Eric Herbert

Mathieu Pernot
email: <mathieu.pernot@espci.fr

Gabriel Montaldo

Mathias Fink

Mickael Tanter

Energy-based adaptive focusing of waves: application to noninvasive

Keywords: Artifacts, Computer-Aided Design, Equipment Design, Equipment Failure Analysis, Image Enhancement, instrumentation, methods, Reproducibility of Results, Sensitivity and Specificity, Ultrasonography, instrumentation, methods

Introduction

In wave physics, adaptive focusing is a research topic that encompasses a wide variety of applications ranging from electromagnetic waves in radar or satellite telecommunications , ultrasonic waves in underwater acoustics or medical ultrasound and finally

[1 ] [2 ] [3 ] [4 ]
optical waves in microscopy or astronomy .

[5 ] [6 ] [START_REF] Gonsalves Ra | Phase retrieval and diversity in adaptive optics[END_REF] Adaptive focusing is based on estimating and then applying time delays (or phase delays for monochromatic waves) on each element of an array to correct differences in the travel path of transmitted or received waves due to medium heterogeneities. Such time delays can be estimated by the echoes from a bright reflector, such as a kidney stone reflecting an incident ultrasonic wave in lithotripsy, or a point like active source such as a star in adaptive astronomy located at the desired focus. In such configurations, the array records the impulse response linking each element in the array to the focal point. From the knowledge of this set of data, the so called Green s function, one ' simply has to time reverse the wavefield (or phase conjugate for monochromatic signals) to focus back on the initial source location. For example, time reversal focusing has encountered a significant success in the ultrasound community because piezo-electric [START_REF] Fink | Time-reversed acoustics[END_REF] [START_REF] Fink | Time-reversal acoustics in biomedical engineering[END_REF] transducer technology enables the exact recording and transmitting of broadband signals (i.e. transmission and recording of both amplitude and phase of the wavefield). Time reversal focusing was also extended to heterogeneous and dissipative media by applying the concept of spatio-temporal inverse filters .

[10 ] [11 ] [12 ] When it is not possible to rely on the presence of a point-like active or passive source at the desired focus, the spatial coherence of echoes reflected by randomly distributed Rayleigh scatterers can be used to assess such time delays . The coherence between [START_REF] Nock | Phase aberration correction in medical ultrasound using speckle brightness as a quality factor[END_REF] [START_REF] Odonnell | Phase aberration measurements in medical ultrasound -human studies[END_REF] signals received on two neighbouring elements of the array can be used in order to estimate the delays. The Van cittert Zernike theorem, developed for optics and later applied to medical ultrasound by Mallard et al. enables an estimation of the focusing quality in [START_REF] Mallart R | Adaptive focusing in scattering media through sound-speed inhomogeneities -the vancittert-zernike approach and focusing criterion[END_REF] [START_REF] Mallart R | The vancittert-zernike theorem in pulse echo measurements[END_REF] such speckle environment from the spatial autocorrelation of the wavefield recorded on the array. However, in some practical situations and application domains, the measurement of the Green s function between the array and the targeted focus is not possible. In optics, one ' has unfortunately only access to the wave intensity so that the phase differences between each receiver or transmitter cannot be estimated.

Recently, in the optics community, Vellekoop et al. proposed an interesting way to overcome this technological limitation . They [START_REF] Vellekoop | Focusing coherent light through opaque strongly scattering media[END_REF] demonstrated the refocusing of optical beams through a multiple scattering medium by estimating only the optical intensity of the beam at the focus. The technique was based on successive modifications of spatial phase distribution of the incident beam using a spatial light modulator (SLM). The implication and potential applications of this method are important in optics because they demonstrated precise control and refocusing of diffuse light through opaque objects.

In the ultrasound community, Urban et al. recently proposed the use of the ultrasonic radiation force as a method for improving [START_REF] Urban | Phase aberration correction using ultrasound radiation force and vibrometry optimization[END_REF] the focusing of a phased array through a heterogeneous medium. The basic principle relies on the idea that maximizing tissue displacements induced at the focus optimizes the focusing at this point. Indeed, it is well accepted that local displacements induced at the focus by the ultrasonic radiation force of the beam is proportional to the beam intensity . They proposed an iterative optimization [START_REF] Sarvazyan | Shear wave elasticity imaging: A new ultrasonic technology of medical diagnostics[END_REF] scheme on an element by element basis. While they attempted to focus the beam on a small spherical target through a strong aberrator, they applied a set of phase corrections to each element of the phased array. The phase shift correction was varied for one element of the phased array and the motion of the target was measured consecutively for each correction. The optimized phase correction for this element was assumed to be the one that maximized the amplitude of the target displacement. Once this element was optimized, the optimization of the other elements was then achieved iteratively using the same technique.

This technique may present several drawbacks in a practical implementation. First the relative contribution of a single element to the total acoustic intensity varies as for large N. For example, for a ten-element array a relative increase of approximately 20 of the % total intensity is obtained at the focus through the optimization of one element, whereas an increase of only 0.4 can be obtained using a % 500 element array. Such small variations of the intensity are barely detectable. For in vivo applications, this method may become completely unusable with large phased arrays that are commonly used in diagnostic ultrasound or in HIFU therapy. A second important issue is the large number of iterations required for the optimization. To achieve the correction, 20 phase changes were necessary for each element of the transducer. As the number of element increases, the number of iterations, and therefore the duration of the optimization, increases rapidly and becomes an important time-limitation. Finally, this focusing technique is not fully optimal beacause the accuracy of the phase correction might not be sufficient after the optimization process for an array with a large number of channels. Two or more correction processes might be necessary to determine the phase with more accuracy.

In this paper, we propose a general concept of energy-based adaptive focusing approach that could be applied in various domains of strongly improve the sensitivity of previous work. In the context of medical ultrasound, the approach proposed in this paper presents the advantage to be applicable in-vivo using arrays with a large number of channels and provides a direct estimation of the optimal aberration correction. The principle relies on the use of the radiation force generated by the ultrasonic beam at the target location. This ultrasound radiation force has been used in soft tissue elasticity imaging to image the viscoelastic properties of tissues through the generation of shear waves or the motion induced at the focal zone .

[18 ] [19 ] [START_REF] Nightingale | Acoustic radiation force impulse imaging: In vivo demonstration of clinical feasibility[END_REF] Contrary to the previous techniques, we propose to optimize the phases of each element of the transducer array all at the same time by using spatially coded emissions. This is done by defining virtual transducers from linear combinations of the array elements. Based on measurements of tissue motion (indirectly linked to the local beam energy), the phase aberration between two of these virtual transducers can be measured using a limited number of acquisitions. Because all elements of the array are used for each illumination, a first advantage for in vivo applicability of the method lies in the strong improvement of signal to noise ratio (SNR) at the measurement location. Once the phase shifts are obtained for all the virtual transducers, the phase aberrations as well as the amplitude aberrations can be derived. Based on a direct and non iterative inversion algorithm, this technique allows an optimal focal spot to be restored non-invasively and can be applied to a large number of elements with a good accuracy while drastically reducing the acquisition time. In this paper, the optimization technique is described as well as the experimental setup. Experimental results demonstrate the potential of this technique using a transducer composed of 64-elements focusing through strong phase aberrations. In this paper, the local displacements induced by the radiation force are estimated by the ultrasonic array via speckle tracking of backscattered echoes.

The theoretical framework of our technique is presented in the first section of the paper. In the second section, this new focusing approach is applied in a first experimental setup to learn how to autofocus the ultrasonic beam onto a point-like reflecting target. The goal of this roof of concept experiment is to demonstrate the efficiency of the technique in a simple configuration. The third section deals with " the application of the technique in a much more realistic configuration for medical applications in which the backscattering medium is made of very small Rayleigh scatterers randomly distributed in space, i.e. speckle environment and does not contain any bright specular reflector. Finally, potential applications and limitations of the technique are discussed in the last section.

Theory

In this section, a theoretical framework is developed to estimate directly the phase and amplitude aberrations for multielement ultrasonic arrays using indirect measurements of the beam intensity. This method relies on the assumption that maximizing the radiation force at the focal point optimizes the acoustic intensity at the same point. Indeed, the acoustic radiation force generated in biological tissues is assumed here to be linearly related to the acoustic intensity. The local force per unit of volume generated by an ultrasound beam in tissues can be written as:

where is the acoustic intensity, is the attenuation coefficient of the tissue, is the speed of sound and e is the direction of the wave I α c z propagation.

Here we assume that the ultrasonic array is composed of N elements working at the same frequency in continuous mode and the amplitude aberration and is the angular frequency. Therefore, the total intensity at the focal point is given by: ω where is the local acoustical impedance of the medium at the focal point.

shows that the total intensity at the focus is Z Equation 2given by the modulus of the sum of the complex amplitudes . We will show now how to determine and based on intensity A n e i Φ n

Φ n A n measurements.

Determination of a phase shift between two transducers

In the case of only two transducer elements, the acoustic intensity at the focus through the aberrator is:

with When an additional phase shift in introduced in the wave transmitted by the second transducer, the intensity

Φ = Φ 2 -Φ 1 x
measured at the focus is modulated by a cosine function of x:

In order to determine , this problem can be written as a linear equations system with 3 unknows: cos( ) and sin( ).

Φ B, C Φ C Φ
This system is solvable if the intensity is measured at least for three different values of in the range 0 2 . Then is derived x

[ -π] Φ
immediately, solving the phase aberration problem.

Generalisation to N-element array using spatially coded emissions

The technique described previously for two elements cannot be applied directly to the case of multielement arrays. The main reason is that the intensity generated by the combination of two small elements of a large phased array is too weak to generate any detectable radiation force effect. We propose here to solve this problem and generalize the previous technique by defining virtual elements that are The transformation basis matrix H must be chosen such as that each vector V is a combination of all the phased array elements. In this paper we propose to use a basis transformation based on the Hadamard decomposition. Taking advantage of the properties of the Hadamard matrix (composed only of 1 and 1), each virtual transducer of this new basis can transmit the full power capability of the array and thus, the effects of the radiation force generated by one of these virtual elements can be detected. Using the same method than for two transducers, two virtual transducers are used with an additional phase shift We propose the following combinations of virtual x.

transducers:

and ( ( where is a virtual transducer used as reference, is the virtual

V 0 + V , V m 0 -V m V 0 + V )e m ix + V 0 -V )e m ix - V 0 V m m th
element and is a phase shift arbitrary chosen. The choice of these combinations was based on the fact that the signals transmitted by the x real transducers have the same amplitude on each channel.

The pressure field generated at the focus by the virtual transducer is:

P m V m
Let us first consider a transformation basis matrix H that is based on the Hadamard decomposition: for 1 the combinations are : m = One can notice that the transmit amplitudes in are either equal to 0 or 1. This allows the use of a simple transmit electronics with Eq. 9 binary efficient transmitters.

The intensity at the focal point induced by the first combination is: I with the complex conjugate of . This gives: V̅ m V m

Proceeding with the same method for the second combination, , we find that:

I 2 m
Here, exp( ) is defined as the reference vector, and we fix arbitrarily its phase and amplitude 1 and 0. Therefore

P B 0 = 0 i Ψ 0 B 0 = Ψ 0 =
we obtain:

For the third combination ( ) ( ) 2( cos sin ), the intensity at the focus is: P 0 + P m e ix + P 0 -P m e ix -= P 0 x + iP m x Two mesurements of ( ) can be performed with two different phase shifts If we fix the two phase shifts as following: /4 I 3 m x x.

x 1 = π and /4, we find the imaginary part of :

x 2 = -π P m
Finally, once the real and imaginary parts of pressure field at the focus are measured by this method, the pressure P m = B m e i Ψ m field received at the focus for each transmit element can be derived using the inverse of the matrix H:

p n T n
One should notice that using this method both the phase aberrations and the relative amplitudes of the wave through the aberrator Φ n are retrieved. Therefore, in order to correct the phase aberrations, one has to correct the transmitted signals with an additional phase A n φ n This enables to create constructive interferences at the focus.

= -Φ n

Basic concept experiment

Experimental setup

In order to show the feasibility of this technique a first proof of concept experiment was performed using a strong reflective target at " " the desired focus location.

A 64 elements phased array, previously described was used for both the generation and the measurement of the target [START_REF] Pernot | High power transcranial beam steering for ultrasonic brain therapy[END_REF] displacement. This array was driven by a transmit/receive multichannel electronic platform. The electronics allowed us to transmit arbitrary signals waveforms (sampling rate 30 MHz, Peak Amplitude 40 Volts) individually and simultaneously on each of the 64 elements. The array was fixed on the side of a tank (450 mm both for width and depth, 400 mm high, see ). The phased array was Fig. 1 spherically curved and was geometrically focused at a 140 mm focal depth. Tansducers were working at a central frequency of 1 MHz.

The reflecting target was a small stainless steel sphere (diameter 1.5 mm) embedded in the center of a soft phantom. This phantom consisted of a 10 polyvinyl alcohol (PVA) cryogel (75 50 20 mm), which was formed by freezing and thawing an aqueous PVA % × × solution . The phantom was homogeneous and underwent one cycle of freezing and thawing. The average shear modulus of the [START_REF] Kenneth | Polyvinyl alcohol cryogel: An ideal phantom material for MR studies of arterial flow and elasticity[END_REF] background material was 40 kPa estimated using Transient Elastography which agreed with values from literature for a 10

[19 ] [23 ] %
PVA solution after one cycle of freezing and thawing. The phantom was fixed to a 3D micro-control plates which allowed a good positioning accuracy (about 1 m). Displacement and position of the micro-control plates were computer controlled.

μ

In this proof of concept experiment a numerical aberrator was introduced by adding electronically supplementary individual time delays (i.e. phase delays for long bursts) on each transmit signal. This set of time delays mimicks a real and infinitely thin aberrating layer located very close to the array, known as phase screen aberator . These phase aberrations were generated randomly in the range 0 2

" " [ -π]
for the 64 channels.

Ultrasonic Sequences

The sphere embedded in the phantom was positioned in the vicinity of the geometrical focus using the 3D micro-control plates. A simple trick was used in order to avoid very precise and fastidious positioning of the target at the exact geometrical focus. The Green s ' function of the spherical target was obtained by successive time reversal operations: first a short impulse (2 cycles, 1 MHz) was emitted by a central element of the array. The resulting wavefront is widely spread in space during propagation and the backscattered echo of the spherical target was received by each transducer of the array. Then, the received signals were normalized, time reversed and finally emitted again. This process was repeated typically 4 or 5 times in order to reach the optimal set of transmit signals focusing on the sphere in a homogeneous medium configuration.

In order to obtain a measurable target displacement from the ultrasonic radiation force, bursts of ultrasound were transmitted during several hundreds of micro-seconds. To optimize the mechanical excitation on the spherical target, the push transmit signal (TRpush) was

" "
obtained from the convolution of a sinusoidal signal at 1 MHz with the transmit signals corresponding to the Green s function of the sphere ' and acquired during the iterative time reversal calibration.

The timing of the transmit sequence was composed of one push transmit signal interleaved between two pulses-echo signals ( ).

" " Fig. 2 The target s initial position was given by the echo of a first pulse. The target was then pushed thanks to the radiation force of the push

' " "
transmit signal. The burst duration was 400 s and the amplitude was 36 V leading to a peak pressure at the focus of 5 MPa. The last step μ consisted of measuring the target s new position using a second pulse echo right after the push, typically after 25 s.

' μ

Displacement measurement

The displacement of the spherical target was estimated using a conventional cross-correlation algorithm performed on consecutive pulse-echo RF signals. An important advantage of this method was the possibility of measuring the motion of the target using the same transducers than for the radiation force sequence. All the elements were used in pulse-echo mode using the optimized focusing law. The received signals were beamformed using the same focusing law. A window of 500 points that corresponds to 10 s was selected around the μ echo of the target on the beamformed RF signal. A cross-correlation was performed on the signals aquired before and after the push to find the relative displacement (see ). Fig. 3 Adaptive focusing experiment

Transmit codes

As described before, this method was based on spatially coded emissions. It required the emission of 4 signals for each virtual transducer that were calculated by linear combinations of columns of the Hadamard s matrix, and ( (

' V 0 + V , V m 0 -V m V 0 + V )e m ix + V 0 -V
where was the first vector used as a reference and was set successively to /4 and /4. For one given combination, the phase )e m ix -

Vo x

-π π and the amplitude of the emission signal of each element were computed based on the value of the complex amplitude of the vector and the phase aberration. Each combination was successively emitted and the received signals were averaged 5 times using the pulse/push sequence (see above).

Data processing

For each virtual transducer, the displacement of the target was measured for the 4 combinations. The real and imaginary part of the complex amplitude associated to this virtual transducer was then calculated as explained in II-A and II-B. The complex B m e i Ψ m § amplitudes of the elements were obtained by the multiplication of the inverse of the hadamard matrix H and (see

). B m e i Ψ m
Eq. 17

Assessment of the efficiency of the adaptive focusing process

Hydrophone scan

In order to test the efficiency of this method, we compared the focus quality at different stages of the optimization process. The focal area was scanned in the XY directions (see ) using a needle hydrophone (diameter of the sensitive surface is 400 m, Speciality Fig. 1 μ Engineering Associates, PZT-Z44 0400). It should be noted that the speed of sound in PVA phantoms is very close to the speed of sound in water at 20 C so that no difference could be observed when we replaced the phantom by the hydrophone. The phantom was

°[25 ]
replaced by the hydrophone at the focal point with a good accuracy (better than hydrophone s diameter) thanks to successive linear scans ' following XYZ directions. Once the hydrophone was in the focal area we scanned a map around the focal plane (10 mm side, 0.5 mm step).

Results

Displacement Displacement dependence of the excitation voltage

Estimation of the displacement

Fast transmit of successive imaging pulses after the push emission allowed us to follow the complete motion of the spherical target until it returned to its initial position. 12 imaging pulses were performed at a Pulse Repetition Frequency (PRF) of 4000 Hz. Results are shown . We observe a maximum displacement of 0.2 s for the fourth pulse echo acquisition. It corresponds to a displacement Fig. 5 Δt = μ of using the speed of sound at 20 C, 1 bar. Displacement increases rapidly and briefly after the push. The target then °slowly returns to its equilibrium position. In the further part of this study we choose to measure the displacement in this increasing part, using pulses n 2 and 3.

°Correction of the phase aberration

Phase correction

Even though the phase aberrations led to a displacement less than 5 times smaller than without aberrator, it was always measurable with a good accuracy. Typical experimental displacements measured during the correction process were in the range of 0 15 m and the -μ average value was 5 m.

μ

We tested this method with a value of the phase aberration amplitude of 2 . We show on the phase difference between the π Fig. 6 D 2 π numerical aberration introduced in the experiment and the phase correction estimated through our technique after 2 successive experiments. We observe that is small (standard deviation 0.53) compared to the phase aberration (standard deviation of the D 2 π σ D 2 π = phase is 1.89). Comparison between phase aberration and phase correction is reported on . σ = Fig. 6 Optimized focusing

shows several hydrophone scans map of the pressure field performed at different stages of the experiment (time reversed, Fig. 7 through the phase aberrator and with the energy-based adaptive focusing corrections) in the focal area, normalized by the value of the maximum acoustic intensity measured using the time reversed phase. The acoustic intensity in the focal area was almost completely restored after correction and the relative weight of the secondary lobes decreases.

The phase aberrator leads to a drop of the peak pressure at the focus of 7.69 dB, which gives a main lobe of about the same amplitude than the secondary lobes. The focusing quality is strongly degraded and the spatial peak is shifted by 1.5 mm. After a first energy based adaptive focusing correction, the focal area was restored, with the main lobe, slightly shifted ( 0.49 mm) and the pressure peak at 3.72 --dB (3.75 mm at 6 dB). The main lobe is clearly discernable from the secondary lobes. A second correction based on the result of the first correction, led to a complete restoration of the focusing quality with a main lobe centered and a peak that reaches the value of 0.89 dB and 3.5 mm width at 6 dB.

-

Sensitivity of the autofocusing process to the noise level

The robustness of the autofocusing process to the experimental noise was investigated. A gaussian random noise was added numerically to the measurements of the displacement acquired previously. The standard deviation of the gaussian distribution

σ noise σ noise
was varied within the range 0 (no additional noise) to the maximum measured displacement. The noise was applied to the successive experiments performed. Then for each value of the noise level the phase correction and finally the pressure restored at the focal point were computed.

shows the normalized pressure at the focal point as a function of the normalized standard deviation of the noise (for Fig. 8 σ̅ noise a perfect correction we would obtain a normalized pressure of ( ) 1.12). P σ̅ noise = This results allows us to assess the robustness and the stability of this technique:even with a noisy measurements of the intensity (for instance, 0.1 i.e 10 of the displacement), the pressure restored at the focal point remains 80 of the optimal value, which still

σ̄n oise = % %
represents a strong improvement of the focusing.

Focusing without reflective target using the speckle noise

Beyond this proof of concept experiment performed on a reflective target, a second experiment in a more realistic setup for in vivo

" "
application is presented here. The adaptive focusing approach is evaluated through a physical aberrator in a speckle environment, in other words without any bright reflector. A tissue-mimicking phantom was used in order to estimate tissue displacements at the focus.

Experimental setup

We used 64 of a 128 element linear imaging array (7.5 MHz central frequency, pitch 0.3 mm, Vermon, France) driven by a multichannel fully programmable electronics (sampling rate in E/R 50 MHz, Amplitude 80 Volts peak to peak). A cylindrical phase shift law was added to the transmit signal in order to set the focal length at 29 mm so that 1.5 (with the focal distance and the f/D = f D aperture). All the array elements were used in transmit and receive for the radiation force generation and also for the motion estimation.

The motion of the target zone was estimated using similar pulse echo sequences than described in the previous section. The received signals were beamformed and the motion of the target zone was computed using the cross-correlation between the signals before and after push. A window of 2 s was selected around the target zone for that purpose.

μ

Target

In this experiment, two successive phase optimisations were made for comparisons. First, using a reflective target located at the focal point and the same phantom as described above (see III-A), we determined the phase aberration and operated a phase optimisation process similar to what was done in the previous section. Secondly, the reflective target was replaced by a random distribution of Rayleigh scatterers contained in the phantom (cellulose powder, 1 w/w). The intent was here to mimick speckle noise backscattering occurring in % medical ultrasound. In this case displacement was directly estimated at the desired focus from the cross correlation of speckle backscattered echoes corresponding to successive insonications.

Physical aberrator

A real phase aberrator was used in this experiment. This aberrator consisted into a 1 mm thickness slice of latex placed directly at the " "

surface of the linear array. Only the half central part of the array was covered by the latex sample (elements 14 to 47).

Results

Phase and amplitude aberration

A first experiment using the physical aberrator was performed using a small stainless steel sphere ( 1 mm) embedded in the phantom ∅ and positioned at the focal point. Using this strong reflective target, an optimisation experiment was first perfomed using the energy based adaptive focusing technique. We also acquired an echo of the reflective target in order to characterize quantitatively the physical aberrator (see ). The phase distribution exctracted from these signals was considered as a gold standard. Fig. 9 In a second experiment, the phased array was moved to another part of the phantom and we performed an energy based adaptive focusing experiment in speckle.

In these experiments one should notice that both transmit and receive are strongly aberrated, both for radiation force pushing and for displacement tracking. The same transmit parameters (amplitude and duration) were used for the pushing emission signals in both

" "
experiments. After performing the optimisation process, the estimated phase correction was substracted from the reference phase aberration. The estimated phases and normalized amplitudes corrections are shown on and in Tab. IV-B0f. Fig. 9 When applying the optimisation process to the speckle environment, the standard deviation of the phase error decreased to σ spec = 0.35, if we exclude the 5 firsts and the 5 lasts elements for which the relative intensity was extremely weak.

The normalized acoustical intensity was estimated at the focal point for both focusing corresponding to the optimization process I applied respectively using the speckle noise environement ( ) or using the spherical target ( ) Through the aberrator, the intensity I spec I sphe was estimated to be 3.02 dB. Using the sphere, we were able to restore almost completely the focus with an intensity of I aber = -I sphe = -0.21 dB. Finally, using the speckle, the result was in between 1.49 dB. I spec = -

The aberration profile of the real aberrator estimated using the optimisation process was very well recovered using the small target configuration. The optimisation process applied in speckle was less efficient but the correction allowed us to reduce the effect of the aberration and increase the acoustical intensity at focus by a factor of 2.

Discussion and conclusion

The work presented in this paper corrects phase aberrations for ultrasonic multichannel arrays using a method based on the maximization of radiation force induced displacement coupled to spatially coded emissions. By transmitting Hadamard coded signals, this technique uses all the array elements for each transmission and therefore uses the full power capability of the array. A strong numerical aberration (up to 2 ) was first corrected by the remote detection of the radiation force induced motion of a small target embedded in π homogeneous soft tissue. Then, using a smaller physical aberrator we have shown the feasibility of using this method in a tissue mimicking phantom without strong reflective targets.

shows that the phase aberration is very well recovered, even without spatial Fig. 7 correlation of the phase aberration between closest elements. To measure the three unknowns in , different measurements of the Eq. 5 3 intensity at the focal point are performed for each element. In our experiments, we added a constraint that accounts for the 2 bit transmit dynamics (i.e. only able to transmit emissions only with normalized amplitudes of 0, 1 or 1). Although only three insonifications are theoretically needed for estimating the optimal phase delay between two Hadamard transmit vectors, this constraint requires a fourth measurement of the intensity to ensure the robustness of the estimation. As a consequence, the phase aberration correction of an N elements array requires 4N successive transmissions. The proposed method strongly reduces the time needed to perform the entire correction process compared to iterative approaches because it is not necessary to explore the full range of possible phase values for each element.

This adaptive focusing method offers major advantages. Indeed, beams (corresponding to sonic, ultrasonic, optical or electromagnetic waves) transmitted by an array can be corrected and refocused at a desired location using the direct or indirect measurement of the beam intensity. Direct estimation can be performed easily using a single probe at the focal point. Indirect measurements can be performed with a measurement of the beam s effects on the medium following insonification, such as, for example, tissue displacement due to the beam ' radiation force or local temperature increase due to beam absorption. For many configurations, this indirect estimation at the focus can be done remotely with, for example, an echographic probe or a MR system in the case of ultrasound beam focusing. Therefore, depending on the application and potential configuration, there is no need for a probe at the focal point.

In this study, wave propagation was assumed to occur in a linear regime. However, for high pressure fields used in HIFU and ultrasonic imaging at high Mechanical Index, wave propagation becomes nonlinear. Signal distortions becomes non negligible mainly in the focal area. This non-linearity affects the acoustic radiation force as higher harmonics generated by the nonlinear propagation results in higher ultrasonic absorption effects at the focal spot. This leads to an overestimation of the acoustic intensity based on obtained in Eq. 1 the linear regime. Therefore, it is important to work within a linear range of acoustic power. This can be ensured by performing the optimization process using transmit signals with lower amplitude and by increasing the transmit voltage only after the aberration correction process. Another solution consists of measuring tissue displacement as a function of the transmitted power or voltage over a large power range, as it was shown in . Fig. 4 In our experiments the duration of the successive transmits required for the optimization was less that 130 ms. However, due to data transfer limitations of the hardware, the entire process took approximately 45 minutes. The pushing emission signals was a matrix of 400 " "

s 30 Ms length and 64 elements width on 8 bits and the low data transmission of our electronics slowed down the experiment μ ×

considerably. The duration of the experiment could be deeply reduced by increasing the transmission ratio and by computing the different signals before the experiment. With a fully optimized hardware, the complete adaptive focusing process would take less than some seconds.

Safety issues should also be considered for medical applications because the number of ultrasonic shots required for adaptive correction can be large in the case of arrays with many elements. In our experiments we did not try to optimize the total energy of the acoustic beam. Nevertheless, the largest negative pressure found in the study was 3.2 MPa which corresponds to a Mechanical Index of 1.1 at a 7.5 MHz central frequency. This value is far below the F.D.A requirements for cavitation issues. In terms of acoustic energy deposition, at least 40 insonications of 50 s burst duration could be performed per second while remaining below the F.D.A. requirements μ for the Ispta value (spatial peak time average Intensity < 720 mW/cm ). This number of insonications per second could even be increased 2 because pressure field at the focus is only at its maximum level value for a limited number of shots during the optimization process.

Another advantage of this method is that the quality of the phase correction does not depend on the timing of the experimental acquisitions as is the case for an iterative inversion process. In iterative approaches, the phase of the elements are corrected one by one with direct adjustment of the phase. The accuracy of the correction is better when most of the array is already corrected than when most of the array is defocused. In our case, the optimization is based on a direct inversion that avoids such problems.

We have seen in that the optimization method allows us to determine both phase and amplitude corrections. In this paper we Eq. 17 used the phase correction to improve the contrast of the focal point in an imaging experiment and to balance the distortion of the focusing due to the assymetrical contribution of the elements of the phased array. The optimization process gives also access to the amplitude correction that also permit to optimize the transmit amplitude on each channel for an optimal ratio between acoustic energy deposition at the focus and transmitted energy from the array. This optimization process is closely linked to the concept of time reversal focusing which provides a spatio-temporal matched filter of wave propagation between the array and the focal point.

[10 ]

This technique has several applications in medical ultrasound, such as aberration correction in diagnostic imaging or in therapeutic ultrasound. In diagnostic imaging, the added value of this aberration correction technique has yet to be proven. The major issue in diagnostic imaging is the time requested for correcting the aberrations which needs to be compatible with real time imaging. To date, we need 3xN ultrasonic insonifications to fully correct any kind of aberration whatever its spatial complexity. However this number could be strongly reduced in the case of aberrating layers with large correlation lengths. Indeed if the correlation length of the aberrator is large compared to the array s pitch, it could be possible to restrict the method to the use of the Hamadard vectors that vary slowly in space. For a ' 128 imaging array focusing at a typical 40 mm depth, the number of degrees of freedom of the propagation operator is much lower than

(40 in refs and

). So, there is an emission vector basis with a fewer degrees of freedom that would be better adapted to the

[24 ] [22 ]
case of medical imaging with slowly varying aberrating layers (as is the case in breast or liver). For these cases the number of insonifications would be strongly decreased and the correction could be performed in a few tens of milliseconds.

An immediate and extremely interesting application of this aberration correction method is in therapeutic ultrasound especially in the case of strongly aberrating layers. In conventional therapeutic applications, such as fibro-uterin, breast or liver HIFU, the HIFU array can both transmit pushing pulses and record tissue displacement, thus providing a complete autocorrection of the ultrasonic beam. In complex cases, such as transcranial ultrasonic focusing, one cannot rely on the backscattered echoes due to strong absorption in the skull bone.

Moreover the speckle tracking technique used for tissue displacement estimation cannot be performed because strong skull aberrations dramatically degrade the beam quality. In these cases, a MRI system could be used to overcome these limitations. Since MR scanners will be coupled to HIFU for the application of transcranial brain therapy, this would have some important clinical advantages such as the ability to determine the focusing zone under MR guidance and the real time control of the phase correction process.

In the case of large aberrations we demonstrated that radiation force induced displacements are large enough to be detected by ultrasound pulse echo sequences. In the case of smaller aberrations and tissue displacements the auto-focusing process was still efficient. It provides an efficient way to autofocus the ultrasonic beam using transmit/receive ultrasonic arrays. Current work is focused on evaluating the proposed approach for complex in vivo cases where radiation force induced displacements cannot be estimated with ultrasonic speckle tracking. In such cases, the ultrasonic measurement of tissue displacement can be replaced a motion sensitive MRI sequence capable of 10 15 measuring small tissue displacements (of the order of 1 m). For example, in medical ultrasound, transcranial focusing is a very complex μ task for two reasons: first, the skull induces strong aberrations. Second, the speckle noise backscattered by brain tissue micro-heterogeneities is strongly absorbed by the skull and consequently cannot be recorded and used by the ultrasonic array for adaptive corrections. For such configurations, the array can be driven only in the transmit mode as is currently the case in most ultrasound therapy systems. This adaptive focusing technique may potentially be used for HIFU brain or liver therapy applications under MRI guidance.

Finally, beyond ultrasound applications, the proposed optimization process could be transposed to optics for focusing light in diffuse medium.

A 64 element phased array was used to perform the focusing optimization on a spherical target embedded in a phantom.

Figure 2

Typical pulse push pulse experimental sequence. The positions of the spherical target was determined after and before pushhanks to pulse

+ + "
echo acquisitions. All transmit signals were aberrated using the random phase shift screen (phase shift for each element), in the range 0 [ -2pi ] .

Figure 3

Pulse echoes of the spherical target recorded on the array, respectively before (solid line) and after (dashed) the emission of the pushing " "

beam. Amplitude is normalized, t 0 corresponds to the emission of the pulse. We observe a time shift ( 0.12 s) which corresponds to a

  wave physics. Similarly to Vellekoop et al. and Urban et al., it is base on the idea of estimating the beam intensity at the focus for successive and different illuminations. This is the semantic origin of the energy-based terminology. In optics, this technique should " "

  generating the same acoustic pressure at the focal point. If an aberrator is placed between the focal point and the ultrasonic array the p 0 contribution of the element number n at the focal point becomes: ( ) where and are respectively the phase and p n t = A p

  combinations of all the elements of the phased array. Note the complex amplitude of the monochromatic signal transmitted by the element of the phased array. The contribution of T n n th this element to the pressure field received at the focal point is where is the complex amplitude of the Green s p n = G T n n = A n e iΦ n G n ' function relating element and the focus. A basis transformation is performed and a new basis of virtual transducers is defined which are n represented by their complex amplitudes:

First, the displacement

  amplitude of the target was investigated as a function of the excitation voltage. The experimental emission sequence was composed of 2 imaging pulse used for the motion estimation and an optimized push function (TRpush). Its duration was 400 s. We increased the excitation voltage of the push emission signal in the range of 4 to 80 the amplitude of the imaging pulse μ are shown on : the displacement amplitude is a quadratic function of the voltage and there is V pp Fig. 4 no saturation at high voltage. It shows that the measured displacement is proportional to the the acoustic intensity in the full range of avalaible power. This confirmed the linear assumption: made from and with represents the stiffness of the phantom, the displacement and the pressure on the relationship Eq. 1 K d Δ P between acoustic radiation force and pressure. We choose 0.8 as a typical voltage amplitude for the push signal in all the other experiments.

Figure 4 Figure 5

 45 Figure 4Displacement as a function of the normalized excitation voltage. The push duration was 400 s. Displacements were measured as explained in μ

Figure 6

 6 Figure 6Phase aberration (thick solid line), phase correction ( , dashed line) and sum of them (thick dotted line) for the 64 elements. We show here ★

Figure 7 Figure 8 Figure 9

 789 Figure 7 Left part, pressure field map of the focal plane measured by a hydrophone at different stages of the correction, of the scans are 10 mm by 10 mm, with a spatial step of 0.5 mm. Axes from are represented. From top to bottom: The pressure field obtained using time reversed Fig. 1 focusing (used here as a reference), through the aberrator and through the aberrator with the second phase correction. Relatives amplitudes are shown on the right part: the ideal time reversal focusing pattern (thick line), through the aberrator (thick dotted line), after the first correction ( ), and after a second correction ( ) ◇ •

Figure 10

 10 Figure 10 Displacement range ( m), average displacement and standard deviations of the phase difference after the energy-based focusing using the μ speckle noise and the reflecting target

[START_REF] Mallart R | The vancittert-zernike theorem in pulse echo measurements[END_REF]

[START_REF] Mallart R | The vancittert-zernike theorem in pulse echo measurements[END_REF]

[START_REF] Mallart R | The vancittert-zernike theorem in pulse echo measurements[END_REF]

[START_REF] Mallart R | The vancittert-zernike theorem in pulse echo measurements[END_REF]

[START_REF] Mallart R | The vancittert-zernike theorem in pulse echo measurements[END_REF]

[START_REF] Mallart R | The vancittert-zernike theorem in pulse echo measurements[END_REF]

[START_REF] Mallart R | The vancittert-zernike theorem in pulse echo measurements[END_REF]

[START_REF] Mallart R | The vancittert-zernike theorem in pulse echo measurements[END_REF]

[START_REF] Mallart R | The vancittert-zernike theorem in pulse echo measurements[END_REF]

[START_REF] Mallart R | The vancittert-zernike theorem in pulse echo measurements[END_REF]

[START_REF] Mallart R | The vancittert-zernike theorem in pulse echo measurements[END_REF]

[START_REF] Mallart R | The vancittert-zernike theorem in pulse echo measurements[END_REF]

[START_REF] Mallart R | The vancittert-zernike theorem in pulse echo measurements[END_REF]


Dana Cohen

Sylviane R Schwer

Proximal Deictic Temporal Reference with Calendar Units *

The talk centres on deictic reference to temporal segments of the near future or past using of the fundamental calendar units (days, years, weeks, months) and their divisions (days of the week, parts of the day).

• The global aim of the study: to identify language specific and cross linguistic patterns in the linguistic use of calendar units.

• A more specific goal: determining to what extent temporal reference can be achieved through linguistic calendar expressions independently of other elements-how much of the necessary information is directly encoded in them and how much is supplied by additional linguistic and extra-linguistic elements.

Today's talk presents initial results of ongoing research. We will consider here some of the properties of three types of expressions employing linguistic calendar terms: the fundamental units (day, year, week, month), parts of the day and the (named) days of the week. The fundamental units have been examined (to varying degrees of depth) in some 20 languages of various language families. The other units have only been examined in a more limited set of languages, at this stage.

As will be shown, the three types of expressions reflect temporal reference to different levels or different cycles and their linguistic behaviour reveals differences in the temporal information they encode and in their ability to function independently as temporal markers.

* We would like to extend our thanks to all our colleagues, friends and informants who supplied us with data and judgements on their own languages.

The Fundamental units

The principal of a calendar is the division of the abstract / metaphorical image of time into more or less identical segments so that every punctual event can be assigned to a unique segment. Formally, temporal identification of a segment is determined by three components: the name of the temporal unit, the orientation (+, -) and the sequential number of the unit, starting with the deictic anchor.

Proximal Deictic Temporal Reference with Calendar Units Temptypac 2010

2/9 - ← → + N … 3 2 1 S 1 2 3 … n Fig. 2
Linguistically, the segment that contains S, the time of speech, functions as the deictic anchor whichever the size of the unit selected. In many languages, reference to Day S is lexicalised, and reference to larger units is formed of the unit modified by a determiner or demonstrative. This type of modification can produce a complex or lexicalised expression, depending on properties of the specific language. (?ci) this week see nädal ha.šavua

The formal three-part process of identification seems to be in operation from a certain number of units (e.g., 2 and higher, depending on granularity and language). However, this method is not employed for more proximal units (±[1-3], depending on language), where this system is frequently replaced by lexicalised or frozen constructions that do not provide these components as independent information. 1 These frozen expressions enable the identification of the intended temporal reference regardless of the linguistic context (such as verbal time reference). The languages examined demonstrate a correlation between the diachronic process of freezing, whether to a level of lexicalization or to a larger unit, and the level of temporal proximity to the present anchor. The more lexicalised forms, most typically DAY S and its immediate +/-1 neighbours, are least transparent in the identification of the temporal reference.

Based on the languages examined, we propose that the anchor of the present (S) conceptually determines 3 zones in the temporal domain -the present, the proximal and the distant -extending both to the future and the past -and separated by a fuzzy boundary.

Fig. 3: Zones of deictic temporal expression

Linguistic reference to the distal zone employs the regular rules of sequential reference (e.g., D+/-n in table 2), while reference to the proximal zone employs specialised constructions.

The languages examined reveal that the three zones vary in size and are not necessarily symmetrical in the present/past orientation. Numbered structures are always possible ≥3, while lexicalisation is typically limited to the 0-1 range. The components needed for temporal reference are more transparent and compositional, while the inner zones use frozen expressions that are more opaque in this respect.

The effect of the type of unit is evident here -'DAY being the primary unit, which most frequently takes the most frozen, lexicalised forms, that are least transparent.

The size of the intersection (and overlap) between the numbered and non-numbered forms varies from language to language, but also among speakers. The distributional properties of these expressions in a number of languages examined indicate a nominal categorisation, although their functions in the clause may be adverbial (as evident in their traditional categorisation in some languages). 3 Consequently, we consider them nominal and referential expressions on par with other nominal forms.

Consequently, the zoning pattern can be correlated with the scale of referential activation/ givenness/ accessibility that is associated with referring expressions in pragmatic theory (cf., Ariel 1990;[START_REF] Gundel | Cognitive Status and the Form of Referring Expressions in Discourse[END_REF]. 4 Accordingly, more opaque (more central) forms refer to more activated, relatively accessible referents in the addressee's mind, while the very transparent numbered structures refer to temporal referents of low accessibility. Thus, the proximal and central DAY units are also the highest in accessibility among these forms.

The Parts of Day

The parts of the day are a decomposition of time which is not in a part-whole relation with the fundamental units of the calendar which it supposedly partitions; that is, the partition is composed of different recurring segments: (Fr. matin, midi, après-midi, soir, nuit; Eng. morning, noon, afternoon, evening, night). Parts of day can be used to refer to two series -either by type (this morning vs. other mornings) or by DAY (this morning vs. this afternoon). The different segments are sequential, but the boundaries between them remain vague and may vary with each token. Consequently, the same point in time can belong to deferent segments, even within the same discourse, as illustrated in (1).

( The parts of day are also referred to by lexicalised and frozen expressions as well, although not necessarily parallel to those used for the fundamental units (*last/next morning vs. last night /*next night; *matin dernier/prochain vs. la nuit 3 The categorial classification of these expressions may vary in different languages. In the set of languages examined these expressions appear with linguistic markers that indicate a nominal nature (plurality, gender, case, demonstratives, definite determiners, quantification, prepositions). Thus, the calendar unit heads a nominal phrase, whatever the function of this phrase in the clause. Marc Wilmet (2003: pp 467) classifies such expressions in French as temporal indefinite pronouns.

4 Functional pragmatic theories of referential expressions argue that the form of referring expressions linguistically encode or signal the discourse functions / cognitive statuses that the intended referent is assumed to have in the mind of the addressee, thereby constraining possible interpretations. (e.g., pronouns are 'active' in [START_REF] Chafe | Givenness, contrastiveness, definiteness, subjects, topics, and point of view[END_REF], 'highly accessible' in Ariel 1990; 'in focus' in Gundel, Hedberg and Zacharski 1993, while new referents are 'inactive', low accessible' and 'type identifiable' respectively).

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dernière/prochaine). These units also display differences with respect to the three proximity zones. In contrast to the fundamental units, these linguistic expressions cannot provide independent temporal identification and require addition linguistic cues, through the verb or other means, to identify the intended segment. Thus, cet après-midi can refer to a present, past or future orientation, although being a part of the present day.

(2)a. Cet après-midi, j'irai chercher Claire à la sortie de l'école. (S=10:00) b. Cet après-midi, nous travaillons sur les calendriers. (S=15:15, 18h15) c. Cet après-midi, nous avons eu des exposés très intéressants. (S=21:00)

This lack of independence is particularly evident with respect to the day and night division. Words referring to day and night refer to an alternation which does not correspond to calendrical dates, but is the mundane division in which night overlaps the calendrical day. A calendrical day includes two successive but disjoint night segments. In some of the languages examined (Fr., Eng., Heb.,) the relevant linguistic expressions (e.g., tonight, cette nuit) cannot independently refer to these segments, leading to potential ambiguity. Thus, ce matin / this morning refers to the morning of the current day while cette nuit / tonight refers to one of two nights that are part of the current day. Within a specific utterance, the intended reference is generally non-ambiguous, thanks to the verbal cues that the missing orientation (affected also by the exact place of S within the Day0 anchor).

Days of the Week

The third type of expression, the names of week days, is an aggregation of consecutive occurrences. Formally, the names of days (e.g., Friday, vendredi) define a disjoint sequence, each element of which is associated with a unique occurrence within the unit of the next granularity, the week.

The part-whole relation between a day and a week is a complete calendar part-whole relation, but linguistically, this relation is less evident. This is exemplified in temporal reference with unmodified names of days in French. identifies the nearest occurrence of the day relative to S (except for Day S itself), but orientation is necessarily provided by the verbal temporal reference as it is with the parts of the day.

S in Monday0: *lundi; mardi = mardi0 or mardi-1; jeudi = jeudi0 or jeudi-1.

Cohen & Schwer

Proximal Deictic Temporal Reference with Calendar Units Temptypac 2010

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The same effect is evident with days of the week introduced by a demonstrative. Thus, the temporal identification of ce jeudi depends on the proximity of Day 0 rather than on its inclusion in Week 0. This is not the case for the expressions that mark lower proximity, specifically days of the week modified by dernier / prochain. These expressions explicitly encodes the orientation of reference and the lower proximity, the exact identification of the intended temporal referent is still unclear. The data provided by our informants suggest that these construction are underspecified as to whether the intended reference is the nearest occurrence or the second in the sequence. Although the resulting ambiguity is affected by the relative proximity of Day S, it is not resolved in specific occurrences.

This ambiguity and the effect of the proximity of Day S are reflected in the judgements shown below (collected from a homogenous group of over 10 people):

(Parallel results were obtained with a past orientation.

S on Monday0, mardi prochain refers to mardi+1 for all informants; mercredi prochain refers to mercredi+1 for some 75% of the group; jeudi prochain is considered ambiguous by 50%, vendredi prochain refers to vendredi0 for some 75% of the group.

S on Friday0, jeudi dernier refers to jeudi-2 for all informants; mercredi dernier refers to mercredi-2 for some 75% of the group; mardi dernier is considered ambiguous by 50%; lundi dernier refers to lundi0 for some 75% of the group.

Typological indexing from a theoretical point of view:

Cognitive Spatialisation of Time :

Temporal expressions in language frequently employ terms that encode a spatial orientation along one dimension inside of the three-dimensional universe : vertical, horizontal, depth. horizontal : e.g. Romance languages vertical: e.g. Chinese, but also found in some expressions in European languages (sous huitaine, sous huit jours, under a week) depth: So far not found basic units that employ depth spatialisation, but it is found for more extensive segments (e.g., the depth of winter). mixed: e.g. Malgasy (Malgache) : horizontal past, vertical future

Stages in the Knowledge of Numeration (in Western culture)

Before the XVII° century: arithmetic versus geometry (Euclid, Aristotle):

• numbers versus magnitudes:5 discrete entities versus mass entities.

• 1 is the generator of all numbers, which means multiplicity of the unit of one kind (5 horses, 5 tables, indicate "five occurrences of the unit horse, table ). So, 1 acts as a classifier, but it is not a number in itself. 6By 1500, zero becomes a number.

• The point generates the line (by a movement)

• the line is part of geometry • calendar units can be numbered from 2

Theoretically, reference points along the lines can be expressed in the following ways:

• using numbers starting with 2 (line 1 in the table below)

• Use of frozen or lexicalised expressions that indicate the alternation of reference by single steps without the use of numbers (line 2).

- … 3U- 2U- U- S U+ 2U+ 3U+ … nU … U--- U-- U- S U+ U++ U+++ …
How are the latter theoretical options realised in language?

• S, U+ and U-are expressed as lexicalized (U-) and (U+) or as modified (U)-and (U)+. Note that the modifiers used in (U-)and (U+)+ constructions enable productive recursion, which is attested, although the frequency is reduced with the increase in number of recursions. This is illustrated in (3), showing a x5 recursion of the modifier ((((((U-)-)-)-)-)-).

(3) GODOT: Avant-avant-avant-avant-avant-hier, je passe au loin devant un arbre et une route de campagne.

(Tribute to Beckett's En attendant Godot; http://desencyclopedie.wikia.com/wiki/En_attendant_Godot)

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In some languages, the recursive pattern is preferred in the spoken language (e.g., Moldavian/Romanian), while in other languages, although frozen forms of the recursive pattern exist for the mid-range proximity (+/-2-3 and rarely for higher), these forms appear to be used less in present day, and speakers show a preference for the compositional enumerated structures that are the only option from more distal reference (e.g., Estonian, Hebrew, Persian (in which the enumerated form was apparently borrowed from Arabic).

After the XVII century (Stevin, Wallis, …)

• step 1 : Magnitudes can be handled as numbers • step 2 : Magnitudes are numbers • 1 is a number • 0 is the natural correspondent of the point • 0 is the generator of numbers • the line is in correspondence with the set of all "real" numbers

- … 3U- 2U- U- S U+ 2U+ 3U+ … nU … U--- U-- U- S U+ U++ U+++ … nS,- … 3S,-2S,-1S,- 0S 1S,+ 2S,+ 3S,+ … nS,+
The options in the last line are found in anaphoric temporal reference rather than in deictic, eg. un jour avant vs la veille.

Knowledge About Sequences/Orders

Expressions that involve non-lexicalised forms, employ modification, frequently using ordinal adjectives (last, dernier), sequential relational adjectives (next, prochain), or numerals (cardinal or ordinal).

The use of cardinals in place of ordinals is frequent (le 2 du mois, 12 du mois de mars), but this is not unique for temporal expressions, and is common in numbered sequences in general (e.g., Louis XIV rather than *Louis le 14ème).

Summary:

This study clearly indicates that deictic temporal reference is far from a simple task, and requires not only knowledge of the appropriate frozen expressions, but also reference to other linguistic and extra-linguistic factors, and even then, the identification of the intended reference may not be absolutely clear.

Several patterns have been highlighted in this talk:

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• Relation to abstract calendars -Different types of linguistic calendar expressions show different relations with the concepts and segmentations of the abstract calendar. This is particularly evident with proximal reference using days of the week, that shows sensitivity to DAY S, but not the containing segment WEEK S. This highlights the fact that linguistic encoding of time is more complex and one cannot assume simple direct reference to abstract calendar systems. • Distance -The temporal domain can be divided into 3 (overlapping or vaguely bounded) zones around the anchor of the present (S), which extend both to the future and the past. Reference to the central and proximal zones uses specialised constructions, while distal reference employs (numbered) sequential constructions. • Transparency -Formally, temporal identification requires three components: unit size, orientation and distance from the central anchor. In language, constructions that refer to distal units (≥+/-3 depending on language) are typically compositionally transparent in this way, but constructions indicating more proximal reference, particularly the 0, +/-1 range, these components may be more opaque or lack altogether. • Referential (In)dependence -Some temporal expressions (most notably reference with the fundamental units) can identify the intended temporal referent on their own, while others cannot function independently and require temporal cues from the rest of the clause (e.g., parts of the day) or extralinguistic information (like the identity of DAY S for the use of days of the week). Interestingly, even this extra information cannot guarantee accurate identification of the intended time.

Fig. 1

 1 Fig. 1

  (U-) Fr. hier; (U+) Persian farda; (U)année dernière; (U)+ année prochaine • U++ and U--can be realized as a modified form (U+)+ or as a different lexicalised form (U++) or U(++). (U+)+ Fr. après-demain; (U++) maxratayim (Heb. 'day after tomorrow', a dual plural lexicalisation of 'tomorrow'). (U--) šilšom (Heb. 'day before yesterday', non-transparent lexicalisation) • U+++ can be realized as a reduplicated modification ((U+)+)+, as a simple duplication (U++)+ , (U+)++ or with a new modifier (U+++). (((U+)+)+) Fr. après-après-demain; ((U+)++) Persian passimfarda; ((U++)+ None found in our data; (U+++) None found in our data.

Table 1 :

 1 Expressions referring to the current S anchor of the various units

		French	English	Estonian	Hebrew
	Day	aujourd'hui	today	täna	ha.yom
	Year	cette année (ci)	this year	see aasta	ha.šana
	Month	ce mois (ci)	this month	see kuul	ha.xodeš
	Week cette semaine

Table 2 .

 2 1: Expressions referring to the proximal sequence of days 2

		French	English	Hebrew
	D+n	dans n jours	in n days	od n yamim
	D+3	après-après-demain	in 3 days	od 3 yamim
	D+2	après-demain	the day after tomorrow	maxratayim
	D+1	demain	tomorrow	maxar
	D0	aujourd'hui	today	ha.yom
	D-1	hier	yesterday	etmol
	D-2	avant-hier	the day before yesterday	šilšom
	D-3	avant-avant-hier	3 days ago	lifney 3 yamim
	D-n	il y a n jours	n days ago	lifney n yamim

1 By 'frozen expression', we refer to expressions such as last year, the day before yesterday, which are phrasal and cannot be referred to as lexicalised, but whose use for a specific temporal reference is standardised (cf., with the non-frozen two days before yesterday). 2 In all these languages, a numbered construction is possible for D+/-2. Hebrew uses a dual plural of `day`in this case lifney/od yomayim.

Table 2 .

 2 2: Expressions referring to the proximal sequence of days

		Estonian	Polish	Malgasy
	D+n	n päeva pärast	za n dni	afaka (n-1) andro
	D+3	üle-ülehomme	popojutrze	afaka 2 andro
	D+2	ülehomme	pojutrze	raha afaka rahampitso
	D+1	homme	jutro	rahampitso
	D0	täna	dzisaj	androany
	D-1	eile	wczorai	omaly
	D-2	üleeile	prezedwczoraj	1 andro lasa
	D-3	üle-üleeile	przed-przedwcoraj	2 andro lasa
	D-n	n päeva tagasi	n dni temu	-(n-1) andro lasa

Table 3 :

 3 Expressions referring to proximal nights(French, English, Taiwanese)

	-2		1		0		+1	+2
	D	N-2,-1	D	N-1,0	S	N0,1	D	N1,2	D
		la nuit derniere		cette nuit la nuit derniere		cette nuit la nuit prochaine		la nuit prochaine
		*yesterday night		?tonight		tonight		tomorrow night
		last night		last night		*next night		*next night
				tsaham		amsi		miamsiamsi

Table 4

 4

	.1
	orientation (backwards):-←	orientation (forwards) : +→
	nU

Table 4

 4

	.2
	orientation (backwards):-←	orientation (forwards) : +→
	nU

According to Pascal and Barrow, Newton (1707), √2 can be understood only as a geometric magnitude.

Except for the Pythagoreans.