Loïc B

Philippe L

Jean Pierre L

Ultrasonic tomography of green wood using a non-parametric imaging algorithm with reflected waves

Keywords: tomography, ultrasound, wood imaging tomographie, ultrasons, imagerie

Ultrasonic computed tomography in reflection was used to assess the integrity of green wood. Qualitative reflectivity images were obtained from back-scattered measurements by reflection tomography, like ultrasound in medical applications.

• The reconstruction algorithm was designed using, in particular, a linear approximation of the forward problem (Born approximation) and based on the assumption that a transversal cross section of wood is isotropic. The experimental device was composed of only one rotating emitter-receiver transducer to record and compute the projections. In this specific case, a tomographic projection was directly associated with a recorded signal. The qualitative aspect of this imaging technique was validated by performing a numerical simulation and tested on a small diameter green wood (Picea abies) log.

• The images obtained were geometrically accurate considering the internal inclusions. It was, however, not possible in the simulation to differentiate the object shape from the background (coupling medium) because the reflectivity value associated with the object was too low. The image obtained with the spruce sample mainly showed the position of the bark as indicated by a very high contrast area. The proportion of transmitted energy was, however, sufficient to reconstruct the artificial inclusion within the sample.

Mots-clés :

INTRODUCTION

Acoustic properties such as ultrasonic attenuation or velocity have been reported to reflect the strength of wood [START_REF] Bucur | Acoustics of wood[END_REF] and predict in situ decay processes [START_REF] Dunlop | Testing of poles by using acoustic pulse method[END_REF]. However, ultrasound methods currently available for assessing the green wood status are limited to obtaining measurements in transmission mode through a section. These methods therefore give average values recorded along the ultrasound pathway through the section, but they are not suitable for assessing spatially heterogeneous acoustic properties of wood. Ultrasonic computed tomography (UCT) could thus be used in this setting to achieve more accurate wood quality assessment. UCT has a very large field of applications in decay diagnosis [START_REF] Martinis | Nondestructive techniques for decay diagnosis on standing trees[END_REF] in standing trees in parks and public gardens, in lumber structure imaging [START_REF] Neuenschwander | Studies for visualizing wood defects using ultrasonics in reflection and transmission modes[END_REF] to reveal knots, grain deviations, cracks and compression wood, and in imaging defects in wood composites, such as delaminations between layers and voids [START_REF] Bucur | Nondestructive Characterization and Imaging of Wood[END_REF]. Other wood imaging techniques exist, ranging from ionizing radiation, thermal techniques, microwave, ultrasound to nuclear magnetic resonance. The most relevant wood imaging technique depends upon the specific application [START_REF] Bucur | Nondestructive Characterization and Imaging of Wood[END_REF].

Although UCT is a valuable tool for imaging soft biological materials [START_REF] Mensah | Enhanced Diffraction Tomography[END_REF], little attention has been paid to using this method for imaging harder media such as green wood. Dikrallah published preliminary results of an analysis of the anisotropy effect on acoustic wave propagation in green wood [START_REF] Dikrallah | Experimental analysis of acoustic anisotropy of green wood by using guided waves[END_REF]. In an important study, Maurer also highlighted the anisotropy effect and proposed a correction procedure for quantitative tomography [START_REF] Maurer | A simple anisotropy correction procedure for acoustic wood tomography[END_REF]. Interactions between ultrasound and wood, especially green wood, make the imaging process particularly intricate. For instance, classical reconstruction algorithms are based on the assumption that straight ray propagation is involved, whereas the incident beam is liable to be refracted at the interface between the soft background (coupling gel, water) and the wood, depending on the incident angle, due to the marked difference between the acoustic properties of the two media. Diffraction -which is closely related to refraction -can also result from the irregular shape of the samples. Moreover, specific reconstruction tools are sometimes required due to the anisotropy of ultrasound propagation through green wood [START_REF] Maurer | Application of nonlinear acoustic tomography for non-destructive testing of trees[END_REF]2006).

Many difficulties may be encountered when attempting to assess wood by quantitative ultrasonic tomography. However, UCT imaging is an attractive non-invasive tool that can be used to perform quantitative 3-D reconstruction of the tree geometry and determine acoustic properties such as ultrasound velocity and attenuation, depending on the type of measurements (time-of-flight or amplitude) performed. In previous studies [START_REF] Martinis | Tomographie ultrasonore pour les arbres sur pied[END_REF][START_REF] Nicolotti | Application and comparison of three tomographic techniques for detection of decay in trees[END_REF], projection data were obtained in transmission mode through a wood specimen, which could be inversed by using classical tomographic reconstruction algorithms [START_REF] Kak | Principles of computerized tomographic imaging[END_REF]. Previously published preliminary results showed that tomographic reconstruction of an academic test object and green wood was feasible using ultrasonic transmission tomography techniques [START_REF] Martinis | Tomographie ultrasonore pour les arbres sur pied[END_REF][START_REF] Maurer | Application of nonlinear acoustic tomography for non-destructive testing of trees[END_REF][START_REF] Socco | Feasibility of ultrasonic tomography for nondestructive testing of decay on living trees[END_REF]. The authors therefore hypothesized that reconstruction of the outer boundary of samples and the inner boundaries of different decay regions would provide information on the wood strength.

However, these studies were seriously hampered by several factors. First, the boundaries of the object could not be easily reconstructed from datasets acquired in transmission mode, but they could from datasets acquired in reflection mode, which gives an improved tomographic view of the scatter shape. Furthermore, the mesh in transmission-UCT is often too large, and cavities of less than 50 mm diameter are generally imperfectly identified [START_REF] Martinis | Tomographie ultrasonore pour les arbres sur pied[END_REF][START_REF] Nicolotti | Application and comparison of three tomographic techniques for detection of decay in trees[END_REF]. Secondly, when inversing the dataset acquired in transmission mode to recover the sound velocity image of a cross-section, the signals have to be clearly identified, and the time-of-flight must be measured accurately from a signal with a low signal-to-noise ratio. Complex interactions between the incident ultrasonic wave field and wood give rise to a complex transmission field with several output signals, which can make it difficult to identify the signals.

The main objectives of our study were first to take full advantage of UCT in diffraction and transmission modes by developing a technique for cross-sectional imaging of wood, and secondly to refine the signal processing technique by developing a suitable signal filtering process based on wavelet decomposition to improve assessments based on time-of-flight measurements.

The present paper focuses on the qualitative (nonparametric) image obtained using reflection-UCT to detect decay in green wood. Reflection-UCT, which is derived from the general diffraction-UCT approach, involves forming an image which highlights the interfaces between media of different acoustic impedance from backscattered measurements. One interesting aspect of reflection tomography in comparison with transmission tomography is that it is not necessary to encompass the object with transmitters and receivers for collection of projection data collection. Transmission and reception of signals are performed by the same transducer. However, since most of the ultrasonic energy is scattered in the forward direction, the transducer must have high sensitivity in order to measure backscattered signals at high signal-to-noise ratios.

We perform a numerical simulation to study the effects of several decay patterns and "artificial" wood configurations on the reflection-UCT data obtained. The experimental test performed on a green wood sample using a tomographic scanner revealed the advantages and limitations of the classical algorithms.

MATERIALS AND METHODS

Experimental set-up

The experimental setup was designed to perform all types of ultrasonic measurements (in reflection, diffraction and transmission). Only one emitter-receiver transducer was used in reflection tomography. It included an acoustic device that could be moved with various degrees of freedom and used to analyze samples in all directions (Fig. 1). Ultrasounds were generated using a Panametric pulse/receiver and Imasonic piezo-composite wide-band transducers with a nominal frequency of 0.5 MHz. The object under investigation was placed in the center of the measuring system. The complete circle area was scanned using a 5 • sampling angular increment, so 72 projections were obtained. Each projection, equal to each backscattered signal in reflection tomography, was composed of 4096 samples regularly spaced at intervals of 50 ns (20 MHz). The upper limit of the number of pixels N (1) that is possible to use for reconstruction was thus 384 × 384 pixels [START_REF] Socco | Feasibility of ultrasonic tomography for nondestructive testing of decay on living trees[END_REF].

N = N Projection × N Sample 2 (1)
with N Pro jection : the number of projections; N Sample : the number of samples per projection. The chosen size of the tomographic image was 300 × 300 pixels, which corresponds to the inner square of the scanned area with an associated spatial resolution of 1 mm 2 scanned per pixel (the rotation radius of the transducer is equal to 22 cm with the object situated in the far field of the ultrasonic emission). The longitudinal resolution (along the emission axis) is the wavelength in the object and was 3 mm. The lateral resolution (perpendicular to the emission axis) was 18 mm using the Fresnel zone formula (2) [START_REF] Socco | Feasibility of ultrasonic tomography for nondestructive testing of decay on living trees[END_REF].

R F = S d c 2 f (2)
with R F : the Fresnel zone radius; S d : the scanned depth (the rotation radius of the emitter-receiver transducer); c: the celerity; f :t h e nominal frequency of the transducer.

Ultrasonic computed tomography

The aim of UCT is to reconstruct the spatial distribution of some geometrical and physical parameters of an object from scattered ultrasonic measurements. These measurements are carried out on sets of transmitter and receiver positions spaced at different densities, with various interrogating wave frequencies. We are then faced both with a forward scattering problem, i.e. predicting the pressure field when the scattering medium and incident field are assumed to be known, and with the inverse scattering problem, i.e. retrieving parameters of the medium from the measured incident and scattered fields. This inverse problem is non-linear and ill posed and there is generally no single solution. It is essential to find a way of eliminating solutions that do not reflect the real situation.

Forward problem modeling

Let A be the operator that describes elastic wave propagation or scattering phenomena in the heterogeneous medium that is to be imaged (3), including boundary and Sommerfeld conditions [START_REF] Morse | Theoretical Acoustics[END_REF].

A. = - 1 c 2 ∂ 2 . ∂t 2 + ρ div( 1 ρ grad.) (3)
with ρ: the variable density, and c: the variable celerity.

Let S be the ultrasonic source (transducer emission), which is assumed to be known. The resulting ultrasonic field ϕ satisfies Equation ( 4):

Aϕ = S (4)
The heterogeneous medium is assumed to be composed of a known part, i.e. the coupling medium associated with the wave operator A 0 (5), and an unknown part, i.e. the object associated with the operator A ′ . The relation between A, A 0 and A ′ is given by Equation ( 6).

A 0 . = - 1 c 2 0 ∂ 2 . ∂t 2 +∆. ( 5
)
with c 0 : the constant celerity in the coupling medium. ∆.: the Laplacian operator.

A = A 0 + A ′ (6)
Solution ϕ 0 of the non-perturbed problem is assumed to be known:

A 0 ϕ 0 = S (7)
Field ϕ ′ induced by perturbation of the object can be thus written as the difference between ϕ and ϕ 0 (ϕ ′ = ϕϕ 0 ). Then ϕ'i st h e solution of Equation ( 8): 5) is known (G 0 solution of A 0 G 0 = δ), the field induced by the object is given by Equation ( 9).

A 0 ϕ ′ = -A ′ ϕ (8) If Green's function G 0 of Equation (
ϕ ′ = -G 0 * A ′ ϕ (9)
In this equation, the operator " * " is the convolution product. Equation ( 9) is the well-known Lippmann-Schwinger equation [START_REF] Morse | Theoretical Acoustics[END_REF].

Linearization and inverse problem in reflection

The inverse problem is the solution of the nonlinear Equation ( 9) and a solution can be found by means of a perturbation scheme (based on a linear approximation). The first-order Born approximation is one of these schemes (Born series) and the first-order solution ϕ ′(1) can be written:

ϕ ′(1) = -G 0 * A ′ ϕ 0 (10)
The far-field solution of Equation ( 10) in direction n for plane wave insonification in direction n 0 is: In the back-scattering configuration used in reflection mode n = n 0 so that the transfer function becomes:

ϕ ′ (1) (x,ω) = - e ik| x| 4π| x| h (1) ( n 0 , n,ω) (11) With h (1) ( n 0 , n,ω) = k 2 [(1 + n 0 . n)α( K) + (1 -n 0 . n) ξ( K)] and K = k( n -n 0 ) • k = ω/
h R(1) ( n 0 ,ω) = 2k 2 ξ( K) with K = -2k n 0 (12)
Equation ( 12) shows that the only parameter that we can hope to attain by back-scattering measurements is the logarithmic fluctuation of the acoustic impedance. Parameter ξ is directly linked to the reflectivity of the object using the following first-order development ( 13). The acoustic impedance contrast is defined as (zz 0 )/z 0 .

ξ(x) = ln z(x) z 0 ≈ z(x) -z 0 z 0 ≈ 2r(x) ( 13
)
with r(x) = z(x)-z 0 z(x)+z 0 : being the spatial reflectivity coefficient of the object.

The aim of ultrasonic reflection tomography is thus to obtain qualitative reflectivity images from back-scattered measurements. The situation is identical to that of X-ray tomography (the source rotates around the object and the projections are recorded after each rotation step), but with a band-pass spectral filter instead of a low-pass spectral filter (because piezoelectric ultrasonic transducers cannot transmit low frequencies). The reconstruction is performed using the summation of ramp-filtered back-projections [START_REF] Franceschini | An optimization method for quantitative impedance tomography[END_REF][START_REF] Mensah | Near-field ultrasound tomography[END_REF]:

|∆r(x,y)| = 1 4π 2π 0 π φ (l) dφ (14)
with (x, y): the spatial coordinates along the horizontal and perpendicular axis; l: the coordinate along the emission axis n 0 ; φ :th eangle formed by the emitter -receiver transducer (emission axis) and the horizontal axis. The vector π φ (l) results from filtering of the projection P φ (l) by a ramp filter. Each projection P φ (l)i sf o r m e db y each received signal; a reflected signal is indeed formed by amplitudes directly linked with the local reflectivity coefficients (along with the local attenuations). Equation ( 14) shows that the inversion procedure computes the modulus of the Laplacian of the reflectivity; only outlines of the object areas (interfaces between media of different acoustic impedance) that induce variations in impedance contrast are thus reconstructed. This last remark explains why this specific reflection tomographic technique produced qualitative and not quantitative images.

Assumptions adopted in UCT for green wood imaging

Weak scattering approximations are generally used when imaging soft low-contrast media such as biological tissues, e.g. the approximations of Born or Rytov [START_REF] Greenleaf | Introduction to Computer Ultrasound Tomography[END_REF]. Contrary to what occurs with soft media, the acoustic contrast between wood and the surrounding reference medium (coupling gel or water) is so great that the problem is intrinsically non-linear in terms of the unknown physical and geometrical parameters, and there is generally no single solution. One solution therefore involves finding the homogenized or "equivalent" material parameters of a spatially bounded heterogeneous medium, and adapting the acquisition and reconstruction software initially developed for use with soft media.

Trees consist of long circular cylinders surrounded by an irregular thick bark. These cylinders are generally assimilated to orthotropic materials with a principal axis of symmetry oriented along the vertical axis [START_REF] Goodman | Orthotropic Elastic Properties of Wood[END_REF]. In this study, transversal cross-sections of the wood to be imaged were taken to be isotropic and therefore assumed to mediate the propagation of more complex waves, similar to volume waves. Since the typical compression bulk wave velocity recorded in the transverse direction was 1400-1600 m/s, the wavelength at the frequency used here (∼0.5 MHz) was around 2.8mm, which is much larger than the microscopic structural heterogeneities (∼10 µm). Reflections induced by annual growth rings cannot be neglected (∼1 mm) and can form a specific geometric pattern in the tomographic image. The first-order Born approximation solution could therefore bias the tomographic reconstruction.

The wavelength in water (∼3 mm) was much shorter than the diameter of the object (approximately 100-200 mm). Ray theory was and straight lines inside the object were taken to approximate the wave propagation paths. Since signal contributions associated with compression waves were only of interest here, a weak nonhomogeneous fluid could be used to model wood.

Methodology

Reflection-UCT was applied to a numerically simulated test target and to a green wood sample. The numerically simulated test target was defined as a homogeneous and isotropic circular 30 mm diameter cylinder (Fig. 2,5 0× 50 pixels). The mean velocity of the compression wave was set at c 0 = 1500 m/s in water (associated acoustic impedance z 0 = 1.5 × 10 6 N.s.m -3 )a n dc = 1600 m/si nt h e material (z = 1.60 × 10 6 N.s.m -3 ). This latter velocity corresponds to the same scale as in a transversal cross section of a tree (in dry condition). The sample contained two 7 mm and 14 mm diameter The spruce sample (Picea abies) was green, surrounded by bark, and measured 125 mm diameter and 200 mm thick. A mixture of glue and sawdust was placed in a 30 mm hole drilled in the center of the sample to create artificial decay (Fig. 3). The mean moisture content, measured on a disk from the same specimen, was 110%. The velocity decreases with increasing moisture content. The maximum velocity corresponds to the dry condition and the decreasing trend is linear up to the fiber saturation point (due to the combined effects of the decreased modulus of elasticity and the increased density). After this point, there is a trend inflection and the velocity continues to decline linearly due only to the increased density [START_REF] Bucur | Acoustics of wood[END_REF]. Considering the influence of moisture content on the sample acoustic impedance, the z = 0.55 × 10 6 N.s.m -3 value measured in dry condition (12% moisture content, 1400 m/s mean velocity and 390 kg/m 3 density) is quasi-equal to the z = 0.59 × 10 6 N.s.m -3 value measured in wet condition (110% moisture content, 860 m/s mean velocity and 690 kg/m 3 density). The velocity and density measurements were obtained on a prismatic sample from the same specimen. The z = 0.59×10 6 N.s.m -3 value was then used as the apriorivalue for the specimen.

RESULTS AND DISCUSSION

Numerical simulation

In order to perform the reconstruction, 180 projections of 70 samples were simulated (2 • angular increment). The upper limit of the reconstructed image size was thus 79 × 79 pixels, but the final tomographic image size was equal to the original size of the test target (50 × 50 pixels). Each projection was set to be sampled at 1 MHz (1 pixel per µsinFig.4). The nominal frequency of ultrasonic waves was 0.5 MHz (the 1 MHz sampling frequency was equal to the limit of the Nyquist -Shannon sampling theorem). The simulated duration of acquisition was 70 µs, which corresponded to the maximum propagation time in reflection.

Figure 4 gives the ultrasonic sinogram formed by 180 simulated signals in reflection (ray theory was applied without taking attenuation in the different media into account; to simplify Figure 4, only the signal envelopes are shown). Figure 5 presents the associated reconstructed image of the numerically simulated test target. In this figure, the transition areas highlighting the interface positions were 6 pixels thick (3.6 mm associated spatial resolution). This resolution was in accordance with the 5 pixel theoretical resolution value (3 mm wavelength). The diameters of the reconstructed inclusions were 12 pixels (12 pixel original diameter of inclusion 1) and 27 pixels (24 pixel original diameter of inclusion 2), while taking the center of the transition area as the interface position. Figure 5 was thus geometrically accurate with respect to the actual inclusion dimensions.

The result was acceptable since the Born approximation was used (resolution close to the theoretical value), and the background consisted of water with an a priori wave velocity of 1500 m/s. The perturbation in this case was introduced by the two holes simulating decay. The magnitude of the reflectivity coefficient associated with the two inclusions was |r 1 | = 0.36 (7 mm diameter inclusion) and |r 2 | = 0.07 (14 mm diameter inclusion). The most highly contrasted area was therefore naturally focused on the inner holes rather than on the external boundaries of the target. In this example, the bark was not simulated, and there was very little difference in velocity between the water and the material (100 m/s). The reflectivity coefficient value associated with the water-material interface was r = 0.03, which meant that only 0.1% of the wave energy was reflected by this interface. It was therefore not possible for the algorithm to discriminate between the two domains. The reference background used for the inversion procedure was finally the set composed of the water and the main material. Since the difference in impedance contrast between the material and the holes increased, they could be easily identified in this example.

Wood sample

Figure 6 shows the ultrasonic sinogram formed by 72 signals in reflection. The difficulties encountered were mainly due to the high wave attenuation during propagation (Fig. 6). Signals were therefore amplified but not saturated. Figure 7 shows the reconstructed reflection-UCT of a spruce sample (Picea abies). Besides the wave attenuation during propagation, note that the bark had very marked effects on the signals.

A large amount of air was probably sandwiched between the healthy tree and the bark. The acoustic impedances were very different, which affected the wave propagation by a sudden interruption. Given the surface texture irregularity, it is likely that only a small part of the incident energy penetrated into the wood, and a large proportion was directly backscattered without penetrating into the healthy wood. This assumption is in line with the strong echoes, which could be taken to reflect the bark thickness when not attenuated between the healthy wood and the bark.

A number of 36 tomographic profiles were extracted from the reconstructed image (Fig. 7) by rotation around the center of the image with a 5 • angular increment. Figure 8 shows an example of a tomographic profile obtained. The wooden sample diameter was correctly assessed (125 mm, Fig. 8), and the inclusion was well positioned in the center, but it was impossible to measure its diameter very precisely with a resolution close to the wavelength (Fig. 8). The measurement of the inclusion diameter was carried out by an operator on the 36 profiles. The average diameter was 48.1 mm (minimum value: 39.1 mm, maximum value: 57.8 mm and standard deviation: 4.0 mm) instead 30 mm (the true value). The difference between the measured value and true value allowed us to estimate the spatial resolution obtained; the resolution value was around 20 mm, which was very close to the theoretical Fresnel zone radius of 18 mm (lateral resolution). The image obtained in Figure 7 and the profile in Figure 8 were blurred by several cumulated effects: the Born approximation was used despite the strong echo produced by the bark, the cross section was assumed to be isotropic and the lateral resolution was 6-fold higher than the longitudinal resolution.

CONCLUSION

A wood qualitative imaging method based on ultrasonic computed tomography analysis of soft tissues was presented. Due to the acoustic properties of wood (attenuation and impedance contrast), working with this material is more complex than dealing with soft tissues. Wood was assumed to be equivalent to an internally weakly contrasted object (local fluctuations in the acoustic characteristics of a cross-section are weak) immersed in a homogeneous reference medium (water). The first-order Born approximation was used to solve the inverse problem of image reconstruction. The main result of the theoretical inversion was that only the interface outlines in the object were reconstructed; this specific reflection tomographic technique produced qualitative but not quantitative images. The transversal cross-sections of the wood to be imaged were taken to be isotropic. Consequently, ray theory was applied and straight lines within the object were considered to approximate the wave propagation paths.

The reconstruction algorithm was tested on an academic object produced by numerical simulation. The image obtained was geometrically accurate considering the two simulated inclusions. The computed inclusion positions and diameters were equal to those of the original object, with a spatial resolution that was in accordance with the theoretical resolution. It was, however, not possible to differentiate the object shape from the background (coupling medium). This phenomenon could be explained by the fact that the reflectivity value of the interface associated with the transition object-background was too low. A green spruce sample with bark was also tested. Artificial decay made with glue and sawdust was created at the center of the sample. The image obtained mainly showed the position of the bark indicated by very high contrast. It seemed that a large proportion of the incident energy was directly backscattered by the bark without going deeper into the material. Despite this phenomenon, the proportion of transmitted energy was sufficient to reconstruct the artificial inclusion within the sample. This inclusion was well positioned at the center, but an 18 mm spatial resolution evaluation made the diameter estimation very inaccurate.

This research could now be continued in two possible areas. The first approach could be based on the Born iterative method, where iterations are calculated numerically from experimental data [START_REF] Lasaygues | Distorted Born Diffraction Tomography: limits and applications to inverse the ultrasonic field scattered by an non-circular infinite elastic tube[END_REF]. The second possible approach could involve modeling the forward problem using the intercepting canonical body approximation method, which reduces the calculation cost. The latter method involves taking the wood covered with bark and the decay it contains to be a non-circular multilayer cylinder in which every layer (bark, growth rings, decay) is homogeneous. Contrary to approaches based on straight ray theory, this method requires no limited approximations and no a priori information. The scattered field is completely integrated [START_REF] Franceschini | An optimization method for quantitative impedance tomography[END_REF].

Figure 1 .

 1 Figure 1. Details of the experimental device for measurements in reflection, diffraction and transmission. Only one emitter-receiver transducer was used in reflection tomography.

  c 0 : representing the wave number in the reference medium. α( K)and ξ( K): the spatial Fourier transforms of α = c 2 -c 2 0 c 2 , the quadratic fluctuation of celerity, and ξ = ln Z Z 0 , the logarithmic fluctuation of acoustic impedance z = ρ c. h (1) ( n 0 , n,ω) : the first order frequency response (or transfer function) of the medium for incidence n 0 and direction of observation n.

Figure 2 .

 2 Figure 2. Numerically simulated test target. The bar indicates the wave velocity value (in m/s).

Figure 3 .

 3 Figure 3. Spruce (Picea abies) sample geometry.

Figure 4 .

 4 Figure 4. Simulated ultrasonic sinogram. The color corresponds to the signal envelope magnitude. A recorded signal corresponds to a projection.

Figure 5 .

 5 Figure 5. Simulated ultrasonic tomogram. The color difference corresponds to the contrast between areas of different acoustic impedance (modulus of the Laplacian of the reflectivity).

Figure 6 .

 6 Figure 6. Reflection-UCT of a spruce (Picea abies) sample with centered artificial decay, ultrasonic reflection sinogram.

Figure 7 .

 7 Figure 7. Reflection-UCT of a spruce (Picea abies) sample with centered artificial decay, ultrasonic reflection tomogram.

Figure 8 .

 8 Figure 8. Reflection-UCT of a spruce (Picea abies) sample with centered artificial decay, y axis profile of the ultrasonic reflection tomogram (modulus of the Laplacian of the reflectivity at x = 0 mm).


Pierre-Marie Chiaroni

Xavier Guerra

Jonathan Cortese

Julien Burel

Thomas Courret

Pacôme Constant

Dit Beaufils

Thibault Agripnidis

Ian Leonard-Lorant

Cédric Fauché

Nourou Dine Bankole

Géraud Forestier

Vincent L'allinec

Peter B Sporns

Gaelle Gueton

Nico Lorena

Marios Psychogios

Jean- Baptiste Girot

Dr Aymeric Rouchaud

Kevin Janot

Nicolas Raynaud

Raoul Pop

Jean-François Hak

Basile Kerleroux

Romain Bourcier

Dr Gaultier Marnat

Chrisanthi Papagiannaki

Dr Nader-Antoine Sourour

Frédéric Clarençon

Dr Eimad Shotar
email: eimad.shotar@aphp.fr

Jeni Research Collaborative

Johnson & Phenoc

Bracco. Dr Kevin Johnson

Janot

Location-Specific Rupture Risk of Intracranial Aneurysms: the Case of Ophthalmic Aneurysms

Keywords: Intracranial Aneurysm, Ruptured Aneurysm, Epidemiology Digital Subtraction Angiography -RIA: Ruptured Intracranial Aneurysm -UIA: Unruptured Intracranial Aneurysm

The study received approval from the CERIM institutional review board (IRB): CRM-2201-220. The need for patient signed consent was waived by the IRB.

Abstract Background

Aneurysm location is a key element in predicting an intracranial aneurysm's rupture risk. A common impression suggests that pure ophthalmic aneurysms are underrepresented in ruptured intracranial aneurysms (RIAs). The purpose of this study is to specifically evaluate the risk of rupture of ophthalmic aneurysms and to compare it to the other aneurysm locations.

Methods

This multicentre study compared the frequency of ophthalmic aneurysms in a prospective cohort of RIAs admitted in 13 neuroradiology centres between January 2021 and March 2021 with a retrospective cohort of patients having unruptured intracranial aneurysms (UIAs) who underwent cerebral angiography at the same neuroradiology centres during the same time period.

Results

Six hundred and four intracranial aneurysms were included in this study (355 UIAs and 249 RIAs; mean age 57 years, IQR [49; 65]; female 386/604 [64%]). Mean aneurysm size was 6.0 mm (5.3 mm for UIAs; 7.0 mm for RIAs, p-value < 0.0001).

Aneurysm shape was irregular for 37% UIAs and 73% RIAs (p-value < 0.0001).

Ophthalmic aneurysms frequency was 14.9% of UIAs (2 nd most common aneurysm location) and 1.2% of RIAs (2 nd least common aneurysm location); OR 0.07 (95% CI [0.02; 0.23], p-value < 0.0001).

Conclusions

Ophthalmic aneurysms seem to have a low risk of rupture compared to other intracranial aneurysm locations. This calls for a re-evaluation of the benefit-risk balance when considering preventive treatment for ophthalmic aneurysms.

Key points:

-What is already known on this topic: Internal carotid artery (ICA) aneurysms (including ophthalmic, carotid-cave, superior hypophyseal, anterior choroidal and ICA termination aneurysms) have a lower risk of rupture compared to other aneurysms. However, the specific relative rupture risk of ophthalmic aneurysms is not known.

-What this study adds: Ophthalmic aneurysms are under-represented among ruptured aneurysms compared to unruptured aneurysms, suggesting a relatively lower risk of rupture -How this study might affect research, practice or policy: the benefit-risk balance should be carefully re-evaluated before deciding on the preventive treatment of unruptured ophthalmic aneurysms

INTRODUCTION

Intracranial aneurysms have an estimated prevalence of 3-8% in a population with a mean age of 50 years without major cardiovascular comorbidities [START_REF] Vlak | Prevalence of unruptured intracranial aneurysms, with emphasis on sex, age, comorbidity, country, and time period: a systematic review and meta-analysis[END_REF][START_REF] Johnsen | Prevalence of unruptured intracranial aneurysms: impact of different definitions -the Tromsø Study[END_REF]. With the increased use of non-invasive brain vessel imaging in the last decades, incidental detection has become more and more frequent.

Deciding preventive treatment of asymptomatic unruptured intracranial aneurysms (UIA) is challenging. It relies on assessing the projected balance between the expected benefit (lowering the risk of neurological impairment or death by rupture and subarachnoid haemorrhage, associated with a 44% mortality rate in Europe) and the risks of treatment-related complications [START_REF] Nieuwkamp | Changes in case fatality of aneurysmal subarachnoid haemorrhage over time, according to age, sex, and region: a meta-analysis[END_REF]. To improve the estimation of the rupture risk of untreated aneurysms, scoring systems incorporating clinical and imaging criteria have been proposed [START_REF]The Natural Course of Unruptured Cerebral Aneurysms in a Japanese Cohort[END_REF][START_REF] Greving | Development of the PHASES score for prediction of risk of rupture of intracranial aneurysms: a pooled analysis of six prospective cohort studies[END_REF][START_REF] Backes | ELAPSS score for prediction of risk of growth of unruptured intracranial aneurysms[END_REF]. Aneurysm location is one of the main risk factors predictive of rupture [START_REF] Rousseau | Location of intracranial aneurysms is the main factor associated with rupture in the ICAN population[END_REF]. Ophthalmic aneurysms, superior hypophyseal aneurysms, carotid cave aneurysms, internal carotid artery termination aneurysms, anterior choroidal artery aneurysms and sometimes posterior communicating artery aneurysms are often grouped under the "internal carotid artery aneurysms" label [START_REF]The Natural Course of Unruptured Cerebral Aneurysms in a Japanese Cohort[END_REF][START_REF] Greving | Development of the PHASES score for prediction of risk of rupture of intracranial aneurysms: a pooled analysis of six prospective cohort studies[END_REF][START_REF] Backes | ELAPSS score for prediction of risk of growth of unruptured intracranial aneurysms[END_REF][START_REF] Rousseau | Location of intracranial aneurysms is the main factor associated with rupture in the ICAN population[END_REF]. However, a widespread clinical impression suggests that ophthalmic aneurysms (i.e.: aneurysms close to the ophthalmic artery, or with the ophthalmic artery arising from their neck) are under-represented among ruptured intracranial aneurysms (RIAs) compared to UIAs. In parallel, ophthalmic aneurysms are more often treated with flow diversion or stent-assisted coiling because of anatomical considerations (especially wide neck) and a better long-term occlusion rate [START_REF] Lylyk | Pipeline Embolization Devices for the Treatment of Intracranial Aneurysms, Single-Center Registry: Long-Term Angiographic and Clinical Outcomes from 1000 Aneurysms[END_REF]. Despite discrepancies in the literature, stent-assisted coiling and flow diversion seem to have higher complication rates compared to coiling: 4-8% versus 1-4%, respectively [START_REF] Kunert | Flow-diverting devices in the treatment of unruptured ophthalmic segment aneurysms at a mean clinical follow-up of 5 years[END_REF][START_REF] Durst | Vision Outcomes and Major Complications after Endovascular Coil Embolization of Ophthalmic Segment Aneurysms[END_REF][START_REF] D'urso | Coiling for Paraclinoid Aneurysms: Time to Make Way for Flow Diverters?[END_REF][START_REF] Maria | Flow Diversion versus Standard Endovascular Techniques for the Treatment of Unruptured Carotid-Ophthalmic Aneurysms[END_REF]. Taken together, these two facts call for an appraisal of the specific risk of rupture of carotid ophthalmic aneurysms.

The purpose of this study was to compare the frequency of ophthalmic aneurysms in patients with aneurysmal subarachnoid haemorrhage and patients with UIAs, in order to estimate the relative rupture risk of ophthalmic aneurysms.

MATERIAL AND METHODS

This was a multicentre, observational study performed according to the STROBE guidelines [START_REF] Von Elm | The Strengthening the Reporting of Observational Studies in Epidemiology (STROBE) statement: guidelines for reporting observational studies[END_REF]. A national institutional review board (IRB) approved the study protocol (IRB approval # CRM-2201-220). This work adheres to the World Medical Association Declaration of Helsinki.

Study population: ruptured intracranial aneurysms

Prospective selection of patients with aneurysmal subarachnoid haemorrhage from January 2021 to March 2021 has previously been reported [START_REF] Lognon | Prospective assessment of aneurysmal rupture risk scores in patients with subarachnoid hemorrhage: a multicentric cohort[END_REF]. Academic centres having contributed to the original study were invited to participate. Thirteen centres [START_REF] Maria | Flow Diversion versus Standard Endovascular Techniques for the Treatment of Unruptured Carotid-Ophthalmic Aneurysms[END_REF] French and 1 Swiss) participated in the present study.

Study population: unruptured intracranial aneurysms

Patients having undergone a cerebral digital subtraction angiography (DSA) performed in all participating centres from January 2021 to March 2021 were screened. Clinical charts and DSAs of all eligible patients were reviewed including age, sex, origin (Finnish, Japanese or other), hypertension, history of subarachnoid haemorrhage, multiple aneurysms and smoking, and aneurysm characteristics, including irregular shape, size and location. Ophthalmic aneurysms were defined as aneurysms with the ophthalmic artery arising from the neck or in the immediate vicinity of the aneurysm, and whose dome pointed superior in agreement with Rhoton's 3 rd rule of aneurysms (illustrated in Supplemental Fig. 1) [15]. Superior hypophyseal aneurysms and carotid cave aneurysms were grouped together, as well as posterior inferior cerebellar artery aneurysms and V4 segment of the vertebral

Statistical analysis

Categorical variables were compared using the χ 2 test. Continuous variables with a normal distribution were compared among groups using the Student's t-test.

Continuous variables without a normal distribution were compared among groups using the Mann-Whitney U test. Odds-ratios for aneurysm rupture were calculated for each aneurysm location, as is recommended for case-control studies [START_REF] Bland | Statistics Notes: The odds ratio[END_REF]. A multivariable logistic regression was applied to identify confounding factors, using age, sex, history of subarachnoid haemorrhage, hypertension, smoking, multiple aneurysms, aneurysm size, irregular aneurysm shape, and aneurysm location as explicative variables. Statistical analysis was performed using MedCalc® Statistical Software version 20.218 (MedCalc Software Ltd, Ostend, Belgium; https://www.medcalc.org; 2023).

RESULTS

Patients' demographics

From January 2021 to March 2021, a total of 604 intracranial aneurysms (355 unruptured and 249 ruptured) were included in this study (Fig. 1

Aneurysm location frequency

The frequency for every aneurysm location is shown in Fig. 2 and reported in Table 1 for all aneurysms (Fig. 2A), UIAs (Fig. 2B) and RIAs (Fig. 2C). Odds-ratios for aneurysm rupture is given for each aneurysm location (Fig. 2D and Table 1).

Ophthalmic aneurysms represented 9.3% (56/604) of intracranial aneurysms, 14.9% (53/355) of UIAs (second most common) and 1.2% (3/249) of RIAs (second least common, tied with anterior choroidal artery aneurysms). Initially, 4 ruptured ophthalmic aneurysms and 57 unruptured ophthalmic aneurysms were reported. Two of these aneurysms were consensually reclassified as a carotid cave aneurysm upon review during manuscript preparation (one ruptured illustrated in Supplemental Fig. 2 and one unruptured). Ophthalmic aneurysms were incidentally discovered 12.4 times more frequently than they were detected during haemorrhage. The unadjusted odds-ratio was 0.07 (95% confidence interval [0.02; 0.23], p value < 0.0001).

Adjusted odds-ratio in the multivariable logistic regression for aneurysm rupture for ophthalmic aneurysms was 0.12 (95% confidence interval [0.03; 0.43], p-value = 0.001).

Aneurysm size

Mean size was 6.0 mm for intracranial aneurysms, 5.3 mm for UIAs, and 7.0 mm for RIAs (Fig. 3A and3B), p < 0.0001. Mean size of unruptured ophthalmic aneurysms was 5.4 mm (6 th highest out of 11 locations), and mean size of ruptured ophthalmic aneurysms was 12.7 mm (highest among all locations). Median size of ruptured ophthalmic aneurysms was 12.0 mm, while median size of ruptured non-ophthalmic aneurysms was 6.0 mm, p-value = 0.11. Ratio between mean size of ruptured aneurysm and mean size of unruptured aneurysm (Fig. 3C) was 2.4 for ophthalmic aneurysms (3 rd highest). Only anterior choroidal artery and posterior cerebral artery aneurysms, representing respectively 2.5% and 0.8% of intracranial aneurysms, were found to have a higher ratio.

DISCUSSION

Our study compared the proportions of each aneurysm location in a prospective cohort of ruptured intracranial aneurysms and a retrospective population of unruptured intracranial aneurysms. Ophthalmic aneurysms were under-represented in the RIA population by more than an order of magnitude. Furthermore, ruptured ophthalmic aneurysms seemed to have a higher mean size compared to other ruptured aneurysms in our study population, while the mean size of unruptured ophthalmic aneurysms is average compared to other unruptured intracranial aneurysms. Although no statistically significative difference was observed regarding the second point, this warrants further study with a larger population, as we only had 3 ruptured ophthalmic aneurysms in our study population. As previously described in the scientific literature, we found that anterior communicating artery aneurysms (and to a lesser extent posterior circulation aneurysms and posterior communicating artery aneurysms) were over-represented in the RIA population, suggesting a higher relative rupture risk compared to other intracranial aneurysms.

Estimating the rupture risk of an UIA is a key element of the decision-making process when considering preventive treatment. Our results are in accordance with existing literature, as internal carotid artery aneurysms have been described as having a lower risk of rupture compared to other intracranial aneurysms [4-7, 17, 18]. To our knowledge, no previous study has estimated the relative risk of rupture for different internal carotid artery aneurysm subgroups and specifically the carotid ophthalmic aneurysm location. Under-representation of carotid cave / superior hypophyseal aneurysms in our ruptured intracranial aneurysm population was not an expected result, and also deserves to be explored, as unruptured carotid cave aneurysms make up roughly 10% of unruptured intracranial aneurysms. As mentioned in the introduction section, ophthalmic aneurysms are routinely treated by stent-assisted coiling and flow diverter stents, with better long-term occlusion rate than regular coiling [START_REF] Lylyk | Pipeline Embolization Devices for the Treatment of Intracranial Aneurysms, Single-Center Registry: Long-Term Angiographic and Clinical Outcomes from 1000 Aneurysms[END_REF]. However, despite some divergent data in the literature, stent-assisted coiling and flow-diverter stenting may have a higher than average complication rate, estimated at 4-8% [START_REF] Kunert | Flow-diverting devices in the treatment of unruptured ophthalmic segment aneurysms at a mean clinical follow-up of 5 years[END_REF][START_REF] Durst | Vision Outcomes and Major Complications after Endovascular Coil Embolization of Ophthalmic Segment Aneurysms[END_REF][START_REF] D'urso | Coiling for Paraclinoid Aneurysms: Time to Make Way for Flow Diverters?[END_REF][START_REF] Maria | Flow Diversion versus Standard Endovascular Techniques for the Treatment of Unruptured Carotid-Ophthalmic Aneurysms[END_REF]. Further studies with a prospective cohort are necessary to confirm that unruptured ophthalmic aneurysms have a low rupture risk, and that they may rupture at higher size compared to other aneurysm locations. Further studies with larger cohorts would enable in-depth analysis of ruptured ophthalmic aneurysms, as only 3 were present in our study. If the results are reproduced, serious reappraisal of carotid ophthalmic unruptured aneurysm preventive exclusion treatment is required. To put it plainly, if, on the one hand, stent-assisted coiling or flow diversion of an ophthalmic aneurysm presents twice as much serious adverse events as standard coiling and, on the other hand, ophthalmic aneurysms have a reduced rupture risk by an order of magnitude, then complication rates of ophthalmic aneurysm stent-based treatments need to be divided by 20 to match the benefit-risk ratio of say, an MCA aneurysm coiling. And this does not even take into account the discrepancy in rupture sizes between ophthalmic and non-ophthalmic aneurysms. These results do not affect the decision-making process for large and giant aneurysms with compressive symptoms, for which treatment is indicated. However, these are an uncommon subgroup of ophthalmic aneurysms. Compressive aneurysms of the internal carotid artery in the literature tend to have a much higher diameter than asymptomatic aneurysms, over 15 mm [START_REF] Boulouis | Flow diversion for internal carotid artery aneurysms with compressive neuro-ophthalmologic symptoms: clinical and anatomical results in an international multicenter study[END_REF][START_REF] Sahlein | Neuroophthalmological outcomes associated with use of the Pipeline Embolization Device: analysis of the PUFS trial results[END_REF]. Meanwhile, the mean size of unruptured ophthalmic aneurysms in our study population was 5.4 mm. No unruptured ophthalmic aneurysm in our study had a diameter over 15 mm, and only 4/53 (8%) had a diameter between 10 mm and 15 mm.

Our study suffers from several limitations. (i) It is a comparison of 2 populations (UIA and RIA), not a direct evaluation of the risk of rupture through a longitudinal follow-up study of a prospective cohort over time, falling short of the gold standard to evaluate risk of rupture statistics. (ii) The UIA populations is comprised only of aneurysms explored by DSA, while not all UIA undergo DSA. We expect that aneurysms that were not explored by angiography, and thus not included in our study, to have been evaluated as having a low risk of rupture based on non-invasive imaging and clinician preconceptions. Therefore, patients with unruptured ophthalmic aneurysms that did not undergo angiography are expected to be overrepresented compared to other locations, especially locations outside "internal carotid artery" aneurysms. If anything, the bias is expected to artificially increase the odds-ratio for ophthalmic aneurysm rupture provided in this study. Nevertheless, this limitation does impact our data. (iii) Counter to expectation, patients in the RIAs subgroup were found to be less prone to have blood hypertension as compared to patients in the UIA subgroup. Patient history data comes from patient interview, either during hospital stay or outpatient consultation. Patients with a RIAs are seen in an emergency setting, often comatose, with medical history often difficult to collect, whereas patients with an unruptured intracranial aneurysm are usually seen as outpatients in a consultation setting, making medical history collection easier and more thorough. Bias in data collection could therefore potentially explain the surprising lower frequency of arterial hypertension in the RIA group. However, this does not impact results regarding aneurysm location. (iv) Carotid cave aneurysms are sometimes misclassified as ophthalmic aneurysms. However only one ruptured carotid cave aneurysm was observed in our study population (which was initially misclassified as an ophthalmic aneurysm). While it was not the purpose of the study, our results suggest that carotid cave / superior hypophyseal aneurysms also have a very low relative risk of rupture compared to other aneurysm locations (with the exception of ophthalmic aneurysms).

Therefore, misclassifications between these 2 subgroups, if they have persisted, would not have a significative impact on the results.

In summary, our study suggested that ophthalmic aneurysms have a very low risk of rupture, and that ophthalmic aneurysms reach a higher-than-average size before they rupture. Operators should take this into account to critically re-evaluate the indication of preventive exclusion treatment of unruptured ophthalmic aneurysms, especially if treatment options with potentially higher complications rates, such as stent-assisted coiling or flow-diversion, are considered.

  artery aneurysms. Inclusion criteria in the UIA subgroup were (a) baseline exploration of an intracranial aneurysm by DSA performed between January 2021 and March 2021; (b) patients aged 18 or more. Exclusion criteria were (a) clinical charts unavailable; (b) imaging data unavailable; (c) dissecting aneurysms and (d) intracavernous aneurysms.

Figure 1 -

 1 Figure 1 -Flowchart of patient selection.

Figure 2 -

 2 Figure 2 -Frequency of different locations for all intracranial aneurysms (A), unruptured intracranial aneurysms (B) and ruptured intracranial aneurysms (C). (D) Forest plot of odds-ratios for aneurysm rupture, comparing aneurysm locations. AChoA: Anterior choroidal artery; MCA : middle cerebral artery; ICA: internal carotid artery; ACA: anterior cerebral artery; Pcom: posterior communicating artery; V4: fourth segment of the vertebral artery; PICA: posterior inferior cerebellar artery; PCA: posterior cerebral artery; Acom: anterior communicating artery.

Figure 3 -

 3 Figure 3 -Mean size of unruptured intracranial aneurysms (A) and ruptured intracranial aneurysms (B) by location. (C) Mean size ratios between ruptured intracranial aneurysms and unruptured intracranial aneurysms by location. AChoA: Anterior choroidal artery; MCA : middle cerebral artery; ICA: internal carotid artery; ACA: anterior cerebral artery; Pcom: posterior communicating artery; V4: fourth segment of the vertebral artery; PICA: posterior inferior cerebellar artery; PCA: posterior cerebral artery; Acom: anterior communicating artery.

Table 1 .

 1 No Finnish or Japanese patients were included.

	). Median age was 57

Table 1 -

 1 Frequency of every aneurysm location for each group, and odds-ratios for aneurysm rupture based on aneurysm locations.

	Adjusted odds-ratio [95%CI] p-value for
	unadjusted OR

UIA: unruptured intracranial aneurysm; RIA: ruptured intracranial aneurysm; 95%CI: 95% confidence interval. * p-value < 0.05.

Acknowledgements:

The authors acknowledge Dr Curado Adelya, Dr Lefebvre Margaux, Dr Papaxanthos Jean, Pr Mounayer Charbel, Dr Saleme Suzana, Dr Nguyen-Kim Dan Linh, Dr Boulouis Grégoire, Dr Ifergan Héloïse, Dr Bala Fouzi, Dr Bibi Richard, Dr Barrot Valère, Pr Herbreteau Denis, Dr Velasco Stéphane, Dr Allard Julien, Dr Lenck Stéphanie, Dr Premat Kévin for their contribution to this original research.

Ethical approval:

All procedures performed in studies involving human participants were in accordance with the ethical standards of the institutional research committee and with the 1964 Helsinki declaration and its later amendments or comparable ethical standards.


NO É DE RANCOURT Tomasz Kania

CONTINUITY OF COORDINATE FUNCTIONALS OF FILTER BASES IN BANACH SPACES

Keywords: . 2020 Mathematics Subject Classification. Primary 46B15, 46B20, 54A20; Secondary 03E15 Filter bases in Banach spaces, generalised bases, automatic continuity, definable filters

We prove that the coordinate functionals associated with filter bases in Banach spaces are continuous as long as the underlying filter is analytic. This removes the large-cardinal hypothesis from the result established by the two last-named authors ([Bull. Lond. Math. Soc. 53 (2021)]) at the expense of reducing the generality from projective to analytic. In particular, we obtain a ZFC solution to Kadets' problem of continuity of coordinate functionals associated with bases with respect to the filter of statistical convergence. Even though the automatic continuity of coordinate functionals beyond the projective class remains a mystery, we prove that a basis with respect to an arbitrary filter that has continuous coordinate functionals is also a basis with respect to a filter that is analytic.

Introduction

Filter bases generalise the familiar concept of a Schauder basis in a Banach space in a very natural way: let F be a filter of subsets of the natural numbers (a family of subsets that is closed with respect to finite intersections and is upwards-closed with respect to set inclusion; in order to avoid trivialities, we assume that every filter contains the filter of cofinite sets) and let X be a Banach space (we consider Banach spaces over the fixed field K of real or complex numbers). A sequence (e n ) ∞ n=1 in X is an F-basis for X whenever for each x ∈ X there exists a unique sequence of scalars (e * n (x)) ∞ n=1 such that x = n,F e * n (x)e n , where the series is interpreted as the F-limit of the sequence of the corresponding partial sums. (A sequence (x n ) ∞ n=1 in a normed space X F-converges to x ∈ X whenever for all ε > 0 there is A ∈ F such that for all n ∈ A one has ∥x -x n ∥ < ε; thus, when this notion is applied to the partial sums filter convergence, x = n,F a n renders to almost all partial sums are close to x, rather than x is the sum of almost all a n 's.)

Unlike for more familiar Schauder bases (i.e., filter bases with respect to the filter of cofinite sets), the question of continuity of the coordinate (linear) functionals x → e * n (x) (x ∈ X) is non-obvious. This very problem was posed by Kadets and revived in 2011 with a particular emphasis on the filter of statistical convergence, that is the filter of subsets of N having density 1 (see, e.g., [START_REF] Connor | A characterization of Banach spaces with separable duals via weak statistical convergence[END_REF][START_REF] Ganichev | Filter convergence in Banach spaces and generalized bases[END_REF]). Certainly, the problem is equivalent to the problem of continuity of the initial F-basis projections (P n ) ∞ n=1 given by P n x = n i=1 e ⋆ i (x)e i for n ∈ N and x ∈ X.

The present paper subsumes (at least in the analytic case) the results obtained in the recent study by the two last-named authors [START_REF] Kania | Large cardinals and continuity of coordinate functionals of filter bases in Banach spaces[END_REF], where it was proved that, assuming the existence of a supercompact cardinal, the coordinate functionals associated with a filter basis are continuous as long as the filter is projective. The aim of the paper is twofold. First, we remove the extra assumption of the existence of supercompact cardinals used for proving automatic continuity of the evaluation functionals of a filter basis at the expense of restricting to analytic ideals, which appear to be quite sufficient from the point of view of applications. In particular, it is sufficient to answer Kadets' question concerning the filter of statistical convergence in ZFC. Secondly, we contribute to (and hopefully trigger) further development of the theory of filter bases in Banach spaces, that may serve as a proxy in the situation where genuine Schauder bases are unavailable. For instance, we prove the principle of small perturbations for such bases (Theorem C) that is a basic result in the context of Schauder bases.

We wish to emphasise that filter bases, despite striking similarities to Schauder bases, may behave quite differently.

• Indeed, there is a filter basis (with respect to the filter of statistical convergence) in a separable Hilbert space whose initial projections are bounded, yet not uniformly ([3, Example 1]). • Kochanek proved in [START_REF] Kochanek | F-bases with brackets and with individual brackets in Banach spaces[END_REF] that for filters generated by fewer sets than the pseudointersection number (in particular, for countably generated filters) one indeed retains the automatic continuity of initial basis projections. Moreover, for 'filtermany' of them there is a uniform bound on the norm of these. However, the filter of statistical convergence is generated by precisely continuum many sets, so Kochanek's result does not apply to it. • Avilés, Kadets, Pérez Hérnandez, and Solecki [START_REF] Avilés | Baire theorem for ideals of sets[END_REF] distilled the key property from Kochanek's proof that makes it work and termed filters having it Baire filters. However analytic Baire filters are necessarily countably generated ([1, Proposition 4.1]).

A word of explanation behind topological properties of filters is required. Since there is a natural correspondence between subsets of the set of natural numbers and 0-1 sequences, one may treat filters on this set as subsets of the Cantor set {0, 1} N equipped with the product topology and consider properties such as being Borel, analytic (continuous image of a Polish space), etc. that we shall freely do. For more information, we refer to [START_REF] Farah | Analytic quotients. Theory of lifting for quotients over analytic ideals on integers[END_REF], where the dual notion of an ideal is considered. All unexplained terminology from Descriptive Set Theory (such as the definitions of the Σ i n , Π i n , and ∆ i n -classes, i = 0, 1, n ∈ N, and of Baire-measurability) is in line with [START_REF] Kechris | Classical descriptive set theory[END_REF].

Our first main result, Theorem A, subsumes the main result of [START_REF] Kania | Large cardinals and continuity of coordinate functionals of filter bases in Banach spaces[END_REF].

Theorem A. Let F be an analytic filter on N. Then for every F-basis the corresponding coordinate functionals are continuous.

As the filter of statistical convergence is Borel ([5, Example 1.2.3(d)]), hence analytic, we obtain a solution in ZFC to Kadets' problem.

Corollary 1. Let F st be the filter of statistical convergence. Then for every F st -basis the corresponding coordinate functionals are continuous.

Theorem A has a natural generalisation in the case where all ∆ 1 n -subsets of Polish spaces are Baire-measurable. Since this assumption cannot be proved in ZFC alone, we decided to present the result as a separate theorem to highlight that Theorem A is a theorem of ZFC, even though Theorem A follows from Theorem A ′ , since, by Souslin's Theorem [START_REF] Kechris | Classical descriptive set theory[END_REF]Theorem 14.11], ∆ 1 1 is the family of all Borel sets and Borel sets are Baire measureable in ZFC.

Theorem A ′ . Let n ⩾ 1. Assume that all ∆ 1
n -subsets of Polish spaces are Baire-measurable. Suppose that F is a Σ 1 n -filter on N. Then for every F-basis the corresponding coordinate functionals are continuous.

A word of explanation on the assumption of Baire-measurability of ∆ 1 n -sets is required. A classical result due to Banach and Mazur ensures that for n ⩾ 1 Baire-measurability of all ∆ 1 n+1 -subsets (and even all Σ 1 n+1 -subsets) of Polish spaces follows from determinacy of Σ 1 n -games on integers (this is an easy consequence of [10, Theorem 21.5]). The latter determinacy assumption has been shown by Martin [START_REF] Martin | Measurable cardinals and analytic games[END_REF] to follow for n = 1 from the existence of a measurable cardinal and by Martin and Steel [START_REF] Martin | A proof of projective determinacy[END_REF] to follow for n ⩾ 2 from the existence of n Woodin cardinals and a measurable cardinal above them. Knowing that supercompact cardinals are measurable (see [START_REF] Jech | Set theory. The third millennium edition, revised and expanded[END_REF]Lemma 20.16]) and limits of Woodin cardinals (see [START_REF] Jech | Set theory. The third millennium edition, revised and expanded[END_REF]Exercise 34.1]), it follows that the existence of a supercompact cardinal implies the determinacy of all projective games on integers and hence the Baire-measurability of projective subsets of Polish spaces. In particular, from Theorem A ′ we can recover the main result of [START_REF] Kania | Large cardinals and continuity of coordinate functionals of filter bases in Banach spaces[END_REF].

Corollary 2 ([9]

). Assume the existence of a supercompact cardinal. Let F be a projective filter on N. Then for every F-basis the corresponding coordinate functionals are continuous.

Nonetheless large cardinals are actually not required to get Baire-measurability of projective sets. It was proved by Martin and Solovay [START_REF] Martin | Internal Cohen extensions[END_REF] that under Martin's Axiom and the negation of the Continuum Hypothesis all Σ 1 2 -subsets of Polish spaces are Baire-measurable. Moreover, it was proved by Shelah [START_REF] Shelah | Can you take Solovay's inaccessible away?[END_REF] that ZFC + 'all projective subsets of Polish spaces are Baire-measurable' is equiconsistent with ZFC alone, thus completely removing the need for large cardinals. Results as those above are often proved in the literature for specific Polish spaces (e.g., the Cantor space), but here we freely use the fact that Baire-measurability results, if established for certain Σ 1 n -classes (n ∈ N) of subsets of the Cantor space, can be elementarily transferred to arbitrary Polish spaces using, e.g., [2, Theorem 3.15], which asserts that any two uncountable Polish spaces without isolated points are Borel-isomorphic via a map that preserves meagre sets.

We remark in passing that, assuming the consistency of one of the above-mentioned largecardinal assumptions, the conclusion of Theorem A ′ is consistent with results obtained by forcings that do not destroy Woodin cardinals and measurable cardinals. It follows from [START_REF] Hamkins | Small forcing creates neither strong nor Woodin cardinals[END_REF]Corollary] and [START_REF] Lévy | Measurable cardinals and the continuum hypothesis[END_REF]Theorem 3] that forcings of cardinality smaller than the first measurable cardinal satisfy this condition.

Our second main result asserts that a filter basis with respect to a filter that is not analytic is also a filter basis with respect to a filter that actually is analytic, provided the coordinate functionals are continuous.

Theorem B. Let F be a filter on N (not necessarily projective). Let (e n ) ∞ n=1 be an F-basis with continuous coordinate functionals. Then there exists an analytic filter

F ′ ⊂ F such that (e n ) ∞
n=1 is also an F ′ -basis.

Observe that, in the above result, the coordinate functionals of the basis (e n ) ∞ n=1 are the same when considered with respect to the filter F or the filter F ′ .

Proofs of Theorems A and A ′

Proof of Theorem A. Let X be a Banach space with an F-basis (e n ) ∞ n=1 . We denote by (e ⋆ n ) ∞ n=1 the corresponding coordinate functionals. We start by proving that coordinate functionals e ⋆ n (n ∈ N) are Σ 1 1 -measurable. Indeed, note that for a fixed open set U ⊂ R and n ∈ N we get

e ⋆ n (x) ∈ U ⇐⇒ ∃ (α i )∈K N i,F α i e i = x ∧ α n ∈ U ⇐⇒ ∃ (α i )∈K N ∀ l∈N ∃ A∈F ∀ m∈A m i=1 α i e i -x < 1 l ∧ α n ∈ U By [10, Proposition 3.3], the space X × K N × {0, 1} N is Polish. For m ∈ N let us define Φ m : X × K N → R by Φ m (x, (α i )) = m i=1 α i e i -x x ∈ X, (α i ) ∈ K N ,
which is a continuous map. We then conclude that for every l ∈ N the set S l given by

S l = (x, (α i ), A) ∈ X × K N × {0, 1} N : ∀ m∈A Φ m (x, (α i )) < 1 l ∧ α n ∈ U = (x, (α i ), A) : ∀ m∈N m / ∈ A ∨ Φ m (x, (α i )) < 1 l ∧ α n ∈ U = m∈N (x, (α i ), A) : m / ∈ A ∨ Φ m (x, (α i )) < 1 l ∧ α n ∈ U = m∈N {(x, (α i ), A) : m / ∈ A} ∪ (x, (α i ), A) : Φ m (x, (α i )) < 1 l ∩ {(x, (α i ), A) : α n ∈ U } is a G δ subset of X × K N × {0, 1} N (the condition 'm / ∈ A' encodes a set that is both closed and open in {0, 1} N ).
Recall that countable intersections of analytic subsets of Polish spaces are analytic, and that images and preimages of analytic sets by a continuous maps between Polish spaces are analytic (see [START_REF] Kechris | Classical descriptive set theory[END_REF]Proposition 14.4]). Also recall that Borel subsets of Polish spaces are analytic (see [START_REF] Kechris | Classical descriptive set theory[END_REF]Theorem 13.7]). Since the filter F is analytic, we deduce that its preimage X × K N × F by the projection X × K N × {0, 1} N → {0, 1} N is also analytic, thus the set

S := l∈N proj X×K N X × K N × F ∩ S l
is analytic. Observe that, for x ∈ X and (α i ) ∈ K N , we have

(x, (α i )) ∈ S ⇐⇒ ∀ l∈N ∃ A∈F (x, (α i ), A) ∈ S ⇐⇒ ∀ l∈N ∃ A∈F ∀ m∈A m i=1 α i e i -x < 1 l ∧ α n ∈ U .
Thus we have (e * n ) -1 (U ) = proj X [S], and we conclude that (e ⋆ n ) -1 (U ) is analytic. On the other hand, e ⋆ n is also Π 1 1 -measurable. Indeed, writing U c =: j∈N V j , where the V j 's are open subsets of K, we obtain

(e ⋆ n ) -1 (U ) c = j∈N (e ⋆ n ) -1 (V j ),
an analytic set.

Once we know that for any open U ⊂ K the set (e ⋆ n ) -1 (U ) is both analytic and coanalytic, by Souslin's Theorem ([10, Theorem 14.11]), it is Borel, so e ⋆ n is Borel-measurable. Since each Borel set is Baire-measurable, we conclude that e ⋆ n is a Baire-measurable homomorphism between Polish groups X and K. Consequently, by the Banach-Pettis Theorem ([10, Theorem 9.10]) e ⋆ n is a continuous map. □

It was not essential for the proof of Theorem A to prove Π 1 1 -measurablity of the coordinate functionals as all analytic sets are already Baire-measureable (see [START_REF] Kechris | Classical descriptive set theory[END_REF]Theorem 21.6]). Nontheless, we decided to follow this way of reasoning in order to emphasise the analogy with the proof of Theorem A ′ whose proof is modelled on the proof of Theorem A and the latter follows from the former. However, for reader's convenience, we present a sketch of the proof with only certain details that are different.

Proof of Theorem A ′ . As in the proof of Theorem A, we fix an open U ⊂ K and we get that

e ⋆ n (x) ∈ U ⇐⇒ ∃ (α i )∈K N ∀ l∈N ∃ A∈F ∀ m∈A m i=1 α i e i -x < 1 l ∧ α n ∈ U
The family of Σ 1 n -sets contains Borel sets, is closed under countable intersections, and under images and preimages with respect to continuous maps between Polish spaces (see [START_REF] Kechris | Classical descriptive set theory[END_REF]Theorem 37.1]). So we can deduce similarly as in the proof of Theorem A that the set

(e ⋆ n ) -1 (U ) is Σ 1 n . Writing U c =: n∈N V n , where the V n 's are open sets, we have (e ⋆ n ) -1 (U ) c = n∈N (e ⋆ n ) -1 (V n ),
an Σ 1 n -set. We deduce that (e ⋆ n ) -1 (U ) is Π 1 n , so taking into account first part of proof we see that it is ∆ 1 n . By the hypothesis, all ∆ 1 n -sets are Baire-measurable, so e ⋆ n (n ∈ N) is a Baire-measurable group homomorphism. Consequently, by the Banach-Pettis Theorem e ⋆ n (n ∈ N) is continuous. □

Proof of Theorem B

We are ready to prove Theorem B.

Proof of Theorem B. Set

A := A ⊂ N : ∃ x∈X ∃ ε>0 A ⊃ n ∈ N : n i=1 e ⋆ i (x)e i -x ⩽ ε .
It follows from the fact that (e n ) ∞ n=1 is an F-basis that A ⊂ F. Note that (e n ) ∞ n=1 remains an F ′ -basis for any filter F ′ on N such that A ⊂ F ′ . Now, for every n ∈ N, consider the set

B n = (x, ε, A) ∈ X × (0, ∞) × {0, 1} N : n ∈ A ∨ n i=1 e ⋆ i (x)e i -x > ε .
Using the continuity of e ⋆ i (i ∈ N) and the fact that {0, 1} N comes equipped with the topology of pointwise convergence, we observe that B n is open. Observing that

A = proj {0,1} N n∈N B n ,
we deduce that A is analytic.

As A ⊂ F, finite intersections on elements of A are readily non-empty. Consequently, A generates a filter F ′ ⊂ F. We have:

F ′ ={A ⊃ N : ∃ n∈N ∃ A 1 ,A 2 ,...,An∈A A ⊃ A 1 ∩ A 2 ∩ . . . ∩ A n } = n∈N proj ({0,1} N ) 1 [{(A, A 1 , . . . , A n ) ∈ ({0, 1} N ) n+1 : A ⊃ A 1 ∩ . . . ∩ A n ∧ A 1 , . . . , A n ∈ A}],
where proj ({0,1} N ) 1 is the projection onto the first coordinate. Consequently, F ′ is an analytic filter and (e n ) ∞ n=1 is an F ′ -basis since A ⊂ F ′ ⊂ F. □

Closing remarks

Having established continuity of basis projections (at least for analytic filters as proved in Theorem A, which in the light of Theorem B appears to be sufficient for applications), it is possible to develop a filter basis theory analogous to the theory of Schauder bases. For instance, we may recover the Bessaga-Pe lczyński small-perturbations principle.

Theorem C. Let X be a Banach space with an F-basis (e n ) ∞ n=1 , with continuous coordinate functionals (for example, F is analytic).

If (f n ) ∞ n=1 is a sequence in X such that ∞ n=1 ∥f n -e n ∥ • ∥e * n ∥ ⩽ δ < 1, for some δ ∈ (0, 1), then (f n ) ∞ n=1 is congruent to (e n ) ∞ n=1 , that is, (f n ) ∞
n=1 is an F-basis and the assignment e n → f n (n ∈ N) extends to a surjective isomorphism of X onto itself.

Proof. The proof are analogous to the familiar case of Schauder bases. Indeed, we observe that the operator T : X → X given by T ( i,F a i e i ) = i,F (a i (f i -e i )) has norm strictly less than one. Indeed, for x = i,F a i e i we have

∥T x∥ = i,F a i (e i -f i ) ⩽ i ∥a i (e i -f i )∥ = i ∥e ⋆ i (x)(e i -f i )∥ ⩽ i ∥e ⋆ i ∥∥x∥∥e i -f i ∥ = i ∥e ⋆ i ∥∥e i -f i ∥ ∥x∥ ⩽ δ∥x∥.
Consequently, I X + T is invertible and maps e i to f i (i ∈ N), which proves that (f n ) ∞ n=1 is an F-basis too. □ Interestingly, general F-bases are different from Schauder bases as, for example, we do not have the basis constant at the disposal. Indeed, it follows from Grunblum's criterion ([4, Theorem V.1]) that if the basis projections are uniformly bounded, then the F-basis is already a Schauder basis.

We conclude our note with two open problems.

Question 1. Does there exists a Banach space X, a filter F on N and an F-basis (e n ) ∞ n=1 for X such that not all coordinate projections (e ⋆ n ) ∞ n=1 are continuous?

Theorem B demonstrates that filter bases with respect to analytic filters provide a very natural framework for the theory. In [3, Example 1] the authors provided an example of an F st -basis in ℓ 2 that is not a Schauder basis. However, F st has a relatively low complexity being F σδ ([5, Example 1.2.3(d)]). It is thus natural to ask whether being analytic in Theorem B can be strengthened to being Borel. Question 2. Let F be a filter and let (e n ) ∞ n=1 be an F-basis for a Banach space X with continuous coordinate functionals. Does there exists a Borel filter F ′ such that (e n ) ∞ n=1 is also an F ′ -basis? Should it be the case, what is the smallest complexity of such F ′ ?

Acknowledgements. The authors are grateful to Jiří Spurný and Miroslav Zelený for organisning the 49 th Winter School in Abstract analysis in Sněžné, Czech Republic in January 2022 during which the research discussed in the present note was initiated. The second-named author thus cordially acknowledges support received from SONATA 15 No. 2019/35/D/ST1/01734 that allowed him to travel to this conference.

support received from GA ČR project 19-07129Y; RVO 67985840.

Email address: swaczyna@math.cas.cz, jaroslaw.swaczyna@p.lodz.pl