Let d p be the diameter of the seeding particle. The particle image diameter can be calculated from

d τ * = (Md p ) 2 + d diff. 2 + d B 2 ∆ pix , (2)
where

d diff. = 2.44λ f # (1 + M)
is the diffraction limited minimum image diameter and d B is the blur circle diameter [START_REF] Raffel | Particle image velocimetry: a prac-tical guide[END_REF][START_REF] Scarano | Tomographic PIV: principles and practice[END_REF]. Let us take the example of the experiments performed in [START_REF] Scarano | On the use of helium-filled soap bubbles for large-scale tomographic PIV in wind tunnel experiments[END_REF] with HFSB bubbles. The illuminated volume was such that L = H = 200 mm, W = 120 mm, l * = 1024 px, ∆ pix = 20 µm and f # = 32 to ensure focus along all the depth W (∆ Z ∼ 300 mm). Assuming d p = 400 µm and neglecting d B , the resulting particle image diameter is d τ * = 3 px, a consistent value found by the authors in their experiments of helium bubbles. Now, decreasing the domain size to L = H = 50 mm while keeping all other parameters the same, d τ * = 9 px due to the change of optical magnification resulting in ∆ Z ∼ 30 mm, i.e. an equivalent volume of approximately V = 75 cm 3 when compared to V = 4800 cm 3 from [START_REF] Scarano | On the use of helium-filled soap bubbles for large-scale tomographic PIV in wind tunnel experiments[END_REF]. To obtain nearly optimal particle image diameters of d τ * = 3 px, the tracer size should be reduced to about d p = 30 µm or less considering the diffraction limit. Here, the value of three pixels is near optimal for minimizing both bias and random error in calculating PIV particle displacement [START_REF] Adrian | Particle image velocimetry[END_REF]. These examples clearly demonstrate that seeding tracers with diameter an order of magnitude smaller than HFSB are preferable for intermediate volume domains. At the same time, the tracers must be larger than DEHS droplets to scatter more light. Additionally, one should note that d τ * = 9 px for a helium-filled soap bubble image would introduce a doublet pattern resulting from the presence of two glare spots [START_REF] Scarano | On the use of helium-filled soap bubbles for large-scale tomographic PIV in wind tunnel experiments[END_REF]. This aspect makes the helium filled soap bubbles less suitable for such a field-of-view. Specifically, pairs of glare points add complexity and uncertainty to identification of particle centers in tracking methods and ambiguity in determination of the correct displacement peak in PIV correlation methods.

Another point of interest is the final resolution of the velocity measurements. Consider now the same domain V = L × H ×W divided in voxels. We define l IB as the length of the interrogation box (IB), i.e. the spatial resolution of the particle-based velocimetry. The number of seeding particles within the interrogation box is given by:

N p IB = n v IB ×C ∆ pix M 3 . (3)
Here, n v IB is the number of voxels per interrogation box, and C = N p /V is the volumetric particle concentration. The tracer concentration C can be written as a function of the source density N s and the particle image diameter d τ [START_REF] Scarano | Tomographic PIV: principles and practice[END_REF]:

C = N s W M ∆ pix 2 4 πd τ * 2 . (4)
Then, the number of particles per interrogation box reads

N p IB = n v IB N s W ∆ pix M 4 πd τ * 2 . ( 5
)
Assuming n v IB = (l IB /l v ) 3 , where l v is the voxel length, and noting that l v M = ∆ pix , we obtain l IB ∝ d τ 2 3 for a desired number of particles per IB (usually between 5 and 10 particles) and optimal source density [START_REF] Scarano | Tomographic PIV: principles and practice[END_REF]. Thus, for intermediate sized volumes, a substantial gain in spatial resolution is expected by decreasing d p from 400 µm to 30 µm, for example. The size of these tracers allows for a higher number density per unit volume which enables them to follow the turbulent structures of the flow and achieve higher spatial resolution measurements.

3 Description of the micro-soap bubbles

Bubble generator

Common methods to generate soap bubbles include pushing or blowing a gas stream through a liquid film. The support of the liquid film determines the dimension of the produced bubble. The method described here overcomes this limitation by using the process of cavitation in liquid media [START_REF] Payri | Study of cavitation phenomena based on a technique for visualizing bubbles in a liquid pressurized chamber[END_REF][START_REF] Gharib | Method and apparatus for the production of microscale bubbles by depressurization cavitation[END_REF]. The mechanism governing the bubble generation is the rapid depressurization of a pressurized solution of liquid containing a certain amount of surfactant.

A schematic of the current device is seen in figure 2. It consists of a 57 liter reservoir filled approximately onethird full with a 5% surfactant-water solution. The surfactant used was a commonly available dish-washing detergent (Palmovile Pure+Clear) without any added dyes or fragrance and density of approximately 1.06 g/cm 3 . The product ingredients were ammonium lauryl sulfate, lauramidopropylamine oxide, and isodeceth-6.

A tube at the bottom of the reservoir feeds the solution through a set of particulate filters followed by a reciprocating piston pump that pressurizes the fluid to approximately Nozzle 70 bars. The pressurized liquid then travels through highpressure tubing to a set of 10 nozzles, each with outlet diameter of 0.2mm. Within each nozzle, the liquid is accelerated through an area ratio of 156:1 to decrease the local pressure. Gas volumes formed by cavitation then collapse and implode fragmenting into many small bubbles.

The bubble streams exiting the nozzles are directed downward toward the free surface of the reservoir bath. Pressurized air is introduced into the reservoir to generate a mild cyclonic flow that causes larger bubbles and droplets to impact the sidewalls and be reintroduced into the solution, while the remaining bubbles and the air stream exit through an opening at the top of the reservoir. Most of the bubbles created in this way remain suspended for long periods of time.

Bubble size characterization

In order to characterize the bubble size, a phase Doppler particle analysis (PDPA) system was used [START_REF] Bachalo | Method for measuring the size and velocity of spheres by dual-beam light-scatter interferometry[END_REF]. PDPA uses a set of crossed laser beams to create an interference fringe pattern at the point where the beams cross. Spherical particles passing through the beam crossing scatter light collected by a receiver with an array of three detectors. The phase lag of the scattered light between detectors is proportional to the particle diameter. A flexible tube 10 cm in diameter and 3 meters long was attached to the exit on top of the bubble generator. The PDPA system was positioned to measure the bubbles as they exited the tube. The generator produced a flux on the order of 10 7 particles per second, resulting in a concentration of 10 5 tracers/cm 3 at the outlet of the flexible tube. Figure 3 shows the measured diameter distribution of the bubbles generated by the device. The mean diameter was 14.7 µm with a standard deviation of 7.3 µm. The inset shows a high-magnification shadowgraph image of a micro bubble obtained using a PowerView CCD 8MP camera (∆ pix = 5.5 µm) equipped with a long distance microscope Model K2 DistaMax yielding a digital resolution of 1.5 µm/px.

Theoretical estimation of the bubble response time

A primary factor that affects the ability of the tracer to faithfully follow the flow is the sum of forces acting on the particle. A complete account for the motion of small particles in a given flow field can be found in [START_REF] Maxey | Equation of motion for a small rigid sphere in a nonuniform flow[END_REF]. Following [START_REF] Maxey | Equation of motion for a small rigid sphere in a nonuniform flow[END_REF] and [START_REF] Mei | Velocity fidelity of flow tracer particles[END_REF], [START_REF] Adrian | Particle image velocimetry[END_REF] proposed the following expression for the particle response time τ p :

τ p = d p 2 |ρ p -ρ f | 18µφ , (6)
where ρ p is the particle density, ρ f the fluid density and µ the dynamic viscosity of the ambient fluid. The factor φ corrects for the particle Reynolds number Re p (see [START_REF] Clift | Bubbles, drops, and particles[END_REF] for a detailed account). To estimate the response time of seeding tracers commonly used in airflow experiments, we compute τ p using equation 6 and present the values in table 1 for various particles considering the Stokes regime (φ = 1). We consider here air at 20 • C and 1 atm. It is important to note that φ = 1 provides a conservative estimation of τ p since the slip flow and resulting finite particle Reynolds number increases the value of φ . Micron-sized DEHS droplets are recognized to faithfully follow the flow due to their reduced size and can be considered as a reference particle. The characteristics of the DEHS particles used for the calculation of τ p were the same as presented in [START_REF] Kähler | Generation and control of tracer particles for optical flow investigations in air[END_REF]. The response time of DEHS particles obtained is about 3µs, in very good agreement when compared to the experimental results of [START_REF] Ragni | Particle tracer response across shocks measured by piv[END_REF], where τ p = 2µs for shock accelerated tracers.

The same estimation was performed for Helium and air filled soap bubbles (HFSB and AFSB) with properties matching those listed in [START_REF] Faleiros | Helium-filled soap bubbles tracing fidelity in wallbounded turbulence[END_REF]. Our calculation of τ p using equation 6 for HFSB bubbles slightly lighter than air results in τ p = 90µs. This lag is at least fifteen times smaller than the estimated response time of the AFSB bubbles, which have a much larger particle-fluid density difference. Measurements of HFSB response time performed by [START_REF] Faleiros | Helium-filled soap bubbles tracing fidelity in wallbounded turbulence[END_REF][START_REF] Faleiros | Generation and control of helium-filled soap bubbles for PIV[END_REF] in the flow upstream of a circular cylinder yielded experimental magnitudes of τ p of 10 -50µs with a dispersion of approximately 40µs. They also reported a response time for AFSB of approximately τ p = 400µs. In both cases, these experimental values of τ p are smaller than our theoretical estimates. This can be explained by the significant slip velocity yielding a correction factor φ larger than unity. Meanwhile, in the absence of direct slip velocity measurements, equation 6 appears to be a good estimator of the considered response time. According to [START_REF] Kerho | Neutrally buoyant bubbles used as flow tracers in air[END_REF] and [START_REF] Afanasyev | Measuring soap bubble thickness with color matching[END_REF], the wall-thickness of soap bubbles lies in the range 0.1-0.3 µm. Indeed, from the characteristics of HFSB and AFSB tracers described above, a wall-thickness in the range 0.1-0.2 µm was calculated to match the value of ρ p indicated by [START_REF] Faleiros | Helium-filled soap bubbles tracing fidelity in wallbounded turbulence[END_REF]. Considering the bubble wall thickness in the range 0.1-0.3 µm and the particle diameter 10-30 µm , a response time range 20 -160µs was estimated for the micro soap bubbles of the current work, suggesting tracing fidelity capabilities comparable to the HFSB tracers. The response time of the micro soap bubbles was obtained experimentally following the protocol described in previous investigations on HFSB bubbles [START_REF] Scarano | On the use of helium-filled soap bubbles for large-scale tomographic PIV in wind tunnel experiments[END_REF][START_REF] Faleiros | Helium-filled soap bubbles tracing fidelity in wallbounded turbulence[END_REF][START_REF] Faleiros | Generation and control of helium-filled soap bubbles for PIV[END_REF][START_REF] Gibeau | A modular, 3D-printed heliumfilled soap bubble generator for large-scale volumetric flow measurements[END_REF][START_REF] Gibeau | Evaluation of a fullscale helium-filled soap bubble generator[END_REF]. In steady flows, the particle response time can be computed as the ratio between the slip velocity and local fluid acceleration:

τ p = |u -u p | u p ∂ u p ∂ x . (7)
Here, u is the fluid velocity, u p is the particle velocity and the local particle acceleration is kept the same as the fluid acceleration.

We consider the decelerating flow upstream of a circular cylinder. The experiments were performed in a closedcircuit wind-tunnel with test section dimension 1.3×0.96 m 2 and free-stream velocity U ∞ = 15 ms -1 . The turbulence intensity at this free-stream condition is of order 1%. A circular cylinder with diameter D = 70 mm and length 10D is mounted mid height, yielding a nominally two-dimensional flow at the cylinder midspan with a Reynolds number Re = ρ f U ∞ D/µ = 6.8 × 10 4 . This geometry gives a blockage ratio of 4% with negligible blockage effects [START_REF] West | The effects of tunnel blockage and aspect ratio on the mean flow past a circular cylinder with reynolds numbers between 10 4 and 10 5[END_REF].

PIV measurements were performed to compute the fluid and particle (bubble) velocities u and u p respectively. The local fluid velocity u is approximated by the velocity field obtained from PIV measurements using oil droplets of diameter ∼ 1 µm generated by a Laskin nozzle generator. This is justified by their very small response time and high tracing fidelity. An Evergreen 200 mJ laser was used to illuminate a planar field of view limited by x ∈ [-80, 0] mm and y ∈ [-30, 30] mm, where (x, y) = (0, 0) is the stagnation point of the cylinder. A TSI Powerview 8MP camera (3320 × 2496 pixels, ∆ pix = 5.5 µm) equipped with a 135mm objective at numerical aperture f # = 5.6 was used to acquire 3000 double-frame images with time separation of ∆ t = 20 µs and sampling rate 3 Hz. The resulting image magnification is M = 0.165. The images were processed using TSI INSIGHT4G, with a recursive, cross-correlation, ensemble PIV scheme with interrogation window of 64 pixels and 50% overlap yielding the final vector spacing of 1.06mm. These parameters were used for both u and u p measurements. Oil-droplet and bubble measurements were performed on separate days to avoid any cross-contamination of tracers.

The mean streamwise velocity and acceleration decay for bubbles (u p ) along the stagnation streamline is depicted in figure 4. The reference fluid velocity u obtained from the oil droplets measurements is also displayed for comparison. The entire flow field (see inset) represents with fidelity the nominal potential flow upstream of the circular cylinder, and a similar flow topology was obtained using the reference oil droplets as seeding tracers. Along the streamline, the mean slip velocity |uu p | is 0.12 ms -1 , which corresponds to 0.8% of the freestream speed and 1.4% mean local velocity deviation.

To compute the bubble response time τ p , we first performed a local regression filter (loess in MATLAB) on both u and u p to remove residual waviness of the data and to obtain a smooth velocity acceleration decay. Then, τ p was calculated using equation 7. The mean response time computed along the stagnation streamline in the region 0.5D < x < 0 within a strip of 4 mm is τ p = 40 µs with standard deviation σ τ p = 30 µs. This selected region of large deceleration close to the stagnation point was also considered in [START_REF] Scarano | On the use of helium-filled soap bubbles for large-scale tomographic PIV in wind tunnel experiments[END_REF] and [START_REF] Gibeau | A modular, 3D-printed heliumfilled soap bubble generator for large-scale volumetric flow measurements[END_REF] to obtain τ p for HFSB bubbles. The measured τ p = 40 µs lies within the theoretical range 20-160 µs estimated using equation 6, where both bubble film thickness and diameter ranges were considered. A response time of 40 µs would correspond to bubbles with diameter 15 (25) µm and film thickness of 0.15 (0.1) µm, respectively. These values correspond well to the measured particle diameter distribution presented in §3.2.

It is important to consider that our calculation of τ p and σ τ p considered velocity values obtained from a cross correlation algorithm within the PIV interrogation window, in which tracers with multiple properties exist. These effective time scales allows one to compute ideally minimum flow time scales to be measured by the current tracers. As discussed by [START_REF] Gibeau | Evaluation of a fullscale helium-filled soap bubble generator[END_REF], the resulting minimum measurable flow time scale reads τ f = 10 × (τ p + 2σ τ p ) = 1000µs with 95% confidence, accounting for a Stokes number S k = τ p /τ f < 0.1 for tracer reliability [START_REF] Tropea | Springer handbook of experimental fluid mechanics[END_REF].

Volumetric measurements

Experiments were conducted in two wind-tunnels of the Department of Aerospace Engineering and Mechanics of the University of Minnesota to characterize the performance of the micro soap-bubbles at two volumetric scales.

Bubble number density and robustness

Experiments were performed in a large recirculating wind tunnel driven by a frequency-controlled P-38 propeller (same as that used for the experiments in Section 3.4) in order to assess bubble quality in images and bubble robustness over different recirculation times. The flow conditioning upstream of the closed test section includes a honeycomb flow straightener with cell size of 8 mm followed by a set of four screens with openings of 700 µm. The contraction is threedimensional with a 6.5:1 area ratio. The test section measures 1.37 m wide, 0.96 m high, and 2.9 m in length. A volumetric imaging system consisting of four Phantom VEO340 4MP cameras (2560 x 1600 pixels, ∆ pix = 10 µm) fitted with 135mm lenses and f # = 22 operating at 800 Hz were aligned on the measuring volume illuminated by a 30 mJ/pulse dualhead Nd:YLF laser positioned on top of the wind tunnel, with a mirror directing the illumination down into the test section through a window on the top of the tunnel. Light volume optics were used to illuminate a volume of 490 cm 3 with dimensions 130 × 75 × 50 mm 3 . This volume lies between the traditional volumetric domains investigated using either DEHS droplets or HSFB bubbles (see Figure 1). A synchronizer timing device with timing resolution of 0.2 ns coordinated the laser pulses and camera image capture, and the images of the bubbles were processed with INSIGHTV3V-4G software. Details of the processing routines used can be found in [START_REF] Boomsma | Time-resolved particle image identification and reconstruction for volumetric 4D-PTV[END_REF].

The tests demonstrated the capability of performing largescale measurements with a free-stream flow velocity of 2.3 ms -1 . The distribution of the tracers in the volume of interest is shown in figure 5. Here, 14000 bubbles were tracked, and the bubble digital image diameters were distributed across 3-4 pixels on average. After filling the tunnel with bubbles, the bubble generator was turned off, allowing the bubbles to recirculate for 5 minutes. During this period, there was a reduction of 32% in the number of tracers imaged. After an additional 5, 10, and 20 minutes, 64%, 47%, and 33% of the initial number of bubbles remained, respectively. The measurements used to compute the response time in section 3.4 were performed 10 minutes after the generator was turned off. The initial relatively rapid decrease in concentration can be attributed to two potential factors. First, one may assume the existence of some droplets emanating from the bubble generator. Any droplets with diameter within the range 7-35 µm will tend to settle out in a period not exceeding 5 minutes after the bubble generator is turned off. In addition, water droplets in the range 1-4 µm would tend to evaporate completely over the same period, while larger ones would decrease in diameter. Second, some fraction of micro soap bubbles (possibly the largest ones) may disappear in the first minutes due to impacts with surfaces and screens in the recirculating wind tunnel. The bubbles were also tested for larger free-stream speeds up to 25 ms -1 . The bubbles were still robust, recirculating for long periods, although the concentration in the images was lower after a comparable time delay.

Measurements in turbulent flow

Another set of experiments was performed in an open-circuit wind-tunnel to assess the capability of bubbles to track relatively small scale motions in turbulence. Two flows were considered: a canonical turbulent boundary layer and the near wake downstream of a sphere embedded within the boundary layer.

Experimental setup

A sketch of the test section and experimental setup is shown in figure 6(a). The test section has a cross section of 600 × 600 mm 2 and a maximum flow speed of 40 ms -1 . A trip wire was placed at the entrance of the test section to generate a turbulent boundary layer. The wind tunnel was driven by a frequency controlled variable speed motor and operated here with a fixed free-stream velocity U o of 3 ms -1 . The estimated friction Reynolds number of the developing turbulent boundary layer at the measurement location was Re τ = 500 with a viscous length scale 0.1 mm.

For volumetric velocity measurements, a V3V-Flex traverse system from TSI Incorporated was employed and displaced according to the sketch of figures 6(a,b). Four 4MP cameras (Cam1 to Cam4, 2560×1600 pixels, ∆ pix = 10 µm) equipped with Scheimpflug adapters were aligned in a planar configuration to image the illuminated volume. Each camera was equipped with a 135 mm lens with numerical aperture f # = 16 and f # = 11 for forward and back-scattering cameras, respectively. The acquisition frequency was 400 Hz. The mean resulting optical magnification considering the four cameras was M = 0.28. The measuring volume was illuminated by a 30mJ/pulse dual-head Nd:YLF laser positioned at the open outlet of the wind tunnel. The volume of interest of 75cm 3 (VOI, dimensions 90 × 56 × 15 mm 3 ) was located 1650 mm downstream of the trip wire with its side plane at the center line of the wind tunnel. A timing box was used to synchronize the laser pulses and the cameras. A sample raw image of the bubbles is depicted in figure 7. The bubbles display image diameters in the range d τ * = 1-4 pixels.

Hereafter, the streamwise, wall-normal, and spanwise directions are denoted by x, y, and z, respectively. The instantaneous velocity fluctuations in the corresponding directions are denoted by u, v and w, while the associated mean velocities are defined by U, V and W , respectively. All quantities normalized by the boundary layer friction velocity u τ and the kinematic viscosity ν are denoted with a superscript (+). The root mean square value of a quantity is denoted by a prime ( ) and its ensemble average is represented by an angle bracket ( ) .

Measurements were conducted for two flow configurations: unperturbed and sphere-perturbed (SP) turbulent boundary layer. For the SP case, a sphere (diameter D = 25 mm, D + = 240) held in place by a spanwise rod through its center was mounted inside the test section 109 mm upstream of the VOI center location (see red cross in figure 6(a)). Two SP cases were considered by changing the gap distance G between the bottom of the sphere and the wall, G =0 and R, where R is the sphere radius. These two configurations are denoted as SP-G0 and SP-GR, respectively. In all cases, the time interval between successive frames was 150 µs and two sampling frequencies 60 Hz and 400 Hz were selected.

Both the calibration and image processing steps prior to volumetric velocity computation were performed in Insight V3V 4G. The 3D-particle tracking and reconstruction method is fully described in [START_REF] Boomsma | Time-resolved particle image identification and reconstruction for volumetric 4D-PTV[END_REF]. For image processing, the raw images were pre-processed using the following steps to reduce background and noise effects: 1) apply a 3 × 3 pixel Gaussian filter to remove the noise; 2) subtract a local median (9 × 9 pixels) to remove the background; 3) subtract a constant intensity of 10 and multiply by a constant value of 5 to increase the contrast; 4) apply a 7 × 7 pixel Gaussian filter to facilitate the particle identification process. For each snapshot, the number of identified bubbles within the VOI was in the range 5000-8000.

For instantaneous velocity fields presented herein, the 3D velocity vectors associated with identified bubbles were binned into a uniform grid. Bad vectors (those exceeding the local mean value ± three times the standard deviation in a given bin) were removed to obtain fields with a uniform vector spacing of 2 × 2 × 2 mm 3 . In the binned flow fields, any local outliers (exceeding the local mean value ± β times the standard deviation, where β = 3 for u, v and w velocity components) were removed. Figure 8 shows a binned velocity vector field for one snapshot.

Results

The capability of the micro soap bubbles was first investigated under the unperturbed condition by comparing the resulting statistical velocity profiles with results from a direct numerical simulation (DNS) of a turbulent boundary layer at a similar Reynolds number (Re τ = 580). The numerical data set is extracted from [START_REF] Jiménez | Turbulent boundary layers and channels at moderate reynolds numbers[END_REF]. The experimental statistics were obtained based on ensemble averages of vectors present in 1500 snapshots. For the averaged statis- tics, all vectors were inserted into bins according to their location. Mean and rms values were computed at each location. Any outliers beyond three standard deviations in any component were removed, and the mean and rms values were recalculated. As presented in figure 9, the profiles of the mean streamwise velocity U, in-plane velocity fluctuations (u , v ) and the related Reynolds shear stress uv agree quite well with those from the DNS, thus demonstrating ex-cellent measurement quality. The values of spanwise velocity fluctuations (w ), however, are higher than in the DNS reference data reflecting the larger uncertainty in the out-ofplane component which is of order 0.5 pixels. Considering the inner length scale l = ν/u τ = 0.1 mm obtained from the estimated friction velocity u τ = 0.142 ms -1 , the associated flow time scale in the boundary layer is τ = l/u τ ∼ 700 µs.

Although this time scale is smaller than the measurable time scale estimated in section 3.4 (τ f = 1000 µs), we note that the tracing fidelity of the employed bubbles is clearly sufficient to reproduce the Reynolds stresses as presented in figure 9.

Figure 10 depicts two flow snapshots each of the sphereperturbed configurations (SP-G0 and SP-GR). In the case where the sphere is close to the wall (SP-G0), the majority of vortical structures induced by the sphere are located within y < D. Also, these shed structures present mainly a negative vorticity sign (clockwise rotation), while alternating vortices are shed from the sphere for the SP-GR configuration shown in figures 10(b,d). In this case, the vortical activity is spread across the region within 0.5D < y < 2D. The complexity of the vorticity field can be explained by the combination of vortices convected within the turbulent boundary layer, the ejected fluid below the sphere and the vortex shedding downstream of the sphere.

Concluding remarks

The current experiments demonstrate the feasibility of using air-filled micro soap bubbles to obtain high-quality planar and volumetric velocity measurements in air flows with intermediate-sized volumes in the range 75 -500 cm 3 . This seeding methodology thus complements existing methods of 1 µm liquid droplets optimized for smaller volumes and 400 µm Helium-filled soap bubbles appropriate for larger volumes. The air-filled microbubble tracers, which have mean diameters of approximately 15 µm, offer much stronger scattering capability than 1 µm droplets, yet also have relatively small response time. The measured mean response time was 40 µs with a dispersion of 30 µs. The bubbles can be generated at high rates to fill large facilities or with relatively large local concentration (50 tracers per cm 3 ) that can be sustained over time. When introduced into a recirculating wind-tunnel, the bubbles were robust and long lasting. When employed as tracers in an illuminated volume of a turbulent boundary layer, the bubbles yielded averaged mean and rms statistics of high quality. Also, instantaneous volumetric measurements allowed resolution and characterization of vortical structures in the wake of a sphere immersed in the same boundary layer. The capabilities of the seeding method described suggest that it is well suited for various additional particle tracking and time-resolved velocimetry techniques [START_REF] Schanz | Shake-the-box: Lagrangian particle tracking at high particle image densities[END_REF].

Fig. 2

 2 Fig. 2 Sketch of the bubble generator (top) and nozzle geometry (bottom).

Fig. 3

 3 Fig.3Size distribution of the micro soap bubbles. The inset is a highmagnification shadowgraph image of a micro bubble with digital resolution of 1.5 µm/px

Fig. 4

 4 Fig. 4 Mean streamwise particle (u p ) and fluid (u) velocity (a) and acceleration (b) decay along the stagnation streamline . The velocity is represented by the colormap of p and streamlines in the inset. The rectangle indicates the region where the velocity and the acceleration were computed to calculate the response time.

Fig. 5

 5 Fig. 5 Tracked particles in the large-scale volume (490 cm 3 ). The particles are colored with their vertical y coordinate to illustrate the seeding concentration in illuminated volume.

Fig. 6 Fig. 7 A

 67 Fig. 6 Sketch of the experimental setup. (a) Top view of the wind tunnel; (b) 3D view of the coordinate system and arrangement of cameras, sphere, and VOI.

Fig. 8

 8 Fig.8Binned vectors in a uniform grid for the unperturbed case.

Fig. 9

 9 Fig. 9 Wall-normal velocity profiles of the inner scaled streamwise velocity (top) and turbulent intensities and Reynolds shear stress for the free-stream condition (bottom).

Fig. 10

 10 Fig. 10 Instantaneous flow fields for sphere-perturbed configurations. (a,c): SP-G0; (b,d): SP-GR. The top and bottom rows are two instants with time separation ∆t + = 13. The vectors on each plane are the in-plane fluctuating velocities while the color denotes the out of plane component of vorticity. The yellow iso-surface depicts flow structures identified with the Q-criterion where Q + = 0.001.

Table 1

 1 Estimated response time for various tracers using equation 6.Seeding tracer ρ p (kg.m -3 ) d p (µm) τ p (µs)

	DEHS	912	1	3
	HFSB	1.1	550	90
	AFSB	4.4	400	1590
	Current study	20-170	10-30	20-160
	3.4 Measurement of the bubble response time

Acknowledgements This work was supported by TSI Incorporated. This work was partially supported by the U.S. National Science Foundation (NSF CBET-1510154). The authors gratefully acknowledge the insightful comments from the three referees.


Salim Khazem

Sylvain Chevallier

Quentin Barthélemy

Karim Haroun

Camille Noûs

Minimizing Subject-dependent Calibration for BCI with Riemannian Transfer Learning

Calibration is still an important issue for user experience in Brain-Computer Interfaces (BCI). Common experimental designs often involve a lengthy training period that raises the cognitive fatigue, before even starting to use the BCI. Reducing or suppressing this subject-dependent calibration is possible by relying on advanced machine learning techniques, such as transfer learning. Building on Riemannian BCI, we present a simple and effective scheme to train a classifier on data recorded from different subjects, to reduce the calibration while preserving good performances. The main novelty of this paper is to propose a unique approach that could be applied on very different paradigms. To demonstrate the robustness of this approach, we conducted a meta-analysis on multiple datasets for three BCI paradigms: motor imagery, eventrelated potentials (P300) and SSVEP. Relying on the MOABB open source framework to ensure the reproducibility of the experiments and the statistical analysis, the results clearly show that the proposed approach could be applied on any kind of BCI paradigm and in most of the cases to significantly improve the classifier reliability. We point out some key features to further improve transfer learning methods.

I. INTRODUCTION

Brain-computer interfaces allow to interact with a computer or a robotic system without relying on muscular capacity. As EEG measurements could be made with relatively inexpensive or easily available devices when compared to other kind brain imaging techniques, it is a common choice to conduct studies on many subjects. While the BCI research is very active and diverse, the EEG-based BCI still suffers from important limitations [START_REF] Thompson | Critiquing the Concept of BCI Illiteracy[END_REF]. Thus, they are mainly used for lab experiment and practical applications are scarce. Indeed, it is possible to enhance the preprocessing or the classification algorithms to better handle the individual variability [START_REF] Lotte | A Review of Classification Algorithms for EEG-based Brain-Computer Interfaces: A 10-year Update[END_REF]. Advanced signal processing and machine learning techniques are available but could not reduce the EEG-based BCI limitations completely.

An important leverage for improving the reliability of BCI is to improve the human-computer interfaces aspects [START_REF] Lotte | A BCI challenge for the signal processing community: considering the user in the loop[END_REF]. The existing protocols are suboptimal and could be enhanced [START_REF] Jeunet | Why standard brain-computer interface (BCI) training protocols should be changed: An experimental study[END_REF]. To address these issues, it is possible to build better model of the user and to better characterize what the user is learning [START_REF] Lotte | A BCI challenge for the signal processing community: considering the user in the loop[END_REF]. One of the main sources of EEG variability is the user, due to his/her fatigue, mood change or attention variations. 1 Salim Khazem, Sylvain Chevallier and Karim Haroun are with LISV, Université de Versailles Saint-Quentin, University Paris-Saclay, Saclay, France sylvain.chevallier@uvsq.fr 2 Quentin Barthélemy is with Foxstream, Vaulx-en-Velin, France q.barthelemy@foxstream.fr 3

Camille

Noûs is with Cogitamus, CNRS, France camille.nous@cogitamus.fr

The BCI protocols generate an important cognitive fatigue, especially if the user is maintaining a high attentional focalization during the interactions. To enforce the BCI reliability, it is common to include a calibration task for training preprocessing and machine learning algorithms. This calibration could be long; it is a source of fatigue and distraction, even before the starting to use the BCI. While setting up a subject independent calibration or a calibrationfree system [START_REF] Fazli | Subject independent EEG-based BCI decoding[END_REF] is possible, the performances are not as good as subject-dependent calibration systems.

To reduce -and ideally to remove -calibration it is possible to rely on transfer learning. This is an important topic in machine learning and many approaches have been adapted and investigated in BCI [START_REF] Lotte | A Review of Classification Algorithms for EEG-based Brain-Computer Interfaces: A 10-year Update[END_REF]. As for classification algorithms, Riemannian approaches that rely on covariance matrices, have achieved robust and efficient discrimination [START_REF] Yger | Riemannian approaches in braincomputer interfaces: a review[END_REF], [START_REF] Congedo | Riemannian geometry for EEG-based brain-computer interfaces; a primer and a review[END_REF], [START_REF] Xu | Feature extraction from the Hermitian manifold for brain-computer interfaces[END_REF]. This geometrical approach of the EEG feature representation allows to define metric spaces with chosen invariances. One main interest of the so-called Riemannian BCI is that the same classification algorithms could be applied to virtually any type of BCI. Indeed, transfer learning using covariance matrices has been investigated and achieves high performance [START_REF] Rodrigues | Riemannian procrustes analysis: Transfer learning for brain-computer interfaces[END_REF], but nonetheless requires calibration data.

The proposed algorithm named MDWM, grounded in the manifold of covariance matrices, extends the works presented in [START_REF] E K Kalunga | Transfer learning for SSVEP-based BCI using Riemannian similarities between users[END_REF] on SSVEP. Our contributions are twofold: MDWM works with small number of training data and it could be applied for different BCI paradigms. In this paper, we generalize the MDWM transfer learning approach to different paradigms and we conduct a meta-analysis on P300 datasets, Motor Imagery datasets and SSVEP datasets. After a short presentation of the method in Section II, this approach is evaluated on multiple datasets in Section III. Section IV concludes this report.

II. METHODS

A. Riemannian BCI

Common preprocessing and classification algorithms for EEG-based BCI rely on estimation of covariance matrices. These matrices are symmetric positive-definite and are thus restricted on a specific region of the space of real matrices. Furthermore, these matrices have a characteristic geometry and, endowed with a proper metric, it is possible to benefit from invariance properties to make robust classification [START_REF] Congedo | Riemannian geometry for EEG-based brain-computer interfaces; a primer and a review[END_REF].

The most simple yet effective Riemannian classifier for BCI is the minimum distance to mean (MDM). This algorithm brings a significant improvement when compared to Euclidean metrics [START_REF] Chevallier | Brain Computer Interfaces Handbook: Technological and Theoretical Advances, chapter 19 -Riemannian classification for SSVEP based BCI : offline versus online implementations[END_REF] but does not achieve the highest classification performance amongst the Riemannian approaches [START_REF] Jayaram | MOABB: trustworthy algorithm benchmarking for BCIs[END_REF]. Most notably, MDM is robust to noise and works well with online implementation [START_REF] Kalunga | Online SSVEP-based BCI using Riemannian geometry[END_REF]. This is thus a very good choice to serve as a basis for a transfer learning approach that should be used with minimal number of calibration samples.

From a set {X 1 , . . . , X m } of EEG epochs with c channels extracted from m trials, we can estimate the associated covariance matrices {Σ 1 , . . . , Σ m }, using

Σ i = 1/c X i X i .
It is advised to use more advanced covariance estimators, such as described in [START_REF] Chevallier | Brain Computer Interfaces Handbook: Technological and Theoretical Advances, chapter 19 -Riemannian classification for SSVEP based BCI : offline versus online implementations[END_REF]. In the space of covariance matrices, the shortest curve between two matrices Σ i and Σ j is known as the geodesic:

Σ i # λ Σ j = Σ 1 2 i Σ -1 2 i Σ j Σ -1 2 i λ Σ 1 2 i , with λ ∈ [0, 1]. (1)
The length of this geodesic is the Riemannian distance.

The first step of the MDM is to compute the average covariance matrix Σk for each of the k = 1, . . . , K classes [START_REF] E K Kalunga | Transfer learning for SSVEP-based BCI using Riemannian similarities between users[END_REF]. Then, it is possible to assign a newly seen covariance matrix Σ to the class with the minimum distance to mean, in the sense of the Riemannian distance.

B. Transfer Learning

The proposed transfer learning approach consists in using as little as possible calibration data from a user, relying on augmented models that make use of information available from other users. As the proposed approach combine difference source EEG to estimate any target EEG signal, this approach is called Minimum Distance to Weighted Mean (MDWM) [START_REF] E K Kalunga | Transfer learning for SSVEP-based BCI using Riemannian similarities between users[END_REF]. This is accomplished by combining the estimated centers of class, using the data available from the dataset of source subjects { Sk } averaged in Dk , and the data available from the target subject Tk as: Āk = T k # λ Dk , using Eq. ( 1). The parameter λ controls the trade-off between target and source data. For λ = 0, only the few data T k from the target subject are used (ie, no transfer scheme). For λ = 1, the calibration of the target subject is completely discarded and we rely entirely on Dk data extracted from the source subjects (ie, calibration free scheme).

The proposed algorithm MDWM, illustrated in Fig. 1, uses only a few calibration data by combining them with existing data from the source subjects. A weight can be added for each source subjects, defined for example as the similarity with the target subject [START_REF] E K Kalunga | Transfer learning for SSVEP-based BCI using Riemannian similarities between users[END_REF]. In this study, uniform weights are chosen throughout the experiments.

C. MOABB

The choice of MOABB framework [START_REF] Jayaram | MOABB: trustworthy algorithm benchmarking for BCIs[END_REF] is grounded in the need for reproducibility: this open-source project allows to download many EEG datasets and to easily evaluate any algorithm on those datasets [14]. The objective of MOABB is to yield reliable machine learning framework and sound statistical analysis to compare classification algorithms. Working on publicly available datasets, it is thus possible to reproduce results. While MOABB is a powerful tool, it still lacks the support to do cross-subject transfer learning. It is a more global issue as no existing framework allows to fairly evaluate transfer learning algorithms. This works is a first attempt to improve MOABB with transfer learning support and we will contribute to this open source project. Our analysis focuses three kinds of paradigms, SSVEP, Motor Imagery (MI) and P300 Speller. For the MI, the choice was based on two typical paradigms signal, left-hand vs right-hand imagery. The number of channels and subjects for each paradigm is described in Table I.

We have benchmarked MDWM on seven different datasets (2 for SSVEP, 2 for MI and 3 for P300) and shows that it performs significantly better than reference machine learning pipelines (CCA, CSP+LDA and xDAWN+LDA).

D. Analysis on transfer learning

Our objective being to minimize the calibration step of the target subject by using data from source subjects. To evaluate the influence of the number of samples used from the target subject, we use different numbers of training samples from the target subject n ∈ {5, 30, 55}. To evaluate the influence of the trade-off between source and target subjects, we report MDWM performances for different values of λ ∈ {0, 0.1, 0.3, 0.7}.

For each dataset, the evaluation is conducted as follows: the target subject data is separated from the other source subjects in a given dataset (following a leave-one-subject-out scheme), the trials from the target subject are separated into n trials for training (considered as calibration data) and the rest of the trials are used for testing. The model is trained on all the data available from the source subjects and the n trials from the target subject. The model is evaluated on the rest of the trials from the target subject. This process is

E. Meta-analysis on classification accuracy

In order to understand if the proposed method (MDWM) is consistently better than the state-of-the-art method (i.e., CSP+LDA for MI, xDAWN+LDA for P300, and CCA for SSVEP) across datasets and paradigms, we use a metaanalysis. It formulates a null hypothesis H 0 , which we are trying to reject, that there is no difference between the two methods. Then, we want to answer the following question: under H 0 , what is the probability to obtain the difference in the means that we observed in the experimental data?

To obtain results with comparable metrics across paradigms, all evaluations are performed with the same classification metric: the balanced accuracy, more appropriate when classes are imbalanced like in P300. Meta-analysis gives two values: a standardized mean difference (SMD) and a p-value p. SMD is a normalized measure to assess the difference between the means of the classification metrics obtained by the two methods. It allows to understand the classification delta between both methods. The p-value is the probability of observing such difference considering the null hypothesis H 0 as true. It allows to reject H 0 when it is very low. Statistical significance is for p < 0.05, for p < 0.01 and for p < 0.001, using Stouffer's method to combine p-values obtained with Wilcoxon signed-rank test on each separate dataset.

III. RESULTS AND DISCUSSIONS

In these experiments, we have used MDWM with λ = 0.7 and n that is two times the number of classes. Results of meta-analysis are shown in Fig. 2. The top plot shows that MDWM outperforms CSP+LDA for MI in all three datasets, with a sizable SMD and a good statistical significance. On middle plot, for the case of P300 speller paradigm, we can notice that MDWM outperforms xDAWN+LDA on two datasets over 3, with a noticeable SMD and a high statistical significance. On the bottom plot, MDWM outperforms CCA for both SSVEP datasets, with a high SMD and a high statistical significance. These results are a clear validation of the proposed transfer learning method, on 8 datasets containing a total of 72 subjects.

The Fig. 3 shows that by increasing the number of samples, the accuracy increases. We notice that by having a λ equal to 0.7, which correspond to rely mostly on data from source subjects, we obtain fairly good results. It is thus possible to reduce the calibration burden without losing to much precision.

As a discussion from these transfer learning results, we could report that not all subjects have an equivalent contribution for reducing the calibration. The "transfer-ability" of a subject is an open question and there are interesting investigation on this matter in [START_REF] Rodrigues | Riemannian procrustes analysis: Transfer learning for brain-computer interfaces[END_REF]. What is interesting here is that the MDWM is sufficiently robust to aggregate results from all subjects without distinction. Indeed, a proper selection of source subjects to train the MDWM could really improve the performances.

We also investigate if the calibration could be completely bypassed, by make use of the resting state EEG. It is common to start the EEG recording by asking the subject to close his/her eyes or to relax a little bit with eyes closed or opened. This resting EEG could be used to characterize the similarity between two subjects and thus to select best matching subject to train MDWM. We have investigated if resting EEG convey as much information as task related data for selecting similar subjects and it is the case. This subject will be the focus of another study.

The performances of MDWM could be improved by carefully selecting the λ value. A possible choice is to set λ as a function of the n, the number of calibration trial. Thus, the more data is available from the subject, the lower could be λ. We plan to investigate different function to set up automatically λ.

As we have pointed out, the reproducibility of results is a crucial issue for signal processing and machine learning approaches in BCI. MOABB, or other open source frameworks, are important to ensure that a fair evaluation of existing algorithms is available. While no framework proposes to benchmark transfer learning approaches yet, we will contribute improving MOABB on this point.

IV. CONCLUSION

In this contribution, we have described a transfer learning approach called MDWM and we have demonstrated that it could be applied on three different BCI paradigms, namely motor imagery, P300 and SSVEP. The MDWM uses a combination of a few calibration data from a target user and data available from other subjects. We have benchmarked MDWM on 7 different datasets with different number of electrodes and positions, and we show that it performs significantly better than reference machine learning pipelines (CSP, xDAWN and CCA). This is, to the authors' knowledge, the first attempt to systematically evaluate one classification pipeline on three different BCI paradigms. These results are promising and are a first step towards a reliable transfer learning for reducing the calibration phase.

ENVIRONMENTAL IMPACT

The approach taken in this submission tried to reduce the number of experiments made on servers, in order to reduce its environmental impact. The authors communicated through Slack, git and overleaf. Based on Shift Project and ADEME reports, we estimate that the impact of publication is roughly equivalent to 73 gCO 2 .

Fig. 2 .

 2 Fig. 2. Meta-analysis on several datasets, for different BCI paradigms comparing MDWM against CSP+LDA on MI (top), with xDAWN+LDA on P300 (middle), and with CCA on SSVEP (bottom).

Fig. 3 .

 3 Fig. 3. F1-score depending on the number of samples available for P300 datasets.

  Dataset of source subjects { Sk }Target subject T k

		Transfer Learning
	Dk			Āk
	Fig. 1. Illustration of the cross-subject transfer learning: combining small
	calibration from the target subject with data extracted from source subjects.
	Dataset name	Paradigm	#Classes #Channels	#Subj.
	SSVEP Exo [13]	SSVEP	4	8	12
	Nakanishi 2015 [15]	SSVEP	12	8	10
	Zhou 2016 [16]	MI	3	14	4
	BNCI-004-2014 [17]	MI	2	22	9
	BNCI-001-2014 [18]	MI	2	22	9
	BNCI-008 2014 [19] P300	2	8	8
	BNCI-009 2014 [20] P300	2	16	10
	BNCI-003 2015 [21] P300	2	8	10
		TABLE I
	LIST OF DATASETS: SSVEP, MOTOR IMAGERY AND P300 SPELLER.

ACKNOWLEDGMENT

QB and SC would like to thank Emmanuel Kalunga for the discussions about this project.


Marie-Constance Corsi

Sylvain Chevallier

Florian Yger

Riemannian geometry on connectivity for clinical BCI -RIGOLETTO A participation to the IEEE WCCI BCI Clinical Challenge

. Despite its clinical applications, one of the main drawbacks is the high inter-subject variability that could be noticed for performance. This is sometimes referred in the literature as the « BCI inefficiency » phenomenon [2] and affects its usability. Among the approaches adopted to tackle these issues are the search for neuromarkers, that potentially capture better the neurophysiological mechanisms underlying the BCI performance and the optimization of the classification pipelines, that could be robust enough to be applied to any subject.

In this work, we proposed an original approach that combines functional connectivity estimators, Riemannian geometry and ensemble learning to ensure a robust classification. This methods was ranked First in the Clinical BCI Challenge-WCCI (within-subject category).

The authors would like to thank the organizers of the WCCI Clinical BCI competition for giving them the opportunity to kickstart this promising collaboration. FY thanks F. Lotte for suggesting several years ago the use of connectivity features in lieu of covariances in Riemannian classifiers.

Proof of concept

Contact information: marie.constance.corsi@gmail.com @MConstanceCorsi

Riemannian geometry

The diffusion of Riemannian geometry for motor imagery BCI has strongly increased in the past decade [START_REF] Lotte | A Review of Classification Algorithms for EEG-based Brain-Computer Interfaces: A 10-year Update[END_REF]. This method consists in the extraction of Symmetric Positive Definite (SPD) matrices, usually the covariance matrices among sensors, for each trial and in considering this as a curved space.

Here, to estimate the distance between two matrices A & B, we used the LogEuclidean distance via the Froebenius norm:

  Our approach got the first position on this task with a substantial margin, the following teams having respectively kappa scores of 0.49 and 0.47 and accuracies of 74.69% and 73.75%. The kappa score obtained on validation is close to 0.68 and to the value obtained on training data with a 5-fold crossvalidation. Our hypothesis is that this result is due to the Functional connectivity Two main estimates were computed within the alpha-beta frequency band (8-30Hz): the coherence (Coh) and the phase-locking value (PLV). We also computed two other FC estimates: the amplitude envelope correlation (AEC) and the imaginary coherence (ICoh). For the sake of completeness, we report the results of AEC and ICoh although those features were not used in the final submission.

  obtained by averaging the FC estimators in each task separately in data obtained from one patient.

2 s 1

 21 𝛿 !" 𝐴, 𝐵 = log 𝐴 -log(𝐵) 𝔉 Coherence (Coh) Phase-Locking value (PLV) Df(t), relative phase between s 1 and s (t) s 2 (t) !"ℎ !" (%) = ( !" (%) " ( !! (%)( "" (%) )*+ = , #∆%(') S 12 (f), cross-spectral density S 11 (f), auto-spectral density Correlation Spatial covariance (Cov), coherence (Coh) and phase-locking value (PLV) are estimated from electroencephalographic (EEG) signals. A first level of classification was performed by FgMDM classifiers that yielded output decision probabilities to train a second level classifier, a ridge regression classifier, that provided the final decision. The associated code is available at https://github.com/sylvchev/wcci-rgcon. associated with an ensemble classifier gives the possibility to take into account the users' specificity. After participating to the competition, we identified different approaches to improve our method. Further investigation should be done regarding the followup to the MDM and the dimensionality reduction.. We plan to refine the selection of the frequency band of interest. Another promising lead would be to extract for each epoch several PSD matrices, each on a different frequency band, and to consider this set as a trajectory on the manifold. robustness provided by a classification based on multiple sources.

Acknowledgements

To assess the performance of our approach we conducted a comparison of the estimators both at the individual and at the group level. For that purpose, we directly applied our approach to an EEG dataset obtained from 8 stroke patients, in the frame of the IEEE WCCI