Nadhir Mesbahi

Ahmed Ouari

Djaffar Ould Abdeslam
email: djaffar.ould-abdeslam@uha.fr

Tounsia Djamah

Amar Omeiri

Direct power control of shunt active filter using high selectivity filter (HSF) under distorted or unbalanced conditions

Keywords: Shunt active power filter (SAPF), Instantaneous powers, Direct power control (DPC), Switching table, High selectivity filter, Distorted or unbalanced conditions

This paper describes the design of a new configuration of direct power control (DPC) based on high selectivity filters (HSF) to achieve near-sinusoidal source current waveforms under different source voltage conditions. The proposed method uses the high selectivity filters instead of the classical extraction filters (low pass filters). The basic idea of the proposed DPC is to choose the best inverter voltage vector in order to minimize instantaneous active and reactive power errors using two hysteresis comparators. Their outputs associated with a switching table, control the active and reactive powers by selecting the optimal switching states of the inverter. Simulation results have proved excellent performance, and verify the validity of the proposed DPC scheme, which is much better than conventional DPC using low pass filters.

Introduction

Nowadays, the widespread use of non-linear loads leads to degradation of power quality of the electrical distribution systems due to generation of high harmonic currents. This results in multiple disagreements such as: increase of line losses, saturation of distribution transformers, and interference to adjacent communication systems. These effects can be worse in the case where the voltages and/or loads are unbalanced.

To improve the power quality, traditional solutions like passive LC filters have been widely used for a long time [START_REF] Akagi | New trends in active filters for power conditioning[END_REF]. However, bulk passive components, series and parallel resonance and fixed compensation characteristics for fixed values of L and C are the main drawbacks of passive LC filters. In order to overcome all these problems simultaneously, various active power filters (APFs) have been developed in recent years [START_REF] Akagi | New trends in active filters for power conditioning[END_REF][START_REF] Singh | A review of active filters for power quality improvement[END_REF] due to the development in power converters and digital signal processors. The objective of these APFs is to reduce the voltage and current harmonic distortion [START_REF] Grady | Survey of active power line conditioning methodologies[END_REF]. The active power filter topology can be connected in series or parallel and combinations of both (unified power quality conditioners). Passive filters combined with active shunt and series are some typical (hybrid) configurations. Most of the industrial applications need current harmonic compensation, so the shunt active filter is more popular than over APFs [START_REF] Singh | A review of active filters for power quality improvement[END_REF][START_REF] Barrero | Active power filters for line conditioning: a critical evaluation[END_REF]. The shunt active power filter (SAPF) is based on the pulse width modulation voltage source inverter topology. Its basic function consists in the injection of currents which are load currents in opposite-phase, so that the load harmonic currents are cancelled [START_REF] El-Kholy | Three-phase active power filter based on current controlled voltage source inverter[END_REF][START_REF] Zaveri | Control strategies for harmonic mitigation and power factor correction using shunt active filter under various source voltage conditions[END_REF].

The performances of the SAPF depend on the voltage-source inverter design, the extraction method used to generate the reference signal (current or power), and the control method used to generate the filter current [START_REF] El-Kholy | Three-phase active power filter based on current controlled voltage source inverter[END_REF][START_REF] Zaveri | Control strategies for harmonic mitigation and power factor correction using shunt active filter under various source voltage conditions[END_REF].

In this context, the most popular control technique is usually based on current controller which forces the SAPF current to follow its reference [START_REF] Buso | Comparison of current control techniques for active filters applications[END_REF][START_REF] Malesani | Robust dead-beat current control for PWM rectifiers and active filters[END_REF][START_REF] Matavelli | A closed-loop selective harmonic compensation for active filters[END_REF][START_REF] Shyu | Model reference adaptive control design for a shunt active-power-filter system[END_REF][START_REF] Rahmani | Experimental design of a nonlinear control technique for three-phase shunt active power filter[END_REF][START_REF] Bojoi | Current control strategy for power conditioners using sinusoidal signal integrators in synchronous reference frame[END_REF][START_REF] Lascu | Frequency response analysis of current controllers for selective harmonic compensation in active power filters[END_REF][START_REF] Limongi | Digital current-control schemes[END_REF][START_REF] Lascu | High performance current controller for selective harmonic compensation in active power filters[END_REF][START_REF] Trinh | An advanced current control strategy for three-phase shunt active power filters[END_REF].

A number of control methods have been reported in the literature such as proportional-integral (PI) control [START_REF] Buso | Comparison of current control techniques for active filters applications[END_REF],

hysteresis control [START_REF] Buso | Comparison of current control techniques for active filters applications[END_REF], dead-beat control [START_REF] Malesani | Robust dead-beat current control for PWM rectifiers and active filters[END_REF], repetitive-based control [START_REF] Matavelli | A closed-loop selective harmonic compensation for active filters[END_REF], adaptive control [START_REF] Shyu | Model reference adaptive control design for a shunt active-power-filter system[END_REF], and nonlinear control [START_REF] Rahmani | Experimental design of a nonlinear control technique for three-phase shunt active power filter[END_REF]. Also, there has been tremendous progress during the last decade in current control techniques for active power filters [START_REF] Bojoi | Current control strategy for power conditioners using sinusoidal signal integrators in synchronous reference frame[END_REF][START_REF] Lascu | Frequency response analysis of current controllers for selective harmonic compensation in active power filters[END_REF][START_REF] Limongi | Digital current-control schemes[END_REF][START_REF] Lascu | High performance current controller for selective harmonic compensation in active power filters[END_REF][START_REF] Trinh | An advanced current control strategy for three-phase shunt active power filters[END_REF] including of a proportional controller plus multiple sinusoidal signal integrators [START_REF] Bojoi | Current control strategy for power conditioners using sinusoidal signal integrators in synchronous reference frame[END_REF], a PI controller plus a series of resonant controllers [START_REF] Lascu | Frequency response analysis of current controllers for selective harmonic compensation in active power filters[END_REF][START_REF] Limongi | Digital current-control schemes[END_REF], or vector PI (VPI) controllers [START_REF] Lascu | High performance current controller for selective harmonic compensation in active power filters[END_REF]. This is due to the development of powerful and fast microprocessors. In recent work [START_REF] Trinh | An advanced current control strategy for three-phase shunt active power filters[END_REF], the authors presented an advanced control strategy to improve the shunt active power filter compensation characteristics, which is based on a PI plus VPI controllers. This control is capable of mitigating harmonic currents as well as reactive power without the demand of a load current measurement and harmonic detector [START_REF] Trinh | An advanced current control strategy for three-phase shunt active power filters[END_REF].

In conventional control strategies applied for SAPF, these current controllers (usually proportional integral (PI) controllers) are employed for controlling output filter currents by an internal current control loop. However, the main drawbacks for this control method have resulted in steady-state errors, and their bandwidth limitation turns into a decrease of quality compensation [START_REF] De Araujo Ribeiro | A robust adaptive control strategy of active power filters for power-factor correction, harmonic compensation, and balancing of nonlinear loads[END_REF]. Due to these facts, many new control strategies are being developed in recent decades. One of the most efficient is the direct power control (DPC), which presents the advantages of fast dynamic performance and simple control implementation, compared to other strategies.

The DPC method is derived from the direct torque control (DTC) used commonly in the high performance induction motor drive [START_REF] Lai | New hybrid fuzzy controller for direct torque control induction motor drives[END_REF]. The DPC controls active and reactive powers, while the DTC is able to control torque and flux [START_REF] Malinowski | A comparative study of control techniques for PWM rectifiers in AC adjustable speed drives[END_REF][START_REF] Noguchi | Direct power control of PWM converter without power-source voltage sensors[END_REF]. The DPC strategy is based on the instantaneous reactive power theory introduced by Akagi et al. [START_REF] Akagi | Instantaneous reactive power compensators comprising switching devices without energy storage components[END_REF] and on the evaluation of the instantaneous active and reactive power error values. In DPC, there are no internal current control loops and no pulse width-modulation (PWM), because the converter switching states are appropriately selected by a switching table. This last is based on the instantaneous errors, between the commanded and estimated values of the active and reactive power, and the power-source voltage vector position (classic DPC [START_REF] Noguchi | Direct power control of PWM converter without power-source voltage sensors[END_REF]) or virtual flux vector position (VF-DPC [START_REF] Malinowski | Virtual-flux-based direct power control of three phase PWM rectifiers[END_REF]). First of all, DPC is a widely used control strategy for three-phase PWM rectifiers [START_REF] Noguchi | Direct power control of PWM converter without power-source voltage sensors[END_REF][START_REF] Malinowski | Virtual-flux-based direct power control of three phase PWM rectifiers[END_REF]. In recent years, several authors have studied and analyzed the shunt active filter behavior under ideal conditions [START_REF] Chen | Direct power control of active filters with averaged switching frequency regulation[END_REF][START_REF] Chaoui | DPC controlled three-phase active filter for power quality improvement[END_REF]. Many control techniques have been used to improve the performance of the active filters in the situation of unbalanced conditions, but they most need to get synchronous rotation angle by using PLL and are difficult to implement. Besides the problems of harmonic distortion, there exist also low power factor and unbalanced load currents at the point of common coupling (PCC) due to the power delivered by the nonlinear loads [START_REF] De Araujo Ribeiro | A robust adaptive control strategy of active power filters for power-factor correction, harmonic compensation, and balancing of nonlinear loads[END_REF][START_REF] Zaveri | Comparison of control strategies for DSTATCOM in three-phase, four-wire distribution system for power quality improvement under various source voltage and load conditions[END_REF]. To overcome the limitations mentioned before, this paper presents a modified version of direct instantaneous power control strategy for three-phase shunt active power filter. In this novel scheme the high selectivity filters (HSF) have been utilized instead of the classical extraction filters (low pass filters). This allows to regulate directly the instantaneous active and reactive powers injected by the active filter, without any current control loops and phase locked loop involved. Moreover, this algorithm works effectively not only under balanced supply voltages, but also under distorted or unbalanced conditions.

The remainder of this paper is organized as follows. Section 2 presents a power circuit configuration of the SAPF and its modelization. In Section 3, the basic principles of proposed DPC are outlined. In Section 4, the validity and effectiveness of the proposed SAPF is tested for several cases. Finally, the main contributions and significant results of this paper are summarized in the conclusion.

System description and modeling

SAPF topology

SAPF can suppress the current harmonics in the distribution networks by generating and injecting current harmonics (compensating current) at the PCC which have the same magnitude but opposite in phase of the current drawn by the nonlinear loads. Thus, the resulting total current drawn from the ac mains is sinusoidal. Fig. 1 shows the fundamental building block of the shunt APF. The SAPF system is made of a standard three-phase IGBT based voltage source inverter (VSI) bridge with the input ac inductors ( f L , f R ) and a dc bus capacitor ( C ) to obtain a self-supporting dc bus for an effective current control. A three-phase ac mains with line impedance ( s L , s R ) is feeding power to a three-phase diode bridge rectifier with a resistive-inductive load.

SAPF model

The considered SAPF is an ideal two-level three-phase inverter which is made of three branches ( a ; b ; c ) with two switches ( The three-phase three-wire shunt active power filter is shown in Fig. 1. It is assumed that the three-phase source voltages are balanced as: [START_REF] Akagi | New trends in active filters for power conditioning[END_REF] where m E and ω are the amplitude of the phase voltage and the angular frequency of the ac source, respectively.

) 3 4 sin( ) ( ) 3 2 sin( ) ( ) sin( ) ( π ω π ω ω - = - = = t E t e t E
Taking into account the absence of the zero sequence in the three-wire system and assuming that the ac supply voltages are balanced, the following assumptions are deduced:

i i i i i i i i i e e e
(2) Fig. 2 shows the equivalent circuit of the SAPF system considering impedances in both the power grid and the load. In this system, the power grid is represented by the internal voltage s e connected in series with an impedance s Z . The load is represented by an equivalent circuit, where the current generator L i represents the purely distorting current load. The active filter is composed of a voltage source f v connected to the PCC by means of the filter impedance f Z .

From the equivalent circuit of Fig. 2, considering the ideal case in which the power-grid impedance is negligible,

that means 0 = s Z
, the relationship between the supply, inverter voltages, and filter current is given as:

f f f f s f i R dt di L e v - - = (3)
Considering a symmetrical and balanced three-phase system and applying Kirchhoff laws to the three-phase equivalent circuit, the voltages supplied by the inverter are obtained in the three-phase coordinates:

fc c fb b fa a dc nk dc c fc f fc f sc fc nk dc b fb f fb f sb fb nk dc a fa f fa f sa fa i S i S i S dt dv C v v S i R dt di L e v v v S i R dt di L e v v v S i R dt di L e v + + = - = - - = - = - - = - = - - = (4)
where nk v : the voltage between the nodes n and k .

The summation of the three first equations of (4) gives:

dc c b a nk fc fb fa v S S S v v v v 3 0 + + = = + + (5)
Substituting Eq. ( 5) to Eq. ( 4), the simplified differential equations can be obtained:

[ ] fc c fb b fa a dc sc f c b a c f dc fc f f fc sb f c b a b f dc fb f f fb sa f c b a a f dc fa f f fa i S i S i S C dt dv e L S S S S L v i L R dt di e L S S S S L v i L R dt di e L S S S S L v i L R dt di + + = + + + - - - = + + + - - - = + + + - - - =                   1 1 3 1 3 1 3 (6)
where fa i , fb i and fc i are VSI compensating currents, and C is the capacitance of the VSI dc-side capacitor.

From the system of equations ( 4), a new control functions ( a u , b u , c u ) can be defined as follows:

                  + + - = + + - = + + - = 3 3 3 c b a c c c b a b b c b a a a S S S S u S S S S u S S S S u (7)
where the matrix of switching functions is expressed as:

abc s abc S T u = (8) with [ ] T c b a abc u u u u , , = [ ] T c b a abc S S S S , , =         - - - - - - = 2 1 1 1 2 1 1 1 2 3 1 s T (9)
3. Proposed formulation for DPC using HSF

Instantaneous power references generation

The operation of the DPC technique is based on the control of instantaneous active ( p ) and reactive ( q ) powers, which are defined by the so-called instantaneous reactive power theory (IRPT) and the evaluation of the active and reactive instantaneous power error values [START_REF] Noguchi | Direct power control of PWM converter without power-source voltage sensors[END_REF][START_REF] Akagi | Instantaneous reactive power compensators comprising switching devices without energy storage components[END_REF].

According to the instantaneous reactive power theory, the system voltage and the load current are transformed from c b a -coordinates into β αcoordinates by using the transformations (10) and (11) [26, 27]:

                          - - - = sc v sb v sa v v v 2 3 2 3 0 2 1 2 1 1 3 2 β α (10)                           - - - = Lc Lb La i a i i i i 2 3 2 3 0 2 1 2 1 1 3 2 β (11)
In fact, instantaneous real power ( p ) is equal to following equation:

Lc sc Lb sb La sa i v i v i v p + + = (12)
The possibility of calculating them from β α -coordinates, as follows

                  - = β α α β β α i i v v v v q p ( 13
)
The instantaneous power p can be decomposed to p p p + = , with p the continuous component and p ~ the harmonic component. The dc and ac components in these instantaneous active and reactive powers are due to fundamental and harmonic currents of the load, respectively. The power values of the dc components are filtered out by two low-pass filters (LPF) or high pass filters (HPF) as described in Fig. 3a. However, the instantaneous reactive power theory, in its standard form, does not allow to work under unbalanced conditions. In this paper, we propose a new method for the computation of instantaneous power reference based on HSFs (Fig. 3b), so the power reference can be calculated. By using the HSF filter, one can operate under any distorted and/or unbalanced conditions. On the other hand, the obtained instantaneous power terms in this method are the instantaneous active power reference ref p and the instantaneous reactive power reference ref q . The first one is generated based on the instantaneous active power ripples drawn by the nonlinear load, plus the power requirement of the dc bus voltage control loop to compensate switching loss, and the second one is equal to the instantaneous reactive power ripples consumed by the nonlinear load [START_REF] Chen | Direct power control of active filters with averaged switching frequency regulation[END_REF].

High selectivity filter

HSFs are used in the harmonic extraction instead of classical extraction filters (HPF or LPF). Hong-sock Song in [START_REF] Hong-Seok | An instantaneous phase angle detection algorithm under unbalanced line voltage condition[END_REF][START_REF] Abdusalam | New digital reference current generation for shunt active power filter under distorted voltage conditions[END_REF] had presented that the integration in the synchronous reference frame is defined by: [START_REF] Limongi | Digital current-control schemes[END_REF] where xy U and xy V are the instantaneous signals, respectively before and after integration in the synchronous reference frame. The equation ( 14) can be expressed by the following transfer function after Laplace transformation: [START_REF] Lascu | High performance current controller for selective harmonic compensation in active power filters[END_REF] We think of introducing a constant K in the transfer function ) (s H

∫ = - dt t U e e t V xy t j t j xy ) ( ) ( ω ω
2 2 ) ( ) ( ) ( ω ω + + = = s j s s U s V s H xy xy
, to obtain a HSF with a cut-off frequency c

ω . So the previous transfer function becomes:

2 2 ) ( ) ( ) ( ) ( ) ( c c xy xy K s j K s K s U s V s H ω ω + + + + = = (16)
By replacing ) (s Vxy by ) (s Xαβ and ) (s Uxy by ) ( ˆs Xαβ , the following expressions can be obtained:

) ( ) ( ) ( ) ( ) ( ) ( ˆ2 2 2 2 s X K s K s X K s K s K s X c c c β α α ω ω ω + + - + + + = ( 17
) ) ( ) ( ) ( ) ( ) ( ) ( ˆ2 2 2 2 s X K s K s K s X K s K s X c c c β α β ω ω ω + + + + + + - = (18)
where X can either be a current or a voltage.

The equations ( 17) and ( 18) can be expressed as follows:

[ ]

) ( ) ( ) ( ) ( ˆs X s s X s X s K s X c β α α α ω - - = (19)
[ ]

) ( ) ( ) ( ) ( ˆs X s s X s X s K s X c α β β β ω + - = (20)
The block diagram of the HSF for extracting the fundamental component ) ( ˆs Xαβ from the signal ) (s Xαβ in the β α -reference frame is shown in Fig. 4.

Control Block Diagram

The block diagram of the proposed DPC technique is shown in Fig. 5. Initial voltages at the coupling point and load currents values are required in order to compute the initial active and reactive powers ( p , q ). The objective of the inner control loop is to minimize the errors between reference and actual values in each sampling step. This is done by selecting the appropriate inverter output voltage vector to push the state of the system towards the reference values. In this case the injected values of instantaneous active and reactive powers are compared with their reference values and in proportion to the error signals the proper switching state is selected from a switching table.

The instantaneous active and reactive powers injected by the active filter are expressed:

fc sc fb sb fa sa inj i v i v i v p + + = (21)
3

)) ( ) ( ) ( ( fb fa sc fa fc sb fc fb sa inj i i v i i v i i v q - + - + - - = (22)
Therefore, we define the active power error p e and the reactive power error q e as the difference between the power references ref p , ref q and the injected powers inj p , inj q respectively:

inj ref q inj ref P q q e p p e - = - = (23)
The error signals of the powers are compared in a two-level hysteresis comparators, which provide two digital signals ( p S and q S ), where:

     - ≤ - = ≥ - = p inj ref p p inj ref p h p p S h p p S if 0 if 1 (24)      - ≤ - = ≥ - = q inj ref q q inj ref q h q q S h q q S if 0 if 1 (25)
Furthermore, the outputs p S and q S are used to access the voltage vector lookup table, along with the corresponding sector. The selection of the voltage vector depends on the location of the PCC voltage in the β αplane. Therefore this plane is divided into 12 sectors as shown in Fig. 6. The table has as outputs the gating signals that generate the corresponding voltage vector. Table 1 shows the proper switching table to control the mentioned SAPF. In our work, all of the simulations tests are based on the switching table proposed in [START_REF] Chen | Direct power control of active filters with averaged switching frequency regulation[END_REF] (Table 1).

Simulation results

To confirm the effectiveness of the proposed DPC for SAPF, simulations are conducted in the environment of Matlab/Simulink. The data used for the simulation studies are given in Appendix B. For the two HSFs, we chose 80 = K to satisfy filter selectivity [START_REF] Abdusalam | New digital reference current generation for shunt active power filter under distorted voltage conditions[END_REF].

Unbalanced sinusoidal source voltage (case B)

For the case B, source voltages are imbalance in magnitude. The magnitude of source voltage of phase-a is less than source voltage of phase-b and phase-c by 20%. Simulation results of case B are presented in Fig. 9 and tabulated in Table 4. The effect of unbalance in source voltage can be observed in load currents. It can be noted from the FFT spectrum of load current that third harmonic and its multiple are present in load current due to unbalance source voltages. It can be noted from Table 4 that the THD of the source current is restricted as per IEEE-519 standard only with proposed scheme. Therefore, source currents are very close to sinusoidal waveforms and are in phase with their corresponding source voltages, thus guaranteeing operation with a power factor very close to unity. It can be seen from these simulation results that the proposed DPC is much better than DPC using LPF.

Balanced distorted source voltage (case C)

For case C, source voltages are balanced and distorted. The seventh harmonic component with amplitude of 1/14 of the fundamental component is present in source voltage. For this case, both source voltage and load currents are harmonically contaminated. The performance of SAPF is shown in Fig. 10 and summarized in Table 5. The harmonics are almost eliminated by the proposed control. It can be noted from Table 5 that only the proposed control is able to limit the THD as per IEEE-519 standard and power factor correction at the same time.

However, source currents for conventional DPC are highly distorted due to the presence of the 3rd harmonic, caused by the 7th harmonic voltage component.

Analysis of results

Tables 3, 4 and 5 show the simulation results obtained using two different control techniques for three different cases, respectively. With reference to Tables 3, 4 and 5, following observations are made for compensated system under various source voltage conditions:

(i) For case A, both control techniques give almost same results. Balanced and sinusoidal source currents, harmonic mitigation and reduction in rms value of source currents are achieved.

(ii) For case B, balancing of source currents is provided by two methods. Among the two techniques, the proposed DPC is outperforms than the conventional one. Moreover, it is also capable to reduce the THD of compensated source current below IEEE-519 standard harmonic current limits.

(iii) For case C, it can be observed that only the proposed DPC guarantees near-sinusoidal source current waveforms.

From the above discussion, it is clear that the proposed control scheme satisfactorily compensates the distorted or unbalanced conditions in three phase system.

In order to demonstrate superiority and performance of the proposed control over the current control method introduced in [START_REF] Abdusalam | New digital reference current generation for shunt active power filter under distorted voltage conditions[END_REF], various comparative evaluation points can be noted under the same conditions. With regard to the control system shown in [START_REF] Abdusalam | New digital reference current generation for shunt active power filter under distorted voltage conditions[END_REF], it must be calculate that the current reference of SAPF is based on the use of HSF. In [START_REF] Abdusalam | New digital reference current generation for shunt active power filter under distorted voltage conditions[END_REF], analysis, simulation and experimental results were given to illustrate the harmonic cancellation using SAPF, and a modulated hysteresis current controller was used to fast track the current reference. A thorough investigation of the simulation results reported in both techniques reveals that the THD in the source currents cannot be brought down below 5% to satisfy the IEEE-519 standard. Therefore, the experimental results presented in [START_REF] Abdusalam | New digital reference current generation for shunt active power filter under distorted voltage conditions[END_REF] have conforted the effectiveness of using HSF in the SAPF control for various conditions. In addition, proposed DPC provides two advantages, compared to other approach. First, there are no internal SAPF current loops or modulator block because the inverter switching states are selected via a switching table. The second advantage is related to reduce the number of hysteresis comparators. Unfortunately, there have been few

publications on DPC applied to the shunt active power filter [START_REF] Chen | Direct power control of active filters with averaged switching frequency regulation[END_REF][START_REF] Chaoui | DPC controlled three-phase active filter for power quality improvement[END_REF]; however, it considers the only in case of sinusoidal source voltages.

Conclusion

This paper has discussed a novel approach for DPC in order to control shunt active power filter. A comparative evaluation of two different control techniques for SAPF has been presented in this paper. A new proper active and reactive power reference generation strategy based on high selectivity filter have been developed. The performance of these control techniques has been evaluated under various source voltage conditions. Under ideal mains conditions, these two control techniques give almost similar results. Under unbalanced or non-sinusoidal source voltage conditions, only one proposed DPC technique shows excellent behaviour in terms of harmonic mitigation, balanced and sinusoidal source currents, and low THD of source currents. Simulation results have proven excellent performance of the proposed DPC scheme which is largely better than conventional DPC.

Table 1

The switching table.

Table 2

Source voltage data and load current data along with THD under various source voltage conditions.

Table 3

Simulation results of two different control strategies under case A.

Table 4

Simulation results of two different control strategies under case B.

Table 5

Simulation results of two different control strategies under case C.

Appendix

- + + + s 1 + - K K c ω c ω α X β X α X ˆ β X ˆ

Fig. 1 .

 1 Fig. 1. Configuration of the SAPF.

Fig. 2 .

 2 Fig. 2. Equivalent circuit of the SAPF.

Fig. 3 .Fig. 4 .

 34 Fig. 3. Block diagram of the power references generation: (a) conventional based LPF and (b) proposed based HSF. Fig. 4. Block diagram of the HSF.

Fig. 5 .

 5 Fig. 5. Configuration of the proposed DPC.

Fig. 6 .

 6 Fig. 6. Sectors on stationary coordinates.

Fig. 7 .

 7 Fig.7. Simulation results of case A: (i) three-phase source voltages and frequency spectrum of source voltage of phase-a, (ii) three-phase load currents and frequency spectrum of load current of phase-a, (iii) three-phase source currents and frequency spectrum of source current of phase-a for the conventional DPC, (iv) three-phase source

Fig. 8 .

 8 Fig. 8. dc capacitor voltage.

Fig. 9 .

 9 Fig.9. Simulation results of case B: (i) three-phase source voltages and frequency spectrum of source voltage of phase-a, (ii) three-phase load currents and frequency spectrum of load current of phase-a, (iii) three-phase source currents and frequency spectrum of source current of phase-a for the conventional DPC, (iv) three-phase source currents and frequency spectrum of source current of phase-a for the proposed DPC, (v) The compensator current of phase-a and source voltage and source current of phase-a for the conventional DPC, (vi)The compensator current of phase-a and source voltage and source current of phase-a for the proposed DPC.

Fig. 10 .

 10 Fig.10. Simulation results of case C: (i) three-phase source voltages and frequency spectrum of source voltage of phase-a, (ii) three-phase load currents and frequency spectrum of load current of phase-a, (iii) three-phase source currents and frequency spectrum of source current of phase-a for the conventional DPC, (iv) three-phase source currents and frequency spectrum of source current of phase-a for the proposed DPC, (v) The compensator current of phase-a and source voltage and source current of phase-a for the conventional DPC, (vi) The compensator current of phase-a and source voltage and source current of phase-a for the proposed DPC.

  Fig. 1.

Fig. 5

 5 Fig. 5.

Table 1

 1

Table 2

 2

Table 3 Table 4 Table 5

 345

	S	p	S	q	1 θ		θ
	1		0 1			110 100	110 100	010 110		010 110		011 010		011 010	001 011	001 011	101 001	101 001	100 101	100 101
	0		0 1			010 001	011 101	011 101		001 100		001 100		101 110	101 110	100 010	100 010	110 011	110 011	010 001
	Case	Source voltage									Load current
			e	sa			e	sb			e	sc		i	La	i	Lb	i	Lc
			rms (V)	THD (%)	rms (V)	THD (%)	rms (V)		THD (%)	rms (A)	THD (%)	rms (A)	THD (%)	rms (A)	THD (%)
	A		220	0	220	0	220		0	16.03	27.86	16.02	27.82	16.02	27.83
	B		176	0	220	0	220		0	14.01	31.69	15.47	25.95	15.46	26.15
	C		222.2	14.29	222.2	14.29	222.2		14.29	15.82	29.07	15.76	29.55	15.79	29.12
	Control strategies	Three-phase source currents					Three-phase compensator currents
						i	sa			i	sb			i	sc	i	fa	i	fb	i	fc
						rms (A)	THD (%)	rms (A)	THD (%)	rms (A)	THD (%)	rms (A)	rms (A)	rms (A)
	Conventional DPC	15.44	1.28	15.45	1.28		15.45	1.28	4.57	4.57	4.57
	Proposed DPC		15.36	0.47	15.35	0.45		15.38	0.43	4.56	4.57	4.56
	Control strategies	Three-phase source currents					Three-phase compensator currents
						i	sa			i	sb			i	sc	i	fa	i	fb	i	fc
						rms (A)	THD (%)	rms (A)	THD (%)	rms (A)	THD (%)	rms (A)	rms (A)	rms (A)
	Conventional DPC	14.90	9		14.80	9.82		14.07	10.33	4.85	4.69	4.60
	Proposed DPC		14.38	1.54	14.53	2.06		14.33	2.61	4.64	4.02	4.46
	Control strategies	Three-phase source currents					Three-phase compensator currents
						i	sa			i	sb			i	sc	i	fa	i	fb	i	fc
						rms (A)	THD (%)	rms (A)	THD (%)	rms (A)	THD (%)	rms (A)	rms (A)	rms (A)
	Conventional DPC	15.38	16.34	15.40	16.38	15.41	16.36	6.17	6.23	6.19
	Proposed DPC		15.00	4.63	15.01	4.46		15.02	4.08	5.64	5.63	5.63

Simulation study has been carried out with non-linear load under three different source voltage conditions:

Case A: balanced sinusoidal source voltage.

Case B: unbalanced sinusoidal source voltage.

Case C: balanced distorted source voltage.

Source voltage and load current data along with total harmonic distortion (THD) for different cases are given in Table 2. Load currents are balanced and highly distorted. During uncompensated condition, source currents are same as load currents. Hence, uncompensated source currents are highly distorted and as a result the percentage of THD exceeds above IEEE-519 standard harmonic current limits.

Following information have been presented in the simulation results for the two control algorithms (DPC based LPF and DPC based HSF) under above mentioned cases:

(i) Three-phase source voltages and frequency spectrum of source voltage of phase-a;

(ii) Three-phase load currents and frequency spectrum of load current of phase-a;

(iii) Three-phase source currents and frequency spectrum of source current of phase-a for the conventional DPC;

(iv) Three-phase source currents and frequency spectrum of source current of phase-a for the proposed DPC;

(v) The compensator current of phase-a and source voltage and source current of phase-a for the conventional DPC;

(vi) The compensator current of phase-a and source voltage and source current of phase-a for the proposed DPC.

In order to plot source voltages and currents on the same scale, source voltages are scaled down by a factor of 10.

Balanced sinusoidal source voltage (case A)

For this case, balanced sinusoidal source voltage condition has been considered. Simulation results of case A are presented in Fig. 7 and summarized in Table 3. From Fig. 7(iii) it can be observed that sinusoidal source currents are achieved after compensation. The magnitude of compensated source current has reduced compared to that of uncompensated condition. The THD of source currents after compensation is restricted as per IEEE-519 standard harmonic current limit. Hence, harmonic mitigation is achieved effectively. Furthermore, the two control strategies converge to similar results under balanced sinusoidal source voltage conditions. Fig. 8 shows simulation results for the dc bus voltage. It can be seen that the voltage dc v is stable and regulated around its


Antoine Deleforge

Vincent Drouard

Laurent Girin

Radu Horaud

MAPPING SOUNDS ONTO IMAGES USING BINAURAL SPECTROGRAMS

We propose a novel method for mapping sound spectrograms onto images and thus enabling alignment between auditory and visual features for subsequent multimodal processing. We suggest a supervised learning approach to this audiovisual fusion problem, on the following grounds. Firstly, we use a Gaussian mixture of locally-linear regressions to learn a mapping from image locations to binaural spectrograms. Secondly, we derive a closed-form expression for the conditional posterior probability of an image location, given both an observed spectrogram, emitted from an unknown source direction, and the mapping parameters that were previously learnt. Prominently, the proposed method is able to deal with completely different spectrograms for training and for alignment. While fixed-length wide-spectrum sounds are used for learning, thus fully and robustly estimating the regression, variable-length sparse-spectrum sounds, e.g., speech, are used for alignment. The proposed method successfully extracts the image location of speech utterances in realistic reverberant-room scenarios.

INTRODUCTION

The association of auditory and visual data has been proved to considerably improve the robustness of speech processing whenever close-distance and frontal recordings of the speaker's face or lips are available, for example to separate speech from noise [START_REF] Girin | Audio-visual enhancement of speech in noise[END_REF][START_REF] Almajai | Visually derived wiener filters for speech enhancement[END_REF][START_REF] Segev | Examplebased cross-modal denoising[END_REF], from competing sources [START_REF] Rivet | Mixing audiovisual speech processing and blind source separation for the extraction of speech signals from convolutive mixtures[END_REF][START_REF] Naqvi | A multimodal approach to blind source separation of moving sources[END_REF], or for speech recognition [START_REF] Heckmann | Noise adaptive stream weighting in audio-visual speech recognition[END_REF][START_REF] Potamianos | Recent advances in the automatic recognition of audiovisual speech[END_REF][START_REF] Barker | Energetic and informational masking effects in an audiovisual speech recognition system[END_REF]. However, the performance of these methods rapidly degrades with far distance (2-3 meters) cameras and microphones. The problem becomes more difficult for two reasons: several objects, e.g., faces, are present at once, and both the visual and auditory data are degraded. Vision is useful, among others, for detecting faces and facial behaviors. Nevertheless, the field of view and the range of a camera are limited and the image features are perturbed by occlusions, perspective effects, illumination, and the relative position between camera and scene. In contrast, auditory signals have the potential to enable verbal interactions but they Support from the European Research Council (ERC) through the Advanced Grant VHIA (#340113) is greatly acknowledged. The loud-speaker emits full-spectrum sounds which are recorded with a microphone pair while the visual marker is used to estimate its corresponding image positions. Right: A speaking face is detected and predicted from a sparse-spectrum sound.

are spatially ubiquitous, subject to overlap, and corrupted by the acoustic environment.

Hence, vision and audition have both strengths and limitations and exploiting their complementarity is still a challenging problem. In this paper we propose a method that aligns auditory and visual data based on mapping auditory spectrograms onto images. We start by learning a sound-toimage transformation from a training set of audio/visual pairs (Fig. 1-left); this mapping is then applied to a sound source that emits from an unknown image location (Fig. 1-right). Several problems need to be addressed for such a method to be effective: (i) the sound representation should encode its location with respect to the microphones while it should be independent of the spectral content of the sound, (ii) in order to properly and fully estimate all the sound-to-image mapping parameters, the training should be performed with full-spectrum signals, and (iii) natural sounds such as speech signals, which are of particular interest, have sparse spectrograms and it is not clear how to map them onto images, based on the sound-to-image transformation parameters that were estimated with full-spectrum sounds.

We propose to use a setup composed of one camera and two microphones. The microphones are embedded into an acoustic dummy head, yielding non-linear filtering effects that depend on the sound direction (azimuth and elevation). We stress that with such a setup one can build interaural spectrograms that encode two-dimensional (2D) directional information and which are independent of the spectral con-tent of the emitted sound [START_REF] Deleforge | 2D sound-source localization on the binaural manifold[END_REF]. In order to estimate a mapping from sounds to images, we propose to learn the parameters of a Gaussian mixture of locally linear regression functions using pairs of full-spectrum-data and pixel coordinates of associated image landmarks.

However, estimating such a function directly, from highdimensional spectrograms to low-dimensional image locations, is problematic for two reasons. Firstly, the large number of parameters that need to be estimated in this case is prohibitive. Secondly, as detailed below, it is not possible to apply this direct function to natural sounds. Instead, we train an image-to-audio regression (Section 2) using full-spectrum (white noise) signals. Then, we derive an analytic expression for the conditional posterior distribution of an image location, given a sparse spectrogram and the learned regression parameters (Section 3). We show that this posterior distribution is a Gaussian mixture whose parameters (priors, means, and covariances) can be computed in closed-form, from the parameters of the regression function. Prominently, the proposed solution can deal with missing data in the observed acoustic vectors, for example, it is able to map natural sounds, such as speech, onto images of faces.

The proposed method is strongly linked with the problem of sound-source localization. This is traditionally done using time difference of arrival (TDOA) or intensity-level differences (ILD) between microphones, e.g., [START_REF] Viste | On the use of spatial cues to improve binaural source separation[END_REF][START_REF] Mandel | Modelbased expectation-maximization source separation and localization[END_REF][START_REF] Woodruff | Binaural localization of multiple sources in reverberant and noisy environments[END_REF] to cite just a few. However these methods perform only one-dimensional (azimuth) source localization. In the framework of audiovisual alignment, 2D estimation is needed in order to associate sound-source directions with image locations, which rules out azimuth-only source localization. Another possibility is to estimate the geometry of the camera-microphone setup, but this turns out to be difficult because of the nonlinear nature of the sound-propagation model [START_REF] Khalidov | Alignment of Binocular-Binaural Data Using a Moving Audio-Visual Target[END_REF]. TDOAbased 2D sound-source localization needs non-coplanar microphone arrays and the solution must solve a complex nonlinear constrained optimization problem [START_REF] Horaud | A geometric approach to sound source localization from time-delay estimates[END_REF].

In [START_REF] Deleforge | 2D sound-source localization on the binaural manifold[END_REF] it was experimentally shown that the space of interaural spectral features is homeomorphic to a 2D manifold that can be parameterized by azimuth and elevation. A dummy head mounted onto a pan-tilt mechanism was used to collect pairs of motor positions and interaural spectrograms, used to train a regression function. However, the emitter was kept static at a single position in all training and test experiments, while the dummy head was rotated onto itself. The method was hence limited to theoretical conclusions rather than practical applications. In this paper, we extend [START_REF] Deleforge | 2D sound-source localization on the binaural manifold[END_REF] by formally stating and proving a theorem showing that natural sound spectrograms can be mapped onto an image plane by inverting a piecewise affine mapping. We subsequently introduce a novel and elegant way of precisely locating a sound source, based on its pixel location, without moving the sensors.

LEARNING WITH FULL SPECTRUM SOUNDS

Let us consider a single, full-spectrum (e.g. white noise) point source, and a static auditory system that captures acoustic vectors in Y ⊂ R D along time. We assume that at any time, these vectors depend on the source position but not on its emitted spectrum. Let X ⊂ R 2 be a set of image locations, asociated to a static camera field of view. We consider N training pairs {(y n , x n )} n=N n=1 ⊂ Y × X that associate acoustic vectors with their corresponding sound source position in the image. These pairs are realizations of observed random variables (Y , X). We consider the following piecewise linear regression from the low-dimensional image plane to the high-dimensional acoustic space:

Y = K k=1 I(Z = k)(A k X + b k ) + E (1)
where matrix

A k ∈ R D×2
Y = y|X = x, Z = k) = N (y; A k x + b k , Σ).
To constrain the affine transformations to be local, we associate the K affine transformations to an equal number of regions {R k } k=K k=1 ⊂ R 2 that define a partitioning of X . The regions are modeled in a probabilistic way by assuming that X follows a mixture of K Gaussians defined by p(

X = x|Z = k) = N (x; c k , Γ k ) with prior p(Z = k) = π k and with c k ∈ R 2 , Γ k ∈ R 2×2 , and K k=1 π k = 1.
To summarize, the model parameters are

θ = {{c k , Γ k , π k , A k , b k } K k=1 , Σ} (2)
and they can be estimated via an EM procedure yielding closed-form expressions for the model parameters [START_REF] Deleforge | High-Dimensional Regression with Gaussian Mixtures and Partially-Latent Response Variables[END_REF].

MAPPING NATURAL SOUNDS ON IMAGES

We now consider the localization of natural sparse-spectrum sounds, e.g., speech. A sound is described by T acoustic vectors forming a time series, namely Y ′ = {y ′ 1 . . . y ′ t . . . y ′ T } ⊂ R D . We assume that these acoustic vectors are emitted from the same location. A time series Y ′ can be viewed as a D × T spectrogram and each entry y ′ dt is referred to as a frequencytime point. Natural sounds are represented by spectrograms that are extremely sparse, i.e., many frequency-time points are null, resulting in an unusable or missing acoustic value at that point. To account for this, we introduce a D × T matrix χ = {χ dt } D,T d,t=1 of binary variables such that χ dt = 1 if the frequency-time point is active and χ dt = 0 otherwise. To summarize, a test sound is described by S = {Y ′ , χ} and we seek the posterior density of the sound's image location, p(x|S; θ). We state and proof a theorem allowing for the full characterization of this density.

Theorem 1 Under the assumption that all the acoustic vectors in S are emitted from the same location, the posterior distribution is a Gaussian mixture model in R 2 , namely

p(x|S; θ) = K k=1 ν k N (x; µ k , V k ).
(

) 3
whose parameters {ν k , µ k , V k } k=K k=1 can be expressed in closed-form with respect to θ and S, namely:

µ k = V k Γ -1 k c k + D,T d,t=1 χ dt σ 2 d a dk (y ′ dt -b dk ) (4)
V k = Γ -1 k + D,T d,t=1 χ dt σ 2 d a dk a ⊤ dk -1 (5)
ν k ∝ π k |V k | 1 2 | Γ k | 1 2 exp - 1 2 D,T d,t=1 χ dt σ 2 d (y ′ dt -b dk ) 2 + c ⊤ k Γ -1 k c k -µ ⊤ k V -1 k µ k ( 6
)
where a ⊤ dk ∈ R 2 is the d th row of A k , bdk ∈ R is the d th entry of b k and ν k is normalized to sum to 1 over k.

The posterior expectation can then be used to estimate the sound's image location:

x = E[x|S; θ] = K k=1 ν k µ k .
Proof of theorem 1. By including the hidden variables Z (section 2) and using the sum rule, we obtain:

p(x|S; θ) = K k=1 p(x|S, Z = k; θ)p(Z = k|S; θ). (7)
Since the proposed model implies an affine dependency between the Gaussian variables X and Y given Z, the term p(x|S, Z = k; θ) is a Gaussian distribution in x. In other words, for each k, there is a mean µ k ∈ R 2 and a covariance matrix

V k ∈ R 2×2 such that p(x|S, Z = k; θ) = N (x; µ k , V k ).
Notice that ν k = p(Z = k|S; θ) is not conditioned by x. With these notations, (7) leads directly to [START_REF] Segev | Examplebased cross-modal denoising[END_REF]. We now detail the computation of the GMM parameters {µ k , V k , ν k } K k=1 . Using Bayes inversion we have:

p(x|S, Z = k; θ) = p(S|x, Z = k; θ)p(x|Z = k; θ) p(S|Z = k; θ) . (8)
Since we already assumed that the measurement noise has a diagonal covariance, the observations in S are conditionally independent given Z and x. Therefore, by omitting the denominator of (8) which does not depend on x, it follows that p(x|S, Z = k; θ) is proportional to

p(x|Z = k; θ) D,T d=1,t=1 p(y ′ dt |x, Z = k; θ) χ dt = N (x; c k , Γ k ) D,T d=1,t=1 N (y ′ dt | a ⊤ dk x + b dk , σ 2 d ) χ dt = C | Γ k | 1 2 exp - 1 2 (A + B) (9)
where

A = D,T d=1,t=1 χ dt e σ 2 d (y ′ dt -a ⊤ dk x -b dk ) 2 , B = (x - c k ) ⊤ Γ -1 k (x -c k )
, and the constant C depends neither on x nor on k. Since p(x|S, Z = k; θ) is a normal distribution in x with mean µ k and covariance V k , we can write:

A + B = (x -µ k ) ⊤ V -1 k (x -µ k ). ( 10
)
By identification between the left-hand and right-hand terms of ( 10), we obtain the formulae ( 4) and ( 5) for µ k and V k respectively. Using Bayes inversion, the mixture's priors ν k = p(Z = k|S; θ) are proportional to π k p(S|Z = k; θ). Unfortunately, we cannot directly decompose p(S|Z = k; θ) into a product over (d, t), as previously done with p(S|x, Z = k; θ). Indeed, while it is assumed that the frequency-time points of the observed spectrogram S are independent given x and Z, this is not true for the same observations given only Z. However, we can use [START_REF] Barker | Energetic and informational masking effects in an audiovisual speech recognition system[END_REF] to obtain

p(S|Z = k; θ) = p(S|x, Z = k; θ)p(x|Z = k; θ) p(x|S, Z = k; θ) . ( 11
)
The numerator is given by ( 9) and the denominator is the normal distribution N (x; µ k , V k ). After simplifying the terms in x, we obtain the desired expression (6) for ν k .

EXPERIMENTS AND RESULTS

Setup. A binaural pair of microphones embedded into an acoustic dummy head is mounted onto a camera system 1 . The audio-visual source used for training is a loud-speaker with a visual marker, e.g., Fig. 1(left). We assume a one-to-one mapping between pixels and sound directions.

Interaural acoustic vectors. Let a sound source emit from a direction corresponding to a pixel position x ∈ R 2 . We denote the complex-valued left-and right-spectrograms with

s L = {s L f t } F,T t=1,f =1 and s R = {s R f t } F,T t=1,f =1
. These spectrograms are built using the short-time Fourier transform with F frequency bands and T windows of size 64ms with 56ms overlap. The resulting spectrograms have 125 temporal bins Fig. 2. A subject counts from 1 to 12 (white numbers) with a normal voice loudness and is static while pronouncing each number. The red circles show the position found with our method. The yellow squares show the results of the Viola-Jones face detector [START_REF] Viola | Robust real-time face detection[END_REF].

per second and 512 frequency bins ranging from 0 to 8KHz. The interaural-level-difference (ILD) and interaural-phasedifference (IPD) spectrograms, α and φ, are defined as the log-amplitude and phase of the ratio between the left and the right spectrograms, i.e., α f t = 20 log 10 (|s

R f t /s L f t |) ∈ R and φ f t = exp(j arg(s R f t /s L f t )) ∈ C ≡ R 2 .
A binaural acoustic vector y is obtained by taking the temporal mean of the concatenation of α and φ, hence we obtain a vector of size D = 1536. Because of the filtering effects from source to microphones, α f t and φ f t are independent of the emitted signal, and depend on the sound source position x if the recorded spectral density at the spectrogram point (f, t) is large enough [START_REF] Deleforge | 2D sound-source localization on the binaural manifold[END_REF]. Time-frequency points for which the total spectral density 10 log 10 (|s

L f t | 2 + |s R f t | 2
) is below a given threshold are treated as missing values.

Training data2 are obtained with a loudspeaker emitting white-noise (WN) from N = 432 different positions lying on a 18 × 24 grid in the camera's field of view, e.g., Fig. 1-left. Since a WN signal have a significant spectral density in all time-frequency points, the associated ILD and IPD spectrograms α n and φ n do not have missing values. Therefore we obtain a training set (y n , x n ), n ∈ [1 . . . N ]. In all our experiments we used K = 32 affine components to learn the regression (Section 2).

Loud-speaker test data 2 are obtained with the loud-speaker emitting 1-second utterances from the TIMIT dataset [START_REF] Garofolo | The DARPA TIMIT acousticphonetic continuous speech corpus CD-ROM[END_REF].The loud speaker is placed at 108 positions on a 9 × 12 grid covering the camera's field of view. The binaural recording associated to each utterance is cut, yielding ILD and IPD spectrograms with 89% missing values on average. These spectrograms are concatenated vertically, resulting in a time series of acoustic vectors S = {Y ′ , χ} (Section 3).

Counting test 2 . The method was also tested on in a more realistic scenario. A participant is asked to count from 1 to 20 in front of the camera. The speaker is required to be roughly static while pronouncing each number, whereas he/she is allowed to move in between numbers. The audio-to-visual mapping method is applied on a 720ms analysis window that is slid over the soundtrack, in order to estimate a speaker position at each video frame for which enough acoustic level is recorded. This is a particularly challenging scenario because the speaker has different direction, distance, orientation and directionality than the loudspeaker used for training, and the speaker emits sparse-spectrum and less loud sounds than in the training set, thus considerably reducing the amount of exploitable spectrogram data.

Results. The average localization error over 108 loud-speaker tests was 21.9±17 pixels horizontally and 23.1±20 vertically. This corresponds to less that 1 • in both azimuth and elevation. The largest ground-truth-to-estimate distance (GTED) error was 89.9 pixels, i.e. ≈ 4 • . For comparison, we used PHAT [START_REF] Aarabi | Self-localizing dynamic microphone arrays[END_REF] as a baseline sound source localization method, with the same test sounds. PHAT estimates the sound's time difference of arrival (TDOA), using cross-correlations at different times and frequency channels. A linear regressor was trained to map TDOA values obtained with PHAT onto the horizontal image axis using the WN training data 3 . Note that the vertical position cannot be estimated from TDOA. The average GTED with PHAT was 64.0±51.5 pixels (3 times larger than with the proposed method) with 21 out of 108 GTED larger than 100 pixels. The average computational time of our method and of PHAT are of respectively 230ms and 400ms for 1s sounds.

Figure 2 shows some frames of the video generated from the counting test 2 . The sound source position estimated by the proposed method is shown by a red circle in the corresponding video frame. The largest mouth-to-estimate distance error is 128 pixels (number 3). This corresponds to ≈ 1.7 • error in azimuth and ≈ 5.3 • error in elevation. For comparison, results obtained with a face detection algorithm [START_REF] Viola | Robust real-time face detection[END_REF] are shown with a yellow square. While this method correctly localized the faces in 10 out of 12 examples, it failed to detect faces #8 and #9 due to partial occlusions. It also featured a number of false detections.

CONCLUSION

We presented a method that maps sound-source directions onto images in order to achieve audio-visual alignment, e.g., speech-to-face association. The direction of a sound is estimated both in azimuth and elevation (2D localization), whereas the vast majority of state-of-the-art techniques estimate only the azimuth (1D localization). 2D localization is absolutely necessary in order to align audio signals with visual features. The method relies on a regression framework [START_REF] Deleforge | High-Dimensional Regression with Gaussian Mixtures and Partially-Latent Response Variables[END_REF] that learns the parameters of a transformation from binaural spectrograms to pixel coordinates. While the offline training needs full-spectrum sounds (white noise), the online localization has a closed-form solution and can be used to locate any sound type, including natural sparse-spectrum sounds. Another advantage of our method is that it requires neither explicit representation of the microphone/camera geometry [START_REF] Khalidov | Alignment of Binocular-Binaural Data Using a Moving Audio-Visual Target[END_REF][START_REF] Horaud | A geometric approach to sound source localization from time-delay estimates[END_REF], nor long audio-visual sequences [START_REF] Segev | Examplebased cross-modal denoising[END_REF]. Proper alignment between acoustic spectrograms and images paves the road towards audio-visual data association for advanced multimodal processing and scene understanding.

Fig. 1 .

 1 Fig. 1. Left: The sound-to-image transformation is learned from an audio-visual target composed of a loud speaker and a visual marker.The loud-speaker emits full-spectrum sounds which are recorded with a microphone pair while the visual marker is used to estimate its corresponding image positions. Right: A speaking face is detected and predicted from a sparse-spectrum sound.

  and vector b k ∈ R D are the parameters of an affine transformation τ k and E ∈ R D is a Gaussian error vector with zero-mean and diagonal covariance Σ = Diag (σ 2 1 . . . σ 2 d . . . σ 2 D ) capturing both the observation noise in R D and the reconstruction error due to the local affine approximation. I is the indicator function and Z is a hidden variable such that I(Z = k) = 1 if and only if Z = k (Y is the transformed of X by τ k ), and 0 otherwise. Consequently we have p(

Full details on the setup at http://team.inria.fr/perception/popeye/

The training data, test data, and video examples are available at http://perception.inrialpes.fr/people/Deleforge/AVASM Dataset/

A linear dependency was observed in practice.


Sylvain Grangeon

Francis Claret

Catherine Lerouge

Bruno Lanson

Structure of nanocrystalline calcium silicate hydrates

teaching and research institutions in France or abroad, or from public or private research centers.

Nanocrystalline calcium silicate hydrates (C-S-H) are the main hydration products and the main binging phases in many types of cement, including ordinary Portland cement. As a result of its ubiquity is these engineered systems, C-S-H controls main cement physical and chemical properties. These properties depend on C-S-H calcium to silicon atomic ratio (Ca/Si), which is commonly described as ranging between 0.6 and 2.3. This means that C-S-H structure dictates cement macroscopic properties. As a consequence, its crystal-chemistry must be understood to be able to understand (and thus predict) cement properties. However, despites decades of study, its crystal structure is still a matter of debate. Depending on the authors and on the calcium to silicon ratio, the structure is described similar to one or more of the following minerals: tobermorite, jennite and possibly portlandite. Such inaccuracy largely results from its X-ray diffraction patterns, which exhibits only a few weak and mostly asymmetrical diffraction maxima and thus cannot be refined using classical methods. By using a specific formalism for the analysis of X-ray diffraction patterns, previously applied to phyllosilicates and phyllomanganates, it is here proposed that C-S-H can be described as a lamellar structure similar to nanocrystalline and turbostratic tobermorite, turbostratism meaning that there is the systematic presence, between adjacent layers, of a random rotation about the normal to the layers and/or a random translation in the layer plane. This model was validated using complementary methods (including transmission electron microscopy, synchrotron X-ray absorption spectroscopy and synchrotron high-energy X-ray scattering). From analysis of literature data, it is proposed that the evolution of C-S-H structure as a function of Ca/Si can be described as interstratification of two different types of layers having calcium to silicon ratios of 0.6 and 1.25, plus discrete Ca(OH) 2 at higher ratios.


Antoine Deleforge

Florence Forbes

Radu Horaud

HYPER-SPECTRAL IMAGE ANALYSIS WITH PARTIALLY-LATENT REGRESSION

Keywords: Hyper-spectral images, Regression, Dimension reduction, Mixture models, Latent variable model

The analysis of hyper-spectral images is often needed to recover physical properties of planets. To address this inverse problem, the use of learning methods have been considered with the advantage that, once a relationship between physical parameters and spectra has been established through training, the learnt relationship can be used to estimate parameters from new images underpinned by the same physical model. Within this framework, we propose a partially-latent regression method which maps high-dimensional inputs (spectral images) onto low-dimensional responses (physical parameters). We introduce a novel regression method that combines a Gaussian mixture of locally-linear mappings with a partially-latent variable model. While the former makes high-dimensional regression tractable, the latter enables to deal with physical parameters that cannot be observed or, more generally, with data contaminated by experimental artifacts that cannot be explained with noise models. The method is illustrated on images collected from the Mars planet.

INTRODUCTION

The analysis of hyper-spectral images often involves to solve an inverse problem to deduce a number of physical parameter values from the observed spectra. This typically requires the estimation of a high-dimensional to low-dimensional mapping, which is challenging. Among others, statistical learning methods have been considered with the advantage that, once a relationship between physical parameters and spectra has been established through training, the learnt relationship can subsequently be used to estimate parameters from new images underpinned by the same physical model [START_REF] Bernard-Michel | Retrieval of Mars surface physical properties from OMEGA hyperspectral images using regularized sliced inverse regression[END_REF]. To obtain training data, radiative transfer models have been developed, that link the chemical composition, the granularity, or the physical state, to the observed spectrum. They are generally used to simulate huge collections of spectra in order to perform the inversion of hyper-spectral images [START_REF] Douté | A comprehensive numerical package for the modeling of Mars hyperspectral images[END_REF].

Support from the European Research Council (ERC) through the Advanced Grant VHIA (#340113) is greatly acknowledged.

Within this framework, we propose a novel regression method. Regression is a fully supervised method that uses input-response pairs of observed data to infer a mapping, and this mapping is then used to predict a response value from an observed input. It is well known that high-dimensional regression is a difficult problem because of the very high number of parameters to be estimated. To address this problem we propose a probabilistic framework, first to learn the parameters of a low-to-high Gaussian mixture of locally-linear regressions, and second to derive the posterior conditional density of the response given an observed input and the estimated model parameters. We show that, by exchanging the roles of input and response variables during training, highdimensional regression becomes tractable and the conditional density of the response has a closed-form solution. Moreover, we propose to incorporate a latent component: while the high-dimensional variable remains fully observed, the lowdimensional variable is the concatenation of an observed vector and of an unobserved (latent) vector. The latent component enables us to deal with those physical parameters that cannot be observed during training, or to prevent the observed parameters to be contaminated by experimental artifacts that cannot be explained with noise models. This model is particularly suited for hyper-spectral imaging, because the computing resources required by radiative transfer models to generate training data increases exponentially with the number of parameters. Thus, they are generally restricted to a small number of parameters, e.g., abundance and grain size of the main chemical components. Other parameters, such as those related to meteorological variability or the incidence angle of the spectrometer are not explicitly modeled and measured. Our method allows to deal with such non-anotated effects by explicitely incorporating a latent part in the low dimensional response variable. This allows some form of slack in the observations due to unknown effects.

Regression has been extensively studied in the literature. To deal with high-dimensional input data, most methods first reduce the dimensionality based on response variables, before performing the actual regression [START_REF] Li | Sliced inverse regression for dimension reduction[END_REF][START_REF] Wu | Kernel sliced inverse regression with applications to classification[END_REF][START_REF] Adragni | Sufficient dimension reduction and prediction in regression[END_REF]. Such two-step approaches cannot be conveniently expressed in terms of a single optimization problem. To estimate non-linear mappings, mixtures of locally linear models have been investigated [START_REF] Xu | An alternative model for mixtures of experts[END_REF][START_REF] Qiao | Mixture of probabilistic linear regressions: A unified view of GMM-based mapping techiques[END_REF][START_REF] Ingrassia | Local statistical modeling via a cluster-weighted approach with elliptical distributions[END_REF], but not in the case of high dimensional inputs and partially-observed responses. An alternative popular approach is to use kernel functions, [START_REF] Wu | Kernel sliced inverse regression with applications to classification[END_REF][START_REF] Tipping | Sparse Bayesian learning and the relevance vector machine[END_REF][START_REF] Lawrence | Probabilistic non-linear principal component analysis with gaussian process latent variable models[END_REF][START_REF] Thayananthan | Multivariate relevance vector machines for tracking[END_REF], with the drawback that these functions cannot be appropriately chosen automatically, and that the mappings learned cannot be inverted. A more detailed description of the proposed approach and its relation to existing methods is available as a research report [START_REF] Deleforge | High-Dimensional Regression with Gaussian Mixtures and Partially-Latent Response Variables[END_REF].

REGRESSION WITH PARTIALLY-LATENT RESPONSE VARIABLES

We describe a regression method that maps a low-dimensional variable X onto a high-dimensional variable Y . We start by describing the GLLiM (Gaussian Locally Linear Mapping) model, in which both X and Y are fully observed. Then we introduce the hybrid GLLiM (hGLLiM) model that treats X as a partially latent variable. Once the parameters of the hybrid GLLiM model are estimated, we provide a closed-form expression for the high-dimensional to low-dimensional mapping. GLLiM relies on a piecewise linear model in the following way. Let and(y, x) is a realization of (Y , X), such that y is the image of x by an affine transformation τ k , among K, plus an error term. This is modeled by a missing variable Z such that Z = k if and only if Y is the image of X by τ k .

{x n } n=N n=1 ∈ R L , {y n } n=N n=1 ∈ R D , L D,
The following decomposition of the joint probability distribution will be used:

p(Y = y, X = x; θ) = K k=1 p(Y = y|X = x, Z = k; θ)p(X = x|Z = k; θ)p(Z = k; θ),
where θ denotes the vector of model parameters. The locally affine function that maps X onto Y is:

Y = K k=1 I(Z = k)(A k X + b k + E k ) (1)
where I is the indicator function, matrix A k ∈ R D×L and vector b k ∈ R D define the transformation τ k and E k ∈ R D is an error term capturing both the observation noise in R D and the reconstruction error due to the local affine approximation. Under the assumption that E k is a zero-mean Gaussian variable with covariance matrix Σ k ∈ R D×D that does not depend on X, Y , and Z, we obtain:

p(Y = y|X = x, Z = k; θ) = N (y; A k x + b k , Σ k ). (2)
To complete the above hierarchical definition and enforce the affine transformations to be local, X is assumed to follow a mixture of K Gaussians defined by

p(X = x|Z = k; θ) = N (x; c k , Γ k ) p(Z = k; θ) = π k where c k ∈ R L , Γ k ∈ R L×L and K k=1 π k = 1.
The parameters of this model are:

θ = {c k , Γ k , π k , A k , b k , Σ k } K k=1 . (3)
The model just described can be learned with standard EM inference methods if X and Y are both observed. The key idea in this paper is to treat X as a partially-latent variable, namely

X = T W , where T ∈ R Lt is observed and W ∈ R Lw is latent (L = L t + L w ).
In hybrid GLLiM, parameter estimation uses observed pairs {y n , t n } N n=1 while it must also be constrained by the presence of the latent variable W . This can be seen as a latent-variable augmentation of classical regression, where the observed realizations of Y are affected by the unobserved variable W . It can also be viewed as a variant of dimensionality reduction since the unobserved low-dimensional variable W must be recovered from {(y n , t n )} N n=1 . The decomposition of X into observed and latent parts implies that some of the model parameters must be decomposed as well, namely c k , Γ k and A k . Assuming the independence of T and W given Z we write:

c k = c t k c w k , Γ k = Γ t k 0 0 Γ w k , A k = A t k A w k .
We devised two EM algorithms to estimate the parameters of the proposed model. The principle of the suggested algorithms is based on a data augmentation strategy that consists of augmenting the observed variables with the unobserved ones, in order to facilitate the subsequent maximum-likelihood search over the parameters. There are two sets of missing variables, Z 1:N = {Z n } N n=1 and W 1:N = {W n } N n=1 , associated with the training data set (y, t) 1:N = {y n , t n } N n=1 , given the number K of linear components and the latent dimension L w . Two augmentation schemes arise naturally. The first scheme is referred to as general hybrid GLLiM-EM, or general-hGLLiM, and consists of augmenting the observed data with both variables (Z, W ) 1:N while the second scheme, referred to as marginal-hGLLiM, consists of integrating out the continuous variables W 1:N previous to data augmentation with the discrete variables Z 1:N . A particularly interesting feature of general-hGLLiM is that both the E-step and M-step can be computed in closed-form for various constraints on the noise covariances {Σ k } K k=1 , including "equal for all k", isotropic, and diagonal constraints. On the other hand, marginal-hGLLiM only has closed-form steps in the isotropic case, but is easier to initialize. Therefore, the latter is used to initialize the former in practice. Full details on the analytical steps of the resulting hGLLiM algorithm are given in [START_REF] Deleforge | High-Dimensional Regression with Gaussian Mixtures and Partially-Latent Response Variables[END_REF], and a Matlab toolbox implementing it is available online 1 .

Once the parameter vector θ has been estimated by the 1 https://team.inria.fr/perception/gllim toolbox/ algorithm, one can derive the following conditional density:

p(X = x|Y = y; θ * ) = K k=1 π * k N (y; c * k , Γ * k ) K j=1 π * j N (y; c * j .Γ * j ) N (x; A * k y + b * k , Σ * k ) (4)
which is a Gaussian mixture with K components. Notice that the parameters associated with this mixture:

θ * = {c * k , Γ * k , π * k , A * k , b * k , Σ * k } K k=1 (5)
are obtained analytically from the learned parameters (3) using the following formulae:

c * k = A k c k + b k , Γ * k = Σ k + A k Γ k A k , π * k = π k , A * k = Σ * k A k Σ -1 k , b * k = Σ * k (Γ -1 k c k -A k Σ -1 k b k ), Σ * k = (Γ -1 k + A k Σ -1 k A k ) -1 . (6)
To predict an output one can use the expectation of ( 4):

E[X|y; θ * ] = K k=1 π k N (y; c * k , Γ * k ) K j=1 π j N (y; c * j , Γ * j ) (A * k y + b * k ). (7)

RETRIEVING MARS PHYSICAL PROPERTIES FROM HYPER-SPECTRAL IMAGES

Visible and near infrared imaging spectroscopy is a key remote sensing technique used to study and monitor planets. It records the visible and infrared light reflected from the planet in a given wavelength range and produces cubes of data where each observed surface location is associated with a spectrum. Physical properties of a planet surface, such as chemical composition, granularity, texture, etc., are some of the most important parameters that characterize the morphology of the spectra. In case of Mars, radiative transfer models have been developed to numerically evaluate the connection between these parameters and observable spectra. Such models allow to simulate spectra from a given set of parameter values [START_REF] Douté | A comprehensive numerical package for the modeling of Mars hyperspectral images[END_REF]. In practice, the goal is to scan Mars's ground from an orbit in order to observe gas and dust in the atmosphere and seek for signs of specific materials such as silicates, carbonates, or ice at the surface. We are therefore interested in solving the associated inverse problem, namely to infer physical parameter values from the observed spectra. Since this inverse problem cannot generally be solved analytically, the use of optimization or statistical methods has been investigated, e.g., [START_REF] Bernard-Michel | Retrieval of Mars surface physical properties from OMEGA hyperspectral images using regularized sliced inverse regression[END_REF]. In particular, supervised statistical learning has been considered with the advantage that, once a relationship between physical parameters and spectra has been established trough training, the learned relationship can then be used for very large datasets and for all new images underpinned by the same physical model.

Within this category of methods, we investigate the potential of the proposed hybrid GLLiM model using a dataset of hyper-spectral images collected from the imaging spectrometer OMEGA instrument [START_REF] Bibring | OMEGA: Observatoire pour la Minéralogie, l'Eau, les Glaces et l'Activité[END_REF] onboard of the Mars express spacecraft. To this end a database of synthetic spectra with their associated physical parameter values were generated using a radiative transfer model. This database is composed of 15,407 spectra associated with five real parameter values, namely, (i) proportion of water ice, (ii) proportion of CO 2 ice, (iii) proportion of dust, (iv) grain size of water ice, and (v) grain size of CO 2 ice. Each spectrum is made of 184 wavelengths. Hybrid GLLiM is used, first to learn a low-to-high dimensional regression function between physical parameters and spectra from the database, and second to estimate the the unknown physical parameters corresponding to a new spectrum using the learned function. Since no ground truth is available for Mars, the synthetic database will also serve as a test dataset to evaluate the accuracy of the predicted parameter values. In order to fully illustrate the potential of hybrid GLLiM, we deliberately ignore two of the five parameters in the database and consider them as latent variables.

During training, we excluded the observed values of the proportion of water ice and the grain size of CO 2 ice. A previous study [START_REF] Bernard-Michel | Retrieval of Mars surface physical properties from OMEGA hyperspectral images using regularized sliced inverse regression[END_REF] revealed that these two parameters were sensitive to the same wavelengths, in comparison with the proportion of dust and are suspected to mix with the other two parameters in the synthetic transfer model. Therefore, not only that they are harder to estimate but they seem to affect the estimation of two other parameters. Moreover, we observed that if these two parameters are treated as observed variables during the learning stage, they tend to degrade the estimation of the other three parameters, which are of particular interest.

The hGLLiM algorithm was compared to JGMM (joint Gaussian mixture model) [START_REF] Qiao | Mixture of probabilistic linear regressions: A unified view of GMM-based mapping techiques[END_REF], SIR (sliced inverse regression) [START_REF] Li | Sliced inverse regression for dimension reduction[END_REF], RVM (multivariate relevance vector machine) [START_REF] Thayananthan | Multivariate relevance vector machines for tracking[END_REF] and MLE (mixture of linear experts) [START_REF] Xu | An alternative model for mixtures of experts[END_REF]. SIR is used with one (SIR-1) or two (SIR-2) principal axes for dimensionality reduction, 20 slices (the number of slices is known to have very little influence on the results), and polynomial regression of order three (higher orders did not show significant improvements in our experiments). SIR quantizes the lowdimensional data X into slices or clusters which in turn induces a quantization of the Y -space. Each Y -slice (all points y n that map to the same X-slice) is then replaced with its mean and PCA is carried out on these means. The resulting dimensionality reduction is then informed by X values through the preliminary slicing. RVM may be viewed as a multivariate probabilistic formulation of support vector regression [START_REF] Smola | A tutorial on support vector regression[END_REF]. As all kernel methods, it critically depends on the choice of a kernel function. Using the authors' freely available code2 , we ran preliminary tests to determine an optimal kernel choice for each dataset considered. We tested An objective evaluation was done by cross validation. We selected 10,000 training couples at random from the training set, tested on the 5,407 remaining spectra, and repeated this 20 times. For all algorithms, training data were normalized to have 0 mean and unit variance using scaling and translating factors. These factors were then used on test data and estimated output to obtain final estimates. This technique showed to noticeably improve results of all methods. We used K = 50 for MLE, hGLLiM and JGMM. MLE and JGMM were constrained with equal, diagonal covariance matrices as they yield the best results. For RVM, the best out of 140 kernels was used. A third degree polynomial kernel with scale 6 showed the best results using cross-validation on a subset of the database. As a quality measure of the estimated parameters, we computed the normalized root mean squared error, or NRMSE 3 , which quantifies the difference between the estimated and real parameter values. NRMSE is normalized enabling direct comparison between the parameters which are of very different range: the closer to zero, the more accurate the predicted values. Table 1 shows NRMSE values obtained for the three parameters considered here. The ground-truth latent variable dimension is L * w = 2, and accordingly, the empirically best dimension for hGLLiM was L † w = 2. hGLLiM-2 outperformed all the other methods on that task, more precisely the error is 36% lower than the second best-performing method, RVM, closely followed by MLE. No significant difference was observed between hGLLiM-2 and hGLLiM-3. Note that due to the computation of the D × D kernel matrix, the computational and memory costs of RVM for training were about 10 times higher than those of hGLLiM, using MATLAB implementations. We also tested the selection of the latent dimension (L w ) based on BIC [START_REF] Deleforge | High-Dimensional Regression with Gaussian Mixtures and Partially-Latent Response Variables[END_REF]. For each train-

3 NMRSE = r P M m=1 ( tm-tm) 2 P M m=1 (tm-t) 2 with t = M -1 P M m=1 tm.
ing set, the hGLLiM-BIC method minimized BIC with the latent dimension in the range 0 ≤ L w ≤ 20, and used the corresponding model to perform the regression. Interestingly, hGLLiM-BIC performed very well on these large training sets (N = 10, 000) as it correctly selected L w = 2 for the 20 training sets used by BIC (the BIC selection could differ with the training set), yielding the same results as those obtained with hGLLiM-2.

Finally, we used an adequately selected subset of the synthetic database, e.g., [START_REF] Bernard-Michel | Retrieval of Mars surface physical properties from OMEGA hyperspectral images using regularized sliced inverse regression[END_REF] to train the algorithms, and test them on real data made of observed spectra. In particular, we focus on a dataset of Mars South polar cap. Since no ground truth is currently available for the physical properties of Mars's South pole regions, we propose a qualitative evaluation using hGLLiM-2 and the three best performing methods, among the tested ones, namely RVM, MLE and JGMM.

We used these four methods in order to extract physical parameters from two hyper-spectral images that correspond, approximately, to the same area but from different view points (orbit 41 and orbit 61). Since we are looking for proportions between 0 and 1, values smaller than 0 or higher than 1 are not acceptable and hence they were set to either one of the bounds. As it can be seen in Fig. 1, hGLLiM-2 estimates proportion maps with similar characteristics for the two view points, which suggests a good consistency of the method. Such a consistency is not observed using the other three tested methods. In addition, RVM and MLE estimate a much higher number of values falling outside the interval [0, 1]. Moreover, hGLLiM-2 is the only method featuring less dust at the South pole cap center and higher concentrations of dust at the boundaries of the CO2 ice, which matches expected results from planetology [START_REF] Douté | Nature and composition of the icy terrains of the south pole of Mars from MEX OMEGA observations[END_REF]. Finally, note that the proportions of CO2 ice and dust clearly seem to be complementary using hGLLiM-2, while this complementarity is less obvious using other methods.

CONCLUSIONS

We proposed a new high-dimensional regression method allowing for partially latent responses. This novel approach outperformed 4 state-of-the-art regression methods in hyperspectral image analysis. A promising extension would be to take into account dependencies between adjacent pixels in images to enforce smoothness. This could be done using Markov Random field. More complex noise models could also be investigated via Student distributions (e.g. [START_REF] Ingrassia | Local statistical modeling via a cluster-weighted approach with elliptical distributions[END_REF]) to allow for outliers accommodation and more robust estimation.

Fig. 1 .

 1 Fig. 1. Proportions obtained with our method (hGLLiM-2) and three other methods. The data correspond to hyper-spectral images grabbed from two different viewpoints of the South polar cap of Mars. Left: orbit 41, Right: orbit 61. White areas correspond to unexamined regions, where the synthetic model does not apply.

Table 1 .

 1 Normalized root mean squared error (NRMSE) for Mars surface physical properties recovered from hyper-spectral images, using synthetic data and different methods.

	Method	Proportion of CO2 ice Proportion of dust Grain size of water ice
	JGMM	0.83 ± 1.61	0.62 ± 1.00	0.79 ± 1.09
	SIR-1	1.27 ± 2.09	1.03 ± 1.71	0.70 ± 0.94
	SIR-2	0.96 ± 1.72	0.87 ± 1.45	0.63 ± 0.88
	RVM	0.52 ± 0.99	0.40 ± 0.64	0.48 ± 0.64
	MLE	0.54 ± 1.00	0.42 ± 0.70	0.61 ± 0.92
	hGLLiM-1	0.36 ± 0.70	0.28 ± 0.49	0.45 ± 0.75
	hGLLiM-2 * †	0.34 ± 0.63	0.25 ± 0.44	0.39 ± 0.71
	hGLLiM-3	0.35 ± 0.66	0.25 ± 0.44	0.39 ± 0.66
	hGLLiM-4	0.38 ± 0.71	0.28 ± 0.49	0.38 ± 0.65
	hGLLiM-5	0.43 ± 0.81	0.32 ± 0.56	0.41 ± 0.67
	hGLLiM-20	0.51 ± 0.94	0.38 ± 0.65	0.47 ± 0.71
	hGLLiM-BIC	0.34 ± 0.63	0.25 ± 0.44	0.39 ± 0.71
	14 kernel types with 10 different scales ranging from 1 to 30,
	hence, 140 kernels for each dataset in total.

http://www.mvrvm.com/Multivariate Relevance Vector