S Dacie

T T Ör Ök

P D Émoulin

M G Linton

C Downs

L Van Driel-Gesztelyi

D M Long

Tibor Török

Typeset using L A T E X preprint2 style in AASTeX61 SEQUENTIAL ERUPTIONS TRIGGERED BY FLUX EMERGENCE -OBSERVATIONS AND MODELING

Keywords: Sun, magnetic fields -Sun, corona -Sun, coronal mass ejections (CMEs)

We describe and analyze observations by the Solar Dynamics Observatory of the emergence of a small, bipolar active region within an area of unipolar magnetic flux that was surrounded by a circular, quiescent filament. Within only eight hours of the start of the emergence, a partial splitting of the filament and two consecutive coronal mass ejections took place. We argue that all three dynamic events occurred as a result of particular magnetic-reconnection episodes between the emerging bipole and the pre-existing coronal magnetic field. In order to substantiate our interpretation, we consider three-dimensional magnetohydrodynamic simulations that model the emergence of magnetic flux in the vicinity of a large-scale coronal flux rope. The simulations qualitatively reproduce most of the reconnection episodes suggested by the observations; as well as the filament-splitting, the first eruption, and the formation of sheared/twisted fields that may have played a role in the second eruption. Our results suggest that the position of emerging flux with respect to the background magnetic configuration is a crucial factor for the resulting evolution, while previous results suggest that parameters such as the orientation or the amount of emerging flux are important as well. This poses a challenge for predicting the onset of eruptions that are triggered by flux emergence, and it calls for a detailed survey of the relevant parameter space by means of numerical simulations.

INTRODUCTION

Coronal mass ejections (CMEs) are eruptions of sheared or twisted magnetic fields from the solar corona. They may contain a filament consisting of dense and cool material initially suspended above the solar surface. Many mechanisms have been proposed to explain how CMEs (and associated flares) are initiated, causing the magnetic structure to rise (see, e.g., [START_REF] Aulanier | Nature of Prominences and their Role in Space Weather[END_REF]. At a critical height, the structure is believed to become torus unstable (Kliem & Török 2006), which causes it to rapidly accelerate upward. At the same time, a current sheet forms between oppositely orientated field lines beneath the unstable structure. Reconnection within the current sheet leads to flaring (as in the CSHKP model [START_REF] Carmichael | [END_REF]Sturrock 1966;Hirayama 1974;Kopp & Pneuman 1976) and additional acceleration of the ejecta.

One possible mechanism for the initiation of eruptions is flux emergence nearby a preexisting, current-carrying magnetic structure, as described by, e.g., [START_REF] Chen | [END_REF]. In their simulation labeled as case B, bipolar field emerges near to a flux rope that is in stable equilibrium with the ambient field. The new flux is orientated "favorably" for reconnection, as defined by Feynman & Martin (1995), meaning that the orientation of the emerging bipole is chosen such that a current layer forms between the flux rope and the bipole. As reconnection occurs across the current layer, two new sets of field lines are created; a small arcade that connects the emerging flux and the ambient field, and long field lines that arch over the flux rope and connect to the other polarity of the emerging flux (see Figures 5b in Chen &[START_REF] Chen | [END_REF]3a in Williams et al. 2005). These latter field lines become somewhat longer due to the displacement of one of their foot points, so their downward acting magnetic tension decreases. This ongoing reduction of magnetic tension leads to a continuous slow rise of the flux rope, which finally leads to loss of equilibrium (or torus instability) and the eruption of the rope. The eruption is potentially facilitated also by an overall expansion of the ambient field due to the changes in the photospheric flux distribution (Ding & Hu 2008).

There have been many observational studies of newly emerging flux occurring before CMEs (Feynman & Martin 1995;Wang & Sheeley 1999;Jing et al. 2004;Xu et al. 2008;Schrijver 2009). These have shown that the most favorable conditions for triggering a CME arise when the orientation of the emerging flux is opposite to that of the existing field and when the emergence occurs close to the polarity inversion line (PIL; e.g., Xu et al. 2008). However, Louis et al. (2015) associated a flare and CME with flux emergence that was neither of favorable orientation nor located close to the PIL, and other exceptions can be found in the studies mentioned above. Other observations have shown magnetic flux emergence apparently causing a filament to split (Li et al. 2015). These contrasting observations suggest that the set of conditions required for emerging flux to initiate an eruption are not yet fully understood (see also Lin et al. 2001).

Many modeling studies have been able to produce eruptions by introducing emerging flux into a pre-existing magnetic field configuration; either containing a potential or sheared arcade field (Notoya et al. 2007;Zuccarello et al. 2008Zuccarello et al. , 2009;;Kusano et al. 2012;Jacobs & Poedts 2012;Roussev et al. 2012;Kaneko & Yokoyama 2014), or a flux rope (e.g. [START_REF] Chen | [END_REF]Lin et al. 2001;Xu et al. 2005;Shiota et al. 2005;Dubey et al. 2006;Ding & Hu 2008). Whether an eruption is produced depends on various parameters such as the strength of the new magnetic flux and its position and orientation. Changing these parameters can lead to cases where the emerging flux acts to additionally stabilize a flux rope rather than triggering its eruption (e.g. the cases shown in Figure 7 of [START_REF] Chen | [END_REF].

In this paper, we study the effects of the emergence of a small bipole nearby a quiescent circular filament on 18 July 2014. The eastern section of the filament is seen to partially split and to form new connectivities, followed by the eruption of its western section shortly after. A few hours later, a second eruption occurs above the PIL segment that has formed between the emerging flux and the pre-existing field, suggesting the formation of nonpotential magnetic fields at this location during the emergence of the bipole. In Section 2, we discuss the observations and propose mechanisms to explain these activities. In Section 3 we present magnetohydrodynamic (MHD) numerical simulations that qualitatively reproduce the filament splitting and the first eruption, and suggest a possible mechanism for the formation of a flux rope between the emerging and preexisting flux. Finally, we discuss the results and draw our conclusions in Section 4.

OBSERVATIONS AND PROPOSED MECHANISMS

Data

A quiescent circular filament and the newly forming active region NOAA 12119, which emerged within a negative polarity area encircled by the filament, close to its eastern section, were studied for the eight hours following the start of the active region's emergence at ≈ 03:30 UT on 18 July 2014 at [-376", -415"] in helioprojective-cartesian coordinates. The partial splitting of the filament and the two eruptions occurred during this time period. Data from the Atmospheric Imaging Assembly (AIA; Lemen et al. 2012) on board the Solar Dynamics Observatory (SDO; Pesnell et al. 2012) were used to identify structures and connectivi- ties in the corona, and photospheric line-of-sight magnetic field measurements, provided by the Helioseismic Magnetic Imager (HMI; Schou et al. 2012) were used to calculate the magnetic flux of the emerging bipole. All images shown in Figs. 1 and 4 below were rotated to the observer's view at 05:55 UT on July 18, which is roughly midway between the respective onset times of the partial filament splitting and the first eruption. The center of this field of view is at -23.7 • longitude (in heliographic coordinates). We refrained from rotating all images to the central meridian, to minimize the interpolation of the data.

Splitting of the Eastern Filament Section

The quiescent filament (shown in Figure 1(b) at 03:30 UT on 18 July 2014) was almost circular in shape and had formed between an area of dispersed negative field (inside the dashed line shown in Figure 1(a),(b)) and positive field (outside). An emerging bipole, which later becomes active region NOAA 12119, began to emerge just to the west of the eastern section of the circular filament. The orientation of its magnetic fields is mostly West to East, although the presence of magnetic tongues leads to some deviation of the PIL orientation from that direction (Figure 1(c)). Following Poisson et al. (2016), these tongues are interpreted as the contribution of the azimuthal field component of the emerging flux tube. They indicate a negative magnetic helicity, which is also suggested by the shear of the loops seen in the corona (Figure 1(d)). 1As the new flux started to emerge, it immediately began to interact with the surrounding magnetic field, as is apparent from the formation of bright loops in the AIA 171 Å images as early as 05:12 UT. These loops are shown in Figure 1(d) and outlined by dashed white curves in the zoom shown in panel Figure 1(e). They are indicative of new magnetic connectivities as a result of reconnection between the magnetic field of the emerging bipole and the magnetic structure supporting the filament, which is likely a highly sheared arcade.

Figure 2 is a top-down diagram showing the field lines of the sheared arcade before reconnection (orange), those of the emerging bipole (blue), and those formed by reconnection (dark green), which are of the same shape as the bright loops outlined in Figure 1(e). This reconnection likely causes the sheared arcade (and thus the filament) to split, at least partially, with some of its flux becoming connected to the emerging bipole.

A schematic side-on view of this phase of the evolution is shown in Figure 3, emphasizing reconnection of the emerging bipole with the field surrounding the highly sheared filament-arcade core.

Here again, the orange field lines depict the filament arcade, blue depict the emerging bipole and green those formed by reconnection. Since the field lines are drawn on a 2D plane, those of the emerging bipole and those surrounding the core of the eastern filament arcade appear to be oriented parallel to each other, but this is not the case in reality.

The dark green field line on the left of Figure 3(c) is equivalent to that on the left of Figure 2. This field line has been shortened by the reconnection (see Figure 2), which increases its magnetic tension and leads to an additional stabilization of the core field. The small dark green line of Figure 3(c) is equivalent to the smaller reconnected line shown in Figure 2. In what follows, we refer to the new magnetic connection associated with the latter line as the "new arcade". Since the new arcade forms by reconnection between the bipole and the original filament arcade, it likely contains a significant amount of shear/twist, which may have been required for powering the second eruption described below.

Due to plasma heating caused by reconnection, it is difficult to follow the evolution of the filament material involved in this reconfiguration. It appears that some of it ended up in the new arcade, as the observations show the presence of a north-south directed, S-shaped filament section that seems to follow that structure (Figures 1(e) and4), albeit some filament material may have been present at that location prior to the emergence (Figure 1(b)).

First Eruption

The western part of the filament is seen to start rising slowly at ≈ 07:00 UT. Around 07:45 UT, the rise accelerates and the western part of the filament fully lifts off. It erupts strongly non-radially eastward, over the eastern part of the filament, and seems to drag the latter with it. It thus appears that the whole filament erupts (Figure 4(e)), except perhaps those sections that were disconnected during the earlier phase of the bipole emergence. The flare loops produced by this eruption can be clearly seen in AIA 171 and 193 Å images, as shown in Figure 4(e),(f). The CME associated with this eruption is first seen in data from the Large Angle and Spec- The bright loops seen in Figure 4(c) before the eruption are interpreted as these small loops, which connect the positive polarity of the emerging bipole and the negative polarity of the filament arcade. They are expected to accumulate above the sheared new arcade that formed earlier on, during the reconnection between the emerging bipole and the eastern filament arcade (Figures 2 and3(c),(d)).

The long overlying loops produced by the reconnection shown in Figure 5(c) have a lower magnetic tension than the field lines that were overlying the filament originally, allowing the western filament arcade to rise. At some point of the evolution the magnetic configuration becomes unstable, possibly due to loss of equilibrium (or torus instability) and erupts. Reconnection beneath the filament (Figure 5(d)) produces the western flare loops shown in Figure 4(f). This is the same process as described in case B of [START_REF] Chen | [END_REF] for an eruption caused by flux emergence nearby a flux rope, and as demonstrated for a fully threedimensional (3D) configuration in Section 3.2.

After the eruption, flare loops can also be seen to the east of the emerging bipole (Figure 4(f)).

The observations indicate that the eruption of the western section of the filament likely destabilized the whole magnetic structure that overlies the PIL shown in dark red in Figure 4(a), which appears reasonable since the highly sheared field carrying the filament was likely extending over the whole PIL. This means that the shortening of the field lines shown in Figure 3(c),(d) was not sufficient to stabilize the configuration, which again appears reasonable as those field lines were relatively large. In this scenario, the eruption is expected to form loops along the entire length of the PIL, but not all of these are observed, probably due to differences in the plasma density associated with the local amount of reconnected flux (less energy is liberated in weaker magnetic fields). As for the reconnection described in Section 2.2, this 3D effect along the circular PIL cannot be depicted in the 2D cartoons of Figure 5, which represents only a 2D cut of the configuration on the western side of the emerging bipole.

Second Eruption

The second eruption originates above the new PIL between the positive polarity of the emerging bipole and a pre-existing negative flux concentration (dashed red line in Figure 4(g)). The eruption occurs just a few hours after the first eruption, at ≈ 10:30 UT. Bright flare loops are seen after this second eruption (Figure 4(h),(i)), showing the relaxation of a highly sheared arcade over a period of about 25 minutes. The CME associated with the second eruption enters the LASCO C2 field of view at ≈ 11:50 UT. It appears to travel faster than the first eruption, which may be due to a removal of some of the overlying coronal field by the first eruption.

The magnetic structure that most likely powers the second eruption is the new arcade that was formed by the reconnection process described in Section 2.2. It is indicated by the small green field lines on the right-hand side of the emerging bipole in Figure 3(d). Magnetic flux is added to this new arcade during the reconnection that triggers the first eruption (Figure 5(c)). The continuous westward motion of the leading positive polarity of the bipole towards the PIL of the new arcade likely concentrated the arcade's shear. Additionally, a highly twisted flux rope may have formed beneath the arcade by the process described in Section 3.3.

How is the second eruption initiated? As shown in Figure 5(d), the first eruption leaves behind a region of reduced magnetic pressure, into which the sheared new arcade (or flux rope) can expand. This induces reconnection between the arcade and the erupting flux to its right-hand side (Figure 5(e)). Note that this reconnection works in the opposite direction as the earlier one shown in Figure 5(c): rather than adding closed flux to the arcade, it opens up flux on its top, thereby reducing the magnetic tension that holds down the arcade's sheared/twisted core. Such behavior has been previously observed, with flare ribbons moving backwards well after a CME was launched: see Figures 11 and12 in Goff et al. (2007) for a similar reversal of the reconnection direction after the launch of a CME. This eventually facilitates the eruption of the core flux, which evolves into the second CME. Behind the eruption the reconnection shown in Figure 5(f) is induced, which creates the flare loops seen in Figure 4(h),(i).

We note that the mechanism described here for the triggering of a second eruption due to a reduction of magnetic tension by a preceding eruption that occurs in an adjacent flux system is basically the same as modeled for "sympathetic" eruptions by Török et al. (2011) and Lynch & Edmondson (2013); see also Gary & Moore (2004)

NUMERICAL MODELING

In this section we compare our interpretations of the observations with MHD simulations of the emergence of a strong and compact bipole in the vicinity of a large coronal flux rope. The simulations we consider here are part of a parametric study that was performed to study the triggering of CMEs by flux emergence (as observed and analyzed by, e.g., Feynman & Martin 1995). This study will be described in a forthcoming publication (Török et al., in preparation); here we restrict ourselves to a brief description of the basic setup.

We emphasize that the simulations of our parametric study were not designed to reproduce the event analyzed in Section 2, which results in a number of differences between the simulations and the observations (see below). Specifically, we are not intending here to reproduce the whole chain of the observed dynamic events (filament split-ting, first and second eruption) in a single simulation. Rather, we choose from our parametric study three independent simulations that start from the same initial state and differ only in the distance between the pre-existing flux rope and the emerging bipole. Each simulation addresses only one of the observed dynamic events. Also, it should be kept in mind that the simulations use idealized configurations, i.e., they are not intended for a quantitative comparison with the observations. Instead, they should be considered merely as "proof-ofconcept", serving to support our interpretations of the observations in terms of different reconnection processes and the resulting dynamics and system reconfigurations. We leave the design of a more realistic simulation of the observed events to a later investigation.

The simulations described here were performed using the MAS (Magnetohydrodynamic Algorithm outside a Sphere) code (e.g., Mikic & Linker 1994;Lionello et al. 1999), which advances the standard viscous and resistive MHD equations. The β = 0 approximation, in which thermal pressure and gravity are neglected, was used here, so that the evolution is driven by the Lorentz force. The use of this approximation is justified here, since the dynamics relevant for our investigation occur in corona, where the plasma beta is low. The spherical simulation domain covers the corona within 1.0-3.5 R , where R is the solar radius. We note that, even though the lower boundary of the MAS domain is associated with the solar surface (r = R ), it should physically be considered here as the bottom of the corona, since we use the β = 0 approximation.

The initial coronal magnetic field consists of a flux rope embedded in a bipolar AR, as shown in a top-down view in Figure 6(a). This configuration was constructed using the modified Titov-Démoulin model (TDm; Titov et al. 2014), such that the flux rope is initially in stable magnetic equilibrium. The center of the TDm configuration is placed at the position (r, θ, φ) = (1., 1.125, 2.46), with r in units of R and θ, φ in radians. The axis of the TDm flux rope is aligned with the φ axis.

After relaxing the system until a sufficiently accurate numerical equilibrium is obtained, the emergence of a strong, compact bipolar AR is modeled "kinematically" (i.e., boundary-driven). To this end, horizontal slices of all three components of the magnetic field and the velocity are extracted at regular time intervals from an MHD simulation that used the Lare3D MHD code (Arber et al. 2001) to model the emergence of a flux rope from the convection zone into a non-magnetized corona (Leake et al. 2013). 2 The slices are extracted at a height of the Lare3D simulation domain that corresponds approximately to the middle of the photosphere-chromosphere layer used in these simulations (see Leake et al. 2013). The velocity components are directly imposed at the lower boundary of the MAS domain and used for the momentum equation in MAS. The radial magnetic field, B r (t), of the Lare3D simulation is superimposed for all slices on B r (r = R ) of the TDm configuration. This superimposed component and the extracted tangential fields and velocities are then used to calculate the electric fields required for the induction equation in MAS (see Lionello et al. 2013 for details).

An extensive parametric study of the resulting evolution was performed by varying the strength, location, and magnetic orientation of the emerging flux. Changing these parameters can change the interaction between the existing and emerging flux system. This leads in some cases to an eruption of the TDm flux rope (for similar studies see, e.g., Chen &[START_REF] Chen | [END_REF]Kusano et al. 2012).

In the simulations presented here, the bipolar AR emerges for about 1.5 hours at an almost constant rate of ≈ 5 × 10 20 Mx h -1 , after which the emergence gradually slows down. After 6 hours, when the emergence has essentially saturated, the total unsigned flux of the AR is ≈ 1.3 × 10 21 Mx, which is about 20 per cent of the total flux of the TDm configuration. At this time, the modeled bipole has reached a size of ≈ 50 Mm (see Figure 6(c)).

We note that in our simulations the orientation of the polarity centers changes in the course of the emergence from east-west to north-south, which can be best seen by comparing panels (a) and (b) of Figure 8. This was not the case for the real bipole, which essentially maintained an east-west orientation throughout the whole observed evolution. This indicates that the twist of the simulated emerging flux is larger than the twist of the real one. We believe that this difference does no affect the essential nature of the reconnection processes described in this section.

The polarity signs of the magnetic configuration and the handedness of the TDm flux-rope were chosen in the parametric study without knowledge of the observed event described in this paper, and it turned out that they are opposite to those observed. Thus, when preparing the simulation data for the visualizations shown in Figures 678, we generated an inverted coordinate, φ (see Figure 6 This transformation changes the handedness of the TDm flux rope and of the emerging flux from negative to positive and from positive to negative, respectively, in agreement with the observations. We finally reverse the sign of B, in order to reproduce the signs of all observed polarities. A corresponding procedure was applied to the current density, j, which is used in Figures 7 and8. These transformations do not affect the evolution of the system, but significantly ease the visual comparison of the simulation results with the observations.

Splitting of the TDm Flux Rope

We first consider the simulation shown in Figure 6, which we call "simulation 1" for further reference. In this run, the bipole emerges centered around (r, θ, φ) = (1., 1.08, 2.46), close to the TDm flux rope (at a distance of 0.045 R in the θ direction), within the negative polarity of the TDm background field. This qualitatively corresponds to the situation shown in Figure 1, namely to the emergence of the bipole close to the eastern section of the circular filament. The orientation of the emerging flux in the simulation is such that the initial axial-field direction of the emerging flux rope is antiparallel to the axial field at the core of the TDm flux rope.

Due to the vicinity of the emerging flux to the TDm rope, the two flux systems start to interact early on in the evolution. Initially, only field lines of the potential field surrounding the TDm flux rope come into contact with the outer field lines of the emerging flux. Since the field direction of these flux systems is essentially antiparallel, a current layer similar to the ones shown in Figures 7 and8 is formed between them. Driven by the expansion of the emerging flux in the corona, reconnection across this layer sets in. Once the outer flux regions have reconnected, the reconnection continues, now involving inner flux regions of the emerging bipole and the TDm flux rope.

Figure 6(c) shows a situation at which a considerable fraction of the TDm flux rope has already reconnected to form new connections between the rope's foot points and the polarity centers of the emerging bipole. Being a result of reconnection, the corresponding field lines should appear bright in emission, just as the two streaks highlighted in Figure 1(e). The morphological agreement between those streaks and the simulated new connectivities supports our interpretation that the emergence of the bipole resulted (at least partially) in the splitting of the flux rope or arcade that was car-rying the eastern section of the circular filament (see Section 2.2).

Due to the initial north-south orientation of the emerging flux rope in the simulation, the field lines of the TDm flux-rope core and of the core of the emerging flux rope are oriented essentially antiparallel when they come into contact and reconnect (Figure 6). This was not the case in the observed event, where the corresponding field directions were approximately perpendicular to each other. Such an orientation should, however, still allow a reconnection of the type shown in Figure 6(c) to occur, as long as the interacting field lines are not close to being parallel (e.g., Linton et al. 2001). Indeed, in another simulation of our parametric study (not shown here), in which the orientation of the emerging flux rope was rotated by 3 π/8 (56 • ) clockwise compared to simulation 1, we still found strong reconnection between the bipole and the TDm rope and the development of new connectivities very similar to those shown in Figure 6(c).

First Eruption

Our second simulation (simulation 2) is shown in Figure 7. In this simulation, the TDm flux rope represents the western section of the circular filament. The orientation of the emerging bipole is the same as in simulation 1, but its center is now located at (r, θ, φ) = (1., 1.04, 2.46), about twice further away (0.085 R in the θ direction) from the TDm rope (just outside of the negative flux concentration of the TDm background field). The larger distance reflects the fact that in the observed case the western filament section was further away from the emerging bipole than the eastern section. The initial configuration of the simulation is shown in panel (a), where the cyan field line represents the potential background field overlying the TDm flux rope. As can be seen in panel (b), the emerging polarities and the TDm background polarities together form a quadrupolar polarity pattern, corresponding to what Feynman & Martin (1995) termed "favorable for reconnection". Note that the view in the figure is chosen such that the bipole emerges to the left (to the east) of the flux rope, as it was the case in the observations.

As the new flux emerges, a current layer forms between the emerging flux and the TDm background field. Reconnection across this layer displaces field-line foot points of the background field from the edge of the negative TDm background polarity to the negative polarity of the emerging flux, i.e., further away from the TDm flux rope (Figure 7(b)). The length of those field lines thus increases and they start to expand, which reduces the magnetic tension above the TDm rope.

However, reconnection is not the only mechanism leading to such expansion. As numerically demonstrated by Ding & Hu (2008), adding a small bipole to a 2D flux-rope configuration changes the configuration in such a way that the magnetic field overlying the flux rope is more expanded, as long as the bipole is placed close to the rope and in an orientation "favorable for reconnection". The expansion is merely due to the change in the boundary condition of the system (see also Wang & Shee-ley 1999); reconnection is not required. This effect takes place in our simulation, as the slowly emerging flux changes the boundary conditions of the coronal magnetic field. Due to the relatively large Alfvén speed in the corona, this information has sufficient time to travel into the domain and to affect the coronal magnetic field.

The combined action of these two mechanisms is visualized in Figure 7(b): the cyan field line has just reconnected with the emerging flux (see the strong kink of the field line at the position of the current layer), and its foot point on the left-hand side of the TDm rope has been displaced further away from the rope. Note that the field line has already expanded at the time it reconnects. This is partly due to the changes at the boundary, and partly due to the fact that field lines above it have reconnected and expanded earlier in the evolution.

As a result of the continuous weakening of the magnetic tension due to field-line expansion, the TDm flux rope eventually cannot be stabilized anymore and erupts (Figure 7(c)). The top of the rope rotates clockwise (when viewed from above), due to its right-handed twist (e.g., Green et al. 2007;Török et al. 2010). Note that the initial opposite axial-field directions of the emerging flux rope and the TDm rope do not fundamentally affect the evolution in this case, since the TDm rope starts to erupt before it would significantly reconnect with the emerging flux.

In the simulation, the eruption sets in about one day after the beginning of the flux emergence, which is much later than in the real event, where the time difference was about four hours. The onset time of the eruption depends on various parameters, predominantly on "how far" the TDm rope is initially from an unstable state, and how efficiently the emergence and associated reconnection act in weakening the stabilizing tension of the overlying flux. Changing the strength and position of the emerging flux and/or changing the initial current in the TDm rope will lead to different onset times. To summarize, simulation 2 demonstrates that the scenario illustrated in Figure 5(a)-(d), and modeled in 2D for an infinitely long flux rope by [START_REF] Chen | [END_REF], can work also in fully 3D simulations, in which the foot points of the coronal flux rope are anchored in the photosphere. Thus, the simulation supports our interpretation put forward in Section 2.3, namely that the first eruption was caused by a reduction of magnetic tension above the western part of the filament, as a result of flux emergence with an orientation "favorable for reconnection".

Flux-Rope Formation Before Second Eruption

As described in Section 2.4, the second eruption originates from the PIL that forms between the positive polarity of the emerging bipole and a neighboring, pre-existing negative flux concentration (Figure 4(g)). In order for the eruption to occur, the flux residing above this PIL must have been non-potential. Since we found no indications for the presence of a PIL at this location prior to the emergence of the bipole, the correspond-ing shear/twist must have accumulated during the emergence process.

We suggested in Section 2.4 that a new sheared arcade formed at this location by the reconnection process described in Section 2.2, and that the shear further concentrated due to the westward motion of the positive polarity. However, it is not clear whether sufficient shear to power an eruption can build up solely by this process. In this subsection we suggest an additional mechanism by which non-potential magnetic fields may have built up at this PIL. To this end, we consider a third simulation (simulation 3). Note that we focus here only on the formation of the pre-eruptive structure; we do not aim to model the eruption itself.

In simulation 3, the orientation of the emerging flux is the same as in the previous simulations, i.e., the configuration is again "favorable" for reconnection. The emergence is now centered around (r, θ, φ) = (1., 1.06, 2.46), at an intermediate distance (0.065 R ) from the TDm rope (Figure 8). This simulation is intended to mimic the emergence of the observed bipole east of the existing negative flux concentration, shown in the left pan-els of Figure 4. We note that the formation of a new flux rope between the emerging and existing polarities, as described below, occurs in a very similar manner also in simulations 1 and 2. The reason why we use simulation 3 here to illustrate this process is that we had already analyzed that particular simulation regarding flux-rope formation prior to writing this article.

As in simulations 1 and 2, the expansion of the emerging flux in the corona leads to the formation of a thin current layer (Figure 8(c)). The layer develops above the "external" section of the PIL, i.e., the section that divides the positive emerging polarity and the pre-existing negative flux (Figure 8(a)). A complex dynamic evolution involving different types of reconnection in the current layer and downward directed flows leads to the formation of a low-lying, highly twisted flux rope (Figure 8(b)). Note that this "external" rope is righthanded, while the less twisted, thicker flux rope that connects the polarity centers of the emerging flux is left-handed (see Leake et al. 2013 and references therein for the formation mechanism of this "central" flux rope).

The accumulation of twisted field lines above the external section of the PIL eventually ceases. However, reconnection in the current layer still continues, now producing sheared, arcade-type field lines that accumulate above the external flux rope (Figure 8 The external flux rope forms in our simulation due to reconnection across the current layer in the corona, so the formation mechanism should be robust with respect to the way in which the flux emergence into the corona is modeled (in our case via kinematic emergence). To check this, we have recently simulated an analogous situation using the Lare3D code, in which the flux emergence into a pre-existing coronal magnetic field is modeled dynamically, i.e., though the buoyant rise of a flux rope through the convection zone. We found the formation of an external flux rope also in this simulation, which will be described in a forthcoming publication.

The mechanisms that lead to the formation of the external rope, the dependence of its formation and properties on parameters such as the amount of twist of the emerging flux, and the implications of this structure for coronal jets and filaments that form between active regions will be discussed in detail in a forthcoming publication (Török et al., in preparation). For our purpose, the important point is that the development of such a flux rope during the emergence of new flux in the vicinity of a preexisting polarity provides an additional explanation for the presence of highly non-potential magnetic fields along the PIL indicated by the dashed lines in Figures 4(g)-(i).

SUMMARY AND CONCLUSIONS

In this study, we investigated a magnetic configuration in which a filament resided above a circular PIL that encircled a dispersed negative polarity. We followed the early evolution of the small, bipolar active region NOAA 12119, which emerged within this polarity, close to the eastern section of the filament. Within eight hours of the onset of emergence, a partial splitting of the filament and two consecutive eruptions, both leading to CMEs, took place in the area. We utilized SDO data and MHD simulations to propose a scenario for the observed chain of events.

Based on the observations, we propose that the bipole initially emerges completely within the arcade-field overlying the eastern section of the filament. Reconnection of the two flux systems leads to a shortening of the field lines surrounding the core field of the filament arcade and stabilizes the core field in this area (Figures 1(d),(e), 2, and 3(c),(d)). This reconnection also causes at least a partial splitting of the field carrying the filament (similar to the case of Li et al. 2015) and thereby produces a new arcade (and S-shaped filament) that connects the bipole with the original filament.

After the western side of the emerging bipole has reconnected through the eastern arcade, it starts to reconnect also with the field of the western arcade (Figure 5(c)). This reconnection adds flux to the previously formed new arcade. Simultaneously, it destabilizes the western arcade, allowing the western section of the filament to rise and eventually erupt (Figures 4 and5(d)).

Meanwhile, the continued emergence and westward motion of the leading polarity of the bipole may have concentrated the shear of the new arcade. Additionally, a highly twisted flux rope may have formed within it, as suggested by our simulation 3 (Section 3.3). Since the first eruption has left behind a region of reduced magnetic pressure and weakened overlying field, the flux rope and surrounding new arcade can expand (Figure 5(e)), eventually leading to the second eruption (as in the sympathetic eruptions modeled by Török et al. 2011).

Our simulations support this scenario. In simulation 1 (Section 3.1), a bipolar flux region is emerged within one of the polarities of the TDm background field, close to the location of the preexisting flux rope. The emerging and pre-existing TDm fields start to reconnect, and the TDm fluxrope field eventually forms new connectivities with the emerging bipole (Figure 6(c)). The shapes and locations of these new connectivities correspond to the bright streaks seen in the observations (Figure 1(d),(e)), suggesting that they were indeed a result of a partial splitting of the magnetic field carrying the eastern section of the filament.

In simulation 2 (Section 3.2), the bipole is emerged further from the TDm flux-rope and with an orientation such that a quadrupolar polarity pattern is formed. This setup mimics the interaction of the emerging flux with the western section of the filament. Both the changes in the boundary conditions caused by the emergence and reconnection between the two flux systems act to reduce the tension of the field overlying the TDm flux rope, which eventually leads to its eruption. This pro-vides an explanation for the first observed eruption, which begins in the western section of the circular filament, relatively far from the emerging bipole (Section 2.3).

In simulation 3 (Section 3.3), we model the formation of a highly twisted flux rope over the PIL between one polarity of an emerging bipolar flux region and a pre-existing flux concentration of opposite polarity. This demonstrates that, in addition to the new arcade, also a highly twisted flux rope may have formed in the source region of the observed second eruption. This addition of nonpotential magnetic field may make it easier to understand how the second eruption could originate in an area where concentrated sheared/twisted flux was not present prior to the flux emergence.

We conclude that the position of the emerging bipole with respect to the background magnetic field configuration is a crucial factor for the interaction of these fields and the resulting evolution. Numerical simulations are able to qualitatively reproduce the various dynamic behavior observed for our case, and the upcoming study of Török et al. will help to characterize the relationship between the position of emerging flux (and of other parameters such as its orientation, helicity sign, and amount of flux) and its interaction with the background field (see also Kusano et al. 2012). Even relatively small amounts of emerging flux may be able to trigger significant changes in the coronal field, increasing the difficulty to predict eruptions. More systematic observations and parametric numerical simulation studies would give us a better idea of the conditions under which it should be possible to predict coronal activity triggered by flux emergence.

Figure 1 .

 1 Figure 1. Apparent splitting of the eastern section of the filament. Length scales for panels (a)-(d) are indicated in (a). Shown are HMI magnetograms (left) and AIA 171 Å images (right), at 03:30 UT (top) and 05:12 UT (bottom) on 18 July 2014. (a) Magnetogram and (b) quiescent filament around the onset of emergence. Red (blue) dashed lines outline the filament location. (c) Region of flux emergence (green oval), slightly to the west of the eastern filament section. (d) Interaction between emerging flux and pre-existing magnetic field. Bright streaks reach both northward and southward, suggesting new magnetic connectivities. (e) Zoom into the area shown as white rectangle in (d). White dashed lines outline the new connectivities. The AIA images shown here and in Figure 4 were processed using the Multi-scale Gaussian Normalization Technique of Morgan & Druckmüller (2014). An animation of this figure is available with the online version of this manuscript.

Figure 2 .

 2 Figure 2. Diagram showing the splitting mechanism. The red dashed circle represents the PIL of the preemergence magnetic configuration, above which the filament resides (Figure 1(b)). The emerging bipole is sketched with + and -signs in black circles, and blue field lines representing its magnetic connectivities. The orange line, which crosses the PIL, represents a highly sheared field line of the eastern filament arcade. It reconnects with the emerging bipole, forming the green field lines.

Figure 3 .

 3 Figure 3. 2D diagram showing the reconnection described in Section 2.2. The black line indicates the photosphere. Dark (light) blue field lines represent newly emerging flux (emerged in the previous panel). Orange field lines show a cross-section of the pre-existing field configuration; with the eastern (western) filament arcade on the left (right). Dark (light) green field lines are formed by reconnection (reconnected in the previous panel). The purple zigzag line represents a current sheet where reconnection takes place. The reconnection produces two new field-line sets, anchored in the negative and positive polarity of the bipole, respectively (cf.Figure2). Note that new flux continues to emerge in panels (c) and (d), but is omitted for clarity.

  Figure 3. 2D diagram showing the reconnection described in Section 2.2. The black line indicates the photosphere. Dark (light) blue field lines represent newly emerging flux (emerged in the previous panel). Orange field lines show a cross-section of the pre-existing field configuration; with the eastern (western) filament arcade on the left (right). Dark (light) green field lines are formed by reconnection (reconnected in the previous panel). The purple zigzag line represents a current sheet where reconnection takes place. The reconnection produces two new field-line sets, anchored in the negative and positive polarity of the bipole, respectively (cf.Figure2). Note that new flux continues to emerge in panels (c) and (d), but is omitted for clarity.

Figure 4 .

 4 Figure 4. First and second eruption. The length scales for all panels is indicated in (a). Shown are HMI magnetograms (left), AIA 171 Å (center) and 193 Å (right) images. (a)-(c) Before the first eruption, at 07:04 UT on 18 July 2014. Dashed lines indicate the location of the filament that undergoes the first eruption. The bright new arcade field lines in (c) are formed by reconnection of the emerging flux with the pre-existing filament arcade, as shown in Figure 5(c). (d)-(f) Shortly after the first eruption, at 08:59 UT. The flare loops produced in the eruption can be seen in (e) and (f). The total unsigned magnetic flux of the emerging bipole was ≈ 4.7 × 10 20 Mx, and it had a size of about 25 × 12 Mm at this time. (g)-(i) Just after the second eruption, at 10:56 UT. The dashed lines indicate the PIL of the erupted arcade. Sheared flare loops produced in the eruption can be seen in (h) and (i). The flux-emergence region is outlined by green ovals in panels (d)-(i). At this time, the emerging bipole has reached a total size of about 28 × 13 Mm and a total flux of ≈ 6.0 × 10 20 Mx. An animation of this figure is available with the online version of this manuscript. The observations are interpreted as shown in Figure 5(a)-(d) and described as follows. After the emerging bipole has 'eaten through' all of the field lines of the eastern arcade, it can start to reconnect with the western arcade. This reconnection (Figure 5(c)) produces two new sets of field lines (dark green); small loops and long overlying ones.The bright loops seen in Figure4(c) before the eruption are interpreted as these small loops, which connect the positive polarity of the emerging bipole and the negative polarity of the filament arcade. They are expected to accumulate above the

  ; DeVore & Antiochos (2005); Joshi et al. (2016); Li et al. (2017).

Figure 5 .

 5 Figure 5. Same as Figure 3, now for the mechanisms believed to produce the two eruptions. (a) Emerging bipole (blue), new arcade (green), and western filament-arcade (orange). The eastern filament-arcade does not participate in the evolution and is omitted here. (b) Continuing bipole emergence and expansion in the corona. (c) Reconnection between bipole and western filament-arcade, forming long overlying field lines and smaller loops above the new arcade. The lengthening of the overlying field lines reduces the magnetic tension on the western filament-arcade, allowing it to expand. (d) Eruption of the western filament-arcade and associated reconnection underneath the filament, cutting its ties to the photosphere and further accelerating it upwards. (e) Expansion of the new arcade induces reconnection with locally open field lines left behind from the eruption, accelerating its rise. (f) Second eruption and flare loops formed by the reconnection in the wake of the eruption.

Figure 6 .

 6 Figure 6. Simulation 1: Emergence of a bipolar flux region close to a pre-existing coronal flux rope. (a) Prior to emergence. Orange field lines depict the core of the TDm rope, which mimics the eastern section of the circular filament shown in Figure 1. (b) Early emergence. Emerging and TDm-background field lines and the current layer that forms between the emerging and pre-existing flux systems are omitted for clarity (see Figures 7 and 8). (c) Later emergence. Reconnection between the emerging flux and the TDm rope has led to the formation of new connectivities similar to the observed ones (cf. Figure 1(e)). Length scales and coordinates are shown in (b).

  (a)), by mirroring the φ coordinate about φ = 2.46 (the center of the TDm configuration). The φ-mirroring transforms the magnetic field from [B r (φ), B θ (φ), B φ (φ)] to [(B r ( φ), B θ ( φ), -B φ ( φ)], keeping ∇ • B = 0 and reversing only the φ component of the Lorentz force.

Figure 7 .

 7 Figure 7. Simulation 2: Flux-rope eruption triggered by emerging flux. Here the TDm flux rope mimics the western section of the circular filament-arcade, so that in this view φ points towards the viewer and θ to the right (cf. Figure 6). Shown are the core of the TDm rope (rainbow-colored field lines), an overlying field line (cyan), and electric currents in a transparent vertical plane perpendicular to the TDm rope axis (shown by |j|/|B| in orange-white colors). (a) Initial configuration. The overlying field line is calculated starting from the positive (red) polarity of the TDm background field. (b) 3.6 h later, after a substantial amount of flux has emerged. The overlying field has expanded and its negative (blue) footpoint has been displaced by reconnection in the current layer that separates the emerging flux from the background flux. (c) Configuration after 24 h, showing the TDm flux rope in the process of eruption.

Figure 8 .

 8 Figure 8. Simulation 3: emergence of a bipolar flux region close to a pre-existing flux concentration (similar to the observations shown in Figure 4). The orientation of the configuration is the same as in Figure 7; only the area containing the negative polarity of the TDm background field is shown. For better visibility, this polarity is highlighted by contour lines of -70, -80, and -90 G in panels (a) and (b). (a) Early phase of emergence. (b) 2.4 hours later. Field lines connecting the emerging-polarity centers outline the core of the emerging flux rope (cyan). A new flux rope has formed at the external PIL between the bipole and the pre-existing flux concentration (yellow). Field lines are colored by the force-free parameter α = j • B/B 2 ; note that the two ropes have opposite handedness. Some arcade-like field lines (with smaller |α|) are shown above the new flux rope. (c) Tilted view of panel (b) along the PIL, showing additionally |j|/|B| in a transparent vertical plane.

  (b),(c)). This ongoing reconnection corresponds exactly to the one sketched in Figure 3(c),(d) and described in Section 2.

  MAS. S.D. acknowledges STFC for support via her studentship. L.v.D.G is partially funded under STFC consolidated grant number ST/N000722/1. L.v.D.G also acknowledges the Hungarian Research grant OTKA K-109276. D.M.L is an Early-Career Fellow funded by the Leverhulme Trust. T.T, C.D, J.E.L, and M.G.L acknowledge support from NASA's LWS and H-SR programs. M.G.L. acknowledges support from the Chief of Naval Research. We also thank the International Space Science Institute (ISSI) team 348 "Decoding the Preeruptive Magnetic Configuration of Coronal Mass Ejections" led by S. Patsourakos and A. Vourlidas.

Luoni et al. (2011) showed that the helicity sign derived from a magnetic-tongue pattern agrees with other proxies, such as loop shear.

The simulation used here is very similar to the cases "ND" and "ND1" described inLeake et al. (2013).

We thank the referee for thoughtful and inspiring comments and Ron Moore for a helpful discussion regarding the second eruption. The authors are thankful to the SDO/ HMI and AIA consortia for the data. We also thank Z. Mikić for assisting in coupling the Lare3D simulations to


We formulate the figure-ground discrimination problem as a combinatorial optimization problem. We suggest a cost function that makes explicit a definition of shape based on interactions between image edges. These interactions have some mathematical analogy with interacting spin systems -a model that is well suited for solving combinatorial optimization problems. We devise a mean field annealing method for finding the global minimum of such a spin system and the method successfully solves for the figure-ground problem.

I n t r o d u c t i o n a n d b a c k g r o u n d

The problem of separating figure from ground is a central one in computer vision. One aspect of this problem is the problem of separating shape from noise. Two-dimensional shapes are the input data of high-level visual processes such as recognition. In order to maintain the complexity of recognition as low as possible it is important to determine at an early level what is shape and what is noise. Therefore one needs a definition of shape, a definition of noise, and a process that takes as input image elements and separates them into shape and noise.

In this paper we suggest an approach whose goal is twofold: (i) it groups image elements that are likely to belong to the same (locally circular) shape while (ii) noisy image elements are eliminated. More precisely, the method that we devised builds a cost function over the entire image. This cost function sums up image element interactions and it has two terms, i.e., the first enforces the grouping of image elements into shapes and the second enforces noise elimination. Therefore the shape/noise discrimination problem becomes a combinatorial optimization problem, namely the problem of finding the global minimum for the cost function just described. In theory, the problem can be solved by any combinatorial optimization algorithm that is guaranteed to converge towards the global minimum of the cost function.

In practice, we implemented three combinatorial optimization methods: simulated annealing (SA), mean field annealing (MFA), and microcanonical annealing (MCA) [START_REF] Hdrault | Figure-ground discrimination: a combinatorial optimization approach[END_REF]. Here we concentrate on mean field annealing.

The figure-ground or shape/noise separation is best illustrated by an example. Fig. 1 shows a synthetic image. Fig. 2 shows the image elements that were labelled "shape" by the mean field annealing algorithm.

The interest for shape/noise separation stems from Gestalt psychologists' figureground demonstrations [START_REF] Kohler | Gestalt Psychology[END_REF]: certain image elements are organized to produce an emergent figure. Ever since the figure-ground discrimination problem has been seen as a side effect of feature grouping. Edge detection is in general first performed. Edge grouping is done on the basis of their connectivity or by using a clustering technique. Noise is eliminated by thresholding.

The connectivity analysis produces edge chains which are further fitted with piecewise analytic curves (lines, conics, splines, etc.). The clustering technique maps image edges into a parameter space and such a parameter space is associated with each curve type: this is the well known Hough transform. There are two problems with the techniques just mentioned: (i) one has to specify analytically the type of curve or curves that are to be sought in the image and this is done at a low-level of the visual process and (ii) the notion of noise is not clearly specified with respect to the notion of shape.

Our work is related to the influential paper of Parent & Zucker [START_REF] Parent | Trace Inference, Curvature Consistency, and Curve Detection[END_REF] who introduced the notion of cocircularity between two edge elements and to the work of Sha'ashua Ullman [START_REF] Sha | Structural Saliency: The Detection of Globally Salient Structures Using a Locally Connected Network[END_REF], and Sejnowski & Hinton [START_REF] Sejnowski | Separating Figure from Ground with a Boltzmann Machine[END_REF].

The work described here has two main contributions. First, we suggest a mathematical encoding of the figure-ground discrimination problem that consists of separating shape from noise using a combinatorial optimization method. Second, we suggest mean field annealing, a deterministic global optimization method.

A c o m b i n a t o r i a l o p t i m i z a t i o n f o r m u l a t i o n

We consider a particular class of combinatorial optimization problems for which the cost function has a mathematical structure that is analog to the global energy of a complex physical system, that is a interacting spin system. First, we briefly describe the state of such a physical system and give the mathematical expression of its energy. We also show the analogy with the energy of a recursive neural network. Second we suggest that the figure-ground discrimination problem can be cast into a global optimization problem of the type mentioned above. The state of an interacting spin system is defined by: (i) A spin state-vector of N elements ~r = [ a l , . . . , ~N] whose components are described by discrete labels which correspond to up or down Ising spins: ai E {-1, +l}. The components o'i may well be viewed as the outputs of binary neurons. (ii) A symmetric matrix J describing the interactions between the spins. These interactions may well be viewed as the synaptic weights be-tween neurons in a network. (iii) A vector 6 = [(~1, 9 9 (~N] describing an external field in which the spins are plunged.

Therefore, the interacting spin system has a '~natural" neural network encoding associated with it which describes the microscopic behaviour of the system. A macroscopic description is given by the energy function which evaluates each spin configuration. This energy is given by:

INN N E(~ ~ "'O'N) = ---2 E E Jij(ri~ -E ~iO'i (1) i=1 j=l i=1
The main property of interacting spin systems is that at low temperatures the number of local minima of the energy function grows exponentially with the number of spins. Hence the adequation between the mathematical model of interacting spin systems and combinatorial optimization problems with many local minima is natural.

We consider now N image elements. Each such element has a label associated with it, Pi, which can take two values: 0 or 1. The set of N labels form the state vector P = ~I,''',PN]. We seek a state vector such that the "shape" elements have a label equal to 1 and the "noise" elements have a label equal to 0. If clj designates an interaction between elements i and j, one may write by analogy with physics an interaction energy:

E N N

saliency(P) = --Ei=I Ej=I cijpipj

Obviously, the expression above is minimized when all the labels are equal to 1. In order to avoid this trivial solution we introduce the constraint that some of the elements in the image are not significant and therefore should be labelled "noise":

Eeonstraint(P) = (E/N_--I pl) 2
The function to be minimized could be something like the sum of these energies: E(p) = Esalieney(P) + )~ Eeon,traint(P) [START_REF] Deriche | Using Canny's criteria to derive a recursively implemented optimal edge detector[END_REF] In this expression ~ is a positive real parameter that has to be adjusted and is closely related to the signal-to-noise ratio. With the substitution Pi = (al + 1)/2 in eq. ( 2), the function to be minimized is given by eq. ( 1) where: Jij = (clj -,~)/2 and 8i = N N)~)/2. Let i and j be two edgels. We want that the interaction between these two edgels encapsulates the concept of shape. That is, if i and j belong to the same shape then their interaction is high. Otherwise their interaction is low. Notice that a weak interaction between two edgels has several interpretations: (i) i belongs to one shape and j belongs to another one, (ii) i belongs to a shape and j is noise, or (iii) both i and j are noise. The interaction coefficient must therefore be a co-shapeness measure. In our approach, co-shapeness is defined by a combination of cocircularity, proximity, and contrast.

The definition of cocircularity is derived from [START_REF] Parent | Trace Inference, Curvature Consistency, and Curve Detection[END_REF] and it constrains the shapes to be as circular as possible, or as a special case, as linear as possible. Proximity restricts the interaction to occur in between nearby edgels. As a consequence, cocircularity is constrained to be a local shape property. The combination of coeireularity and proximity will therefore allow a large variety of shapes that are circular (or linear) only locally. Contrast enforces edgels with high gradient module to have a higher interaction coefficient than edgels with a low gradient module.

Following [START_REF] Parent | Trace Inference, Curvature Consistency, and Curve Detection[END_REF] and from Fig. 3 it is clear that two edgels belong to the same circle if and only if: Ai + )~j = r. In this formula, hi is the angle made by one edgel with the line joining the two edgels. Notice that a circle is uniquely defined if the relative positions and orientations of the two edgels verify the equation above. This equation is also a local symmetry condition consistent with the definition of local symmetry of Brady & Asada. Moreover, linearity appears as a special case of cocircularity, namely when hl = 0 and Aj =Ix.

Fig. 3. The definition of cocircularity between two edgels (i and j).

From this cocircularity constraint we may derive a weaker constraint which will measure the closeness of a two-edgel configuration to a circular shape: Aij =[ Ai + Aj --lr [. Aij will vary between 0 (a perfect shape) and ~ (no shape). Finally the cocircularity coefficient is allowed to vary between 1 for a circle and 0 for noise and is defined by the formula: ci~ ~ = (1-A~j/~r ~) exp (-A~ffk). The parameter k is chosen such that the cocircularity coefficient vanishes rapidly for non-circular shapes.

The surrounding world is not constituted only by circular shapes. Cocircularity must therefore be a local property. That is, the class of shapes we are interested to detect at a given scale of resolution are shapes that can be approximated by a sequence of smoothly connected circular arcs and straight lines. The proximity constraint is best described by multiplying the cocircularity coefficient with a coefficient that vanishes smoothly as the two edgels are farther away from each other: ~fox = exp (-d5/(2~)) where dlj is the distance between the two edgels and ad is the standard deviation of these distances over the image. Hence, the edgel interaction will adjust itself to the image distribution of the edgel population.

A classical approach to figure-ground discrimination is to compare the gradient value at an edgel against a threshold and to eliminate those edgels that fall under this threshold. An improvement of this simply-minded nonlinear filtering is to consider two thresholds such that edgel connectivity is better preserved [START_REF] Canny | A Computational Approach to Edge Detection[END_REF]. Following the same idea, selection of shapes with high contrast can be enforced by multiplying the interaction coefficient with a term whose value depends on contrast: e c.~ = gigj/g2ma~ where gma~ is the highest -S 3 gradient value over the edgel population. Finally the interaction coefficient between two edgels becomes:

cocir fox contrast
4 Mean field annealing (MFA)

The states reachable by the system described by eq. ( 1) correspond to the vertices of a Ndimensional hypercube. We are looking for the state which corresponds to the absolute minimum of the energy function. Typically, when N = 1000, the number of possible configurations is 2 g ~ 10 3~ The problem of finding the absolute minimum is complex because of the large number of local minima of the energy function and hence this problem cannot be tackled with local minimization methods (unless a good initialization is available). We already mentioned that the functional to be minimized has the same structure as the global energy of an interacting spin system. To find a near ground state of such a physical system we will use statistical methods. Two analysis are possible depending on the interaction of the system with its environment: either the system can exchange heat with its environment (case of the canonical analysis) or the system is isolated (case of the microcanonical analysis) [START_REF] Hdrault | Figure-ground discrimination: a combinatorial optimization approach[END_REF]. We will consider here the canonical analysis.

This analysis makes the hypothesis that the physical system can exchange heat with its environment. At the equilibrium, statistical thermodynamics shows that the free energy F is minimized. The free energy is given by: F = E -TS, where E is the internal energy (the energy associated with the optimization problem) and S is the entropy (which measures the internal disorder). Hence, there is a competition between E and S. At low temperatures and at equilibrium, F is minimal and TS is close to zero. Therefore, the internal energy E is minimized. However the minimum of E depends on how the temperature parameter decreases towards the absolute zero. It was shown that annealing is a very good way to decrease the temperature.

We are interested in physical systems for which the internal energy is given by eq. ( 1). The remarks above are expressed in the most fundamental result of statistical physics, the Boltzmann (or Gibbs) distribution:

Pr(E(a) = Ei) = exp (-Ei/(kT)) Z(T) (4)
which gives the probability of finding a system in a state i with the energy Ei, assuming that the system is at equilibrium with a large heat bath at temperature kT (k is the Boltzmann's constant). Z(T) is called the partition function and is a normalization factor: Z(T) = ~, exp (-E,/(kT)). This sum runs over all possible spin configurations. Using eq. ( 4) one can compute at a given temperature T the mean value over all possible configurations of some macroscopic physical parameter A:

(A) = ~ A, Pr(E(~) = E,) = ~" A, exp (-E,/(kT)) n Z(T) (5)
Unfortunately, the partition function Z(T) is usually impossible to compute. Nevertheless, when the system is described by eq. ( 1) one can use eq. ( 5) which, with an additionnal hypothesis, is a basic equation for mean field approximation and for mean field annealing.

In order to introduce the mean field annealing algorithm, we first introduce the somehow more classical mean field approximation method which has been used to solve optimization problems [START_REF] Orland | Mean field theory for optimization problems[END_REF][START_REF] Peterson | A new method for mapping optimization problems onto neural networks[END_REF][START_REF] H6rault | Neural Networks and Graph K-Partitioning[END_REF]: it is a simple analytic approximation of the behaviour of interacting spin systems in thermal equilibrium. We start by developing eq. ( 1) around

cr i: N N N 1 k=l,k#i j=l,j#i k=l,k#i
~i is the total field which affects the spin ai:

Oi ------Jij o'j + ~i \j=l

The mean field (~i) affecting ~i is computed from the sum of the fields created on the spin ~ri by all the other spins "frozen" in their mean states, and of the external field ~i viewed by the spin ai. The mean state of a spin (o'i) is the mean value of the (ri's computed over all possible states that may occur at the thermal equilibrium. We obtain:

We introduce now the following approximation [START_REF] Stanley | Introduction to Phases Transitions and Critical Phenomena[END_REF]: The system composed of N interacting spins is viewed as the union of N systems each composed of a single spin. Such a single-spin system {ai} is subject to the mean field (~i) created by all the other single-spin systems. Let us study such a single spin system. It has two possible states: {-1} or {+1}. The probability for the system to be in one of these states is given by the Boltzmann distribution law, eq. ( 4): exp (-(4,) a~ a0 e {-1, 1}

P(X, = a ~ ~ p()~ ~'+ P"(-~i ~--1)'

Xi is the random variable associated with the value of the spin state. Notice that in the case of a single-spin system the partition function (the denominator of the expression above) has a very simple analytical expression. By combining eq. ( 5), [START_REF] Kohler | Gestalt Psychology[END_REF], and ( 7), the mean state of ai can now be easily derived:

(ai)~ (+i) exp(-(*i)/T)+(-1)exp((~i)/T)=tanh(~=lJij(aj)+~i) exp ( (#i) /T) + exp (-(Oi) /T) T (8)

We consider now the whole set of single-spin systems. We therefore have N equations of the form:

#i = tanh ()-~J N=I Jij pj + 6i) T ( 9
)
where Pi = (ai). The problem of finding the mean state of interacting spin system at thermal equilibrium is now mapped into the problem of solving a system of N coupled -Initially the temperature has a high value and as soon as every spin has been updated at least once, the temperature is decreased to a smaller value. Then the temperature continues to slightly decrease at each step of the convergence process. This does not guarantee that a near equilibrium state is reached at each temperature value but when the temperature is small enough then the system is frozen in a good stable state. Consequently, the convergence time is reduced since at low temperatures the convergence to a stationary solution is accelerated. This strategy was successfully used to solve hard np-complete graph combinatorial problems [START_REF] H6rault | Neural Networks and Graph K-Partitioning[END_REF], [START_REF] H6rault | Neural Networks and Combinatorial Optimisation: A Study of NP-Complete Graph Problems[END_REF];

-Van den Bout & Miller [START_REF] Van Den Bout | Graph Partitioning Using Annealed Neural Networks[END_REF] tried to estimate the critical temperature Tr At this temperature, some of the mean field variables pi begin to move significantly towards either -1 or +1. Hence their strategy consists performing two sets of iterations: one iteration process at this critical temperature until a near state equilibrium is reached and another iteration process at a temperature value that is close to 0. However, the critical temperature is quite difficult to estimate.

We currently use the first of the annealing schedules described above.

Example and discussion

We tested these algorithms over a wide variety of images. The images are preprocessed as follows. Edges are first extracted using the Canny/Deriche operator [START_REF] Deriche | Using Canny's criteria to derive a recursively implemented optimal edge detector[END_REF]. The tangent direction associated with such an edge is computed by fitting a straight line in the leastsquare sense to a small set of connected edges. Then this small set of edges is replaced by an edgel, i.e., the fitted line. The position of the edgel is given by its midpoint, its direction is given by the direction of the line, and its contrast is given by the average contrast of the edges forming the edgel. Fig. 4 shows the input data of our example. Fig. 2 and Fig. 5 show the result of applying mean field annealing. In this paper we attacked the problem of figure-ground discrimination with special emphasis on the problem of separating image data into curve and noise. We proposed a global approach using combinatorial optimization. We suggested a mathematical encoding of the problem which takes into account such image properties as cocircularity, proximity, and contrast and this encoding fits the constraints of the statistical modelling of interacting spin systems.

We conclude that the interacting spin model is well suited for encoding the figureground problem. Moreover, the analogy of the energy of such a model with the energy of ~t t lattl 9 ~Jto Fig. 6. Evolution of the/ti variables in MFA for the image above.

a recursive neural network allows one to assert that the MFA algorithm proposed here is implementable on a fine-grained parallel machine. In such an implementation, a processor is associated with an edgel (or a spin, or a neuron) and each processor communicates with all the other processors.

Figure 2 a

 2 Figure-Ground Discrimination by Mean Field Annealing *

Fig. 1 .

 1 Fig. 1. A synthetic image with 1250 elements. Fig. 2. The result of applying mean field an-Circles, a straight line, and a sinusoid are nealing to the synthetic image. The elements plunged into randomly generated elements, shown on this figure were labelled "shape".

  image array contains two types of information: changes in intensity and local geometry. Therefore the choice of the image elements mentioned so far is crucial. Edge elements, or edgels are the natural candidates for making explicit the two pieces of information just mentioned. An edgel can be obtained by one of the many edge detectors now available. An edgel is characterized by its position in the image (xi, Yl) and by its gradient computed once the image has been low-pass filtered. The x and y components of the gradient vector are: gx(xi, Yi) --OIy(xi, Yi) and gy(Xi Yi) = Ob(Xi' Yi) Ox ' Oy I! is the low-pass filtered image. From the gradient vector one can easily compute the gradient direction and magnitude: Oi and gi, e.g., Fig. 3: f gy(Xi,Yi) ~ 7f 2 2, 0i = arctan t m\~/ + ~ and gi = (gx(zi, Yi) + gytxl, Yl))

Fig. 4 .

 4 Fig. 4. Set of edgels obtained with no noise Fig. 5. The result of applying MFA to the imelimination,

* This research has been sponsored in part by "Commissariat ~ l'Energie Atomique," in part by the ORASIS project, and in part by CEC through the ESPRIT-BRA 3274 (FIRST) project.

non-linear equations, i.e., eq. ( 9) 9 In the general case, an analytic solution is rather difficult to obtain 9 Instead, the solution for the vector/t = [/Jl,"',PN] may well be the stationary solution of the following system of N differential equations:

where r is a time constant introduced for homogeneity. In the discrete case the temporal derivative term can be written as:

-Au where /~ is the value of/Ji at time tn. By substituting in eq. ( 10) and by choosing 7-= At, we obtain an iterative solution for the system of differential equations described by eq. ( 10): Vie{l,...,N},#~ +1 =tanh( ~jN=xJ'jtt:+Si)T' ,n>l_ [START_REF] Stanley | Introduction to Phases Transitions and Critical Phenomena[END_REF] where/~ is an estimation of (o'i) at time tn or at the n'th iteration 9

Starting with an initial solution It o = [/~0,.. ", #~v] the convergence is reached at the n'th iteration such that ttn = ~,...,#~v] becomes stationary 9 The physicists would say that the thermal equilibrium has been nearly reached. When the vector 6 is the null vector, one can start with a vector tt 0 close to the obvious unstable solution [0,..., 0]. In practice, even when the vector 8 in not null one starts with an initial configuration obtained by adding noise to [0,. 9 0]. For instance the 0, 9 , ~i s are chosen randomly in the interval [-10 -5 , +10-5]. This initial state is plausible for an interacting spin system in a heat bath at high temperature 9 In fact the spin values (q-1 or -1) are equally likely at high temperatures and hence, (trl) ---0 for all spin i. During the iterative process, the /~i's converge to values in between -1 or 3-1.

Two convergence modes are possible:

-Synchronous mode 9 At each step of the iterative process all the /t~'s are updated using the/~-1,s previously calculated.

-Asynchronous mode 9 At each step of the iterative process a spin #n is randomly selected and updated using the/z n-l's.

In practice, the asynchronous mode produces better results because the convergence process is less subject to oscillations frequently encountered in synchronous mode. In order to obtain a solution for the vector ~r from the vector it, one simply looks at the signs of the/z~ 's. A positive sign implies that the probability that the corresponding spin has a value of 3-1 is greater than 0.5: if 1/2(1 3-#~) > 0.5 then o'i = 3-1 else ai = -1.

A practical difficulty with mean field approximation is the choice of the temperature T at which the iterative process must occur. To avoid such a choice one of us [START_REF] H6rault | Neural Networks and Combinatorial Optimisation: A Study of NP-Complete Graph Problems[END_REF] and other authors [START_REF] Van Den Bout | Graph Partitioning Using Annealed Neural Networks[END_REF] have proposed to combine the mean field approximation process with an annealing process giving rise to mean field annealing. Hence, rather than fixing the temperature, the temperature is decreased during the convergence process according to two possible annealing schedules: